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{"url":"http:\/\/toc.ilab.sztaki.hu\/articles\/v008a024\/index.html","text":"Volume 8 (2012) Article 24 pp.\u00a0533-565\nSolving Packing Integer Programs via Randomized Rounding with Alterations\nPublished: October 28, 2012\n$\\newcommand{\\eee}{\\mathrm e}$\nOur second result is for the class of packing integer programs (PIPs) that are column sparse, i.e., where there is a specified upper bound $k$ on the number of constraints that each variable appears in. We give an $(\\eee k+o(k))$-approximation algorithm for $k$-column sparse PIPs, improving over previously known $O(k^2)$-approximation ratios. We also generalize our result to the case of maximizing non-negative monotone submodular functions over $k$-column sparse packing constraints, and obtain an $\\smash{\\left(\\frac{\\eee^2k}{\\eee-1} + o(k) \\right)}$-approximation algorithm. In obtaining this result, we prove a new property of submodular functions that generalizes the fractional subadditivity property, which might be of independent interest.","date":"2020-04-07 11:31:03","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7484585046768188, \"perplexity\": 617.6671446500903}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-16\/segments\/1585371700247.99\/warc\/CC-MAIN-20200407085717-20200407120217-00330.warc.gz\"}"} | null | null |
Inquisitions Post Mortem
Abstracts of Inquisitiones Post Mortem For the City of London: Part 3
Inquisitions: 1592
Abstracts of Inquisitiones Post Mortem For the City of London: Part 3. Originally published by British Record Society, London, 1908.
Sebastian Bruskett, Esquire.
William Billinge, Citizen and Waxchandler.
John Kettell alias Wyttye, Citizen and Clothworker.
Edward Orwell, Gentleman.
John Graunge, Citizen and Haberdasher.
Christopher Myers, Gentleman.
Edward Leighe, Gentleman.
Inquisition taken at the Guildhall, 5 September, 34 Eliz. [1592], before William Webb, knight, Mayor and escheator, after the death of Sebastian Bruskett, esq., by the oath of Robert Dickenson, Thomas Russell, Thomas Seawell, John Harrison, John Stevins, William Feake, William Crowche, James Robinson, John Tompson, John Bonde, Peter Noxon, Thomas Wigges, Nicholas Maddox, John Langley, Hugh Ingram, Robert Saunders, Robert Durraunt, Richard Rogers and Christopher Dickenson, who say that
Sebastian Bruskett long before his death was seised in his demesne as of fee of 1 messuage situate in the parish of St. Gabriel Fanchurche, London, now in the tenure of Peter Van Lore, jeweller; and 6 messuages lying within a certain lane called St. Sythes Lane in the parish of St. Benedict Sherehogg in the ward of Cordwayner streete, London, now or late in the several tenures of Jane Bruskett, widow, Thomas Wallwyn, Thomas Cox, Richard Pepper, Richard Carpenter and John Poole.
So seised, the said Sebastian made his will in November, 1591, as follows [here given in English]: I give to my wife Jane Briskett all my lands and tenements in St. Sythes Lane, being 6 houses in number, the one in the occupation of Peter Van Lore, jeweller, the great messuage house in the occupation of [blank], the other tenements in the tenures of [blank]: all the said premises to remain until the marriage of my only daughter and child Elizabeth Bruskett to my said wife.
The messuage in the said parish of St. Gabriel Fanchurch is held of the Queen in chief by the service of the 40th part of a knight's fee, and is worth per ann., clear, 4 marks. The 6 messuages in St. Sythes lane are held of the Queen in free burgage, and are worth per ann., clear, £5 10s.
Sebastian Bruskett died 5 August last past; Elizabeth Bruskett is his only daughter and next heir, and is now aged 12 years, 6 months and 5 days.
Chan. Inq. p. m., vol. 232, No. 9.
Inquisition taken at the Guildhall, 22 June, 34 Eliz. [1592], before William Webbe, Mayor and escheator by virtue of a writ de melius inquirend, after the death of William Billynge, citizen and waxchandler of London, by the oath of Robert Dickensen, Thomas Russell, Thomas Sewell, John Harrison, William Harvye, John Stevyns, William Crowche, John Bonde, James Robinson, George Robertes, Nicholas Hawkesforthe, Edward Swayne, Christopher Dickenson, John Langley, Robert Saunders, Richard Rogers, John Palmer, John Jeninges and Thomas Wigges, who say that
William Billinge long before his death was seized in his demesne as of fee of 1 messuage in the parish of St. Laurence in Old Jewry, London, and so seised made his will 31st October, 1581, and thereby bequeathed the said messuage to Joan Billinge then his wife and to her heirs for ever.
The said messuage is held of the Queen in chief by the 100th part of a knight's fee, and is worth per ann., clear, 33s. 4d.
William Billinge died the last day of May, 1582. Afterwards the said Joan, late the wife of the said William, married Edward Winstanley of London, gent., and they were jointly seized of the said messuage in right of the said Joan.
Chan. Inq. p. m., vol. 232, No. 10.
Inquisition taken at the Guildhall, 19 February, 34 Eliz. [1592], before William Webb, Mayor and escheator, after the death of John Kettell alias Wyttye, citizen and clothworker of London, by the oath of Robert Dickenson, Thomas Sewell, John Harrison, William Harvy, William Crowtche, James Robinson, George Robertes, Edward Pillesworth, Nicholas Hawkesforth, Thomas Wigges, Edward Swayne, Thomas Smith, Christopher Dickenson, Robert Saunders, and John Langley, who say that
John Kettell alias Wyttye, long before his death was seised in his demesne as of fee of 1 messuage lying in a certain street called Candlewickestreete, in the parish of St. Mary Abchurch, London, late in the tenure of the said John Kettell.
So seised the said John in fulfilment of certain covenants specified in certain indenture made between himself of the one part and Francis Stoughton of the Inner Temple, London, gent., and Anthony Marler, citizen and mercer of London, of the other part, agreed as follows [indenture here given in full in English]: Indenture made 31 December, 28 Eliz. [1585] between the said John Kettell of the one part and the said Francis Stoughton and Anthony Marler of the other part. Whereas the said John Kettell is now seised in his demesne as of fee simple to him and his heirs for ever, or in fee tail general or special of all that messuage situate in Candlewicke street, late of William Kettell, late citizen and clothworker of London, deceased, father of the said John and now in the occupation of the said John: whereas also the said John Kettell intends shortly by the grace of God to take to wife Martha Lawrence, daughter of Thomas Laurence, late citizen and draper of London, deceased: these indentures witness that in consideration of the said marriage and for a jointure to be made for the said Martha, it is agreed between the said parties that the said John Kettell shall before the end of Hilary term next following at his own costs suffer the said Francis and Anthony to prosecute him in a writ of Entre in le Poste before the Justices of the Common Pleas at Westminster, in order that a recovery may be had of the said messuage to the intent that the said Francis and Anthony shall stand thereof seised to the use of the said John Kettell and his heirs until the said marriage be solemnized, and afterwards to the use of the said John and Martha and the heirs of the said John for ever.
Afterwards, to wit, in Hilary term, 28 Eliz., a certain recovery was suffered of the said messuage, by pretext whereof and by force of the Statute of Uses the said John Kettell was thereof seised until the said marriage.
The said marriage was afterwards solemnized. The said John Kettell was likewise seised in his demesne as of fee of 1 other messuage, now or late in the tenure of John Pearson, fishmonger, lying in the street called Bridge street in the parish of St. Magnus the Martyr in London.
So seised, the said John made his will 1 January, 1591 [here given in English] as follows: I give the messuage wherein I now dwell and the 2 shops thereto belonging and the reversions thereof immediately after the death of Martha my wife, who has an estate therein for life, to William Kettell my son and to the heirs of his body; for default, and for default of male issue of my body I give the same to my daughter Grace Kettell and to the heirs of her body; for default, to William Kettell my man and to the heirs of his body; and for default to my right heirs for ever.
I give to the said Martha my wife for the bringing up and education of my children my messuage situate in New Fish street, now in the occupation of Edmond Goodwyn, which I lately purchased of Mr. Keeling, and all the rents thereof until the said William my son shall accomplish his full age of 21, or if he die, until my said daughter shall come of age or marry; if they both die (which God forbidd) then my said wife shall have the said messuage for life.
The said messuage in the parish of St. Mary Abchurch is held of the Queen in chief by knight's service, but by what part of a knight's fee the jurors know not, and is worth per ann., clear, £3 6s. 8d. Of whom or by what service the messuage in the parish of St. Magnus the Martyr is held the jurors know not: it is worth per ann., clear, 40s.
John Kettell died 23 January last past; William Kettell is his son and next heir and was aged 3 years on the 25th day of December last past.
The said Martha still survives in the parish of St. Mary Abchurche.
Inquisition taken at the Guildhall, 19 February, 34 Eliz. [1592], before William Webbe, Mayor and escheator, after the death of Edward Orwell of London, gent., by the oath of Robert Dickenson, Thomas Sawill, John Harrison, William Harvey, William Crowche, James Robinson, George Robertes, Edward Pillesworthe, Nicholas Hawkesforthe, Thomas Wigges, Edward Swayne, Thomas Smith, Christopher Dickenson, Robert Saunders, and John Langley, who say that
Long before the death of the said Edward Orwell, a certain Lawrence Husey, Doctor of Laws, was seised in his demesne as of fee of 1 messuage with a garden thereto adjoining, wherein the said Edward Orwell lately dwelt, lying in the parish of Christchurch, London, which was formerly the parish of St. Ewin in Newgate market, London.
So seised, the said Edward by deed dated 23 May, 21 Eliz. [1579], sold the said messuage and garden to the said Edward Orwell and Mary then his wife and to the heirs of the said Edward, by virtue whereof they entered into the said premises and were thereof seised. The said Edward in his demesne as of fee and the said Mary in her demesne as of free tenement for her life. The said Edward and Mary were seised to them and the heirs of the said Edward of 1 marsh containing 15 a. called Wild land marsh lying in Rypley Marsh within the parish of Barking in co. Essex; 4 a. lying within Dyers land in Barking; 4½ a. of marsh lying in Ripley marsh, late of Robert Tirrell, lately purchased of Westan Browne, esq.; 1 messuage in Brenchley in co. Kent, with all those lands, tenements, and hereditaments called Yonges, le Rech, Mayland, Cattesland and Powlehurst; 3 pieces or parcels of land and wood called Sherman Reede, Byrchett and Longland containing 40 a. of land lying in the chapelry of Uckfould within the parish of Buckstead in co. Sussex; 1 parcel of meadow called Fulling mill meade containing 4 a. 1 r. of land there near Bullicatts mill lately purchased of Arthur Longworth.
The said Edward Orwell was also seised of 3 other messuages and 3 gardens with 1 close thereto adjacent in Brenchley, lately purchased of John Alchorne.
The said Edward made his will 5 January, 1591 [here given in English] as follows: I give to my "most kinde wiefe" Mary and her heirs for all, all my lands and tenements at Brenchley in co. Kent to the end that she make sale thereof to the most advantage as soon as may be, and the money thereof coming to go towards the payment of my debts and the education and preferment of my daughters in marriage; but if the said Mary die then I give the said premises to Mr. Doctor Lewyn and to my cousin Mr. Robert Hamond and to their heirs for ever, to sell the same to the uses before mentioned.
The messuage and other the premises within the City of London are held of the Queen in chief by the 200th part of a knight's fee, and are worth per ann., clear, 5 marks.
The 15 a. of marsh called Wildelond and the 4 a. of land within Dyers land are held of the Queen in chief by the 200th part of a knight's fee, and are worth per ann., clear, 55s. of whom the said 4½ a. of marsh, late of Robert Tyrrell are held is not known: they are worth per ann. 11s. 8d. The said premises in Brenchley purchased of William Barrentine, viz., the said messuage, garden, 10 a. of land and 20 a. of pasture are held of the manor of Sallmons in co. Kent by fealty only in common socage, and are worth per ann., 33s. 4d. Ten acres of land, 20 a. of pasture and 30 a. of wood being another parcel of the said premises are held of the manor of Yalding in co. Kent in free socage by fealty only, and are worth per ann., clear, £3 6s. 8d. Twenty acres of land and 40 a. of pasture parcel and residue of the said premises late of William Barentyne are held of the manor of Woldham in co. Kent in free socage by fealty only, and are worth per ann. 50s. The premises purchased of John Alchorne are held of the manor of Yalding by fealty in free socage, and are worth per ann., 20s.
Of whom the said premises in Sussex are held is not known: they are worth per ann. 40s.
Edward Orwell died 5 January last past; Edward Orwell is his son and next heir and was aged 12 years on the 14th day of June last past.
The said Mary, late the wife of the said Edward Orwell still survives.
Inquisition taken at the Guildhall, 19 February, 34 Eliz. [1592], before William Webb, Mayor and escheator, after the death of John Graunge, citizen and haberdasher of London, by the oath of Robert Dickenson, John Harrison, William Harvy, William Crowche, James Robinson, George Robertes, Edward Pillesworthe, Nicholas Hawkesforth, Thomas Wigg, Edward Swayne, Thomas Smith, Christopher Dickenson, Robert Saunders, John Langley and Thomas Sawyll, who say that John Graunge was seised of 7 messuages now made into 8 messuages, with all shops, cellars, sollars, entries, ways, lights, &c., thereto belonging now or late in the several tenures of the said John Graunge, Thomas Wetherall, Henry Taylford, Elizabeth Dryver, widow, John Carter, John Richardson, and John Evans, lying in the parish of St. Martin within Ludgate, London: which said premises he purchased to him and his heirs of Richard Willis and Katherine his wife, daughter and heir of Robert Phillippes, late citizen and leatherseller of London, deceased; also all those lands, tenements, soil or ground late of Thomas Alleyn situate under any part of the houses, buildings, tenements or hereditaments of the said John Graunge being in the said parish of St. Martin near Ludgate, viz., all that land and soil containing in length 18 feet and in width 7½ feet, which adjoin the west part of a certain wall, soil and land of the said Thomas Alleyn, and lies under part of the house and structure of the said John Graunge in the said parish; all that land and soil with a sink or washhouse (sentina sive latrina) there containing in length 12 feet and in width 8 feet adjoining the north part of the said wall, and lies under parcel of the house of the said John in the said parish; all that entry, soil and ground adjoining the north part of the said wall lying under the said house in the said parish, containing in length 10 feet and in width 2½ feet; all that soil and ground adjoining the west part of the said wall lying under the said house in the said parish, containing in length 8 feet and in breadth 5 feet: which said premises last recited the said John Graunge purchased to him and his heirs of Thomas Alleyn, citizen and haberdasher of London; also of divers other messuages with all the houses, buildings, barns, stables, gardens, orchards, &c., thereto belonging, now or late in the several tenures of the most noble William Herbert, knight, late Earl of Pembrook, deceased, [blank] Bryche, Joan Wyse, widow, Anthony Uvedale,Thomas Moore, Henry Hye, [blank] Throwghton and [blank] Wilson, lying in the parish of St Giles in the Fields in co. Middlesex: all which said premises last recited the said John Graunge purchased to him and his heirs of Robert Downes of Acton in co. Suffolk, esq., and Edward Downes, brother of the said Robert, gent, 1 close of land called Newlands, containing about 24 acres, and all that parcel of land or lane to the said close adjoining, now or late in the tenure of George Harrison, gent., lying within the parish of Mariboone in co. Middlesex, all which said premises last recited the said John Graunge purchased to him and his heirs of the said Robert Downes of Acton in co. Suffolk, esq., and George Downes of Sudbury in the said county, gent.; 1 other messuage called Turkses alias Turkes at Wateringes with all the houses, buildings, barns, stables, gardens, &c., thereto belonging; 4 closes of arable land and pasture lying near the said messuage, containing about 30 a.; 1 croft called Swannes Crofte containing about 4 a.; 1 marsh or meadow called Gubbines meade, containing about 10 a.; 1 other marsh or meadow called Thome meade containing about 6 a.; 1 marsh called Redd meade containing about 3 a.; 2 other marshes called Chatterings containing about 4 a.; 4 a. in the common marsh of Havering: all which said premises last recited are in the vills and parishes of Hornchurch and Havering in co. Essex, now or late in the tenure of Thomas Heard deceased, and were purchased by the said John Graunge to him and his heirs of John Page of the Inner Temple London, gent., and John Legatt of Hornechurch Hall in the parish of Hornchurch, Essex, gent.
The 7 messuages now made into 8, lying near Ludgate in the parish of St. Martin, are held of the Queen in chief by knight's service, to wit, by the 20th part of one knight's fee and by the yearly rent of 12s. 4d., and are worth per ann., clear, £4. Of whom the said land, soil or ground in the said parish of St. Martin lately purchased of the said Thomas Alleyn are held the jurors know not: they are worth per ann., clear, 3s. 4d. The messuages lying in the parish of St. Giles in the Fields, the close of land or pasture called Newlandes with the lane thereto adjacent, lying in the parish of Mariboone are held of the Queen in chief by knight's service, but by what part of a knight's fee is not known: they are worth per ann., clear, £5. Of whom the messuage called Turkes with all the parcels of land, meadow and marsh thereto belonging lying in the vills and parishes of Hornechurch and Havering are held the jurors know not: they are worth per ann., clear, 30s.
John Graunge died 28 October, 33 Eliz. [1591]; John Graunge is his son and next heir, and was then aged 30 years and more.
Inquisition taken at the Guildhall, 8 May, 34 Eliz. [1592], before William Webb, Mayor and escheator, after the death of Christopher Myers, gent., by the oath of Robert Dickenson, John Harrison, William Crowche, William Feake, Edward Osborne, John Bonde, John Thompson, Edward Pilsworth, John Adlin, John Dixon, Thomas Wigge, James Robinson, Edward Swayne, Christopher Dickenson, Robert Derant, Robert Saunders, Stephen Porter and Cuthbert Lee, who say that
Christopher Myers long before his death was seised in his demesne as of fee of 1 messuage and 1 garden late in the tenure of Thomas Bramley lying within the parish of St. Margaret in Lothbury, London; 1 other messuage and garden situate in the street of Lothbury in the said parish now in the tenure of Richard Goode; 1 other messuage lying in the parish of St. Margaret Moyses in the street called Friday street, London, late in the tenure of William Hobson; 1 other messuage with all the buildings, gardens, stables, &c., thereto belonging commonly called Ridegate alias Rigate in the street of East Smithefield near the Tower of London in the parish of St. Botolphe without Algate, London, formerly parcel of the lands and possessions of the late Monastery of Coggeshall alias Coxhall in co. Essex, dissolved; 1 other messuage with all houses, &c., &c., situate in the parish of St. Mary, Athill [upon the Hill] next Billingsgate, London, now in the tenure of Roger Staveld, sometime parcels of the lands and possessions of the late College of Pontefract in co. York. So seised, the said Christopher Myers in fulfilment of certain covenants specified in certain indentures dated 12 May, 19 Elizabeth [1577] made between him the said Christopher of the one part and Richard Allington of Westley in co. Cambridge, gent., of the other part, in consideration of a marriage to be had between the said Christopher Myers and Margaret Allington, one of the daughters of the said Richard Allington, to the intent that she may have a sufficient jointure out of the lands, &c., of the said Christopher and in full satisfaction of her dower, agreed by the said indenture that he or his heirs at or before the feast of St. John the Baptist then next following would levy a fine of the said messuage called Ridegate alias Rigate in East Smithfield near Tower Hill, and the messuage in the said parish of St. Mary Athill next Billingsgate to a certain Clement Cisley, Esq., and to the said Richard Allington, gent., and should acknowledge the said tenements to be the right of the said Clement and should remise the same to the said Clement and Richard and the heirs of the said Clement for ever: which said fine should be to the use of the said Christopher Myers and Margaret and of the heirs of their bodies; and after their deceases without issue, to the use of the right heirs of the said Christopher for ever.
Shortly afterwards the said Christopher married the said Margaret, and a fine was levied of the said premises to the said Clement Cisley and Richard Allington to the uses above declared: by virtue whereof and by force of the Statute of Uses the said Christopher and Margaret were jointly seised of the said premises.
The first of the said 2 messuages in the said parish of St. Margaret, Lothbury, is held of the Queen by fealty only in free burgage, and is worth per ann., clear, £6 13s. 4d. The other messuage there is held of the Queen by fealty only in free burgage, and is worth per ann., clear, 40s. Of whom the said messuage in Friday Street in the said parish of St. Margaret Moyses is held the jurors know not: it is worth per ann., clear, 40s. The messuage called Rydgate in East Smithfield is held of the Queen in chief by knight's service, but by what part of a knight's fee is not known, and is worth per ann., clear, 33s. 4d. Of whom the tenement in the parish of St. Mary Athill is held is not known: it is worth per ann., clear, 53s. 4d. Christopher Myers died 24 February, 34 Eliz.; Walter Myers is his son and next heir and was aged 14 years on the 16th day of March last past.
The said Margaret still survives.
Inquisition taken at the Guildhall, 10 June, 34 Eliz. [1592], before William Webb, Mayor and escheator, after the death of Edward Leighe, gent., by the oath of Robert Dickenson, Thomas Sawyll, William Harvye, William Crouche, William Feake, John Bonde, James Robinson, George Robertes, Richard Rogers, Edward Swayne, Christopher Dickenson, Robert Saunders, Robert Durrant and Thomas Russell, who say that
Long before the death of the said Edward Leighe a certain Robert Grace, late citizen and clothworker of London, was seised in his demesne as of fee of 1 messuage lying in the street of Fleetstreete in the parish of St. Brigitte alias St. Brydes, London, now or late in the tenure of [blank] Lodge; and 1 messuage or inn called le Rose with divers tenements thereto adjacent, with stables, cellars, &c., lying in the street of West Smithfield in the parish of St. Sepulchre in the suburbs of the City of London, now or late occupied by William Freeman, Cuthbert Rydley and George Gibson.
So seised, the said Robert Grace made his will dated 16 October, 5 and 6 Philip and Mary [1558] as follows [here given in English]:
I give to Henry Leighe all my lands and tenements in Fleet street and Smithfield for the term of his life; after his death, the same to remain to Garrett Leighe and to the heirs of his body for ever.
By virtue of which said will, the said Henry Leigh immediately after the death of the said Robert Grace entered into the said premises and was thereof seised in his demesne as of free tenement for term of his life, the remainder thereof belonging to the said Garrett Leighe son of the said Henry and the heirs of his body for ever: which said Garrett died in the lifetime of the said Henry, having issue a certain Edward Leigh (named in the writ).
Long before the death of the said Edward Leigh King Henry 8 by his Letters Patent dated 23 September in the 34th year of his reign [1542] gave to John Nashe then one of the pages (pagettorum) of the chamber and to Alice then his wife, inter alia, all that messuage, with cellars, houses, &c., lying in the parish of St. Dunstan in the West in Fleete streete, viz., between the tenement of William Kyrbye on the west and the tenement of Robert Ducke on the east, then in the tenure of the said William Kyrby and late in that of John Bray, and sometime belonging to the late house or priory of the Carmelite brothers in the suburbs of London, lately dissolved. The said John Nashe died some years ago, and the said Alice survived him and afterwards married [blank] Garawey and still survives at Acton in co. Middlesex.
Afterwards King Henry 8 by Letters Patent dated 19 January in the 35th year of his reign [1544] gave inter alia to Thomas Brooke, citizen and merchant tailor of London, the reversion of the said messuage: to hold to him and his heirs for ever.
So seised, the said Thomas Brooke by deed dated 23 February, 35 Hen. 8 [1544] gave to the said Henry Leigh, grandfather of the said Edward Leigh (named in the writ) inter alia the reversion of the said messuage: to hold to the said Henry and to Isabella then his wife and to the heirs of the said Henry for ever: by virtue whereof the said Henry was thereof seised in his demesne as of fee in reversion after the death of the said Alice Garawey.
Before the death of the said Edward Leigh the said Henry Leigh was seised in his demesne as of fee of all that messuage with cellars, houses, &c., situate in the said parish of St. Dunstan in Fleete streete, viz., between the tenement of the said William Kirby on the west and the tenement of the said Robert Ducke on the east, formerly in the tenure of John Wisenden and afterwards in that of Elizabeth Chippingdall, formerly belonging to the House of the Carmelite brothers in the suburbs of London; also all that messuage with houses, shops, &c., lying in the said parish of St. Dunstan, late in the tenure of John Ouley, and afterwards divided into 3 small messuages then in the several tenures of the said Henry Leigh, John Burder and Roger Mellye: which said messuage in the tenure of the said Henry Leighe was known by the name of the Marigolde; and all that piece of land containing 7 feet to the said messuage adjoining, lying in the said street and parish, then in the tenure of the said Henry Leighe, and to the said late house of the Carmelite brothers sometime belonging: which said piece of land was then built upon and was parcel of the said 3 messuages: which said messuages and land the said Henry Leigh purchased to him and his heirs of the said Henry Brooke; 1 other house or messuage called le Flower de Luce, then in the tenure of John Harward and afterwards in that of Anthony Hickman lying in Fleete streete; divers other tenements in Fewter Lane in the parish of St. Dunstan in Fleete streete to the late monastery of St. Mary Overy in co. Surrey formerly belonging: which said house called le Flower de Luce the said Henry Leigh purchased to him and his heirs of Thomas Arundell, knight, and Henry Saunders.
So seised, the said Henry Leigh the grandfather made his will 6 April, 1568, as follows [here given in English]: I give all my lands, tenements, rents, &c., as well within the City of London and the suburbs thereof as elsewhere within the realm of England to Edward Leighe, son and heir of the said Gerard Leigh, for his life; after his decease, then to the first begotten son of the body of the said Edward and to the heirs male of his body; for default, to the 2nd to the 12th sons of the said Edward and to the heirs male of their several bodies; for default, then to Suzan, Elizabeth, Anne, Margaret and Alice Leigh, daughters of the said Gerard my son, and to the several heirs of their several bodies; for default, to the heirs general of the body of the said Edward; for default, to the right heirs of my cousin Henry Leighe, son of Robert Leigh late of Eastwick in co. Hertford and to their heirs for ever, provided always that Margery Nicholson, widow, shall have for her life after the death of Elizabeth my wife 1 tenement of the yearly rent of 20s., now in the tenure of William Blage lying in Fetter Lane in the said parish of St. Dunstans, paying yearly for the same 1 pepper corn.
So seised, the said Henry Leighe died 9 . . ., 10 Eliz., after whose death the said Edward entered into all the said premises and was thereof seised in his demesne as of freehold for the term of his life, with remainders as abovesaid.
The said Edward Leighe was likewise seised in his demesne as of fee of 1 stable in le White Friars in London.
Of whom the said messuage in Fleet street in the said parish of St. Brigitt late of the said Robert Grace is held the jurors know not: it is worth per ann., clear, 20. Of whom the messuage or inn called le Rose in West Smithfield is held the jurors know not: it is worth per ann., clear, 50s. All the said lands, messuages and tenements lying in the said parish of St. Dunstan in the West, in Fleet street, purchased of the said Thomas Brooke, are held of the Queen in chief by the service of the 100th part of a knight's fee, and are worth per ann., clear, £7. The messuage called le Flower de Luce and the said tenement assigned to the said Margery Nicholson lying in the said street and parish are held of the Queen in free burgage by fealty only and not in chief and are worth per ann., clear, £4 13s. 4d. Of whom the stable in le White Fryers is held the jurors know not: it is worth per ann., clear, 10s. 4d.
Edward Leigh died 12th June, 32 Eliz. [1590] without issue; Suzanna now the wife of John Nicolls, Anna now the wife of John Osborne, and Margaret Nixon, widow, relict of Robert Nixon, deceased are the sisters and next coheirs of the said Edward, the said Suzanna being aged 38 years and more, the said Anna 26 years and more and the said Margaret 28 years and more at the time of the death of the said Edward. Elizabeth Leighe, and Alice Leighe 2 of the daughters of the said Garrett Leighe died without issue in the lifetime of the said Edward.
Margery Robertes alias Nicholson still survives. Edward Bearblocke, citizen and goldsmith of London, entered into all the premises late of Robert Grace immediately after the death of the said Edward Leighe and took the issues thereof, but by what title the jurors know not.
John Nicholls in right of the said Suzanna, John Osbourne in right of the said Anne and Robert Nixon and Margaret his wife took the rents and profits of the residue of the premises from the death of the said Edward by virtue of the will of the said Henry Leighe. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,087 |
{"url":"https:\/\/www.r-bloggers.com\/2009\/02\/r-good-practice-%E2%80%93-adding-footnotes-to-graphics\/","text":"[This article was first published on \"R\" you ready?, and kindly contributed to R-bloggers]. (You can report issue about the content on this page here)\nWant to share your content on R-bloggers? click here if you have a blog, or here if you don't.\n\nIn some statistical programs there is the option available to attach a footnote to the graphical output that is created. This footnote may contain the name of the script or the file that produced the graphic, the author\u2019s name and the date of creation. In SAS for example there is a footnote command to achieve this. Ever since I realized that this makes life a lot easier, I wrote a simple three-lines function in R which I use at the end of the construction of any graphic. I suppose, that this is what my professors meant with \u201cgood practice\u201d. The nice thing about implementing this in the grid graphics system is that you can produce multiple graphics [e.g. by par(mfrow=c(2, 2))] and still the footnote will be positioned correctly.\n\n###############################################################\n## ##\n## R: Good practice - adding footnotes to graphics ##\n## ##\n###############################################################\n\n# basic information at the beginning of each script\nscriptName <- \"filename.R\"\nauthor <- \"mh\"\nfootnote <- paste(scriptName, format(Sys.time(), \"%d %b %Y\"),\nauthor, sep=\" \/ \")\n\n# default footnote is today's date, cex=.7 (size) and color\n# is a kind of grey\n\nmakeFootnote <- function(footnoteText=\nformat(Sys.time(), \"%d %b %Y\"),\nsize= .7, color= grey(.5))\n{\nrequire(grid)\npushViewport(viewport())\ngrid.text(label= footnoteText ,\nx = unit(1,\"npc\") - unit(2, \"mm\"),\ny= unit(2, \"mm\"),\njust=c(\"right\", \"bottom\"),\ngp=gpar(cex= size, col=color))\npopViewport()\n}\n\nmakeFootnote(footnote)\n\n## Example ##\nplot(1:10)\nmakeFootnote(footnote)\n\n###############################################################\n\n\nHere an example of a footnote added to the graphical output.\n\nCorrelation matrix with footnote\n\nCheers, Mark","date":"2022-01-28 21:41:56","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.2745666801929474, \"perplexity\": 4316.699920669098}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-05\/segments\/1642320306346.64\/warc\/CC-MAIN-20220128212503-20220129002503-00708.warc.gz\"}"} | null | null |
WEDNESDAY, MARCH 14, 2018 | IN THIS ISSUE
Pappajohn Venture Competition prizes double to $100K
Sean Williams named president, CEO of Mercy Iowa City
ImOn's March Gladness bracket challenge tips off
Final LWV forum to focus on affordable housing
CBJ Movers & Shakers: Week of 3.12.18
Events & CBS2/FOX 28 Headlines
Applications for the John Pappajohn Entrepreneurial Venture Competition are now being accepted, with $100,000 in cash prizes - double the amount from previous years - now on the line.
The annual business competition, now in its 13th year, has historically awarded $50,000 in prizes, funded by a contribution from Iowa business philanthropists John and Mary Pappajohn in an effort to support the state's entrepreneurial ecosystem. This year's competition will include a cash match from the Iowa Economic Development Authority.
Iowa businesses that have been in operation for four years or less, or are not yet cash flow positive, are eligible to apply. This competition is open to businesses in the tech, biotech, green tech, medical, advanced manufacturing, agriculture, engineering and education industries.
Participants will present their business plan through a written submission and, if selected to advance to the final round of the competition, through in-person pitches to competition judges. First place will be awarded $40,000, second place $25,000, and third place $15,000. Awards for other categories will total $20,000.
Applications for the John Pappajohn Entrepreneurial Venture Competition are due May 21. For more information or to apply, visit pappajohnentrepreneurialventurecompetition.com
Sean Williams has been selected to serve as the new president and CEO of Mercy Iowa City, effective May 1.
Mr. Williams currently serves as president and CEO of Mercy Medical Center - Clinton, where he has led the organization for the past eight years. Mercy Iowa City Interim CEO Shane Cerone, along with Mercy Iowa City COO Casey Greene, will assist to ensure a smooth transition, officials said in a statement.
"Sean has made significant contributions to Mercy Health Network and to the excellence of our Clinton ministry," Mercy Health Network President and CEO Bob Ritz said in a press release. "With his proven track record of leadership and deep understanding of our values and mission, he is uniquely qualified and we are pleased to elevate his talents to serve Mercy Iowa City."
During Mr. Williams' tenure at Mercy Medical Center - Clinton, the organization realized greater clinical success, colleague engagement, patient experience and quality metrics. These achievements led to the hospital receiving Magnet recognition and numerous patient satisfaction and industry quality awards. He also previously served as CEO of Jones Regional Medical Center in Anamosa, where he led the planning and building of a replacement hospital.
"I am honored and excited to join the talented team at Mercy Iowa City," Mr. Williams said in the release. "Their reputation for top quality care is well established and I am humbled to be associated with them."
Mr. Williams, a Quad Cities native, earned a bachelor's degree in economics at Loras College in Dubuque and a Master of Healthcare Administration at Des Moines University. He and his wife, Katie, are excited to return to Iowa City where they met and were married 15 years ago.
Cedar Rapids-based ImOn Communications has kicked off its third-annual March Gladness bracket challenge, giving local nonprofits a chance to win hundreds in funding on March 23.
The March Gladness Challenge follows along the lines of the NCAA March Madness Tournament, but instead of basketball teams, eight local nonprofits will compete. Each day of the challenge, two nonprofits will vie for votes on ImOn's Facebook page and customer website. The nonprofit that receives the most votes will move on to the next round.
The competition will continue until only one nonprofit is left standing. That organization will receive a $500 donation from ImOn, while the runner-up will receive $250.
Organizations competing include the Cedar Rapids Library Foundation, Cedar Valley Humane Society, Friends of the Animal Center Foundation, Jane Boyd, Johnson County Crisis Center, Shelter House, the Salvation Army and United Way of East Central Iowa.
Past winners include Arc of East Central Iowa and Last Hope Animal Rescue. For more information visit myimon.com/marchgladness.
Final LWVJC forum to focus on affordable housing
Affordable housing will take center stage at the final League of Women Voters of Johnson County Sunday Speaker Series event of 2018, set for March 18 at 2 p.m.
A panel of speakers will address affordable housing during the event, to be held in the Iowa City Public Library, Room A. Panelists will include Casey Cook, of Cook Appraisal; Crissy Canganelli, of Shelter House; and Tracy Hightshoe, neighborhood and development services director for the city of Iowa City.
The discussion will include two studies: One that reviews area building permits to help determine the rental market and another that reviews the costs associated with homelessness. The panelists will also take questions following the presentation.
For more information, visit the LWVJC website or the organization's Facebook page.
View all of this week's Movers & Shakers in the March 12 edition of the CBJ.
Short-Term Event Planner
BizMix: TRU by Hilton, by Cedar Rapids Metro Economic Alliance and Marion Chamber of Commerce, 4-6 p.m., 3900 Westdale Parkway SW, Cedar Rapids. BizMix brings together area professionals for an evening of casual networking over complimentary hors d'oeuvres and cocktails, and is hosted by a different member business each month. Free.
Coralville Roundtable, by Iowa City Area Chamber of Commerce, noon-1 p.m., Texas Roadhouse, 2520 Corridor Way, Coralville. Roundtables are social lunches over the noon hour. All are invited to network and keep up-to-date with chamber and community events. Free for members. Call the chamber at (319) 337-9637 if interested and not a member.
WEDG Annual Dinner, by Washington Economic Development Group, 5:30 p.m., Riverside Casino, 3184 Highway 22, Riverside. Network with 280-plus community leaders from around greater Washington County. Tickets: $40. To register, visit conta.cc/2EJUsWR.
Forward the FREE CBJ Business Daily newsletter to your friends and colleagues, and share the feeling of being informed! Use our fast, one-minute subscription to the CBJ's newsletters here, or check out our other subscription options here.
See something we missed? Send tips, leads, corrections, etc. to news@corridorbusiness.com.
Headlines from CBS2/FOX 28
These news items are provided by CBS2/FOX 28
The Iowa City Community School District is working on efforts to keep students safe in the classroom. District leaders say they currently use methods like controlled access to secure their facilities, although they don't have a constant law enforcement presence on school grounds. That could soon change, however. "Having officers present in the school will be a very significant change for the district," said Stephen Murley, superintendent of the Iowa City Community School District. Following the deadly mass shooting at Marjory Stoneman Douglas High School in Parkland, Florida last month, school resource officers are now being reconsidered in Iowa City. "One of the things we have experienced in the past couple of weeks is a heightened sense of awareness by our students," said Mr. Murley. With the school district considering a more constant approach, district leaders plan on creating a task force made up of parents and students to weigh in on any future security decisions. Read the full story here.
After months of speculation the Roughriders USHL team might skate down the Corridor to a new sports arena in Coralville, the club signed a deal Tuesday making Cedar Rapids "Rider Town USA" for at least 15 more years. The Cedar Rapids City Council unanimously approved the new long-term lease and Roughriders Hockey Club Co-President Doug Miller says everybody wins. "We couldn't be happier, we're excited to be here for many years to come. Cedar Rapids is a great town, it's an amazing city and it welcomed us for the last two decades with open arms," Mr. Miller said. The new agreement means the hockey team will have rent-free use of the Cedar Rapids Ice Arena and the city will receive all profits from the concessions. It also stipulates both the city and the team will invest more than $1 million into the 19-year-old ice facility over that time for improvements including new seats, new HDTV video capabilities, a new scoreboard, fan decks for group gatherings and more concession stands.
T hese news items are provided by CBS2/FOX 28
CBS2 Chief Meteorologist Terry Swails' Weather First Forecast
With plenty of sunshine and bare ground, temperatures will be climbing well into the 50s today. Winds will be kicking out of the southwest around 15-30 mph. A weak cold front will move in early this evening and switch the winds out of the north and winds will die down. Cooler air will move in as we head toward the end of the week. Temperatures will be in the mid to upper 40s Thursday and mid to upper 30s Friday. Our next disturbance moves in late Friday and will bring the chance for some light rain to a wintry mix to light snow overnight and into Saturday morning. There are still uncertainties as to where exactly the precipitation will set up and how much accumulation there will be. Good news is precipitation moves out early Saturday morning and it will be cool, dry and cloudy for St. Patrick's Day. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 6,649 |
package hr.hrg.watch.build;
import java.io.File;
import java.nio.file.Path;
import io.methvin.watcher.DirectoryChangeEvent.EventType;
public class FileDef {
public EventType eventType;
public Path path;
public File file;
public long lastEventTime;
public long lastModified;
public long length;
public FileDef(Path path, EventType eventType) {
this.path = path;
this.eventType = eventType;
this.file = path.toFile();
lastEventTime = System.currentTimeMillis();
lastModified = file.lastModified();
length = file.length();
}
public void update(EventType eventType) {
this.eventType = eventType;
lastEventTime = System.currentTimeMillis();
lastModified = file.lastModified();
length = file.length();
}
@Override
public int hashCode() {
return path.hashCode();
}
@Override
public boolean equals(Object obj) {
if(obj instanceof FileDef) {
return path.equals(((FileDef)obj).path);
}
return false;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,808 |
A plastic pent shed from The Plastic People is perfect if you need somewhere to store garden clutter, such as the lawn mower or child's toys. Organise the inside of your pent shed with our shed shelving too. Made with durable plastic that requires no maintenance, this is as ideal alternative to a wooden or metal shed.
The Skylight Pent Sheds provide modern, durable, safe and secure storage. The sheds look sleek with a nod to tradition from their standard tongue and groove shiplap cladding (seen on traditional timber sheds) but their real beauty is that they are made from virtually unbreakable, maintenance-free polycarbonate together with a rust resistant aluminium and galvanised steel frame and a non-slip floor. The design and materials mean these sheds provide safe, secure storage and have a very long, maintenance-free life span making them all round easy, excellent value.
The Skylight Pent Shed features an innovative opaque roof skylight in its roof which allows natural light in without the need for any further windows. This design provides maximum security and privacy, keeping your items from view, as well as UV protection. Harmful UV rays are filtered by the opaque skylight allowing contents in the shed protection from possible sun damage / fading. The sheds are also water-tight.
Unlike wooden or metal sheds, they do not crack, fracture, rust, rot, peel, bend or fade and therefore never need painting or staining saving you time and money. They are pretty much maintenance free - cleaning is a simple wipe down with hot, soapy water.
The Skylight Sheds come in size 6 x 4ft.
Home assembly required. Simple sliding panel assembly - please click on our Help & Advice tab above for Installation Instructions. The shed can be secured to the ground with the integrated anchoring system, providing a sturdy structure able to withstand all weather conditions. A video showing assembly can also be viewed here.
Please find links to assembly information for Skylight Sheds. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,518 |
Q: Refactoring for-loop Linq query into a single query I have a set of data that requires some processing but I want to split the work available threads. How can I change this into a single instruction, preferably removing the for-loop?
string[] keysForThread;
IEnumerable<string> allData;
List<string> dataForSingleThreadToProcess;
for (int i = 0; i < keys.length; i++)
dataForSingleThreadToProcess.AddRange(allData.Where(x => x.StartsWith(keys[i])));
I don't mind changing the string[] keysForThread into a List and/or the List dataForSingleThreadToProcessinto an IEnumerable.
A: It sounds like you want:
var data = allData.Where(datum => keys.Any(key => datum.StartsWith(key))
.ToList();
A: This is a direct way of doing what your code does without the for loop.
var data = keys.SelectMany(k => allData.Where(d => d.StartsWith(k)).ToList();
However, Jon skeet's answer will probably be more efficient.
A: This should do the job:
dataForSingleThreadToProcess = allData.Where(x => keys.Any(x.StartsWith)).ToList();
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,595 |
I got preempted for the Notre Dame fire just as Jack was asking Adrienne to see her credentials again.
Re: Did anybody get a full show today?
I lost literally 1-2 minutes at the end.
I only lost the last minute or so, but then at 2:00 p.m. Channel 7 that shows GH lost the entire hour.
I went to the recap page, so I'm caught up.
I got to see the whole show. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,508 |
namespace King.Service.ServiceBus.Wrappers
{
using Microsoft.Azure.ServiceBus;
using System;
using System.Collections.Generic;
using System.Linq;
using System.Threading;
using System.Threading.Tasks;
/// <summary>
/// Bus Subscription Client Wrapper
/// </summary>
public class BusSubscriptionClient : ISubscription
{
#region Members
/// <summary>
/// Subscription Client
/// </summary>
protected readonly ISubscriptionClient client = null;
#endregion
#region Constructors
/// <summary>
/// Default Constructor
/// </summary>
/// <param name="connection">Connection</param>
/// <param name="topic">Topic</param>
/// <param name="subscription">Subscription</param>
public BusSubscriptionClient(string connection, string topic, string subscription)
: this(new SubscriptionClient(connection, topic, subscription))
{
}
/// <summary>
/// Constructor
/// </summary>
/// <param name="client">Subscription Client</param>
public BusSubscriptionClient(ISubscriptionClient client)
{
if (null == client)
{
throw new ArgumentNullException("client");
}
this.client = client;
}
#endregion
#region Properties
/// <summary>
/// Subscription Client
/// </summary>
public virtual ISubscriptionClient Client
{
get
{
return this.client;
}
}
#endregion
#region Methods
/// <summary>
/// On Message
/// </summary>
/// <param name="callback">Call Back</param>
/// <param name="options">Options</param>
public virtual void OnMessage(Func<Message, CancellationToken, Task> callback, MessageHandlerOptions options)
{
this.client.RegisterMessageHandler(callback, options);
}
/// <summary>
/// Add Rule
/// </summary>
/// <param name="name">Name</param>
/// <param name="filter">Filter</param>
public async Task AddRule(string name, Filter filter)
{
await this.client.AddRuleAsync(name, filter);
}
#endregion
}
} | {
"redpajama_set_name": "RedPajamaGithub"
} | 5,321 |
{"url":"https:\/\/labs.tib.eu\/arxiv\/?author=M.%20Engelkemeier","text":"\u2022 ### Proposal for Quantum Simulation via All-Optically Generated Tensor Network States(1710.06103)\n\nWe devise an all-optical scheme for the generation of entangled multimode photonic states encoded in temporal modes of light. The scheme employs a nonlinear down-conversion process in an optical loop to generate one- and higher-dimensional tensor network states of light. We illustrate the principle with the generation of two different classes of entangled tensor network states and report on a variational algorithm to simulate the ground-state physics of many-body systems. We demonstrate that state-of-the-art optical devices are capable of determining the ground-state properties of the spin-1\/2 Heisenberg model. Finally, implementations of the scheme are demonstrated to be robust against realistic losses and mode mismatch.\n\u2022 MAGIC (Major Atmospheric Gamma Imaging Cherenkov) is a system of two 17 m diameter, F\/1.03 Imaging Atmospheric Cherenkov Telescopes (IACT). They are dedicated to the observation of gamma rays from galactic and extragalactic sources in the very high energy range (VHE, 30 GeV to 100 TeV). This submission contains links to the proceedings for the 35th International Cosmic Ray Conference (ICRC2017), held in Bexco, Busan, Korea from the 12th to the 17th of July, 2017.\n\u2022 Spontaneous breaking of Lorentz symmetry at energies on the order of the Planck energy or lower is predicted by many quantum gravity theories, implying non-trivial dispersion relations for the photon in vacuum. Consequently, gamma-rays of different energies, emitted simultaneously from astrophysical sources, could accumulate measurable differences in their time of flight until they reach the Earth. Such tests have been carried out in the past using fast variations of gamma-ray flux from pulsars, and more recently from active galactic nuclei and gamma-ray bursts. We present new constraints studying the gamma-ray emission of the galactic Crab Pulsar, recently observed up to TeV energies by the MAGIC collaboration. A profile likelihood analysis of pulsar events reconstructed for energies above 400GeV finds no significant variation in arrival time as their energy increases. Ninety-five percent~CL limits are obtained on the effective Lorentz invariance violating energy scale at the level of $E_{\\mathrm{QG}_1} > 5.5\\cdot 10^{17}$GeV ($4.5\\cdot 10^{17}$GeV) for a linear, and $E_{\\mathrm{QG}_2} > 5.9\\cdot 10^{10}$GeV ($5.3\\cdot 10^{10}$GeV) for a quadratic scenario, for the subluminal and the superluminal cases, respectively. A substantial part of this study is dedicated to calibration of the test statistic, with respect to bias and coverage properties. Moreover, the limits take into account systematic uncertainties, found to worsen the statistical limits by about 36--42\\%. Our constraints would have resulted much more competitive if the intrinsic pulse shape of the pulsar between 200GeV and 400GeV was understood in sufficient detail and allowed inclusion of events well below 400GeV.\n\u2022 The microquasar Cygnus X-1 displays the two typical soft and hard X-ray states of a black-hole transient. During the latter, Cygnus X-1 shows a one-sided relativistic radio-jet. Recent detection of the system in the high energy (HE; $E\\gtrsim60$ MeV) gamma-ray range with \\textit{Fermi}-LAT associates this emission with the outflow. Former MAGIC observations revealed a hint of flaring activity in the very high-energy (VHE; $E\\gtrsim100$ GeV) regime during this X-ray state. We analyze $\\sim97$ hr of Cygnus X-1 data taken with the MAGIC telescopes between July 2007 and October 2014. To shed light on the correlation between hard X-ray and VHE gamma rays as previously suggested, we study each main X-ray state separately. We perform an orbital phase-folded analysis to look for variability in the VHE band. Additionally, to place this variability behavior in a multiwavelength context, we compare our results with \\textit{Fermi}-LAT, \\textit{AGILE}, \\textit{Swift}-BAT, \\textit{MAXI}, \\textit{RXTE}-ASM, AMI and RATAN-600 data. We do not detect Cygnus X-1 in the VHE regime. We establish upper limits for each X-ray state, assuming a power-law distribution with photon index $\\Gamma=3.2$. For steady emission in the hard and soft X-ray states, we set integral upper limits at 95\\% confidence level for energies above 200 GeV at $2.6\\times10^{-12}$~photons cm$^{-2}$s$^{-1}$ and $1.0\\times10^{-11}$~photons cm$^{-2}$s$^{-1}$, respectively. We rule out steady VHE gamma-ray emission above this energy range, at the level of the MAGIC sensitivity, originating in the interaction between the relativistic jet and the surrounding medium, while the emission above this flux level produced inside the binary still remains a valid possibility.\n\u2022 MAGIC, a system of two imaging atmospheric Cherenkov telescopes, achieves its best performance under dark conditions, i.e. in absence of moonlight or twilight. Since operating the telescopes only during dark time would severely limit the duty cycle, observations are also performed when the Moon is present in the sky. Here we develop a dedicated Moon-adapted analysis to characterize the performance of MAGIC under moonlight. We evaluate energy threshold, angular resolution and sensitivity of MAGIC under different background light levels, based on Crab Nebula observations and tuned Monte Carlo simulations. This study includes observations taken under non-standard hardware configurations, such as reducing the camera photomultiplier tubes gain by a factor ~1.7 (Reduced HV settings) with respect to standard settings (Nominal HV) or using UV-pass filters to strongly reduce the amount of moonlight reaching the cameras of the telescopes. The Crab Nebula spectrum is correctly reconstructed in all the studied illumination levels, that reach up to 30 times brighter than under dark conditions. The main effect of moonlight is an increase in the analysis energy threshold and in the systematic uncertainties on the flux normalization. The sensitivity degradation is constrained to be below 10%, within 15-30% and between 60 and 80% for Nominal HV, Reduced HV and UV-pass filter observations, respectively. No worsening of the angular resolution was found. Thanks to observations during moonlight, the maximal duty cycle of MAGIC can be increased from ~18%, under dark nights only, to up to ~40% in total with only moderate performance degradation.\n\u2022 The large jet kinetic power and non-thermal processes occurring in the microquasar SS 433 make this source a good candidate for a very high-energy (VHE) gamma-ray emitter. Gamma-ray fluxes have been predicted for both the central binary and the interaction regions between jets and surrounding nebula. Also, non-thermal emission at lower energies has been previously reported. We explore the capability of SS 433 to emit VHE gamma rays during periods in which the expected flux attenuation due to periodic eclipses and precession of the circumstellar disk periodically covering the central binary system is expected to be at its minimum. The eastern and western SS433\/W50 interaction regions are also examined. We aim to constrain some theoretical models previously developed for this system. We made use of dedicated observations from MAGIC and H.E.S.S. from 2006 to 2011 which were combined for the first time and accounted for a total effective observation time of 16.5 h. Gamma-ray attenuation does not affect the jet\/medium interaction regions. The analysis of a larger data set amounting to 40-80 h, depending on the region, was employed. No evidence of VHE gamma-ray emission was found. Upper limits were computed for the combined data set. We place constraints on the particle acceleration fraction at the inner jet regions and on the physics of the jet\/medium interactions. Our findings suggest that the fraction of the jet kinetic power transferred to relativistic protons must be relatively small to explain the lack of TeV and neutrino emission from the central system. At the SS433\/W50 interface, the presence of magnetic fields greater 10$\\mu$G is derived assuming a synchrotron origin for the observed X-ray emission. This also implies the presence of high-energy electrons with energies up to 50 TeV, preventing an efficient production of gamma-ray fluxes in these interaction regions.\n\u2022 It is widely believed that the bulk of the Galactic cosmic rays are accelerated in supernova remnants (SNRs). However, no observational evidence of the presence of particles of PeV energies in SNRs has yet been found. The young historical SNR Cassiopeia A (Cas A) appears as one of the best candidates to study acceleration processes. Between December 2014 and October 2016 we observed Cas A with the MAGIC telescopes, accumulating 158 hours of good-quality data. We derived the spectrum of the source from 100 GeV to 10 TeV. We also analysed $\\sim$8 years of $Fermi$-LAT to obtain the spectral shape between 60 MeV and 500 GeV. The spectra measured by the LAT and MAGIC telescopes are compatible within the errors and show a clear turn off (4.6 $\\sigma$) at the highest energies, which can be described with an exponential cut-off at $E_c = 3.5\\left(^{+1.6}_{-1.0}\\right)_{\\textit{stat}} \\left(^{+0.8}_{-0.9}\\right)_{\\textit{sys}}$ TeV. The gamma-ray emission from 60 MeV to 10 TeV can be attributed to a population of high-energy protons with spectral index $\\sim$2.2 and energy cut-off at $\\sim$10 TeV. This result indicates that Cas A is not contributing to the high energy ($\\sim$PeV) cosmic-ray sea in a significant manner at the present moment. A one-zone leptonic model fails to reproduce by itself the multi-wavelength spectral energy distribution. Besides, if a non-negligible fraction of the flux seen by MAGIC is produced by leptons, the radiation should be emitted in a region with a low magnetic field (B$\\lessapprox$100$\\mu$G) like in the reverse shock.)\n\u2022 ### MAGIC observations of the microquasar V404 Cygni during the 2015 outburst(1707.00887)\n\nJuly 4, 2017 astro-ph.HE\nThe microquasar V404 Cygni underwent a series of outbursts in 2015, June 15-31, during which its flux in hard X-rays (20-40 keV) reached about 40 times the Crab Nebula flux. Because of the exceptional interest of the flaring activity from this source, observations at several wavelengths were conducted. The MAGIC telescopes, triggered by the INTEGRAL alerts, followed-up the flaring source for several nights during the period June 18-27, for more than 10 hours. One hour of observation was conducted simultaneously to a giant 22 GHz radio flare and a hint of signal at GeV energies seen by Fermi-LAT. The MAGIC observations did not show significant emission in any of the analysed time intervals. The derived flux upper limit, in the energy range 200--1250 GeV, is 4.8$\\times 10^{-12}$ ph cm$^{-2}$ s$^{-1}$. We estimate the gamma-ray opacity during the flaring period, which along with our non-detection, points to an inefficient acceleration in the V404\\,Cyg jets if VHE emitter is located further than $1\\times 10^{10}$ cm from the compact object.\n\u2022 B1957+20 is a millisecond pulsar located in a black widow type compact binary system with a low mass stellar companion. The interaction of the pulsar wind with the companion star wind and\/or the interstellar plasma is expected to create plausible conditions for acceleration of electrons to TeV energies and subsequent production of very high energy {\\gamma} rays in the inverse Compton process. We performed extensive observations with the MAGIC telescopes of B1957+20. We interpret results in the framework of a few different models, namely emission from the vicinity of the millisecond pulsar, the interaction of the pulsar and stellar companion wind region, or bow shock nebula. No significant steady very high energy {\\gamma}-ray emission was found. We derived a 95% confidence level upper limit of 3.0 x 10 -12 cm -2 s -1 on the average {\\gamma}-ray emission from the binary system above 200 GeV. The upper limits obtained with MAGIC constrain, for the first time, different models of the high-energy emission in B1957+20. In particular, in the inner mixed wind nebula model with mono-energetic injection of electrons, the acceleration efficiency of electrons is constrained to be below ~(2-10)% of the pulsar spin down power. For the pulsar emission, the obtained upper limits for each emission peak are well above the exponential cut-off fits to the Fermi-LAT data, extrapolated to energies above 50 GeV. The MAGIC upper limits can rule out a simple power-law tail extension through the sub-TeV energy range for the main peak seen at radio frequencies.\n\u2022 The extragalactic VHE gamma-ray sky is rich in blazars. These are jetted active galactic nuclei viewed at a small angle to the line-of-sight. Only a handful of objects viewed at a larger angle are known so far to emit above 100 GeV. Multi-wavelength studies of such objects up to the highest energies provide new insights into the particle and radiation processes of active galactic nuclei. We report the results from the first multi-wavelength campaign observing the TeV detected nucleus of the active galaxy IC 310, whose jet is observed at a moderate viewing angle of 10 deg - 20 deg. The multi-instrument campaign was conducted between 2012 Nov. and 2013 Jan., and involved observations with MAGIC, Fermi, INTEGRAL, Swift, OVRO, MOJAVE and EVN. These observations were complemented with archival data from the AllWISE and 2MASS catalogs. A one-zone synchrotron self-Compton model was applied to describe the broad-band spectral energy distribution. IC 310 showed an extraordinary TeV flare at the beginning of the campaign, followed by a low, but still detectable TeV flux. Compared to previous measurements, the spectral shape was found to be steeper during the low emission state. Simultaneous observations in the soft X-ray band showed an enhanced energy flux state and a harder-when-brighter spectral shape behaviour. No strong correlated flux variability was found in other frequency regimes. The broad-band spectral energy distribution obtained from these observations supports the hypothesis of a double-hump structure. The harder-when-brighter trend in the X-ray and VHE emission is consistent with the behaviour expected from a synchrotron self-Compton scenario. The contemporaneous broad-band spectral energy distribution is well described with a one-zone synchrotron self-Compton model using parameters that are comparable to those found for other gamma-ray-emitting misaligned blazars.\n\u2022 In this work we present data from observations with the MAGIC telescopes of SN 2014J detected in January 21 2014, the closest Type Ia supernova since Imaging Air Cherenkov Telescopes started to operate. We probe the possibility of very-high-energy (VHE; $E\\geq100$ GeV) gamma rays produced in the early stages of Type Ia supernova explosions. We performed follow-up observations after this supernova explosion for 5 days, between January 27 and February 2 in 2014. We search for gamma-ray signal in the energy range between 100 GeV and several TeV from the location of SN 2014J using data from a total of $\\sim5.5$ hours of observations. Prospects for observing gamma-rays of hadronic origin from SN 2014J in the near future are also being addressed. No significant excess was detected from the direction of SN 2014J. Upper limits at 95$\\%$ confidence level on the integral flux, assuming a power-law spectrum, d$F\/$d$E\\propto E^{-\\Gamma}$, with a spectral index of $\\Gamma=2.6$, for energies higher than 300 GeV and 700 GeV, are established at $1.3\\times10^{-12}$ and $4.1\\times10^{-13}$ photons~cm$^{-2}$s$^{-1}$, respectively. For the first time, upper limits on the VHE emission of a Type Ia supernova are established. The energy fraction isotropically emitted into TeV gamma rays during the first $\\sim10$ days after the supernova explosion for energies greater than 300 GeV is limited to $10^{-6}$ of the total available energy budget ($\\sim 10^{51}$ erg). Within the assumed theoretical scenario, the MAGIC upper limits on the VHE emission suggest that SN 2014J will not be detectable in the future by any current or planned generation of Imaging Atmospheric Cherenkov Telescopes.\n\u2022 ### Observations of Sagittarius A* during the pericenter passage of the G2 object with MAGIC(1611.07095)\n\nNov. 21, 2016 astro-ph.HE\nContext. We present the results of a multi-year monitoring campaign of the Galactic Center (GC) with the MAGIC telescopes. These observations were primarily motivated by reports that a putative gas cloud (G2) would be passing in close proximity to the super-massive black hole (SMBH), associated with Sagittarius A*, located at the center of our galaxy. This event was expected to give astronomers a unique chance to study the effect of in-falling matter on the broad-band emission of a SMBH. Aims. We search for potential flaring emission of very-high-energy (VHE; $\\geq$100 GeV) gamma rays from the direction of the SMBH at the GC due to the passage of the G2 object. Using these data we also study the morphology of this complex region. Methods. We observed the GC region with the MAGIC Imaging Atmospheric Cherenkov Telescopes during the period 2012-2015, collecting 67 hours of good-quality data. In addition to a search for variability in the flux and spectral shape of the GC gamma-ray source, we use a point-source subtraction technique to remove the known gamma-ray emitters located around the GC in order to reveal the TeV morphology of the extended emission inside that region. Results. No effect of the G2 object on the VHE gamma-ray emission from the GC was detected during the 4 year observation campaign. We confirm previous measurements of the VHE spectrum of Sagittarius A*, and do not detect any significant variability of the emission from the source. Furthermore, the known VHE gamma-ray emitter at the location of the supernova remnant G0.9+0.1 was detected, as well as the recently discovered VHE source close to the GG radio Arc.\n\u2022 ### A search for spectral hysteresis and energy-dependent time lags from X-ray and TeV gamma-ray observations of Mrk 421(1611.04626)\n\nNov. 14, 2016 astro-ph.HE\nBlazars are variable emitters across all wavelengths over a wide range of timescales, from months down to minutes. It is therefore essential to observe blazars simultaneously at different wavelengths, especially in the X-ray and gamma-ray bands, where the broadband spectral energy distributions usually peak. In this work, we report on three \"target-of-opportunity\" (ToO) observations of Mrk 421, one of the brightest TeV blazars, triggered by a strong flaring event at TeV energies in 2014. These observations feature long, continuous, and simultaneous exposures with XMM-Newton (covering X-ray and optical\/ultraviolet bands) and VERITAS (covering TeV gamma-ray band), along with contemporaneous observations from other gamma-ray facilities (MAGIC and Fermi-LAT) and a number of radio and optical facilities. Although neither rapid flares nor significant X-ray\/TeV correlation are detected, these observations reveal subtle changes in the X-ray spectrum of the source over the course of a few days. We search the simultaneous X-ray and TeV data for spectral hysteresis patterns and time delays, which could provide insight into the emission mechanisms and the source properties (e.g. the radius of the emitting region, the strength of the magnetic field, and related timescales). The observed broadband spectra are consistent with a one-zone synchrotron self-Compton model. We find that the power spectral density distribution at $\\gtrsim 4\\times 10^{-4}$ Hz from the X-ray data can be described by a power-law model with an index value between 1.2 and 1.8, and do not find evidence for a steepening of the power spectral index (often associated with a characteristic length scale) compared to the previously reported values at lower frequencies.\n\u2022 We describe and report the status of a neutrino-triggered program in IceCube that generates real-time alerts for gamma-ray follow-up observations by atmospheric-Cherenkov telescopes (MAGIC and VERITAS). While IceCube is capable of monitoring the whole sky continuously, high-energy gamma-ray telescopes have restricted fields of view and in general are unlikely to be observing a potential neutrino-flaring source at the time such neutrinos are recorded. The use of neutrino-triggered alerts thus aims at increasing the availability of simultaneous multi-messenger data during potential neutrino flaring activity, which can increase the discovery potential and constrain the phenomenological interpretation of the high-energy emission of selected source classes (e.g. blazars). The requirements of a fast and stable online analysis of potential neutrino signals and its operation are presented, along with first results of the program operating between 14 March 2012 and 31 December 2015.","date":"2020-02-21 16:33:59","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7060534358024597, \"perplexity\": 1757.3418535619849}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-10\/segments\/1581875145533.1\/warc\/CC-MAIN-20200221142006-20200221172006-00445.warc.gz\"}"} | null | null |
\section{Introduction}
The computer algebra systems, for instance {\it Mathematica} \cite{math}, are useful for making numeric and symbolic computations for ordinary differential equations and to visualize the results.
The Lane--Emden (LE) equation \cite{Mathematica_Lane-Emden} is one of the most important classical equations of mathematical physics. It originally appeared in astrophysics. It originates from studying static star structures and was proposed by Lane in \cite{Lane} and Emden in \cite{Emden}. Later on, it also appeared in kinetic theory, quantum mechanics and other fields (see \cite{Mach_Lane_Emden, Havas} and the references therein).
The LE equation is of the form
\begin{equation}
\triangle \theta + \theta^{p}=0,
\label{Lane-Emden_general}
\end{equation}
where $p$ is a natural number. Here $\theta=\theta(x_1,\ldots,x_d)$ and $d$ is space dimension. To preserve the reflection symmetry the nonlinear term in (\ref{Lane-Emden_general}) is usually rewritten as $|\theta|^{p-1}\theta$, however we will not consider this case. Equation (\ref{Lane-Emden_general}) appears in many physical and mathematical applications even in higher than three space dimensions, see, for instance, \cite{Chandrasekhar}, \cite{Benguria1}, \cite{Benguria2}, \cite{Kycia_PDE} and the references therein.
If the spherical symmetry is assumed, equation (\ref{Lane-Emden_general}) can be formulated on the positive semiline as
\begin{equation}
\theta '' +\frac{d-1}{x}\theta'+\theta^{p}=0,
\label{Lane-Emden_spherical}
\end{equation}
where $\theta=\theta(x)=\theta\left(\sqrt{\sum_{i=1}^d x_i^2}\right)$ and $'=d/dx$. This equation possesses the scaling symmetry, i.e., the scaled solution
\begin{equation}
\theta_{\lambda}(x)=\frac{1}{\lambda^{\alpha}}\theta\left(\frac{x}{\lambda}\right),\qquad \alpha=\frac{2}{p-1}
\label{scaling}
\end{equation}
is a solution as well. It enables us to generate solutions with different initial conditions from the normalized ones.
The simplest solution is of the form
\begin{equation}
\theta(x)=b_{\infty}\,x^{-\alpha}, \qquad b_{\infty}= \left(\frac{2(p(d-2)-d)}{(p-1)^{2}}\right)^{1/(p-1)}.
\label{b_solution}
\end{equation}
It can be obtained by assuming a power type form for $\theta(x)$, substituting it into the equation (\ref{Lane-Emden_spherical}) and equating the coefficients. The solution is singular at the origin and, therefore, it is of no direct physical importance, however it is of crucial importance in our further analysis. Other closed form solutions are known only for a few special values of $d$ and $p$. In general, solutions with the typical initial data $\theta(0)=1$, $\theta'(0)=0$ can be obtained in terms of the power series and they are well known. The discussion of movable singularities of solutions can be found in \cite{Lane_Emden_Hunter} and \cite{Lane_Emden_szeregi}. We also note that this equation falls in the class of equations considered in \cite{GF_RH}.
One of the nontrivial cases when the solution can be found in a closed form is the critical case when
\begin{equation}
p_{Q}=:p=\frac{d+2}{d-2}.
\label{critical_p}
\end{equation}
Note that it is the case when a functional, called the energy functional in physical applications,
\begin{equation}
E[\theta]=\int d^{d}x \left(\frac{1}{2}\theta'^{2}+\frac{1}{p+1}\theta^{p+1}\right)
\end{equation}
is scale invariant under (\ref{scaling}). The closed form solution for this case generalizes the Schuster and Emden solution for $d=3$ and it is sometimes called the generalized Talenti-Aubin solution, especially when one considers nonlinear wave equations \cite{Kycia_PDE}. It is of the form
\begin{equation}
\theta(x)=\frac{1}{(1+ax^{2})^{\alpha}},\qquad a=\frac{p-1}{4d}
\label{Talenti-Aubin_solution}
\end{equation}
where $p$ and $d$ satisfy (\ref{critical_p}).
\section{Main result}
The Lane--Emden equation can be analyzed by the Emden substitution \cite{Emden}
\begin{equation}
\theta=z\,x^{-\alpha},\qquad y=-\ln(x),
\label{Emden_substitution}
\end{equation}
which gives the equation
\begin{equation}
z''+\frac{2p-dp+d+2}{p-1}z'+\frac{2(2-d)p+2d}{(p-1)^{2}}z+z^{p}=0,
\label{equation_Emden_substitution}
\end{equation}
where $z=z(y)$ and $'=d/dy$. However, a slight modification of this substitution which generalizes \cite{Mach_Lane_Emden} in the form
\begin{equation}
\theta=b_{\infty}\,z\,x^{-\alpha},\qquad y=-\ln(x),
\label{Modified_Emden_substitution}
\end{equation}
where $b_{\infty}$ is given by (\ref{b_solution}), symmetrizes the last two terms and we obtain
\begin{equation}
z''+\frac{2p-dp+d+2}{p-1}z'+\frac{2(2-d)p+2d}{(p-1)^{2}}(z-z^{p})=0.
\label{equation_Modified_Emden_substitution}
\end{equation}
Both these equations can be interpreted as equations describing a particle moving with friction (the term with $z'$) in the field of forces given by the last two terms. For the later analysis we consider equation (\ref{equation_Modified_Emden_substitution}).
It is surprising that for the critical case the friction term vanishes and we get
\begin{equation}
z''-\frac{(d-2)^{2}}{4}(z-z^{p})=0,
\label{critical_equation_Modified_Emden_substitution}
\end{equation}
where $p$ and $d$ are connected by (\ref{critical_p}). Due to the lack of the dissipation the energy functional for (\ref{critical_equation_Modified_Emden_substitution}) is preserved \cite{Benguria2}.
The standard method of integration (by multiplying equation (\ref{critical_equation_Modified_Emden_substitution}) by $2z'$ and integrating) gives
\begin{equation}
(z')^{2}=\frac{(d-2)^{2}}{2}\left(\frac{1}{2} z^{2}-\frac{1}{p+1} z^{p+1}+C\right),
\label{critical_equation_integrated}
\end{equation}
where $C$ is a constant of integration. Thus, we see that the general solution of the LE equation in the critical case can be written by using the elliptic functions.
From the physical viewpoint we can only consider critical cases: (i) $d=3,\;p=5$; (ii) $d=4,\;p=3$ and (iii) $d=6,\;p=2$, and we should analyse only real solutions of equation (\ref{critical_equation_Modified_Emden_substitution}). The real solutions fall into the following classes (see \cite{Benguria2} for a simple proof): (1) positive or negative but not constant, or (2) sign changing, or (3) identically zero, or (4) the solution (\ref{b_solution}) and its negative counterpart when $p$ is odd in case the the nonlinear term in (\ref{Lane-Emden_general}) is rewritten as $|\theta|^{p-1}\theta$.
The case (i) is fully studied in \cite{Mach_Lane_Emden}. Our attention is focused here on cases (ii) and (iii). The motivation to study such cases is not only for the mathematical completeness, but also for the physical interpretation: the list of all solutions for the critical LE equation enables us to compare all critical cases and answer the question whether our three dimensional space is distinguished or not.
The form of the real solutions for (\ref{critical_equation_integrated}) depends on the number of zeros of the polynomial on the right hand side of the equation. In addition, all roots are constant solutions. Therefore, by (\ref{Modified_Emden_substitution}) they correspond to the solution (\ref{b_solution}).
\section{$d=4$, $p=3$}
In this case the equation (\ref{critical_equation_integrated}) has the form
\begin{equation}
(z')^{2} = \frac{1}{2}( -z^{4} + 2 z^{2} + C ),
\label{critical_equation_integrated_d=4_p=3}
\end{equation}
where $C$ is a new constant denoted here for simplicity by $C$.
Now the solution can be obtained by integrating
\begin{equation}
\pm \int \frac{dy}{\sqrt{2}}=\int \frac{dz}{\sqrt{-z^{4} + 2 z^{2} + C}}.
\label{differential_equation_d=4_p=3}
\end{equation}
The further analysis relies on the nonnegativeness of the polynomial
\begin{equation}
w_{43}(z)= -z^{4} + 2 z^{2} + C.
\label{polynomial_d=4_p=3}
\end{equation}
A few different cases can be distinguished:
\begin{itemize}
\item { $C < -1$: In this case $w_{43}(z) <0 $ for all real $z$ and, therefore, there are no real solutions.}
\item { $C = -1$: there are only two points $z=\pm 1$ for which $w_{43}(z)=0$, and they give the solutions that correspond to solution (\ref{b_solution});}
\item { $ C \in (-1;0)$: there are two disjoint sets on which $w_{42}(z) \ge 0$, namely $$z \in \left[-\sqrt{1+\sqrt{1+C}};-\sqrt{1-\sqrt{1+C}}\right] \cup \left[\sqrt{1-\sqrt{1+C}};\sqrt{1+\sqrt{1+C}}\right]; $$}
\item { $C = 0$: In this case $w_{43}(z) \ge 0$ when $z \in [-\sqrt{2}; \sqrt{2}]$;}
\item { $C > 0$: for every $ |z| < z_{0}$, where $z_{0}$ is positive real root of $w_{43}(z)=0$. In this case there are two real solutions and two imaginary ones.}
\end{itemize}
The Figure \ref{rysunek_w43_polynomial} illustrates the situation.
\begin{figure}[htp]
\centering
\includegraphics[scale=0.7]{43.eps}
\caption{The polynomial (\ref{polynomial_d=4_p=3}) for different values of $C$. Note the set of $z$ where $w_{43}(z) \ge 0$.}
\label{rysunek_w43_polynomial}
\end{figure}
All the cases will be analysed one by one in the following subsections.
\subsection{$ C \in (-1;0)$}
Let us focus on the case when $z \in \left[\sqrt{1-\sqrt{1+C}};\sqrt{1+\sqrt{1+C}}\right] $. The remaining case can be analysed in a similar way. Equation (\ref{differential_equation_d=4_p=3}) can be factorised as follows:
\begin{equation}
\pm \int \frac{dy}{\sqrt{2}} = hh' \int \frac{dz}{\sqrt{- \left( 1 - h^{2} z^{2} \right) \left( 1 - h'^{2} z^{2} \right) } },
\end{equation}
where
$$ h =\frac{1}{ \sqrt{1-\sqrt{1+C}} } > h'=\frac{1}{ \sqrt{1+\sqrt{1+C}}} > 0.$$
The integral
\begin{equation}
I=hh'\int_{\frac{1}{h}}^{z_{0} \leq \frac{1}{h'} } \frac{dz}{\sqrt{- \left( 1 - h^{2} z^{2} \right) \left( 1 - h'^{2} z^{2} \right) } }
\end{equation}
can be brought to the Jacobian elliptic integral \cite{Wang_Guo}, \cite{NIST_math_functions} by the standard substitution \cite{Fichtenholtz}, \cite{Wang_Guo}:
\begin{equation}
h' z =\sqrt{1-\frac{h^{2}-h'^{2}}{h^{2}}u^{2}},
\label{substitution_Fichtengoltz_C_(-10)}
\end{equation}
where now $0 < u <1$. The substitution gives
$$I=h' \int_{0}^{u_{0}} \frac{du}{\sqrt{ ( 1-u^{2}) (1-k^{2}u^{2}) } }=h' arcsn(u_{0},k),$$
where the elliptic modulus is of the form
$$k=\sqrt{\frac{h^{2}-h'^{2}}{h^{2}}} = \sqrt{\frac{2\sqrt{C+1}}{1+\sqrt{C+1}}},$$
$u_{0}(z_{0})$ can be obtained from (\ref{substitution_Fichtengoltz_C_(-10)}) and $arcsn$ is the inverse of the Jacobian elliptic function $sn()$.
Returning to the original variables $\theta$, $x$ we get the solution
\begin{equation}
\begin{array}{l}
\theta(x) = \pm\frac{1}{x}\sqrt{\sqrt{1+\sqrt{1+C}} (1-k^{2}y^{2}(x))}, \\
y(x)=sn(\pm\sqrt{1+\sqrt{1+C}} \frac{\ln(Bx)}{\sqrt{2}},k),
\end{array}
\end{equation}
where $B$ is a constant of integration.
\subsection{$C=0$}
In this case the Talenti-Aubin solution can be recovered as follows. From
$$\pm \int \frac{dy}{\sqrt{2}}=\int \frac{dz}{\sqrt{-z^{4} + 2 z^{2}}}$$
after some simple manipulations one gets
$$z=\sqrt{2}sech(\pm \ln(Bx) )=\pm 2\sqrt{2} Bx \frac{1}{1+(Bx)^{2}},$$
where, as previously, $B$ is an integration constant and $y=-\ln(x)$. Substituting $2\sqrt{2}B =\frac{1}{\lambda}$ and $\theta (x) = \frac{z}{x}$ we obtain the scaled (see (\ref{scaling})) Talanti-Aubin solution (\ref{Talenti-Aubin_solution})
$$\theta(x)=\pm \frac{1}{\lambda} \frac{1}{1+\frac{1}{8}(x/\lambda)^{2}}.$$
\subsection{$C>0$}
In this case the Jacobian elliptic integrals can also be used. Equation (\ref{differential_equation_d=4_p=3}) can be rewritten in the form
\begin{equation}
\pm \int \frac{dy}{\sqrt{2}} = hh' \int \frac{dz}{\sqrt{ \left( 1 - h^{2} z^{2} \right) \left( 1 + h'^{2} z^{2} \right) } },
\end{equation}
where now
$$ 0< h =\frac{1}{ \sqrt{1+\sqrt{1+C}} } < h'=\frac{1}{ \sqrt{\sqrt{1+C}-1}}.$$
The standard change of variable \cite{Fichtenholtz}
$$hz=\sqrt{1-u^{2}}$$
in the integral
$$I=hh' \int_{\frac{1}{h}}^{z_{0} \leq \frac{1}{h}} \frac{dz}{\sqrt{ \left( 1 - h^{2} z^{2} \right) \left( 1 + h'^{2} z^{2} \right) } }$$
gives ($0 < u \leq 1$)
$$I=-\frac{hh'}{\sqrt{h^{2}+h'^{2}}} \int_{0}^{u_{0}} \frac{du}{\sqrt{(1-u^{2})(1-k^{2}u^{2})}}=-\frac{hh'}{\sqrt{h^{2}+h'^{2}}} arcsn(u_{0},k),$$
where the elliptic modulus is defined by
$$k=\sqrt{\frac{h'^{2}}{h^{2}+h'^{2}}}=\sqrt{\frac{1+\sqrt{C+1}}{2\sqrt{C+1}}}.$$
Therefore, denoting an integration constant by $B$, we obtain
$$\pm \frac{\ln(Bx)}{\sqrt{2}}=-\frac{1}{\sqrt{2\sqrt{C+1}}}arcsn(u_{0}(z_{0}),k),$$
which gives
\begin{equation}
\begin{array}{l}
\theta(x)=\pm \frac{1}{x} \sqrt{(1+\sqrt{C+1})(1-y^{2}(x))}, \\
y(x)=sn(\pm\sqrt{2\sqrt{C+1}}\frac{\ln(Bx)}{\sqrt{2}},k).
\end{array}
\end{equation}
\section{$d=6$, $p=2$}
In this case the equation (\ref{critical_equation_integrated}) is of the form
\begin{equation}
(z')^{2} = \frac{4}{3}( -2z^{3} + 3 z^{2} + C )
\label{critical_equation_integrated_d=6_p=2}
\end{equation}
and we obtain
\begin{equation}
\frac{dz}{\sqrt{-2z^{3} + 3 z^{2} + C}} = \pm \frac{2dy}{\sqrt{3}}.
\label{differential_equation_d=6_p=2}
\end{equation}
The nonnegativeness of the polynomial
\begin{equation}
w_{62}(z)= -2z^{3} + 3 z^{2} + C
\label{polynomial_d=6_p=2}
\end{equation}
determines possible solutions for different $C$ values.
We have to consider the following cases
\begin{itemize}
\item { $C<-1$: the polynomial $w_{62}(z) \geq 0 $ for $z < z_{0}$, where $z_{0}$ is the only real solution of $w_{62}(z)=0$;}
\item { $C=-1$: we have the real root $z_0$ of $w_{62}(z) = 0$ such that for $z < z_{0}< 0$ we get $w_{62}(z) \geq 0$, and the second real root at $z=1$; }
\item { $C \in (-1;0)$: there are three real roots $a<b<c$ of $w_{62}(z)=0$, therefore for $z \in (-\infty,a] \cup [b,c]$ the polynomial $w_{62}(z)$ is nonnegative;}
\item {$C=0$: we have exactly solution (\ref{Talenti-Aubin_solution});}
\item {$C>0$: there is only one real root $z_{0}$ of $w_{62}(z)=0$, and $w_{62}(z)$ is nonnegative for $z <z_{0}$.}
\end{itemize}
The cases above can be illustrated by the plot (\ref{rysunek_w62_polynomial}).
\begin{figure}[htp]
\centering
\includegraphics[scale=0.7]{62.eps}
\caption{The polynomial (\ref{polynomial_d=6_p=2}) for different values of $C$. Note the set of $z$ where $w_{62}(z) \ge 0$.}
\label{rysunek_w62_polynomial}
\end{figure}
In the following analysis the Weierstrass representation of elliptic integrals \cite{Wang_Guo} will be used. However, this representation can be easily transformed into the Jacobian elliptic integral form using transformations described in \cite{Wang_Guo}.
\subsection{$C<-1$, $C >0$ and $C=1$}
By integrating (\ref{differential_equation_d=6_p=2}) we get the equation
\begin{equation}
\pm \frac{2}{\sqrt{3}} y + B = \int_{-\infty}^{z<z_{0}} \frac{dz}{\sqrt{-2z^{3} + 3 z^{2} + C}},
\label{integral_equation_d=6_p=2_case_1}
\end{equation}
where $B$ is an integration constant. The integral on the right-hand side of the (\ref{integral_equation_d=6_p=2_case_1}) can be brought to the standard form of the Weierstrass elliptic integral \cite{Wang_Guo}, \cite{NIST_math_functions}
$$z=\int_{\infty}^{\xi}\frac{du}{\sqrt{4u^{3}-g_{2}u-g_{3}}}, \quad \wp(z,g_{2},g_{3})=\xi,$$
where $\wp$ is the Weierstrass elliptic function, by the following change of variables \cite{Wang_Guo}
$$z=-2u+\frac{1}{2}.$$
The integral then transforms into the form
$$I=-\int_{\infty}^{u_{0}} \frac{du}{\sqrt{4u^{3}-\frac{3}{4}u- \frac{1}{4}(-C-\frac{1}{2})}}.$$
Hence, by the fact that the Weierstrass function is even, the solution is of the form
\begin{equation}
\theta(x)=4x^{-2}\left(\frac{1}{2}- 2\wp \left( \frac{2}{\sqrt{3}} \ln(Bx),\frac{3}{4},-\frac{1}{4} \left( C + \frac{1}{2} \right) \right) \right),
\end{equation}
where $B$ is a different integration constant denoted for simplicity by the same letter.
For $C=1$, there is also $z=1$ value for which $w_{62}(1)=0$. This case corresponds to the solution (\ref{b_solution}).
\subsection{$C \in (-1;0)$}
In this case, for $z < a$ the solution is the same as in the proceeding case. For $ z \in [b,c]$ we can use a simple formula to relate the integrals:
$$\int_{\frac{1}{4}-\frac{b}{2}}^{u_{0}<\frac{1}{4}-\frac{c}{2}} f(x)dx= \int_{\infty}^{u_{0}} f(x)dx - \int_{\infty}^{\frac{1}{4}-\frac{b}{2}} f(x)dx.$$
The last elliptic integral is in general not real ($w_{62}(u)<0$), therefore this case is not interesting in physical applications.
\subsection{$C=0$}
In this case it can be shown that the solution is the scaled solution (\ref{Talenti-Aubin_solution}) in the following way.
The equation can be integrated
$$\int \frac{dz}{\sqrt{-2z^{3} + 3 z^{2}}} = \pm \int \frac{2dy}{\sqrt{3}},$$
which gives after using $y=-\ln(x)$
$$z=\frac{3}{2}( 1 - tanh^{2}(\pm \ln(Bx))=\frac{6(Bx)^{2}}{(1+(Bx)^{2})^{2}}.$$
Setting $2\sqrt{3}B=\frac{1}{\lambda}$ and restoring $\theta(x)=4\frac{z}{x^{2}}$ we obtain the scaled solution (\ref{Talenti-Aubin_solution})
$$\theta(x)=\frac{1}{\lambda^{2}}\frac{1}{(1+\frac{1}{24}( x / \lambda)^{2})^{2}}.$$
\section{Conclusions}
The analysis presented in this paper is an extension of the results presented in \cite{Mach_Lane_Emden}. We see that similar features can be found. The most characteristic are the cases when $C=-1$ when the solution (\ref{b_solution}) appears, and the cases $C=0$ when the solution (\ref{Talenti-Aubin_solution}) is recovered. In other cases elliptic integrals are involved. This shows that the critical $d$, $p$ cases are quite similar to each other.
Plots for $d=4$, $p=3$ and $d=6$, $p=2$ cases are presented in Figure \ref{rysunek_43_62_all_solutions}.
\begin{figure}[htp]
\centering
\includegraphics[scale=0.7]{43all.eps}
\includegraphics[scale=0.7]{62all.eps}
\caption{Left panel presents all solutions of $d=4$, $p=3$ Lane-Emden equation. On the right panel there are all solutions of $d=6$, $p=2$ Lane-Emden equation. Solid line - (\ref{b_solution}) and Talenti-Aubin, dashed lines $C=-2$, $C=-1$, $C=-0.5$, $C=1$, $C=2$. Logarithmic scale on $0X$ axis is used. }
\label{rysunek_43_62_all_solutions}
\end{figure}
One can also observe that when the space dimension $d$ increases then $p$ related by (\ref{critical_p}) decreases, and the polynomial order of the right-hand side of (\ref{critical_equation_integrated}) decreases therefore some solutions that are present in $d=3$, $p=5$, namely the Srivastava solutions \cite{Mach_Lane_Emden}, are missing.
\section{Acknowledgements}
RK is supported by the Warsaw Center of Mathematics and
Computer Science from the founds of the Polish Leading National Research
Centre (KNOW). RK is grateful to Patryk Mach for drawing his attention to paper \cite{Mach_Lane_Emden}.
GF is supported by NCN grant 2011/03/B/ST1/00330.
\par
\small
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{"url":"https:\/\/math.stackexchange.com\/questions\/2268062\/solve-the-following-differential-eqn-with-constant-variables","text":"# Solve the following differential eqn with constant variables\n\nIn this image you can see an Ex 3 I have formed the auxilary Eqn i, but i am not able to solve the auxilary eqn with degree 4 , and find the roots of the eqn\n\n\u2022 There are \"obvious\" roots to $m^4-m^3-9 m^2-11 m-4=0$; at least, $m=-1$ is a double root. Just continue ! \u2013\u00a0Claude Leibovici May 6 '17 at 5:02\n\u2022 @projectilemotion. I know that; I just wanted the user to finish the work ! Cheers. \u2013\u00a0Claude Leibovici May 6 '17 at 6:04\n\nNotice that at one point during your calculations, you somehow 'converted' a minus sign into a plus sign. Your auxilliary equation should be: $$m^4-m^3-9m^2-11m\\color{red}{-}4=0$$ Using the Rational Root Theorem, we know that $m=4$ is a solution to your auxilliary equation.\nUsing polynomial division, we obtain: $$\\frac{m^4-m^3-9m^2-11m-4}{m-4}=m^3+3m^2+3m+1, m\\neq 4$$ Notice that this may be factored to give $(m+1)^3$. Hence, $m=-1$ is a root of multiplicity $3$.","date":"2019-08-22 04:34:54","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8651567101478577, \"perplexity\": 309.4094451063491}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-35\/segments\/1566027316783.70\/warc\/CC-MAIN-20190822042502-20190822064502-00380.warc.gz\"}"} | null | null |
The G&L Roundtable is a private invitation-only G&L Community forum that shapes the landscape of innovation and business processes in gaming and hospitality. The G&L Roundtable hosts the most gaming CIOs in one private forum in North America and presents an amazing opportunity for G&L IT industry leaders.
Join Avatier Vice President of Support Services, Chris Arnold, to discuss the information security and enterprise risks posed by your employees and system administrators.
Avatier will be joining our G&L IT colleagues at the Siena Golf Course for informal knowledge sharing on topics relevant to the industry and information security. As the best identity management solution available to G&L IT operations, Avatier looks forward to participating and gaining from the event.
Through discussions on ways to innovate G&L enterprise risk management and information security, Avatier is grateful to be recognized as an identity management industry leader. As in previous years, we look forward to the opportunity to align our product roadmap with our customers' identity and access management needs.
Avatier Vice President of Support Services, Chris Arnold, is proud to MC the G&L Roundtable Award Reception. Each year the G&L Roundtable recognizes exceptional colleagues who have contributed to the G&L community.
Chris will be presenting award recipients with their plaques during the reception. As a regular participant in the G&L Roundtable and awards ceremony, Avatier is honored and looks forward to its role as MC of this prestigious award's ceremony. | {
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{"url":"http:\/\/math.stackexchange.com\/questions\/116807\/convert-ellipse-parameter-from-general-parametric-form-to-general-polar-form\/116812","text":"# Convert ellipse parameter from General parametric form to General polar form\n\nI am facing problem to convert ellipse standard parameters. Everything I say here is refer to http:\/\/en.wikipedia.org\/wiki\/Ellipse\n\nI know what are the General parametric form parameter . Lets call them $a$,$b$,$\\varphi$, $t_X$, $t_Y$\n\nNow I need to find the general polar form parameter. I follow the equation in wikipedia But I may misunderstand what it says.\n\nHere is what I think\n\n$$r_0 = \\sqrt{tx^2+ty^2}$$ $$\\theta_0 = \\tan^{-1} \\frac{t_Y}{t_X}$$ and $$\\phi = \\varphi$$\n\nThe following is my MATLAB code, the upper section is the general parametric form, I also draw the ellispe. The lower section is my incorrect general polar form\n\n%==================\n\nclose all\n\n data = [0.6397 0.9520 15.9195 1.1430 -0.3844];\n\na = data(1);\nb = data(2);\nang = data(3);\ntranX = -data(4);\ntranY = -data(5);\n\nx = zeros(1,3601);\ny = zeros(1,3601);\n\ncounter = 1;\n\nfor t = 0:.1:360\nx(counter) = tranX + a*cosd(t)*cosd(ang) - b*sind(t)*sind(ang);\ny(counter) = tranY + a*cosd(t)*sind(ang) + b*sind(t)*cosd(ang);\ncounter=counter+1;\nend\n\nfigure;plot(x,y)\n\n%=============================================\n\nr0 = norm([tranX tranY]);\ntheta0 = atand(tranY\/tranX);\nrho = ang;\nrr = zeros(1,3601);\ncounter = 1;\n\nfor t = 0:.1:360\nP(counter) = r0*[(b*b-a*a)*cosd(t+theta0-2*rho)+(a*a+b*b)*cosd(t-theta0)];\nR(counter) = (b*b-a*a)*cosd(2*t-2*rho)+a*a+b*b;\nQ(counter) = sqrt(2)*a*b*sqrt(R(counter)-2*r0*r0*(sind(t-theta0))^2);\nrr(counter) = (P(counter)+Q(counter))\/R(counter);\ncounter = counter + 1;\nend\n\n[XX,YY] = pol2cart((0:.1:360)*2*pi\/180,rr);\n\nfigure; plot(XX,YY,'.')\n\n-\nPlease, write in $\\LaTeX$. You can learn some codes like, ${x}^{2n+1}$ makes ${x}^{2n+1}$, $\\sin \\theta$ makes $\\sin \\theta$ and $T_{2n}$ makes $T_{2n}$ \u2013\u00a0 Pedro Tamaroff Mar 5 '12 at 21:02\n\nMaybe $r=\\sqrt{t_X^2+t_Y^2}$. Else you got a part of a circle.","date":"2014-08-28 03:15:31","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8204132914543152, \"perplexity\": 2430.045495321226}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2014-35\/segments\/1408500830074.72\/warc\/CC-MAIN-20140820021350-00007-ip-10-180-136-8.ec2.internal.warc.gz\"}"} | null | null |
\section{Introduction}
To fault-tolerantly operate a scalable universal quantum computer, one protects logical information using a quantum error-correcting code, and removes errors without disturbing the encoded information \cite{Shor1996, Gottesman1996, Preskill1998}.
This can be achieved with stabilizer codes.
Each stabilizer generator is measured, yielding an outcome $\pm 1$, and a classical decoding algorithm then computes the recovery operator.
Unfortunately, optimal decoding of generic stabilizer codes is computationally hard \cite{Hsieh2011,Iyer2015}.
Thus, to render this task tractable one should restrict attention to codes with some structure.
Topological stabilizer codes \cite{Kitaev2003, Bravyi1998, Bombin2006, Haah2011,Bombin2013book,Kubica2015}, such as the toric and color codes, have a lot of structure due to the geometric locality of their stabilizer generators.
Namely, any stabilizer returning a $-1$ measurement outcome indicates the presence of errors in its neighborhood.
By exploiting this syndrome pattern, many efficient decoders with high error-correction thresholds have been proposed \cite{Dennis2002,Wang2009,Wang2011,Duclos-Cianci2010,Bravyi2013a,Fowler2012a,Anwar2013, Delfosse2014,Delfosse2017a,Delfosse2017,Nickerson2017,Maskara2018}.
However, most of these decoders use global classical information about the measurement outcomes and thus require communication between distant parts of the system.
In any realistic setting, new faults appear during the time needed to collect and process global syndrome data~\cite{Fowler2012, Terhal2015}.
Thus, to avoid error accumulation we desire fast decoders, which ideally use only local information.
A very promising class of topological quantum code decoders is based on cellular automata (CA) \cite{Harrington2004, Herold2015, Dauphinais2017}.
CA decoders are very efficient because they naturally incorporate parallelization and can be implemented on dedicated hardware without any non-local communication.
As initially suggested in Ref.~\cite{Dennis2002}, a simple CA, called Toom's rule \cite{Toom1980, Bennett1985,Grinstein2004}, can successfully protect quantum information encoded into the 4D toric code on a hypercubic lattice.
Moreover, recent numerical simulations \cite{Pastawski2011,Breuckmann2017,Breuckmann2017a} indicate that heuristic decoders based on Toom's rule have non-zero error-correction thresholds for higher-dimensional toric codes.
In this article we address the fundamental question whether using a CA is a viable error-correction strategy for topological quantum codes.
First, we propose a new CA, the Sweep Rule, which is a generalization of Toom's rule to any locally Euclidean lattice in $d\geq 2$ dimensions.
The Sweep Rule shrinks $(k-1)$-dimensional domain walls for any $k = 2,\ldots, d$.
Then, we use the Sweep Rule to design a new local decoder of the toric code in $d\geq 3$ dimensions, the Sweep Decoder, and rigorously prove a lower bound on its performance for perfect syndrome extraction.
Finally, we numerically demonstrate successful error suppression using a noisy version of the Sweep Rule.
In particular, we estimate the sustainable threshold error rate ${p^{\textrm{bcc}}_{\textrm{sus}} = 0.99\pm 0.02\%}$ of the Sweep Decoder for phase-flip errors and imperfect syndrome measurements in the 3D toric code on the body-centered cubic (bcc) lattice; see Fig.~\ref{fig_numerics}.
Our decoding scheme works reliably against Pauli $X$ or Pauli $Z$ errors if the corresponding syndrome is at least one-dimensional and the error rate is below the theshold value; thus it can protect topological quantum memories in $d\geq 4$ dimensions.
Our results also lead to new CA decoders for the color code in $d\geq 3$ dimensions, presented in the accompanying article~\cite{cc_decoder}.
\begin{figure}[h!]
\centering
\includegraphics[width=.9\columnwidth]{figures/fig_numerics}
\caption{
(Inset) The failure probability $p_{\textrm{fail}}(p,L)$ of the Sweep Decoder for the 3D toric code on the bcc lattice $\mathcal{L}$ after $N_{\textrm{cyc}} = 2^8$ correction cycles, where $p$ is the phase-flip error rate and $L$ is the linear size of $\mathcal{L}$.
We estimate the threshold $p_{\textrm{th}}(N_{\textrm{cyc}}) \approx 1.055\%$ from the crossing point of different curves.
(Main) We find the sustainable threshold $p^{\textrm{bcc}}_{\textrm{sus}} = 0.99\pm 0.02\% $ by fitting the numerical ansatz from Eq.~(\ref{eq_ansatz_pheno}) to the data.
}
\label{fig_numerics}
\end{figure}
\sectionprl{Limitations of Toom's rule}
Consider the square lattice with a classical $\pm 1$ spin placed on every face and encode one bit of information by setting all spins to be either $+1$ or $-1$.
We want to protect the encoded bit against random spin flips, $\pm 1\mapsto \mp 1$.
This can be achieved with a CA, which flips certain spins based on \textit{locally} available information.
A simple example is the deterministic Toom's rule which sets the spin $s_C^{(T+1)}$ at time $T+1$ to
\begin{equation}
s^{(T+1)}_C = \sgn\left(s^{(T)}_C + s^{(T)}_E + s^{(T)}_N\right),
\label{eq_toom}
\end{equation}
where $\sgn(\cdot)$ is the sign function, {$s^{(T)}_E$ and $s^{(T)}_N$ are the neighboring spins on faces to the east and north at time $T$; see Fig.~\ref{fig_Toom}(a)}
The update can be simultaneously applied to all the spins in the square lattice.
\begin{figure}[t!]
\includegraphics[width=.95\columnwidth]{figures/fig_Toom}
\caption{
(a)
At time $T$ the spin $s^{(T)}_C = -1$ (green face) differs from its neighbors to the east $s^{(T)}_E = 1$ and north $s^{(T)}_N = 1$ (red faces).{ According to Eq.~(\ref{eq_toom}), Toom's rule sets} $s^{(T+1)}_C = 1$.
(b) A 2D lattice built of three types of parallelograms with a domain wall (red), which cannot be removed by repeated application of a naive generalization of Toom's rule.
(c) The 3D toric code on the bcc lattice \cite{vzome} has qubits on faces and $X$-stabilizers associated with edges.
Any configuration of $Z$ errors (green) results in a 1D loop-like $X$-syndrome (red).
}
\label{fig_Toom}
\end{figure}
We can rephrase Toom's rule as a conditional spin update determined by the local configuration of the 1D domain wall, i.e., the set of all edges of the lattice separating faces with spins of different value.
Let $\epsilon^{(T)}$ and $\sigma^{(T)}$ denote the set of faces with $-1$ spins and the corresponding domain wall at time $T=1,2,\ldots$.
We write $\sigma^{(T)} = \partial_2 \epsilon^{(T)}$ to capture the fact that $\sigma^{(T)}$ is the boundary of $\epsilon^{(T)}$ containing all the edges bounding faces in $\epsilon^{(T)}$.
Then, Toom's rule flips a spin on some face $f$, i.e., $s^{(T+1)}_f = - s^{(T)}_f$, iff the east and north edges of $f$ belong to $\sigma^{(T)}$; see Fig.~\ref{fig_Toom}(a).
If we know $\sigma^{(T)}$ and the set of all spins flipped between time $T$ and $T+1$, which we denote by $\rho^{(T)}$, then the domain wall at time $T+1$ is
\begin{equation}
\sigma^{(T+1)} = \sigma^{(T)} + \partial_2 \rho^{(T)}.
\end{equation}
Note that this update does not require the knowledge of the actual spin values but only the locations of flipped spins, and from that perspective it may be viewed as a local rule governing the dynamics of the domain wall.
Moreover, if the domain wall disappears by time $T$,
i.e., $\sigma^{(T)} = 0$, then $\rho = \sum_{i=1}^{T-1} \rho^{(i)}$ can be viewed as an estimate
\footnote{This strategy, however, is neither guaranteed to terminate nor to return $\rho = \epsilon^{(1)}$.}
of $\epsilon^{(1)}$ with the boundary $\partial_2 \rho$ matching the initial domain wall $\sigma^{(1)}$.
As we will see later, correcting errors in the toric code in $d \geq 3$ dimensions can also be rephrased as estimating $\epsilon^{(1)}$ given its boundary $\sigma^{(1)}$, by exploiting the domain-wall structure of the syndrome.
This version of Toom's rule works for the square lattice, but it is not obvious how to generalize it to other 2D lattices, or to higher dimensions.
To illustrate the difficulty, consider the 2D lattice in Fig.~\ref{fig_Toom}(b).
If one uses a simple update rule \textit{``flip a spin iff east and north edges of the face belong to the domain wall''}, then there exist spin configurations with domain walls which cannot be removed by repeated application of this rule.
For such error syndromes, the Toom's rule decoder fails to correct the erroneous spins.
To define a workable version of Toom's rule, the lattice must have suitable properties, which we now specify.
\sectionprl{Causal lattices}
We consider a lattice $\mathcal{L}$, which is a triangulation (possibly without any symmetries) of the Euclidean space $\mathbb{R}^2$.
We denote by $\face i {\mathcal{L}}$ the set of all $i$-simplices of $\mathcal{L}$.
In particular, $\face 0 {\mathcal{L}}$, $\face 1 {\mathcal{L}}$ and $\face 2 {\mathcal{L}}$ correspond to vertices, edges and triangular faces of $\mathcal{L}$.
We assume that each $\face i {\mathcal{L}}$ contains countably many elements and define the sweep direction as a unit vector $\vec t \in \mathbb{R}^2$ not perpendicular to any edge of $\mathcal{L}$.
We define a path $\pth u w$ between two vertices $u$ and $w$ of the lattice $\mathcal{L}$ to be a collection of edges
{$(u,v_1),\ldots,(v_{n},w) \in\face{1}{\mathcal{L}}$, where $v_i\in\face{0}{\mathcal{L}}$.}
If the sign of the inner product $\vec{t}\cdot (v_i,v_{i+1})$ is the same for all edges in the path $\pth u w$, then we call the path causal and denote it by $\cpth u w$.
We remark that any pair of the vertices of $\mathcal{L}$ is connected by a path but there might not exist a causal path between them; see Fig.~\ref{fig_causal_structure}(a).
Finally, we define the causal distance
\begin{equation}
\label{eq_cdist}
\cdist{u,w} = \min_{\cpth u w} |\cpth u w |
\end{equation}
to be the length of the shortest causal path between $u$ and $w$; if there is no causal path, then $\cdist{u,w} = \infty$.
We observe that the sweep direction $\vec t$ induces a binary relation $\preceq$ over the set of vertices $\face{0}{\mathcal{L}}$.
We say that $u$ precedes $w$, i.e., $u\preceq w$ for $u,v\in \face 0 {\mathcal{L}}$, iff there exists a causal path $\cpth u w$ and $\vec{t} \cdot (v_i,v_{i+1}) > 0 $ for any edge $(v_i,v_{i+1})\in \cpth u w$.
Equivalently, we write $w\succeq u$ and say that $w$ succeeds $u$.
Abusing the notation, we write $v \preceq \kappa$ if all vertices $\face 0 \kappa$ of a $k$-simplex $\kappa\in\face k {\mathcal{L}}$ succeed $v$, i.e., $v\preceq u$ for all $u\in\face 0 \kappa$; similarly for $\kappa \preceq v$.
\begin{figure}[t!]
\centering
\includegraphics[width = .95\columnwidth]{figures/fig_Sweep_Rule}
\caption{
(a) {Vertices} $u$ and $v$ are connected by a path $\pth u v$ (red), but there is no causal path between them; $v$ and $w$ are connected by a causal path $\cpth u v $ (blue).
We shaded in green and blue the future $\uset v$ and past $\dset v$ of $v$.
(b) The causal diamond $\cdia V$ (blue) of a subset of vertices $V = \{ v_1,v_2,v_3,v_4\}$ is defined as the intersection of the future of the infimum of $V$ with the past of the supremum of $V$.
(c) The Sweep Rule is defined for every vertex and locally updates $\pm 1$ spins on neighboring faces.
Since the vertex $v$ is trailing, spins on two green faces will be flipped.
}
\label{fig_causal_structure}
\end{figure}
We can think of the partial order $\preceq$ between vertices of the lattice as a causality relation between points in the discretized $(1+1)$D spacetime with $\vec t$ corresponding to the time
\footnote{We warn the reader that later we use the time $T$ to index how many times the CA rule is applied.}
direction; see Fig.~\ref{fig_causal_structure}(a)(b).
We define the future $\uset v$ and past $\dset v$ of a vertex $v\in\face{0}{\mathcal{L}}$ as the collection of all simplices of $\mathcal{L}$ succeeding and preceding $v$, namely
\begin{eqnarray}
\uset v &=& \{ \kappa\in\face{k}{\mathcal{L}} | \forall k\textrm{ and } v\preceq \kappa \},\\
\dset v &=& \{ \kappa\in\face{k}{\mathcal{L}} | \forall k\textrm{ and } v\succeq \kappa \}.
\end{eqnarray}
Every finite subset of vertices $V\subseteq\face 0 {\mathcal{L}}$ has a unique supremum, the vertex $\sup V$, where $\sup V$ lies in the future of each $u \in V$, and furthermore $\sup V$ lies in the past of each vertex $w$ which is in the future of each $v\in V$.
The infinum $\inf V$ is defined analogously.
Lastly, we define the causal diamond $\cdia V$ as the intersection of the future of $\inf V$ and the past of $\sup V$, i.e.,
\begin{equation}
\cdia V = \uset{\inf V} {\cap} \dset{\sup V}.
\end{equation}
This discussion of causal structure generalizes to lattices embedded in a torus; however, one has to excercise caution since the partial order is well-defined only within local regions.
Also, in case of higher-dimensional lattices we make certain assumptions about their causal structure, such as the existence of unique infimum and supremum of $V$.
To avoid technical details, we simply refer to lattices satisfying those assumptions as causal and defer the discussion to Appendix~\ref{app_lattices}.
\sectionprl{Sweep Rule}
Let $\mathcal{L}$ be a 2D causal lattice with $\pm 1$ spins on triangular faces and $\epsilon\subseteq\face{2}{\mathcal{L}}$ denote the set of all faces with $-1$ spins.
The corresponding domain wall $\sigma$ can be found as the boundary $\partial_2 \epsilon$.
Let $v$ be a vertex of $\mathcal{L}$ and denote by $\sigma \rest v$ the restriction of the domain wall $\sigma$ to the edges incident to $v$.
We say that $v$ is trailing if $\sigma \rest v$ is non-empty and belongs to the future of $v$, namely $\sigma\rest v \subset\ \uset v$; see Fig.~\ref{fig_Sweep_illustrated}.
We propose a new local spin update rule defined for every vertex $v$ of $\mathcal{L}$.
\begin{definition}[Sweep Rule]
\label{def_sweep2}
If a vertex $v$ is trailing, then find a subset of neighboring faces $\phi(v)$ in the future $\uset v$ with boundary locally matching the domain wall, i.e., $(\partial_2 \phi(v))\rest v = \sigma\rest v$, and flip spins on faces in $\phi(v)$.
\end{definition}
\noindent This Rule is deterministic and there is a unique choice of $\phi(v)$.
The spin update results in the domain wall being locally pushed away from any trailing vertex $v$; see Fig.~\ref{fig_Sweep_illustrated}.
Note that nothing happens if a vertex is not trailing.
We can, however, consider a very similar CA, the Greedy Sweep Rule, which always tries to push the domain wall away from $v$ in the sweep direction $\vec t$, irrespective of $v$ being trailing; see Appendix~\ref{app_sweep}.
In Lemma~\ref{lemma_properties} we present properties of the Sweep Rule (proven in Appendix~\ref{app_properties}) needed to establish a non-zero threshold of the Sweep Decoder.
\begin{figure}[t!]
\centering
\includegraphics[width=.95\columnwidth]{figures/fig_Sweep_illustrated}
\caption{
{For each trailing vertex $v$ (black) at time $T=1,2,3$ the Sweep Rule finds} a subset $\phi(v)$ of neighbouring faces (green) in the future $\uset v$, whose boundary $\partial_2 \phi(v)$ locally matches the domain wall $\sigma^{(T)}$ (red), i.e., $(\partial_2 \phi(v))\rest v = \sigma^{(T)}\rest v$.
Flipping spins in $\phi(v)$ pushes $\sigma^{(T)}$ away from $v$ in the sweep direction $\vec t$.
Note that $\phi(v)$ and $\sigma^{(T)}$ are always in the causal diamond $\cdia{\sigma^{(1)}}$ (blue) of the initial domain wall $\sigma^{(1)}$.
}
\label{fig_Sweep_illustrated}
\end{figure}
\begin{lemma}[Sweep Rule Properties]
\label{lemma_properties}
Let $\sigma$ be a domain wall in the causal lattice $\mathcal{L}$.
If the Sweep Rule is simultaneously applied to every vertex of $\mathcal{L}$ at time steps $T=1,2,\ldots$, then
\begin{enumerate}
\item (Support) the domain wall $\sigma^{(T)}$ at time $T$ stays within the causal diamond $\cdia \sigma$, i.e.,
\begin{equation}
\sigma^{(T)} \subset \cdia \sigma,
\label{eq_support}
\end{equation}
\item (Propagation) the causal distance between $\sigma$ and any vertex $v$ of $\sigma^{(T)}$ is at most $T$, i.e.,
\begin{equation}
\cdist{v,\sigma} \leq T,
\end{equation}
\item (Removal)
the domain wall is removed by time $T^*$, i.e., $\sigma^{(T)} = 0$ for all $T> T^*$, where
\begin{equation}
T^* = \max_{\cpth{\inf\sigma}{\sup\sigma}} |\cpth{\inf\sigma}{\sup\sigma}|.
\label{eq_removal}
\end{equation}
\end{enumerate}
\end{lemma}
The Sweep Rule may also be defined for vertices of a $d$-dimensional causal lattice $\mathcal{L}$ with spins placed on $k$-simplices $\face k {\mathcal{L}}$, where $k =2, \ldots, d$.
However, for $k \neq d$ the local choice of spins to flip $\phi(v)$ may not be unique (this does not happen in 2D).
Thus, we consider a family of rules corresponding to different ways of choosing $\phi(v)$ in such a way that, roughly speaking, the local causal structure of the domain wall is preserved after flipping spins on $k$-simplices in $\phi(v)$; see Appendix~\ref{app_sweep}.
\sectionprl{Sweep Decoder}
We may use the $d$-dimensional version of the Sweep Rule to decode the toric code on the $d$-dimensional causal lattice $\mathcal{L}$.
Recall that the toric code of type $k=1,\ldots,d-1$ {is} defined by placing qubits on $k$-simplices of $\mathcal{L}$, and associating $X$- and $Z$-stabilizers with $(k-1)$- and $(k+1)$-simplices.
Then, $Z$-stabilizers, $Z$-logical operators and $X$-syndromes correspond to, respectively, the elements of $\im\partial_{k+1}$, $\ker\partial_k$ and $\im\partial_k$, where $\partial_i$ denotes the $i$-boundary operator; see Appendix~\ref{app_lattices}.
If $\epsilon\subseteq\face k {\mathcal{L}}$ is the set of qubits affected by $Z$ errors, then the corresponding $X$-syndrome is $\sigma = \partial_k \epsilon$.
Thus, for $k \geq 2$, decoding of $Z$ errors can be phrased as the already discussed problem of estimating locations of $-1$ spins given the corresponding domain wall.
{Note} that for $k \leq d-2$ decoding of $X$ errors is analogous but in the dual lattice $\mathcal{L}^*$ with the $Z$-syndrome forming a $(d-k-1)$-dimensional domain wall.
\begin{algorithm
\caption{Sweep Decoder}
\SetKwInOut{Require}{Require}
\KwIn{$X$-syndrome $\sigma\in\textrm{im }\partial_{k}$, {$k=2,\ldots,d-1$}}
\KwOut{$k$-dimensional correction $\rho\subseteq\face{k}{\mathcal{L}}$}
initialize $T$ = 1, $\sigma^{(1)} = \sigma$\\
unless $T> T_{\mathrm{max}} $ or $\sigma^{(T)} = 0$ repeat:\\
\begin{enumerate}
\item apply the Sweep Rule simultaneously to every\\ vertex of $\mathcal{L}$ to get $\rho^{(T)}$
\item find $\sigma^{(T+1)} = \sigma^{(T)} + \partial_k\rho^{(T)}$
\item update time step $T\leftarrow T+1$\\
\end{enumerate}
if $T\leq T_{\mathrm{max}}$
\footnote{$T_{\mathrm{max}}$ is of the order of the linear size $L$ of the lattice $\mathcal{L}$; see Appendix~\ref{app_proof_together}.}, then $\rho = \sum_{i=1}^{T-1}\rho^{(i)}$, otherwise $\rho = \texttt{FAIL}$\\
\KwRet{$\rho$}
\end{algorithm}
This \textit{Sweep Decoder} may fail for either one of two reasons.
First, it might not terminate within time $T_{\mathrm{max}}$,
which results in $\rho = \texttt{FAIL}$.
Second, the correction $\rho$ combined with the initial error $\epsilon$ may implement a non-trivial logical operator, i.e., $\rho + \epsilon \not\in \im \partial_{k+1}$.
{However}, the Sweep Decoder a has non-zero error-correction threshold --- if the $Z$ error rate is below threshold, then the failure probability rapidly approaches zero as the code's block grows.
We establish this fact by deriving a lower bound $p^*_{\mathrm{th}} > 0$ on the threshold error rate (which explicitly depends on the local structure of $\mathcal{L}$).
\begin{theorem}[Threshold]
\label{thm_thres}
Consider a family of causal lattices $\mathcal{L}$ of growing linear size $L$ on the $d$-dimensional torus, and define the toric code of type $k=2,\ldots,d-1$ on $\mathcal{L}$.
Then, there exists a constant $p^*_{\mathrm{th}} > 0$, such that for any phase-flip error rate $p < p^*_{\mathrm{th}}$ the failure probability of the Sweep Decoder for perfect syndrome extraction goes to zero as $L\rightarrow\infty$.
\end{theorem}
\noindent In Appendix~\ref{app_proof} we present a rigorous proof of Theorem~\ref{thm_thres} based on renormalization group ideas \cite{Gacs1988, Harrington2004, Bravyi2013a}; here we only outline the proof strategy.
\begin{proof}
First, we decompose each error configuration into recursively defined ``connected components,'' where a ``level-$n$'' connected component has a linear size growing exponentially with $n$.
The probability of a level-$n$ connected component is doubly-exponentially small in $p / p^*_{\mathrm{th}}$.
The connected components are well isolated from other errors; therefore, using Lemma~\ref{lemma_properties} and some modest assumptions about the lattice family, we can show that a connected component with linear size small compared to $L$ will be successfully removed by repeated application of the Sweep Rule.
Therefore, the Sweep Decoder fails only if the contains a level-$n$ connected component with size comparable to $L$, which is very improbable for large $L$ and $p < p^*_{\mathrm{th}}$.
\end{proof}
\sectionprl{Numerical simulations}
In Theorem~\ref{thm_thres} we assumed that the Sweep Rule is applied flawlessly, but in a realistic scenario the Rule itself is noisy; the noise degrades the effectiveness of error correction and reduces the threshold.
We have numerically investigated the performance of the Sweep Decoder for the 3D toric code on the bcc lattice with qubits on faces.
We consider a phenomenological noise model such that in each error correction cycle Pauli $Z$ errors on qubits occur with probability $p$, and in addition measured syndrome bits are flipped with probability $p$.
Using Monte Carlo simulations we find the threshold $p_{\textrm{th}}(N_{\textrm{cyc}})$ for a fixed number $N_{\textrm{cyc}}$ of noisy correction cycles followed by perfect syndrome extraction and full decoding.
Note that $p_{\textrm{th}}(1)$ is the threshold for perfect syndrome extraction.
We are, however, interested in the so-called sustainable threshold $p^{\textrm{bcc}}_{\textrm{sus}} = \lim_{N_{\textrm{cyc}}\rightarrow\infty} p_{\textrm{th}}(N_{\textrm{cyc}})$ \cite{Brown2015, Terhal2015}.
We observe that the threshold $p_{\textrm{th}}(N_{\textrm{cyc}})$ is very well approximated by the numerical ansatz
\begin{equation}
p_{\textrm{th}}(N_{\textrm{cyc}}) \sim p^{\textrm{bcc}}_{\textrm{sus}} (1 - (1- p_{\textrm{th}}(1)/p^{\textrm{bcc}}_{\textrm{sus}}) N_{\textrm{cyc}}^{-\gamma}),
\label{eq_ansatz_pheno}
\end{equation}
with the fitting parameters ${p^{\textrm{bcc}}_{\textrm{sus}} = 0.99\pm 0.02\%}$ and ${\gamma = 0.855\pm 0.010}$; see~Fig.~\ref{fig_numerics}.
These numerical results were actually obtained for a variant of the Sweep Decoder based on the Greedy Sweep Rule, which has a higher threshold than the decoder based on the Sweep Rule.
In Appendix~\ref{app_sweep} we discuss the Greedy Sweep Rule, explain how it generalizes to locally Euclidean lattices, and use it to estimate the sustainable threshold of the 3D toric code on the cubic lattice $p^{\textrm{cubic}}_{\textrm{sus}} = 1.98\pm 0.02\% $.
\sectionprl{Discussion}
We have presented a new CA, the Sweep Rule, which generalizes Toom's rule to any locally Euclidean $d$-dimensional lattice.
This Rule can be used to decode a topological quantum code whose error syndrome is at least one dimensional, including the color code; see the accompanying article~\cite{cc_decoder}.
We proved that a decoder based on the Sweep Rule has a non-zero accuracy threshold for the toric code, and we numerically studied its performance against a phenomenological noise model.
Our results provide a rigorous justification for using CA error-correction strategies for topological quantum codes.
We hope that our proof techniques will lead to new CA decoders with provable thresholds for codes on lattices with boundaries, hyperbolic lattices or other quantum low-density parity-check codes.
The Sweep Rule may also be of independent interest for defining statistical-mechanical problems inspired by quantum information \cite{Kubica2014,Yoshida2014,Kubica2017}.
As for Toom's rule, one can consider a non-deterministic variant of the Sweep Rule and study the evolution of spins generated by this probabilistic CA.
We conjecture that the resulting spin dynamics is non-ergodic and that the phase diagram contains regions with multiple coexisting stable phases, as established in 2D by Toom~\cite{Toom1980}.
\sectionprl{Acknowledgements}
A.K. thanks Nicolas Delfosse for invaluable feedback throughout the project, and Ben Brown and Mike Vasmer for stimulating discussions.
A.K. acknowledges funding provided by the Simons Foundation through the ``It from Qubit'' Collaboration.
Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation.
J.P. acknowledges support from ARO, DOE, IARPA, NSF, and the Simons Foundation. The Institute for Quantum Information and Matter (IQIM) is an NSF Physics Frontiers Center.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 33 |
Le château de Philipshof (Schloß Philipshof) est un petit château de l'arrondissement de Prignitz dans le Brandebourg, situé dans la ville de Putlitz.
Histoire
Le domaine est mentionné comme bien de la famille des chevaliers Gans von Putlitz. Le château est reconstruit en 1895 dans un style éclectique, mêlant des éléments Renaissance nordique et des éléments romantiques. Il reste dans les mains de la famille Gans von Pulitz, jusqu'à l'expulsion de celle-ci à la fin de la Seconde Guerre mondiale.
Le château et ses terres sont nationalisés et les autorités locales de la république démocratique allemande décident d'y installer une école en 1949. La demeure est restaurée en 1955. Elle sert de bâtiment scolaire pour une école élémentaire jusqu'à l'été 2005.
Le château est donc vide depuis lors, et nécessite des travaux de rénovation.
Source
Château en Brandebourg | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 621 |
Le Colibri à gorge rubis (Archilochus colubris) est une espèce d'oiseau-mouche du genre Archilochus.
Description
Les adultes ont un plumage vert iridescent sur le dessus tandis qu'ils sont blanc grisâtre sur le dessous. Ils ont le bec long, droit et très mince. Le mâle adulte a la gorge iridescente de couleur rouge rubis et la queue échancrée. La femelle a une queue arrondie et foncée avec l'extrémité blanche. Elle n'a pas de zone colorée sur la gorge. Elle se distingue très difficilement de la femelle du Colibri à gorge noire. L'immature est semblable à la femelle.
Comportement
Le colibri à gorge rouge est solitaire. Les adultes de cette espèce ne sociabilisent pas (ou seulement quelques minutes). Aussi bien les mâles que les femelles, sont agressifs envers les autres oiseaux-mouches. Ils défendent leur territoire et chassent les autres oiseaux-mouches qui y pénètrent. Les femelles défendent aussi leur progéniture.
Alimentation
Ces oiseaux se nourrissent du nectar d'une trentaine d'espèces de fleurs grâce à une longue langue extensible, ou capturent des insectes en vol. Dans certaines parties de l'aire d'hivernage, des individus arrivent alors que les sources de nectar sont encore rares, et dépendent des insectes pour s'alimenter, jusqu'à ce que la floraison associée à la saison sèche se produise. Même quand les fleurs nectarifères sont disponibles, les insectes peuvent représenter jusqu'à 14 % du régime hivernal.
Répartition et habitat
Cette espèce, que l'on rencontre au sud du Canada et dans la zone orientale des États-Unis, est migratrice et passe la plus grande partie de l'hiver au Mexique ou en Amérique centrale. Les colibris y fréquentent une large gamme de forêts tropicales, principalement les forêts sèches de plaine et les forêts secondaires jusqu'à d'altitude. La majeure partie de la population suit une route migratoire longue de 800 kilomètres qui contourne le golfe du Mexique, aussi bien en automne qu'au printemps.
Reproduction
Ce colibri niche dans les forêts caducifoliées et mixtes, les parcs et les jardins, à travers la plus grande partie de l'Est des États-Unis et du Sud du Canada. Les mâles arrivent sur les sites de nidification de début avril à fin mai, une semaine avant les femelles. La nidification s'étend jusqu'en juillet, exceptionnellement en août. La parade nuptiale consiste en une démonstration aérienne élaborée au cours de laquelle le mâle effectue des va-et-vient en vol près de la femelle, décrivant un demi-cercle de 2 ou de rayon.
La construction du nid commence de mi-avril à début juin, et est réalisée principalement par la femelle en 4 ou 5 jours. Ce nid est une coupe de de diamètre, composée d'éléments végétaux, de lichens et de toiles d'araignées ; l'extérieur est tapissé de lichens et l'intérieur est garni de duvet végétal. Le nid est placé à 4 ou de haut sur une branche horizontale. Les mâles sont polygames, ne s'associant habituellement avec une femelle que le temps de la parade et de la ponte. La ponte se compose de 2 œufs. C'est la femelle seule qui nourrit et prend soin des jeunes. Elle les nourrit une à trois fois par heure par régurgitation. Lorsqu'ils ont de 18 à 22 jours, les juvéniles quittent le nid. Il arrive souvent qu'une couvée de remplacement ait lieu si la première a échoué, mais les secondes nichées sont rares. Le mâle défend avec agressivité les places d'alimentation situées dans son territoire.
Longévité et mortalité
Le plus vieux colibri à gorge rubis connu est une femelle qui a vécu 9 ans et un mois. Presque tous les oiseaux-mouches de 7 ans ou plus sont des femelles, les mâles survivent rarement plus de 5 ans. La haute mortalité des mâles est due à leur perte de poids importante durant la saison de reproduction car ils doivent défendre activement leur territoire, ce qui leur demande beaucoup d'énergie
Cinéma
Le personnage de Flit, dans le dessin animé Pocahontas, est un colibri à gorge rubis.
Galerie
Liens externes
Faune et flore du pays : Colibri à gorge rubis
Projet de recherche sur le Colibri à gorge rubis au Québec
Trochilidae | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,859 |
Q: Cleanup Function is not running in useEffect in React Redux I have two dispatch actions 1. fetchHomepage and 2. removeHomepage. I want to run removeHomepage action when I click other link on header menu. So I added removeHomepage in Cleanup Function in useEffect but my cleanup function is not running. When I change something and save then it runs. I am using Redux in this React project. Please can anybody help me out.
HEADER COMPONENT -
import React, { useEffect, useState } from "react";
import "./header.css";
import { useDispatch, useSelector } from "react-redux";
import {
fetchHomepage,
removeHomepage,
} from "../../redux/actions/HomepageActions";
import { setBgColor } from "../../redux/actions/HomepageActions";
import { Link } from "react-router-dom";
import { useLocation } from "react-router-dom";
import Skeleton from "../other/skeletons/Skeleton";
const Header = () => {
const dispatch = useDispatch();
const isLoading = useSelector((state) => state.homepageReducer.isLoading);
const header = useSelector((state) => state.homepageReducer.data.navLinks);
const bgColor = useSelector((state) => state.homepageReducer.bgColor);
const { bgColour, borderColor, iconColor } = bgColor;
const product = useSelector((state) => state.addToCartReducer.products);
const { pathname } = useLocation();
const location = pathname;
console.log("component loaded");
useEffect(() => {
console.log("useffect ran");
const changeNavBg = (bgColor) => {
if (window.scrollY > 0) {
dispatch(setBgColor(bgColor));
} else {
dispatch(
setBgColor({
bgColour: "bg-transparent",
borderColor: "",
iconColor: "",
})
);
}
};
window.addEventListener("scroll", () => {
changeNavBg({
bgColour: "bg-black",
borderColor: "border-white",
iconColor: "text-white",
});
});
dispatch(fetchHomepage());
return () => {
console.log("cleanup function ran");
dispatch(removeHomepage());
window.removeEventListener("scroll", changeNavBg);
};
}, []);
if (isLoading) {
return <Skeleton header={"header"} />;
}
return (
<>
<h1>HEADER</h1>
{
<div className="dvHeader">
<nav
className={`navbar navbar-expand-lg ${bgColour} ${
location !== "/" ? "bg-black" : ""
} navbar-light fixed-top`}
>
<div className="container">
<div>
<button
className="navbar-toggler"
type="button"
data-toggle="collapse"
data-target="#slideDownMenu"
aria-controls="slideDownMenu"
aria-expanded="false"
aria-label="Toggle navigation"
>
<span className="">
<i
className={`fa fa-bars ${iconColor} ${
location !== "/" ? "text-white" : ""
} `}
></i>
</span>
</button>
</div>
<div className="order-lg-0">
<Link to="/">
<img
width="50"
src={header && header.logo.url}
className="img-fluid"
alt=""
/>
</Link>
</div>
<div className="order-lg-2 d-lg-none">
<span
className="d-flex"
data-toggle="modal"
data-target="#mobileCartModal"
>
<span className="d-inline-block mr-1">
<i
className={`fa fa-shopping-cart ${iconColor} ${
location !== "/" ? "text-white" : ""
} `}
></i>
</span>
<span
className={`d-inline-block ${iconColor} ${
location !== "/" ? "text-white" : ""
} `}
>
{product.length}
</span>
</span>
</div>
<div
className="collapse navbar-collapse order-lg-1"
id="slideDownMenu"
>
<ul className="navbar-nav mr-auto px-3">
<li className="nav-item">
<Link className="nav-link" to="/products">
Shop
</Link>
</li>
<li className="nav-item dropdown">
<Link
className="nav-link dropdown-toggle"
to="#"
id="navbarDropdown"
role="button"
data-toggle="dropdown"
aria-haspopup="true"
aria-expanded="false"
>
Learn
</Link>
<div
className="dropdown-menu"
aria-labelledby="navbarDropdown"
>
<Link className="dropdown-item" to="#">
Process
</Link>
<Link className="dropdown-item" to="#">
About Us
</Link>
<Link className="dropdown-item" to="#">
Blog
</Link>
<Link className="dropdown-item" to="#">
News
</Link>
<Link className="dropdown-item" to="#">
Beyond The Bottle
</Link>
</div>
</li>
<li className="nav-item">
<Link
className="nav-link"
to="#"
data-toggle="modal"
data-target="#loginModal"
>
Login
</Link>
</li>
<li className="nav-item">
<Link
className="nav-link"
to="#"
data-toggle="modal"
data-target="#signupModal"
>
Sign Up
</Link>
</li>
</ul>
<form className="dvSearch my-2 my-lg-0 px-3">
<input
className={`form-control border-top-0 border-right-0 border-left-0 mr-sm-2 ${
window.innerWidth > 991 ? borderColor : ""
} ${location !== "/" ? "border-white" : ""} `}
type="text"
placeholder="Search"
aria-label="Search"
/>
<button className="btn my-2 my-sm-0" type="submit">
<i
className={`fa fa-search ${iconColor} ${
location !== "/" ? "text-white" : ""
} `}
></i>
</button>
</form>
</div>
</div>
</nav>
</div>
}
<div style={{ height: "10000px" }}></div>
</>
);
};
export default Header;
ACTIONS -
//ACTION CREATOR FETCH HOMEPAGE
export const fetchHomepage = () => {
const apiKey = process.env.REACT_APP_API_KEY;
const url = `https://api.json-generator.com/templates/UhZ_20Akrr7T/data?access_token=${apiKey}`;
return async (dispatch, getState) => {
const response = await fetch(url);
const data = await response.json();
dispatch({
type: actionTypes.SET_HOMEPAGE,
payload: data,
});
};
};
// remove homepage
export const removeHomepage = () => {
return {
type: actionTypes.REMOVE_HOMEPAGE,
};
};
//set nav bg color
export const setBgColor = (data) => {
return {
type: actionTypes.SET_NAV_BG_COLOR,
payload: data,
};
};
HOMEPAGE REDUCER-
import { actionTypes } from "../constants/action-types";
const homeiState = {
data: {},
bgColor: {},
isLoading: true,
};
export const homepageReducer = (state = homeiState, { type, payload }) => {
switch (type) {
case actionTypes.SET_HOMEPAGE:
return {
...state,
data: payload,
isLoading: false,
};
case actionTypes.REMOVE_HOMEPAGE:
return {
...state,
isLoading: true,
};
case actionTypes.SET_NAV_BG_COLOR:
return {
...state,
bgColor: payload,
};
default:
return state;
}
};
A: Your dependency array is empty that's why your cleanup function is not running.
A: The useEffect's cleanup function is only called before the component unmounts, your Header component is not unmounting, thus it will not be called.
If you want to call removeHomepage, I suggest calling it in the event function when clicking on a header item or creating a new useEffect hook that reacts to changes to a certain redux value.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,623 |
Aloe schomeri es una especie de planta suculentadel género Aloe. Es originaria de Madagascar.
Descripción
Aloe schomeri crece sin tallo o tronco muy corto y forma grupos. Las hasta 30 hojas lanceoladas estrechas forman densas rosetas. Es de color verde oscuro de 20 a 30 cm de largo y 3 a 5 cm de ancho. El margen es cartílaginoso brillante y casi blanco con dientes casi blancos de 2 mm de largo y de 5 a 8 mm de distancia. La inflorescencia es generalmente simple o de vez en cuando se compone de una o dos ramas. Alcanza una longitud de hasta 65 centímetros. Las brácteas tienen una longitud de aproximadamente 9 milímetros, y 5 milímetros de ancho. Las flores son amarillas de 21 mm de largo y redondeadas en la base.
Distribución
Es una planta herbácea con las hojas suculentas que se encuentra en Madagascar en la Provincia de Toliara.
Taxonomía
Aloe schomeri fue descrita por Werner Rauh y publicado en Kakteen und andere Sukkulenten 17(2): 22–23, abb. 1–4. 1966.
Etimología
Ver: Aloe
schomeri: epíteto otorgado en honor de Menko Schomerus, propietario de una mina en Ampanihy en Madagascar.
Referencias
schomeri
Flora de Madagascar
Plantas descritas en 1966
Plantas descritas por Rauh | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,247 |
\section{Introduction}\label{sec:intro}
The study of symmetries of a physical model and of dualities that
relate it to other points in the relevant moduli space has often
proved an indispensable source of knowledge on the structure of the
physical system of interest -- suffice it to invoke the use of
Ward--Takahashi identities in the derivation of correlation
functions of a quantum field theory, the `duality net' of consistent
(super)string backgrounds, or the AdS/CFT duality that has gained us
insights into a strongly coupled QCD-type field theory through the
study of a weakly coupled string theory. This by now well-ingrained
and widely exploited constatation forms the base of a series of
papers, of which the present one is the first, discussing the r\^ole
played by world-sheet defects in the description of symmetries and
dualities of a large class of two-dimensional field theories with
conformal symmetry known as non-linear $\sigma$-models, first
considered in all generality in \Rcite{Friedan:1985phd}. These are
-- in the simplest setting -- theories of harmonic embeddings
$\,X:\Sigma\to M\,$ of an oriented two-dimensional lorentzian manifold
$\,\Sigma$,\ termed the world-sheet, in a metric manifold
$\,(M,{\rm g})$,\ called the target space and generically equipped with
additional cohomological structure. In the purely bosonic setting,
and in the absence of the dilaton field on $\,M$,\ this additional
structure comes from the 2-category $\,\gt{BGrb}^\nabla(M)\,$ of bundle
gerbes (with connection) over $\,M$,\ introduced in
\Rcite{Stevenson:2000wj} and further elaborated in
\Rcite{Waldorf:2007mm}. Its objects, termed bundle gerbes with
connection, or gerbes for short, were first considered, in rather
abstract terms, in \Rcite{Giraud:1971}. The more intuitive
(hyper)cohomological description was given only in
\Rcite{Brylinski:1993ab}, and an intrinsic geometric definition
followed in Refs.\,\cite{Murray:1994db,Murray:1999ew}. Gerbes are
geometric structures representing classes in the second real Deligne
(hyper)cohomology group of $\,M$.\ As such, they serve to give a
rigorous \emph{global} definition of the topological term in the
action functional of the $\sigma$-model on a closed world-sheet. The
term is determined by a specific Cheeger-Simons differential
character, to wit, the so-called surface holonomy of the gerbe over
$\,\Sigma\,$ and it is locally expressed as the integral of the
pullback, along $\,X$,\ of a primitive of the closed curvature
3-form of the gerbe to the world-sheet. The function of the Deligne
cohomology, first brought into the picture in
Refs.\,\cite{Alvarez:1984es,Gawedzki:1987ak}, transcends the
definition of the classical action functional -- indeed, it gives
rise to a natural classification scheme for $\sigma$-models on a given
target space in terms of equivalence classes of gerbes, which -- for
a given curvature -- span the sheaf-cohomology group
$\,H^2\bigl(M,{\rm U}(1)\bigr)$,\ and it canonically defines the
pre-quantum bundle of the theory, thus establishing the basis of the
geometric quantisation scheme. This was realised already in
\Rcite{Gawedzki:1987ak} and subsequently employed in
\Rcite{Felder:1988sd} in the setting of the Wess--Zumino--Witten
(WZW) $\sigma$-model with a compact simple connected Lie group as the
target space. Over and above these, gerbe theory provides us with
concrete tools for constructing new theories from the existing ones
by way of gauging subgroups $\,{\rm K}\,$ of the isometry group of
$\,(M,{\rm g})\,$ and through orientifolding, both procedures being
founded on the notion of an equivariant (resp.\ twisted-equivariant,
cp.\ Refs.\,\cite{Schreiber:2005mi,Gawedzki:2008um}) structure on
the gerbe. The structure uses distinguished 1- and 2-cells of the
2-category of bundle gerbes over the nerve of the action groupoid
$\,{\rm K}{\hspace{-0.04cm}\ltimes\hspace{-0.05cm}} M$,\ and it descends to a gerbe structure on the quotient
space $\,M/{\rm K}\,$ under suitable conditions. Since, furthermore,
all gerbes on $\,M/{\rm K}\,$ can be obtained in this manner, cp.\
\Rcite{Gawedzki:2010rn}, it gives us a classification of
$\sigma$-models on the quotient space (resp.\ that of $\sigma$-models for
unorientable world-sheets in the case of orientifolding). The
existence and uniqueness theorems established in
Refs.\,\cite{Gawedzki:2002se,Gawedzki:2003pm} and
\Rcite{Gawedzki:2007uz} for the orbifolded resp.\ orientifolded
variants of the WZW model, for which there exists an explicit
construction of the gerbe (worked out, in steps, in
Refs.\,\cite{Gawedzki:1987ak,Chatterjee:1998,Meinrenken:2002}), are
in perfect agreement with known results of the structure-heavy
Conformal Field-Theory (CFT) analyses of modular invariants from
Refs.\,\cite{Felder:1988sd,Kreuzer:1994,Brunner:2002em} and those of
the categorial quantisation of the $\sigma$-model, reported in
\Rcite{Fuchs:2004dz}. Analogous statements from
\Rcite{Gawedzki:2010rn} pertaining to the case of continuous group
actions, the latter presenting an additional complication due to the
coupling between the gerbe and the principal ${\rm K}$-connection on a
(generically non-trivial) principal ${\rm K}$-bundle over $\,\Sigma\,$
which may fail because of global gauge anomalies, go far beyond the
long-established results of both the geometric and algebraic
discussion of
Refs.\,\cite{Hull:1989,Hull:1990ms,Figueroa:1994ns,Figueroa:1994dj},
and the (conformal) field-theoretic analysis of
Refs.\,\cite{Goddard:1984vk,Gawedzki:1988nj,Hori:1994nc,Fuchs:1995tq}.
\medskip
The more general physical meaning of the full-blown 2-categorial
structure associated with bundle gerbes and its naturality in the
context of the two-dimensional field theory have been brought to the
fore by the construction of the multi-phase $\sigma$-model in
\Rcite{Runkel:2008gr}. In this construction, the two-dimensional
spacetime $\,\Sigma\,$ (or its euclidean version) is split into a
collection of domains $\,\Sigma_i,\ i\in\ovl{1,N}\subset{\mathbb{N}}$,\ each
carrying its own phase of the full theory (i.e.\ a choice $\,M_i\,$
of a connected component of the target space with the attendant
structures of the metric and the gerbe) and separated from adjacent
domains by lines of discontinuity of the embedding field $\,X$,\
termed defect lines and mapped by $\,X\,$ into a correspondence
space $\,Q\,$ called the bi-brane world-volume. Defect lines, in
turn, intersect at the so-called defect junctions, sent by $\,X\,$
into another correspondence space $\,T$,\ dubbed the inter-bi-brane
world-volume. The constitutive elements of the construction of
\Rcite{Runkel:2008gr} and the ensuing definition of the action
functional of the $\sigma$-model are recalled in Section
\ref{sec:lagr}.
An important source of inspiration for the construction, with its
assignment of distinguished 1-cells of $\,\gt{BGrb}^\nabla(Q)\,$ to
defect lines and 2-cells of $\,\gt{BGrb}^\nabla(T)\,$ to defect
junctions, were the earlier findings of
Refs.\,\cite{Freed:1999vc,Kapustin:1999di,Carey:2002,Gawedzki:2002se,Gawedzki:2004tu}
and \Rcite{Fuchs:2007fw} in which the relevant cohomological
structures had been identified over the submanifolds of $\,M\,$ and
$\,M_i\x M_j$,\ respectively, defining the codomain (the so-called
D-brane or $\mathcal{G}$-brane world-volume) of the restriction of $\,X\,$
to connected components of the boundary of $\,\Sigma\,$ in the former
case, and determining the discontinuity of $\,X\,$ along a defect
line homeomorphic to $\,{\mathbb{S}}^1\,$ in the latter case. These
cohomological structures, corresponding to vector bundles twisted by
the gerbe in a well-defined manner, contribute their own part to the
classification scheme of consistent $\sigma$-models on world-sheets
with defects and straightforwardly accommodate a variant of the
orientifolding and gauging constructions set up for the bulk gerbes,
as demonstrated in
Refs.\,\cite{Gawedzki:2002se,Gawedzki:2004tu,Gawedzki:2008um,Gawedzki:2010ggd}.
It well deserves to be pointed out that the ensuing explicit
constructions of orbifold and orientifold $\mathcal{G}$-branes in the
controlled setting of the WZW model, presented in
Refs.\,\cite{Gawedzki:2004tu,Gawedzki:2010G}, indicate the presence
of an essentially new species of $\mathcal{G}$-brane, dubbed the non-abelian
brane in the original \Rcite{Gawedzki:2004tu}, over those conjugacy
classes in the target Lie group which are invariant under the action
of non-cyclic components of $\,{\rm K}$.\ These $\mathcal{G}$-branes have
properties suggestive of an interpretation in terms of irresoluble
stacks of fixed-point fractional branes of
\Rcite{Diaconescu:1997br}. Their existence, peculiar to the maximal
orbifold $\,{\rm Spin}(4n)/({\mathbb{Z}}_2\x{\mathbb{Z}}_2)\,$ and ubiquitous on
proper orientifolds of group manifolds, had not been predicted by
the standard CFT methods, and so they provide a tangible example of
a novel string-theoretic insight gained by purely gerbe-theoretic
methods. The intriguing internal (open-string) dynamics of these
branes still awaits an in-depth treatment.
A piece of motivation that is more immediately related to the
subject matter of the present paper, and also of a more
field-theoretic flavour (as seen from the two-dimensional
perspective), comes from the CFT studies of conformal interfaces
(i.e.\ defect lines transmissive to that half of the conformal
symmetries of either of the two phases of the theory supported over
the two domains of the world-sheet separated by the defect line
which preserve that line). The concept originated from the
condensed-matter considerations of \Rcite{Oshikawa:1996dj} and was
later transplanted into the string-theoretic domain in
\Rcite{Petkova:2000ip} (in a purely operator-algebraic language) and
in \Rcite{Bachas:2001vj} (in a more geometric world-sheet terms).
Subsequent studies have diverged into a variety of specialised
directions, including the classificatory analysis and specific
constructions of
Refs.\,\cite{Fuchs:2002cm,Quella:2002ct,Fuchs:2007tx,Fuchs:2007fw,Bachas:2009mc,Gawedzki:2010ggd}
for various distinguished classes of CFT (such as, e.g., the free
boson, the (gauged) WZW model and, more generally, an arbitrary
rational CFT), the discussion of the fusion of conformal interfaces
in the quantum r\'egime in
Refs.\,\cite{Petkova:2000ip,Fuchs:2002cm,Bachas:2007td}, alongside a
description of perturbed defect CFTs and Renormalisation-Group (RG)
flows in their presence advanced in
Refs.\,\cite{Bachas:2004sy,Alekseev:2007in,Runkel:2007wd,Kormos:2009sk,Bachas:2009mc},
and related to certain integrable structures of CFT in
Refs.\,\cite{Runkel:2007wd,Manolopoulos:2009np}. Analogous results
have also been obtained in the context of supersymmetric
two-dimensional field theories, cf., e.g.,
Refs.\,\cite{Brunner:2007qu,Brunner:2008fa,Brunner:2009zt,Brunner:2010xm}.
The studies carried out to date, and in particular those reported in
Refs.\,\cite{Frohlich:2004ef,Frohlich:2006ch,Schweigert:2007wd,Runkel:2008gr,Sarkissian:2008dq,Bachas:2008jd},
bear ample evidence of a prominent r\^ole played by conformal
interfaces in establishing correspondences between phases of CFT, in
encoding order-disorder dualities among various CFTs, and in mapping
into one another their RG flows as well as UV and IR fixed points of
the latter. Finally, they can be associated with the so-called
spectrum-generating symmetries of string theory, relating -- via
fusion with boundary states (a process that has not been fully
understood up to now) -- the D-brane categories of a dual pair of
CFTs. All this leads to a natural question as to a state-space
interpretation of the conformal world-sheet defects of
\Rcite{Runkel:2008gr} and the attendant cohomological structures on
the codomain of $\sigma$-model fields, encompassing the data carried by
both the defect lines and their junctions. This question is at the
core of the present paper, and in our search for an answer, we shall
be guided by insights inferred from the detailed treatment of the
maximally symmetric WZW defects in
Refs.\,\cite{Fuchs:2007fw,Runkel:2008gr,Runkel:2009su,Runkel:2010}.
The latter provide an excellent setting in which to look, in
particular, for conditions necessary and sufficient for the
correspondence between the phases of the $\sigma$-model determined by
the defect to be compatible with the module structure on the
respective state spaces with respect to the action of an extended
current symmetry algebra. This issue is put in a wider
generalised-geometric context and subsequently elaborated at great
length in \Rcite{Suszek:2010b}, forming the second part of the
series opened by the present article.\medskip
A natural framework for establishing the sought-after state-space
interpretation of the conformal defects is provided by the canonical
description of the $\sigma$-model. The description can be derived in
the so-called covariant (or first-order) formalism of
Refs.\,\cite{Gawedzki:1972ms,Kijowski:1973gi,Kijowski:1974mp,Kijowski:1976ze,Szczyrba:1976,Kijowski:1979dj}
which leads to a systematic reconstruction of the symplectic
structure on the state space of the two-dimensional field theory of
interest. The basic tools of the formalism are introduced in Section
\ref{sub:can-gen}. These are subsequently applied, in the remainder
of Section \ref{sec:can}, to the two qualitatively distinct sectors
of string theory present on a generic world-sheet with an embedded
defect, that is the untwisted sector, composed of strings
represented by smooth loops embedded in the respective connected
components of the target space, and the twisted sector, with states
represented by piecewise smooth maps from the unit circle into the
target space, with point-like discontinuities which can be
understood as resulting from transversal intersections with defect
lines. A prototypical example of the latter sector is provided by
the twisted sector of string theory on an orbifold of a smooth
target space, first discussed in
Refs.\,\cite{Dixon:1985jw,Dixon:1986jc}, in which case the
discontinuities are determined by elements of the orbifold group.
The upshot of the analysis carried out in Section \ref{sec:can} is a
full-fledged canonical description of the classical (bosonic) string
with a multi-phase world-sheet.
The key advantage of working with the global geometric structures
from $\,\gt{BGrb}^\nabla(M\sqcup Q\sqcup T)\,$ in the classical setting
is that they actually afford inroads into the quantum r\'egime of
the theory. As mentioned already in the opening paragraph of the
present section, this fact has been known ever since the
introduction of the hypercohomological language into rigorous
studies of the two-dimensional $\sigma$-model in
\Rcite{Gawedzki:1987ak}, which is where the transgression map was
defined. The latter is a cohomology map canonically assigning to the
1-isomorphism class of the gerbe of the $\sigma$-model with target
space $\,M\,$ the isomorphism class of a circle bundle over its
configuration space $\,{\mathsf L} M\equiv C^\infty({\mathbb{S}}^1,M)\,$ (the
free-loop space of $\,M$) with a connection whose curvature yields,
upon pullback to the state space $\,{\mathsf P}_{\sigma,\emptyset}\cong{\mathsf T}^*
{\mathsf L} M\,$ of the theory and correction by a canonical (and
topologically trivial) term, the symplectic form of the
(defect-free) $\sigma$-model. In other words, the gerbe determines a
pre-quantum bundle $\,\mathcal{L}_{\sigma,\emptyset}\to{\mathsf P}_{\sigma,\emptyset}\,$
of the closed string, a prerequisite of its geometric quantisation,
cp., e.g., \Rcite{Woodhouse:1992de}. The important novel result for
strings with multi-phase world-sheets, anticipated by the findings
of Refs.\,\cite{Gawedzki:2002se,Gawedzki:2004tu} and derived in
Section \ref{sec:can} in analogy with the original result for the
untwisted sector, is the existence of a straightforward
generalisation of the transgression map to the twisted sector of the
string. This generalised cohomology map canonically assigns the
isomorphism class of a pre-quantum bundle over the space of twisted
states to the equivalence class of a coupled pair consisting of the
$\sigma$-model gerbe and the associated bi-brane, a fact following
directly from Theorem \ref{thm:trans-tw}.\medskip
The canonical formalism thus reconstructed constitutes an excellent
basis for phrasing the question about the r\^ole of defects in a
rigorous manner. Guided by the simple geometric intuition conveyed
by the world-sheet picture of the cross-defect identification of
states effected by the propagation of the closed string, as
illustrated in Figure \ref{fig:defect-corr},
\begin{figure}[hbt]~\\[5pt]
$$
\raisebox{-50pt}{\begin{picture}(50,50)
\put(-79,-4){\scalebox{0.25}{\includegraphics{defect-corr.pdf}}}
\end{picture}
\put(0,0){
\setlength{\unitlength}{.60pt}\put(-28,-16){
\put(-10,40) { $\ell_{1,2}$ }
\put(-82,40) { $\psi_1$ }
\put(58,40) { $\psi_2$ }
\put(-160,133) { ${\rm CFT}_1$ }
\put(135,133) { ${\rm CFT}_2$ }
\put(-10,250) { $t$ }
}\setlength{\unitlength}{1pt}}}
$$
\caption{The correspondence between states mediated by the defect
line: the state $\,\psi_1\,$ from the phase $\,{\rm CFT}_1\,$ is
transferred to the state $\,\psi_2\,$ from the phase $\,{\rm
CFT}_2\,$ across the defect line $\,\ell_{1,2}$.\ The arrow above
the picture represents the world-sheet time direction.}
\label{fig:defect-corr}
\end{figure}
we are led to investigate conditions under which the data of the
defect (including the gluing condition to be imposed on the
lagrangean fields of the model) define a duality between the phases
of the $\sigma$-model separated by the defect. The point of departure
for these investigations is the identification of a (pre-quantum)
duality with an isomorphism
\begin{eqnarray}\nonumber
{\rm pr}_1^*\mathcal{L}_{\sigma,\emptyset}\vert_{\gt{I}_\sigma}\cong{\rm pr}_2^*\mathcal{L}_{\sigma,
\emptyset}\vert_{\gt{I}_\sigma}
\end{eqnarray}
of the pullbacks of the pre-quantum bundle along the canonical
projections $\,{\rm pr}_\a:{\mathsf P}_{\sigma,\emptyset}\x{\mathsf P}_{\sigma,\emptyset}\to
{\mathsf P}_{\sigma,\emptyset},\ \a\in\{1,2\}\,$ over the graph $\,\gt{I}_\sigma\,$
of a symplectomorphism that preserves the hamiltonian density of the
$\sigma$-model. The relevant conditions are stated in Theorem
\ref{thm:def-dual}, and it is worth underlining that they single out
bi-brane world-volumes $\,Q\,$ that are surjectively submersed onto
the target space $\,M$,\ in keeping with the results of
\Rcite{Fuchs:2009si}. The reverse question as to the circumstances
under which a duality gives rise to consistent defect data is
subsequently examined for an important class of dualities in the
remainder of Section \ref{sec:def-as-iso}, culminating in Theorems
\ref{thm:duali-T-bib} and \ref{thm:duali-N-bib}. Altogether, the
findings of Section \ref{sec:def-as-iso} establish a rather strong
and general correspondence between the so-called topological defects
and dualities of string theory, the latter including, in particular,
symmetries of a single phase induced from distinguished isometries
of the target space and the proper (T-)duality between string models
on topologically non-equivalent principal torus bundles, of the kind
originally discussed in
Refs.\,\cite{Buscher:1987qj,Buscher:1987sk}.\medskip
Just as defect-line data constrain the `tunelling' of the closed
string between images (with respect to the embedding map) of the
supports of adjacent phases of the theory, those carried by defect
junctions are of relevance to the stringy interaction processes,
represented by regions in the world-sheet homeomorphic to a sphere
with (at least) three punctures and an embedded defect subgraph. The
simple world-sheet intuition behind this statement is depicted in
Figure \ref{fig:defect-junct-int}.
\begin{figure}[hbt]~\\[5pt]
$$
\raisebox{-50pt}{\begin{picture}(50,50)
\put(-79,-4){\scalebox{0.25}{\includegraphics{defect-junct-int.pdf}}}
\end{picture}
\put(0,0){
\setlength{\unitlength}{.60pt}\put(-28,-16){
\put(-35,57) { $\ell_{1,2}$ }
\put(48,165) { $\ell_{2,3}$ }
\put(-64,177) { $\ell_{3,1}$ }
\put(-3,162) { $\jmath_{1,2,3}$ }
\put(-35,96) { $\jmath^\vee_{1,2,3}$ }
\put(-86,47) { $\psi_1$ }
\put(25,30) { $\psi_2$ }
\put(-45,210) { $\psi_3$ }
\put(-163,72) { ${\rm CFT}_1$ }
\put(76,40) { ${\rm CFT}_2$ }
\put(-5,260) { ${\rm CFT}_3$ }
\put(160,140) { $t$ }
}\setlength{\unitlength}{1pt}}}
$$
\caption{The basic $\,2\to 1\,$ splitting-joining interaction across
the three-valent defect junction. The arrow to the right of the
picture represents the time direction.} \label{fig:defect-junct-int}
\end{figure}
It can readily be formalised in the canonical language, in which one
is led to expect the emergence of symplectomorphic identifications
among multi-string states in interaction, lifting to isomorphisms of
the associated pullback pre-quantum bundles over an interaction
subspace spanned by these states. Theorems
\ref{thm:cross-def-int-untw} and \ref{thm:cross-def-int-tw} confirm
these expectations independently for each of the two sectors of the
state space of the $\sigma$-model, that is for the untwisted sector and
the twisted sector, altogether giving rise to a canonical picture of
the cross-defect splitting-joining interaction of the string. In
this picture, the data of the 2-isomorphism associated with a defect
junction are shown to transgress, in a canonical manner, to local
data of the expected isomorphism of the pullback pre-quantum
bundles. This result can be viewed as a logical completion of the
transgression scheme for $\,\gt{BGrb}^\nabla(M\sqcup Q\sqcup T)\,$
anticipated by the findings of \Rcite{Gawedzki:1987ak}. A natural,
if also merely implicit in the present treatment, consequence of the
existence of cross-defect identifications among multi-string states
is an interpretation of the defect-junction data in terms of
intertwiners between representations of (current) symmetry algebras
furnished by the multi-string state spaces that enter the definition
of the interaction subspace. This interpretation is substantiated in
\Rcite{Suszek:2010b}.\medskip
The clear-cut canonical interpretation of world-sheet defects, and
-- in particular -- the relation between the latter and dualities of
string theory, in conjunction with the by now rich knowledge --
gathered in
Refs.\,\cite{Gawedzki:2003pm,Gawedzki:2004tu,Schreiber:2005mi,Gawedzki:2007uz,Gawedzki:2008um}
and further enhanced in \Rcite{Gawedzki:2010rn} through the study of
continuous group actions -- on the gerbe-theoretic structure of
multi-phase world-sheets and on their r\^ole in defining string
theory on quotient target spaces suggest a natural generalisation of
the notion of a smooth (pseudo-)riemannian manifold with extra
(cohomological) structure. Inspired by the pioneering work
\cite{Hull:2004in}, but also taking into account the findings of
\Rcite{Jureit:2006yf}, where multi-phase world-sheets were
considered from the point of view of a consistent interaction of the
closed string on the orbifolded target space, we may conceive a
situation in which the string world-sheet is mapped into a target
space modelled on a (pseudo-)riemannian manifold with a gerbe over
it only \emph{locally}, with geometric data (those of the metric and
of the gerbe) from local charts glued together by means of local
data of the bi-brane implementing \emph{bona fide} $\sigma$-model
dualities. This ultimately leads to the concept of a non-geometric
background with the structure of a duality `quotient' (provided that
the latter can be defined in a meaningful manner), or a
D(uality)-fold, generalising the idea of a T-fold based on the
T-duality group. The correspondence between gerbes on ${\rm K}$-spaces
and those on ${\rm K}$-quotients of the latter is a strong indication
that string theory on a would-be D-fold prerequires a
self-consistent hierarchy of cohomological structures, to wit, the
bulk gerbe, the duality bi-brane and the basic inter-bi-brane (for
three-valent defect junctions) from which all components of the
inter-bi-brane associated with defect junctions of valence higher
that 3 could be induced in a well-defined and physically intuitive
manner. The relevant intuition derives from the observation that an
insertion of a local defect field for a defect junction of valence
$\,n>3\,$ can be generated in a suitably regularised limiting
procedure of bringing together a number of insertions of local
defect fields for defect junctions of valence 3. The existence of
such a hierarchy was first discussed (and illustrated with the
explicit example of the central-jump WZW defect) in
\cite{Runkel:2008gr} under the name of defect-junction data with
induction. In Remark \ref{rem:duality-scheme}, the original
discussion is extended and rephrased in the language of simplicial
objects in the category of differentiable manifolds, inspired by the
study of equivariant structures and purely physical considerations,
whereupon the notion of a simplicial string background is
introduced. This is then conjectured to be the point of departure in
any consistent construction of a D-fold, to which we are hoping to
return in the future.\medskip
Prior to concluding this introductory section, let us add a few more
comments on the structure of the present paper. First of all, we
have decided to organise the discourse into a collection of
definitions, propositions and theorems, interspersed with examples
and occasional remarks of a looser nature. Secondly, the more
technical proofs have been relegated to the appendices. Finally,
several open questions have been collected in the closing Section
\ref{sec:out}, alongside a brief recapitulation of the results.
\bigskip
\noindent{\bf Acknowledgements:} The author is much beholden to
S.~Fredenhagen, I.~Runkel and, in particular, to K.~Gaw{\c{e}}dzki
for discussions and their sustained interest in the project reported
in the present paper. He also gratefully acknowledges the kind
hospitality of Laboratoire de Physique de l'\'Ecole Normale
Sup\'erieure de Lyon, Albert-Einstein-Institut in Potsdam, Bereich
Algebra und Zahlentheorie des Departments Mathematik an der
Universit\"at Hamburg and Matematyczne Centrum
Konferencyjno-Badawcze IM PAN in B{\c{e}}dlewo, where parts of this
work was carried out.
\section{Defects in the lagrangean picture}\label{sec:lagr}
In the present paper, we shall be concerned with a theory of bosonic
fields on an oriented two-dimensional space-time in the presence of
domain walls that split the space-time into domains supporting the
respective phases of the field theory. Fields of the theory take
values in differentiable manifolds with additional geometric
structure, captured neatly by gerbe theory, and a working knowledge
of the rudiments thereof is assumed throughout this paper. The
reader unfamiliar with the theory is referred to, e.g., the
literature cited in the Introduction. Thus, let us begin with
\begin{Def}\label{def:bckgrnd}
A \textbf{string background} is a triple $\,\gt{B}=(\mathcal{M},\mathcal{B},\mathcal{J} )\,$
composed of the following geometric structures:
\begin{itemize}
\item the \textbf{target} $\,\mathcal{M}=(M,{\rm g},\mathcal{G})\,$ consisting of
a manifold $\,M$,\ termed the \textbf{target space}, with a metric
$\,{\rm g}$,\ a closed 3-form $\,{\rm H}\,$ and an abelian gerbe $\,\mathcal{G}\,$
(with connection) of curvature $\,{\rm H}$;
\item the \textbf{$\mathcal{G}$-bi-brane} $\,\mathcal{B}=\bigl(Q,\iota_\a,\omega,\Phi\
\vert\ \a\in\{1,2\}\bigr)\,$ consisting of a manifold $\,Q$,\ termed
the \textbf{$\mathcal{G}$-bi-brane world-volume}, with a 2-form $\,\omega$,\
termed the \textbf{$\mathcal{G}$-bi-brane curvature}, and a pair of smooth
maps $\,\iota_\a:Q\to M,\ \a\in\{1,2\}$,\ and of a gerbe
1-isomorphism (a $(\iota_1^*\mathcal{G},\iota_2^*\mathcal{G})$-bi-module)
\begin{eqnarray}\nonumber
\Phi\ :\ \iota_1^*\mathcal{G}\xrightarrow{\cong}\iota_2^*\mathcal{G}\otimes I_\omega\,,
\end{eqnarray}
written in terms of a trivial gerbe $\, I_\omega\,$ with curving
$\,\omega$,\ obeying the identity
\begin{eqnarray}\nonumber
\Delta_Q{\rm H}=-{\mathsf d}\omega\,,\qquad\Delta_Q:=\iota_2^*-\iota_1^*\,;
\end{eqnarray}
\item the \textbf{$(\mathcal{G},\mathcal{B})$-inter-bi-brane} $\,\mathcal{J}=\bigl(T_n,
\bigl(\varepsilon^{k,k+1}_n,\pi^{k,k+1}_n \ \vert\ k\in\ovl{1,n}\bigr),
\varphi_n\ \vert\ n\in{\mathbb{N}}_{\geq 3}\bigr)$,\ with $\,\ovl{1,n}=\{\ \
k\in{\mathbb{Z}} \quad\vert\quad 1\leq k\leq n \}$,\ consisting of a disjoint
sum of manifolds $\,\bigsqcup_{n\in{\mathbb{N}}_{\geq 3}}\,T_n=:T$,\ termed
the \textbf{$(\mathcal{G}, \mathcal{B})$-inter-bi-brane world-volume}, with a
collection of orientation maps $\,\varepsilon^{k,k+ 1}_n:T_n\to\{-1,+1\}\,$
and smooth maps $\,\pi^{k,k+1}_n:T_n\to Q\,$ subject to the
constraints
\begin{eqnarray}\label{eq:proto-simpl}
\iota_2^{\varepsilon_n^{k-1,k}}\circ\pi_n^{k-1,k}=\iota_1^{\varepsilon_n^{k,k+1}}
\circ\pi_n^{k,k+1}\,,\qquad k\in\ovl{1,n}\,,
\end{eqnarray}
with $\,(\iota_1^{+1},\iota_2^{+1})=(\iota_1,\iota_2)\,$ and
$\,(\iota_1^{-1},\iota_2^{-1})=(\iota_2,\iota_1)$,\ and of
distinguished gerbe 2-isomorphisms
\begin{eqnarray}\label{diag:2iso}
\xy (50,0)*{\bullet}="G3"+(5,4)*{\mathcal{G}_n^3\otimes I_{\omega_n^{1,2}+\omega_n^{2,
3}}}; (25,-20)*{\bullet}="G2"+(-10,0)*{\mathcal{G}_n^2\otimes I_{\omega_n^{1,2}}};
(75,-20)*{\ \vdots}="dots"; (85,-20)*{\,;};
(35,-40)*{\bullet}="G1"+(0,-4)*{\mathcal{G}_n^1};
(65,-40)*{\bullet}="G1add"+(10.5,-4)*{\mathcal{G}_n^1\otimes I_{\omega_n^{1,2}+
\omega_n^{2,3}+\ldots+\omega_n^{n,1}}}; (50,-40)*{}="id";
\ar@{->}|{\Phi_n^{2,3}\otimes{\rm id}} "G2";"G3"
\ar@{->}|{\Phi_n^{3,4}\otimes{\rm id}} "G3";"dots" \ar@{->}|{\Phi_n^{1,2}}
"G1";"G2" \ar@{->}|{\Phi_n^{n,1}\otimes{\rm id}} "dots";"G1add"
\ar@{=}|{{\rm id}_{\mathcal{G}_n^1}} "G1"+(2,0);"G1add"+(-2,0)
\ar@{=>}|{\varphi_n} "G3"+(0,-3);"id"+(0,+3)
\endxy
\end{eqnarray}
written in terms of 1-isomorphisms $\,\Phi_n^{k,k+1}=\pi_n^{k,k+1\,
*}\Phi^{\varepsilon_n^{k,k+1}}$,\ with $\,\Phi^{+1}=\Phi\,$ and $\,\Phi^{-
1}=\Phi^\vee\,$ (the dual 1-isomorphism), between gerbes $\,\mathcal{G}_n^k=
(\iota_1^{\varepsilon_n^{k,k+1}}\circ\pi_n^{k,k+1})^*\mathcal{G}$,\ and the trivial
gerbes with global curvings $\,\omega_n^{k,k+1}=\varepsilon_n^{k,k+1}
\,\pi_n^{k,k+1\,*}\omega$.\ The latter satisfy the Defect-Junction
Identity (DJI)
\begin{eqnarray}\nonumber
\Delta_{T_n}\omega=0\,,\qquad\Delta_{T_n}:=\sum_{k=1}^n\,\varepsilon_n^{k,k+1}\,
\pi_n^{k,k+1\,*}\,.
\end{eqnarray}
\end{itemize}
\begin{flushright}$\checkmark$\end{flushright}\end{Def} \noindent In the subsequent sections, we shall oftentimes
have a need for a more explicit description of the gerbe-theoretic
concepts invoked in Definition \ref{def:bckgrnd}. For this reason,
we recall
\begin{Def}\label{def:loco}
Let $\,\mathscr{M}\,$ be a differentiable manifold, and let $\,\mathcal{S}^q_\mathscr{M},\
q\in\ovl{0,\dim\,\mathscr{M}}\,$ be the following sheaves over $\,\mathscr{M}$:
\begin{itemize}
\item $\mathcal{S}^0_\mathscr{M}:=\unl{\rm U}(1)_\mathscr{M}$,\ the sheaf of locally smooth
${\rm U}(1)$-valued maps on $\,\mathscr{M}$;
\item $\mathcal{S}^q_\mathscr{M}:=\unl\Omega^q(\mathscr{M}),\ q>0$,\ the sheaf of locally
smooth (real) $q$-forms on $\,\mathscr{M}$.
\end{itemize}
Given the differential Deligne complex
\begin{eqnarray}\nonumber
\mathcal{D}(n)_\mathscr{M}^\bullet\ :\ \mathcal{S}^0_\mathscr{M}\xrightarrow{\ {\mathsf d}^{(0)}:=
\frac{1}{{\mathsf i}}{\mathsf d}\log\ }\mathcal{S}^1_\mathscr{M} \xrightarrow{\ {\mathsf d}^{(1)}:={\mathsf d}\
}\mathcal{S}^2_\mathscr{M} \xrightarrow{\ {\mathsf d}^{(2)}:={\mathsf d}\ }\mathcal{S}^3_\mathscr{M}
\xrightarrow{\ {\mathsf d}^{(3)}:={\mathsf d}\ }\cdots\xrightarrow{\ {\mathsf d}^{(n-1)}:
={\mathsf d}\ }\mathcal{S}^n_\mathscr{M}\,,
\end{eqnarray}
denote by $\,\cA^{n,\bullet}( \mathcal{O}_\mathscr{M})\,$ the diagonal sub-complex
of the \v Cech--Deligne double complex $\,\check{C}^\bullet\bigl(\mathcal{O}_\mathscr{M},
\mathcal{D}(n)_\mathscr{M}^\bullet\bigr)\,$ obtained, for a given choice
$\,\mathcal{O}_\mathscr{M}=\{\mathcal{O}^\mathscr{M}_i\}_{i\in\mathscr{I}}\,$ of a good open cover of
$\,\mathscr{M}\,$ (with non-empty multiple intersections of its elements
denoted as $\,\mathcal{O}^\mathscr{M}_{i_1}\cap\mathcal{O}^\mathscr{M}_{i_2}\cap\cdots\cap
\mathcal{O}^\mathscr{M}_{i_n}=: \mathcal{O}^\mathscr{M}_{i_1 i_1\ldots i_n}\,$ and assumed
contractible),\ by extending $\,\mathcal{D}(n)_\mathscr{M}^\bullet\,$ through the
\v Cech complexes
\begin{eqnarray}\nonumber
\check{C}^0(\mathcal{O}_\mathscr{M},\mathcal{S}_\mathscr{M}^q)\xrightarrow{\ \check{\d}^{(0)}\ }\check{C}^1(\mathcal{O}_\mathscr{M}
,\mathcal{S}_\mathscr{M}^q)\xrightarrow{\ \check{\d}^{(1)}\ }\check{C}^2(\mathcal{O}_\mathscr{M},\mathcal{S}_\mathscr{M}^q)
\xrightarrow{\ \check{\d}^{(2)}\ }\cdots
\end{eqnarray}
associated to $\,\mathcal{O}_\mathscr{M}$,\ and with the standard \v Cech
coboundary operators
\begin{eqnarray}\nonumber
\check{\d}^{(p)}\ &:&\ \check{C}^p(\mathcal{O}_\mathscr{M},\mathcal{S}_\mathscr{M}^q)\to\check{C}^{p+1}(\mathcal{O}_\mathscr{M},
\mathcal{S}_\mathscr{M}^q)\cr\cr
&:&\ (s_{i_0 i_1 \ldots i_p})\mapsto\bigl((\check{\d}^{(p)}s)_{i_0 i_1
\ldots i_{p+1}}):=\left(\sum_{k=0}^{p+1}\,(-1)^k\,s_{i_0 i_1
\underset{\widehat{i_k}}{\ldots} i_{p+1}}\vert_{\mathcal{O}^\mathscr{M}_{i_0 i_1
\ldots i_{p+1}}}\right)\,.
\end{eqnarray}
The above is written in the additive notation for local sections $\,
s_{i_0 i_1 \ldots i_p}\in\mathcal{S}_\mathscr{M}^q(\mathcal{O}^\mathscr{M}_{i_0 i_1 \ldots i_p})
\,$ of the sheaves $\,\mathcal{S}_\mathscr{M}^q$,\ in which "+" stands for
multiplication of sections if $\,q=0\,$ and for addition of sections
otherwise, and in which multiplication of a section by a real number
$\,c\,$ stands for the raising of the section to the power $\,c\,$
if $\,q=0\,$ and for the multiplying of the section by $\,c\,$
otherwise. This notation shall be used throughout the paper.
Finally, we write as $\,{}^{\tx{\tiny $\mathscr{M}$}}\hspace{-2pt} D_{(r)}\,$ the Deligne differential
defined component-wise as
\begin{eqnarray}\nonumber
{}^{\tx{\tiny $\mathscr{M}$}}\hspace{-2pt} D_{(r)}\ :\ \cA^{n,r}(\mathcal{O}_\mathscr{M})\to\cA^{n,r+1}(\mathcal{O}_\mathscr{M})\,,
\qquad\qquad{}^{\tx{\tiny $\mathscr{M}$}}\hspace{-2pt} D_{(r)}\vert_{\check{C}^p(\mathcal{O}_\mathscr{M},\mathcal{S}^q_\mathscr{M})}=
{\mathsf d}^{(q)}+(-1)^{q+1}\,\check{\d}^{(p)}\,,
\end{eqnarray}
with the corresponding Deligne (hyper-)cohomology groups denoted as
$\,{\mathbb{H}}^r\bigl(\mathscr{M},\mathcal{D}(n)^\bullet_\mathscr{M}\bigr)$.\ A \textbf{local
presentation of string background $\,\gt{B}=(\mathcal{M},\mathcal{B},\mathcal{J})\,$} consists
of the following data
\begin{itemize}
\item for the gerbe $\,\mathcal{G}\,$ over the target space $\,M$,\ a \v
Cech--Deligne cochain
\begin{eqnarray}\nonumber
\mathcal{G}\xrightarrow{\rm loc.}(B_i,A_{ij},g_{ijk})=:b\in\cA^{3, 2}(\mathcal{O}_M)
\end{eqnarray}
with \textbf{curvings} $\,B_i$,\ \textbf{connections} $\,A_{ij}\,$
and \textbf{transition functions} $\,g_{ijk}$,\ satisfying the
cohomological identity
\begin{eqnarray}\label{eq:DG-is-H}
{}^{\tx{\tiny $M$}}\hspace{-2pt} D_{(2)}b=({\rm H}\vert_{\mathcal{O}^M_i},0,0,1)\,;
\end{eqnarray}
the local data $\,b\,$ are determined up to \textbf{gauge
transformations}
\begin{eqnarray}\label{eq:gauge-trans-gerbe}
b\mapsto b+{}^{\tx{\tiny $M$}}\hspace{-2pt} D_{(1)}\pi\,,\qquad\pi:=(\Pi_i,\chi_{ij})\in\cA^{3,
1}(\mathcal{O}_M)\,;
\end{eqnarray}
thus, gauge equivalence classes of local data correspond to elements
of $\,{\mathbb{H}}^2\bigl(M,\mathcal{D}(2)^\bullet_M\bigr)$;
\item for the $\mathcal{G}$-bi-brane 1-isomorphism $\,\Phi$,\ a \v
Cech--Deligne cochain
\begin{eqnarray}\nonumber
\Phi\xrightarrow{\rm loc.}(P_i,K_{ij})=:p\in\cA^{2,1}(\mathcal{O}_Q)
\end{eqnarray}
satisfying the cohomological identity
\begin{eqnarray}\label{eq:DPhi-is}
{}^{\tx{\tiny $Q$}}\hspace{-2pt} D_{(1)}p=\check{\Delta}_Q b+\ovl\omega\,,
\end{eqnarray}
in which $\,\ovl\omega=(\omega\vert_{\mathcal{O}^Q_i},0,1)\,$ are local data of
the trivial gerbe $\,I_\omega$,\ and $\,\check{\Delta}_Q:=\check{\iota}_2^*
-\check{\iota}_1^*\,$ for the \v Cech-extended $\mathcal{G}$-bi-brane maps
$\,\check\iota_\a=(\iota_\a,\phi_\a)\,$ with index maps $\,\phi_\a:
\mathscr{I}_Q\to\mathscr{I}_M\,$ covering the respective manifold maps $\,\iota_\a
:Q\to M\,$ as per
\begin{eqnarray}\nonumber
\iota_\a(\mathcal{O}^Q_i)\subset\mathcal{O}^M_{\phi_\a(i)}\,,
\end{eqnarray}
and in which we use the shorthand notation
\begin{eqnarray}\nonumber
\check{\iota}_\a(B_i,A_{ij},g_{ijk}):=\iota_\a^*(B_{\phi_\a(i)},
A_{\phi_\a(i)\phi_\a(j)},g_{\phi_\a(i)\phi_\a(j)\phi_\a(k)})\,;
\end{eqnarray}
the local data $\,p\,$ are determined up to \textbf{$\mathcal{G}$-twisted
gauge transformations}
\begin{eqnarray}\label{eq:gauge-trans-bi}
p\mapsto p+\check{\Delta}_Q\pi-{}^{\tx{\tiny $Q$}}\hspace{-2pt} D_{(0)}w\,,\qquad w:=(W_i)\in\cA^{2
,0}(\mathcal{O}_Q)\,,
\end{eqnarray}
with the $\mathcal{G}$-twist $\,\check{\Delta}_Q\pi\,$ ensuring that the
defining identity \eqref{eq:DPhi-is} is preserved under a gauge
transformation \eqref{eq:gauge-trans-gerbe};
\item for the $(\mathcal{G},\mathcal{B})$-inter-bi-brane 2-isomorphisms
$\,\varphi_n$,\ \v Cech--Deligne cochains
\begin{eqnarray}\nonumber
\varphi_n\xrightarrow{\rm loc.}(f_{n,i})=:F_n\in\cA^{1,0}(\mathcal{O}_{T_n}
)
\end{eqnarray}
satisfying the cohomological identities
\begin{eqnarray}\label{eq:Dphin-is}
{}^{\tx{\tiny $T_n$}}\hspace{-2pt} D_{(0)}F_n=-\check{\Delta}_{T_n}p\,,
\end{eqnarray}
in which $\,\check{\Delta}_{T_n}:=\sum_{k=1}^{n+1}\,\varepsilon_n^{k,k+1}\,
\check{\pi}_n^{k,k+1\,*}\,$ for the \v Cech-extended $(\mathcal{G},\mathcal{B}
)$-inter-bi-brane maps $\,\check\pi_n^{k,k+1}=(\pi_n^{k,k+1},
\psi_n^{k,k+1})\,$ with index maps $\,\psi_n^{k,k+1}:\mathscr{I}_{T_n}\to
\mathscr{I}_Q\,$ covering the respective manifold maps $\,\pi_n^{k,k+1}:T_n
\to Q\,$ as per
\begin{eqnarray}\nonumber
\pi_n^{k,k+1}(\mathcal{O}^{T_n}_i)\subset\mathcal{O}^Q_{\psi_n^{k,k+1}(i)}\,,
\end{eqnarray}
and in which we use the shorthand notation
\begin{eqnarray}\nonumber
\check{\pi}_n^{k,k+1\,*}(P_i,K_{ij}):=\pi_n^{k,k+1\,*}(P_{\psi_n^{k
,k+1}(i)},K_{\psi_n^{k,k+1}(i)\psi_n^{k,k+1}(j)})\,;
\end{eqnarray}
the local data $\,F_n\,$ undergo a compensating gauge transformation
\begin{eqnarray}\nonumber
F_n&\mapsto&F_n+\check{\Delta}_{T_n}w
\end{eqnarray}
under a $\mathcal{G}$-twisted gauge transformation
\eqref{eq:gauge-trans-bi}, ensuring that the defining identity
\eqref{eq:Dphin-is} is preserved.
\end{itemize}
\begin{flushright}$\checkmark$\end{flushright}\end{Def} \noindent The reader is urged to consult
\Rcite{Brylinski:1993ab} for a thorough introduction to the
cohomological constructs used in the above definition. Here, we
merely point out an important consequence of the cohomological
description of gerbes and 1- and 2-isomorphisms, which provides us
with a natural classification scheme of string backgrounds.
\bero
The set of 1-isomorphism classes of gerbes with a given curvature
over a manifold $\,\mathscr{M}\,$ is a torsor under a natural action of the
sheaf-cohomology group $\,H^2\bigl(\mathscr{M},{\rm U}(1)\bigr)$.
\end{Prop}
\begin{Prop}\label{prop:2iso-class-1iso
The set of 2-isomorphism classes of 1-isomorphisms between two given
gerbes over a manifold $\,\mathscr{M}\,$ is a torsor under a natural action
of the sheaf-cohomology group $\,H^1\bigl( \mathscr{M},{\rm U}(1)\bigr)$.
\end{Prop}
\begin{Prop}
The set of inequivalent 2-isomorphisms between two given
1-isomorphisms of gerbes over a manifold $\,\mathscr{M}\,$ with $\,|\pi_0(
\mathscr{M})|\,$ connected components is a torsor under a natural action of
the sheaf-cohomology group $\,H^0\bigl( \mathscr{M},{\rm U}(1)\bigr)\cong{\rm U}(1)^{|
\pi_0(\mathscr{M})|}$.
\end{Prop}
\noindent All three statements are simple corollaries of the
relation between the Deligne hypercohomology and sheaf cohomology,
taken in conjunction with the contents of Definition \ref{def:loco},
cf., e.g., \cite{Brylinski:1993ab,Gawedzki:2002se,Gomi:2003}.
Another auxiliary concept of use in the sequel is introduced in the
following
\begin{Def}\label{def:net-field}
Let $\,\Sigma\,$ be a closed oriented two-dimensional manifold with an
intrinsic metric $\,\gamma\,$ of a lorentzian signature\footnote{Note
that -- unlike \Rcite{Runkel:2008gr} -- we are dealing with the
lorentzian version of the world-sheet theory here as we intend to
discuss its canonical structure. It is a classic result in topology,
cf., e.g., \Rxcite{Thm.\,40.10}{Steenrod:1951}, that a global
lorentzian structure can exist on $\,\Sigma\,$ iff $\,\Sigma\,$ is
non-compact or $\,\Sigma\,$ is homeomorphic with a torus or a Klein
bottle. In what follows, we shall mainly be interested in $\,\Sigma
\cong{\mathbb{R}}\x{\mathbb{S}}^1\,$ (an infinite cylinder) with a view to a canonical
interpretation of defects. In this case, there are no obstructions
to the existence of a lorentzian metric. In more general situations,
and -- in particular -- in the case of the trinion (also known as
``pair-of-pants'') geometry representing the basic splitting-joining
interaction of strings, we shall disregard the signature problem,
with the implicit understanding that a proper treatment of the
world-sheet metric may require passing to the euclidean version of
the theory, accompanied by the complexification of the field space.
These manipulations are not going to invalidate our conclusions.
\label{foot:mink-vs-eukl}} $\,(-,+)$,\ termed the
\textbf{world-sheet} and split into patches $\,\wp$,\ forming the
patch set $\,\gt{P}_\Sigma$,\ by an embedded oriented graph $\,\Gamma$,\ to
be termed the \textbf{defect quiver}. The graph is composed of a
collection of oriented lines $\,\ell$,\ termed \textbf{defect
lines}, forming the edge set $\,\gt{E}_\Gamma\,$ of $\,\Gamma\,$ and
intersecting at a number of points $\,\jmath$,\ termed
\textbf{defect junctions} and forming the vertex set $\,\gt{V}_\Gamma\,$
of $\,\Gamma$.\ Furthermore, let $\,\gt{B}=(\mathcal{M},\mathcal{B},\mathcal{J})\,$ be a string
background as in Definition \ref{def:bckgrnd}. A
\textbf{network-field configuration $\,(X\,\vert\,\Gamma )\,$ in string
background $\,\gt{B}\,$ on world-sheet $\,(\Sigma,\gamma)\,$ with defect
quiver $\,\Gamma\,$} is a pair composed of the defect quiver $\,\Gamma\,$
embedded in the world-sheet $\,\Sigma$,\ together with a map
$\,X:\Sigma\to M\sqcup Q\sqcup T\,$ such that
\begin{itemize}
\item $X\,$ restricts to a once differentiable map $\,\Sigma\setminus\Gamma
\to M$,\ a once differentiable map $\,\Gamma\setminus\gt{V}_\Gamma\to Q$,\ and
it sends $\,\gt{V}_\Gamma\to T\,$ in such a manner that a defect junction
$\,\jmath\,$ of valence $\,n_\jmath\,$ is mapped to
$\,T_{n_\jmath}$;
\item for every $\,p\in\Gamma\setminus\gt{V}_\Gamma\,$ and $\,U\subset\Sigma\,$ a
small neighbourhood of $\,p\,$ split into subsets $\,U_\a,\ \a\in\{1
,2\}\,$ by $\,\Gamma\,$ so that the vector $\,\widehat n\,$ normal to
$\,\Gamma\,$ at $\,p\,$ and pointing towards $\,U_2\,$ together with the
vector $\,\widehat t\,$ tangent to $\,\Gamma\,$ at $\,p\,$ and
determining its orientation define a right-handed basis $\,(\widehat
n,\widehat t)\,$ of $\,{\mathsf T}_p \Sigma\,$ as in Figure
\ref{fig:nbdry-tan}, the map $\,X\vert_\a\,$ admits a differentiable
extension $\,X_{|\a}:\ovl U_\a\to M\,$ to the closure of $\,U_\a$,\
with the property $\,X_{|\a}(p)=\iota_\a\circ X(p)$;
\begin{figure}[hbt]~\\[5pt]
$$
\raisebox{-50pt}{\begin{picture}(50,50)
\put(-79,-4){\scalebox{0.25}{\includegraphics{nbdry-tan.pdf}}}
\end{picture}
\put(0,0){
\setlength{\unitlength}{.60pt}\put(-28,-16){
\put(-45,145) { $\widehat t$ }
\put(15,123) { $\widehat n$ }
\put(-35,113) { $p$ }
\put(-145,195) { $\Sigma$ }
\put(-16,225) { $\ell$ }
\put(-80,170) { $U_1$ }
\put(20,170) { $U_2$ }
\put(-80,63) { $U$ }
}\setlength{\unitlength}{1pt}}}
$$
\caption{The right-handed basis $\,(\widehat n,\widehat t)\,$ of the
tangent space $\,{\mathsf T}_p\Sigma\,$ at a point $\,p\,$ on the defect line
$\,\ell\,$ embedded in the world-sheet $\,\Sigma\,$ and splitting the
neighbourhood $\,U\,$ of $\,p\,$ (inside the blue contour) into
subsets $\,U_1\,$ and $\,U_2$.} \label{fig:nbdry-tan}
\end{figure}
\item the Defect Gluing Condition (DGC)
\begin{eqnarray}\nonumber
\tx{DGC}_\mathcal{B}(\psi_{|1},\psi_{|2},X)\equiv{\mathsf p}_{|1}\circ\iota_{1\,*}
-{\mathsf p}_{|2}\circ\iota_{2\,*}-X_*\widehat t\righthalfcup\omega(X)=0\,,\qquad
\qquad\psi_{|\a}=(X_{|\a},{\mathsf p}_{|\a})\,,\\ \label{eq:DGC}
\end{eqnarray}
is satisfied at each $\,p\in\ell\in\gt{E}_\Gamma\,$ for a vector
$\,\widehat t\,$ tangent to $\,\Gamma\,$ at $\,p\,$ and determining its
orientation, and for
\begin{eqnarray}\nonumber
{\mathsf p}_{|\a}={\rm g}(X_{|\a})(X_{|\a\,*}\widehat n,\cdot)\,,
\end{eqnarray}
with a vector $\,\widehat n=\gamma^{-1}\bigl(\widehat t\righthalfcup\Vol(\Sigma,\gamma),
\cdot\bigr)\,$ written in terms of the metric volume form $\,\Vol(
\Sigma,\gamma)\,$ on $\,\Sigma\,$ and defining $\,X_{|\a\,*}\widehat n\,$ in
terms of a (one-sided) derivative;
\item for $\,\jmath\in\gt{V}_\Gamma\,$ an $n_\jmath$-valent defect
junction and $\,\ell_{k,k+1}\,$ a defect line converging at
$\,\jmath$,\ the map $\,X\vert_{\ell_{k,k+1}\setminus\gt{V}_\Gamma}\,$
admits a differentiable extension $\,X_{k,k+1}:\ell_{k,k+1}\to Q\,$
such that $\,X_{k,k+1}(\jmath)=\pi_{n_\jmath}^{k,k+1}\circ X(\jmath
)$;
\item for $\,(\jmath,\ell_{k,k+1})\,$ as above, the orientation map
takes the value $\,\varepsilon_{n_\jmath}^{k,k+1}\bigl(X(\jmath)\bigr)=+
1\,$ if $\,\ell_{k,k+1}\,$ is oriented towards $\,\jmath\,$ (an
incoming defect line), and the opposite value $\,\varepsilon_{n_\jmath}^{k,
k+1}\bigl(X(\jmath)\bigr)=-1\,$ otherwise.
\end{itemize}
\begin{flushright}$\checkmark$\end{flushright}\end{Def} We may now give a precise description of the main object of
our study.
\begin{Def}\label{def:sigmod}
Let $\,(X\,\vert\,\Gamma)\,$ be a network-field configuration in string
background $\,\gt{B}\,$ with field space $\,M\sqcup Q\sqcup T\,$ on
world-sheet $\,(\Sigma,\gamma)\,$ with defect quiver $\,\Gamma$,\ and choose a
local presentation of $\,\gt{B}\,$ with respect to good open covers
$\,\mathcal{O}_\mathscr{M},\ \mathscr{M}\in\{M,Q,T\}$.\ Furthermore, let $\,\triangle(\Sigma
)\,$ be a \textbf{triangulation of $\,\Sigma\,$ subordinate to
$\,\mathcal{O}_\mathscr{M},\ \mathscr{M}\in\{M,Q,T\}\,$ with respect to $\,(X\,\vert\,\Gamma)\,$} in the
following sense:
\begin{itemize}
\item $\triangle(\Sigma)\,$ induces a triangulation $\,\triangle(\Gamma)
\subset\triangle(\Sigma)\,$ of the defect quiver in such a manner that
each defect line $\,\ell\in\gt{E}_\Gamma\,$ is covered by the edges
$\,e\in\triangle(\Gamma)\,$ and $\,\gt{V}_\Gamma\subset\triangle(\Gamma)$;
\item for each plaquette $\,p\in\triangle(\Sigma)$,\ there exists a
\v Cech index $\,i_p\in\mathscr{I}_M\,$ of $\,\mathcal{O}_M\,$ such that $\,X(p)
\subset\mathcal{O}^M_{i_p}$,\ which we fix;
\item for each defect edge $\,e\in\triangle(\Gamma)$,\ there exists a
\v Cech index $\,i_e\in\mathscr{I}_Q\,$ of $\,\mathcal{O}_Q\,$ such that $\,X(e)
\subset\mathcal{O}^Q_{i_e}$,\ which we fix;
\item for each defect vertex $\,\jmath\in\Gamma\,$ of valence
$\,n_\jmath$,\ we fix a \v Cech index $\,i_\jmath\in
\mathscr{I}_{T_{n_\jmath}}\,$ of $\,\mathcal{O}_{T_{n_\jmath}}\,$ such that $\,X(
\jmath)\in\mathcal{O}^{T_{n_\jmath}}_{i_\jmath}$.
\end{itemize}
The \textbf{(two-dimensional) non-linear $\sigma$-model for
network-field configurations $\,(X\,\vert\,\Gamma)\,$ in string
background $\,\gt{B}\,$ on world-sheet $\,(\Sigma, \gamma)\,$ with defect
quiver $\,\Gamma\,$} is a theory of continuously differentiable maps
$\,X:\Sigma \to M\sqcup Q\sqcup T$,\ determined by the principle of
least action applied to the action functional
\begin{eqnarray}\label{eq:sigma}
S_\sigma[(X\,\vert\,\Gamma);\gamma]=-\tfrac{1}{2}\,\int_\Sigma\,{\rm g}_X({\mathsf d} X
\overset{\wedge}{,}\star_\gamma{\mathsf d} X)+S_{\rm top}[(X\,\vert\,\Gamma)]\,,
\end{eqnarray}
in which
\begin{itemize}
\item ${\mathsf d} X=\p_a X^\mu\,{\mathsf d}\sigma^a\otimes\p_\mu$,\ in local coordinates
$\,\{\sigma^a\}^{a\in\{1,2\}}\,$ on $\,\Sigma\,$ and $\{X^\mu\}^{\mu\in
\ovl{1,\dim\,M}}\,$ on $\,M$,\ and the target-space metric is
assumed to act on the second factor of the tensor product;
\item $\star_\gamma\,$ is the Hodge operator on $\,\Gamma(\wedge^\bullet
{\mathsf T}^*\Sigma)\,$ determined by the world-sheet metric $\,\gamma$;
\item the topological term
\begin{eqnarray}\nonumber
S_{\rm top}[(X\,\vert\,\Gamma)]=-{\mathsf i}\,\log{\rm Hol}_\gt{B}(X\,\vert\,\Gamma)
\end{eqnarray}
is given by the generalised surface holonomy
$\,{\rm Hol}_\gt{B}(X\,\vert\,\Gamma )\,$ for the network-field configuration
$\,(X\,\vert\,\Gamma)$,\ which, in a triangulation of $\,\Sigma\,$
subordinate to the good open covers $\,\mathcal{O}_\mathscr{M},\
\mathscr{M}\in\{M,Q,T\}\,$ with respect to $\,(X\,\vert\,\Gamma)\,$ (consisting of
plaquettes $\,p$,\ edges $\,e\,$ and vertices $\,v$) and a local
presentation of $\,\gt{B}\,$ as described in Definition
\ref{def:loco}, takes the form
\begin{eqnarray}\nonumber
-{\mathsf i}\,\log{\rm Hol}_\gt{B}(X\,\vert\,\Gamma)&=&\sum_{p\in\triangle(\Sigma)}
\left[\int_p\,X_p^*B_{i_p}+\sum_{e\subset p}\left(\int_e\,X_e^*
A_{i_p i_e}-{\mathsf i}\,\sum_{v\in e}\,\log X^*g_{i_p i_e i_v}^{\varepsilon_{pe
v}}(v)\right)\right]\cr\cr
&&+\sum_{e\in\triangle(\Gamma\setminus\gt{V}_\Gamma)}\,\left(\int_e\,X_e^*
P_{i_e}-{\mathsf i}\,\sum_{v\in e}\,\log X^*K_{i_e i_v}^{-\varepsilon_{ev}}(v)
\right)\cr\cr
&&-{\mathsf i}\,\sum_{\jmath\in\gt{V}_\Gamma}\,\log X^*f_{n_\jmath,i_\jmath}(
\jmath)\,.
\end{eqnarray}
\end{itemize}
\begin{flushright}$\checkmark$\end{flushright}\end{Def} An extensive discussion of the various components of the
string background $\,\gt{B}\,$ was presented, alongside a derivation
of the local formula for $\,S_{\rm top}[(X\,\vert\,\Gamma)]$,\ in
\Rcite{Runkel:2008gr}, to which we refer the reader for details.
Here, we merely point out the statement of consistency:
\begin{Prop}\cite[Sec.\,2.7]{Runkel:2008gr}
The topological term $\,S_{\rm top}[(X\,\vert\,\Gamma)]\,$ of the action
functional of the non-linear $\sigma$-model for network-field
configurations $\,(X\,\vert\,\Gamma)\,$ in string background $\,\gt{B}\,$
on world-sheet $\,(\Sigma,\gamma)\,$ with defect quiver $\,\Gamma\,$ is
independent of the choice $\,\triangle(\Sigma)\,$ of the triangulation
and invariant under gauge transformations of the local presentation
of the string background $\,\gt{B}$,\ as described in Definition
\ref{def:loco}.
\end{Prop}
\noindent And the statement of symmetry:
\begin{Thm}\cite[Sec.\,2.9]{Runkel:2008gr}\label{thm:conf-def}
The non-linear $\sigma$-model for network-field configurations $\,(X\,
\vert\,\Gamma)\,$ in string background $\,\gt{B}\,$ on world-sheet $\,(\Sigma
,\gamma)\,$ with defect quiver $\,\Gamma\,$ of Definition \ref{def:sigmod}
is invariant with respect to arbitrary (gauge) transformations
\begin{eqnarray}\nonumber
&X\mapsto X\circ D\,,\qquad\qquad\gamma\mapsto D^*\gamma\,,\qquad D\in
{\xcD iff}^+_\Gamma(\Sigma)\,,&\cr\cr
&\gamma\mapsto{\rm e}^{2w}\cdot\gamma\,,\qquad{\rm e}^{2w}\in{\rm Weyl}(\gamma)&
\end{eqnarray}
from the semidirect product $\,{\xcD iff}^+_\Gamma(\Sigma){\hspace{-0.04cm}\ltimes\hspace{-0.05cm}}{\rm Weyl}(\gamma)\,$
of the group $\,{\xcD iff}^+_\Gamma(\Sigma)\,$ of those (orientation-preserving)
diffeomorphisms of $\,\Sigma\,$ that preserve $\,\Gamma$,\ with the group
$\,{\rm Weyl}(\gamma)\,$ of Weyl rescalings of the metric $\,\gamma$. \end{Thm}
\begin{Rem}\label{rem:Weyl-anom} It deserves to be emphasised that a
generic string background $\,\gt{B}=(\mathcal{M},\mathcal{B},\mathcal{J})\,$ does not lead to
a consistent quantum field theory with a non-anomalous realisation
of the gauge symmetries of the classical action functional. Such a
realisation prerequires that the Weyl anomaly of the theory vanish,
which -- in turn -- imposes constraints on the various components of
a consistent string background, cf., e.g.,
Refs.\,\cite{Friedan:1985phd,Braaten:1985is,Gawedzki:1996ias}. \end{Rem}\begin{center}*\hspace{2cm}*\hspace{2cm}*\end{center}
Given the conformal character of the network-field configurations
considered, as stated in Theorem \ref{thm:conf-def}, it is useful to
put the world-sheet metric $\,\gamma\,$ (locally) in the minkowskian
gauge $\,\gamma\equiv\eta=\textrm{diag}(-1,1)$,\ which we impose now for the
remainder of the paper. In this gauge, the theory of maps referred
to in Definition \ref{def:sigmod} is determined by the set of
second-order non-linear differential equations\footnote{The field
equations use the Christoffel symbols
\begin{eqnarray}\nonumber
\bigl\{\begin{smallmatrix} \nu \\ \rho\sigma\end{smallmatrix}\bigr\}=
\tfrac{1}{2}\,\bigl({\rm g}^{-1}\bigr)^{\nu\lambda}\,\bigl(\p_\rho
{\rm g}_{\sigma\lambda}+\p_\sigma{\rm g}_{\rho\lambda}-\p_\lambda{\rm g}_{\rho\sigma}\bigr)
\end{eqnarray}
of the target-space metric.}
\begin{eqnarray}\label{eq:field-eqs}\qquad\qquad
{\rm g}_{\mu\nu}(X)\,\eta^{ab}\,\bigl(\p_a\,\p_b X^\nu+\bigl\{
\begin{smallmatrix} \nu \\ \rho\sigma \end{smallmatrix}\bigr\}(X)\,\p_a
X^\rho\,\p_b X^\sigma\bigr)+3{\rm H}_{\mu\rho\sigma}(X)\,\epsilon^{ab}\,\p_a
X^\rho\,\p_b X^\sigma=0\,.
\end{eqnarray}
In what follows, we shall mainly consider a space-like defect line
$\,\ell\,$ which is the locus of the equation $\,t=0\,$ in the
adapted coordinates $\,(t,\varphi)=(\sigma_1,\sigma_2)\,$
\label{page:adapt} in the vicinity of a point $\,p\in\ell$.\ In
these coordinates, $\,( \widehat n,\widehat t)=(\p_t,\p_\varphi)\,$
and our definition of $\,{\mathsf p}\,$ coincides with the standard
definition of the kinetic momentum of the $\sigma$-model field $\,X$,\
whence also the notation. At such a defect line, we have the (local)
differentiable extensions $\,X_{|\a}\,$ of the patch map
$\,X\vert_{\wp_\a}\,$ to the defect edge
\begin{eqnarray}\nonumber
X_{|\a}\vert_{\wp_\a}=X\vert_{\wp_\a}\,,\qquad\qquad X_{|\a}(0,
\varphi)=\iota_\a\circ X(\varphi)\,,
\end{eqnarray}
with one-sided normal derivatives
\begin{eqnarray}\nonumber
X_{|\a\,*}\widehat n(\varphi)=\lim_{\epsilon\to 0^+}\frac{X_{|\a}^\mu
\bigl((-1)^\a\,\epsilon,\varphi\bigr)-X_{|\a}^\mu(0,\varphi)}{(-1)^\a\,
\epsilon}\,\p_\mu\,.
\end{eqnarray}
Prior to finishing this introductory section by presenting a couple
of examples of bi-branes, let us add the following preparatory
\begin{Rem}\label{rem:states} The world-sheet $\,\Sigma\,$ with an embedded
defect quiver $\,\Gamma\,$ can be understood as a model of multiple
phases, in coexistence and undergoing transitions, of the underlying
CFT, in which the particular phases are represented by the patches
$\,\wp\in\gt{P}_\Sigma$.\ From this point of view, it is natural to set
up the canonical description of the theory in each patch
independently, and only upon completing the task, examine the
relations between the ensuing phase-restricted state spaces imposed
by the defects that separate the phases. In this picture, the defect
lines $\,\ell\in\gt{E}_\Gamma\,$ appear as space-like domain walls of the
two-dimensional field theory carrying the geometric data that effect
the transition. This is the basic setting in which we shall carry
out our analysis in the next section, phrasing our considerations in
terms of Cauchy data of the dynamical evolution, to be localised on
a space-like Cauchy contour $\,\mathscr{C}$.\ In principle, we might attach
the phase (patch) label to the dynamical objects defined over a
given patch but we choose, instead, to shift the dependence on the
phase of the underlying CFT to the definition of the embedding map
$\,X$,\ along the lines of \Rcite{Runkel:2008gr}, which enables us
to develop a unified treatment of all admissible phases at the same
time. The modular invariance of the quantised (euclidean version of
the) CFT leads us to consider a dual of the picture described in
which time-like and space-like contours are swapped. Having set out
with space-like defect lines, we thus end up with time-like ones,
and the obvious question arises as to the nature of the canonical
description of this dual CFT. Motivated by the distinguished example
of boundary defects and the associated $\mathcal{G}$-branes of string
theory, we should be inclined to formulate our description in terms
of Cauchy data localised on open segments stretched transversally
between defect lines, each contained in a single patch of the
world-sheet. The problem with this description, masked by the
triviality of the patch data for the patch `behind' the defect line
in the boundary case, is that consistency of its formulation in a
single patch prerequires the knowledge of the field configuration
across the defect lines to which the open segment is attached, cf.\
\Reqref{eq:DGC}. Geometrically, this is reflected in the inability
to write down gauge invariant functionals using only the space-time
data assigned to a single patch, augmented by those for the defect
lines bounding it. The unavoidable incompleteness of such a
description prompts us to conceive a formulation in which open
segments stretching between pairs of defect lines are -- instead --
joined, consistently with \Reqref{eq:DGC}, to form a closed contour
$\,C_{\{\ell_k\}}\,$ intersecting transversally a (finite) number
$\,I\in{\mathbb{N}}_{>0}\,$ of defect lines $\,\ell_k,\ k\in\ovl{1,I}$,\ with
the dynamical data localised on $\,C_{\{\ell_k\}}$.
Clearly, Cauchy contours which do not intersect the defect quiver
(and the attendant dynamical structures) can be regarded as special
examples of the $\Gamma$-twisted ones, with all defects intersected by
the corresponding Cauchy contour $\,C_\emptyset\equiv C\,$ assumed
trivial. Thus, unless expressly stated otherwise, we shall always
mean both non-trivially $\Gamma$-twisted and untwisted states whenever
we use the notation for the $\Gamma$-twisted ones in the sequel. \end{Rem}\begin{center}*\hspace{2cm}*\hspace{2cm}*\end{center}
\begin{Eg}\label{eg:triv-def}\textbf{The trivial (inter-)bi-brane.}
\\[-8pt]
Given an arbitrary target $\,\mathcal{M}=(M,{\rm g},\mathcal{G})$,\ there always exists
the \textbf{trivial $\mathcal{G}$-bi-brane}
\begin{eqnarray}\nonumber
\mathcal{B}_{\rm triv}=(M,{\rm id}_M,{\rm id}_M,0,{\rm id}_\mathcal{G})\,,
\end{eqnarray}
and the attendant \textbf{trivial $(\mathcal{G},\mathcal{B}_{\rm triv}
)$-inter-bi-brane}
\begin{eqnarray}\nonumber
\mathcal{J}_{\rm triv}=\left(\bigsqcup_{i=1}^{2^n}\,M,\bigl(\varepsilon_n^{k,k+1},
{\rm id}_M \ \vert\ k\in\ovl{1,n}\bigr),{\rm id}_{{\rm id}_\mathcal{G}}\,\vert\,n\in
{\mathbb{N}}_{\geq 3}\right)\,.
\end{eqnarray}
\end{Eg}\medskip
\begin{Eg}\label{eg:WZW-def}\textbf{The maximally symmetric WZW defects.}
\\[-8pt]
\noindent \emph{\textbf{The target.}} We consider here a
distinguished class of defects in the Wess--Zumino--Witten (WZW)
$\sigma$-model of \Rcite{Witten:1983ar}, with target $\,\mathcal{M}_{\mathsf k}=({\rm G},
{\rm g}_{\mathsf k},\mathcal{G}_\sfk)$,\ where
\begin{itemize}
\item[(T.i)] ${\rm G}\,$ is the group manifold of an arbitrary compact
simple 1-connected Lie group, with Lie algebra $\,\gt{g}\,$ and a
trace $\,{\rm tr}_\gt{g}\,$ on $\,\gt{g}\,$ normalised such that the equality
\begin{eqnarray}\nonumber
{\rm tr}_\gt{g}(t_A\,t_B)=-\tfrac{1}{2}\,\delta_{AB}
\end{eqnarray}
holds for generators $\,t_A\,$ of $\,\gt{g}$,\ the latter satisfying
the defining commutation relations
\begin{eqnarray}\nonumber
[t_A,t_B]=f_{ABC}\,t_C\,,
\end{eqnarray}
with $\,f_{ABC}\,$ the structure constants of $\,\gt{g}\,$ -- this
prescription yields the standard matrix trace for, e.g., $\,{\rm G}=
{\rm SU}(2)$;
\item[(T.ii)] ${\rm g}_{\mathsf k}\,$ is the Cartan--Killing metric
\begin{eqnarray}\nonumber
{\rm g}_{\mathsf k}=-\tfrac{{\mathsf k}}{4\pi}\,{\rm tr}_\gt{g}\bigl(\theta_L\otimes\theta_L\bigr)
\,,\qquad{\mathsf k}\in{\mathbb{Z}}_{>0}\,,
\end{eqnarray}
written in terms of the standard left-invariant Maurer--Cartan
1-form $\,\theta_L(g)=g^{-1}\,{\rm d} g\in\Gamma({\mathsf T}^*{\rm G})\otimes\gt{g}\,$ on
$\,{\rm G}$;
\item[(T.iii)] $\mathcal{G}_\sfk=\mathcal{G}_1^{\otimes{\mathsf k}}\,$ is the ${\mathsf k}$-th power of the
basic gerbe\footnote{Meinrenken's construction of the basic gerbe
for a general (compact simple 1-connected) Lie group was preceded by
that of \Rcite{Gawedzki:1987ak} for $\,{\rm SU}(2)\,$ and that of
Refs.\,\cite{Chatterjee:1998,Hitchin:1999} which works for $\,{\rm
SU}(N)$.\ The non-simply connected case was worked out in Refs.\
\cite{Gawedzki:2002se,Gawedzki:2003pm}.} $\,\mathcal{G}_1\,$ of
\Rcite{Meinrenken:2002}, with curvature equal to the Cartan 3-form
\begin{eqnarray}\nonumber
{\rm H}_{\mathsf k}=\tfrac{{\mathsf k}}{12\pi}\,{\rm tr}_\gt{g}(\theta_L\wedge\theta_L\wedge
\theta_L)
\end{eqnarray}
whose cohomology class is the generator of $\,H^3({\rm G})\cong{\mathbb{Z}}$.
\end{itemize}
The action functional for a defect-free world-sheet $\,\Sigma\,$ is
given by
\begin{eqnarray}\nonumber
S_{{\rm WZW},{\mathsf k}}[g]=\tfrac{{\mathsf k}}{8\pi}\,\int_\Sigma\,{\rm tr}_\gt{g}\bigl(
\theta_L(g)\wedge\star_{\rm H}\theta_L(g)\bigr)-{\mathsf i}\,\log{\rm Hol}_{\mathcal{G}_\sfk}(g)
\,,
\end{eqnarray}
and the constant $\,{\mathsf k}\in{\mathbb{Z}}_{>0}\,$ is called the level of the
WZW model. The field equations of the model have the compact form
\begin{eqnarray}\nonumber
\bigl(\eta^{ab}+\epsilon^{ab}\bigr)\,\p_a\bigl(g^{-1}\,\p_b g\bigr)=0\,,
\end{eqnarray}
which can further be rewritten, using the light-cone coordinates
$\,\sigma^\pm=\sigma^2\pm\sigma^1\,$ and the attendant derivatives
$\,\p_\pm=\tfrac{\p\ }{\p\sigma^\pm}$,\ as
\begin{eqnarray}\nonumber
\p_+\bigl(g^{-1}\,\p_-g\bigr)=0\,.
\end{eqnarray}
A general solution to this equation factorises as
\begin{eqnarray}\nonumber
g(\sigma)=g_L(\sigma^+)\cdot g_R(\sigma^-)^{-1}\,,
\end{eqnarray}
for independent ${\rm G}$-valued maps $\,g_L\,$ and $\,g_R\,$ on
$\,{\mathbb{R}}\,$ with equal monodromies (when viewed as maps on $\,{\mathbb{R}}/2\pi
{\mathbb{Z}}$), cf.\ \Rcite{Gawedzki:2001rm}. In addition to the standard
conformal symmetry of Theorem \ref{thm:conf-def}, realised, on the
infinitesimal level, by two (chiral) copies of the Lie algebra of
diffeomorphisms of the circle, the bulk theory enjoys a level-${\mathsf k}$
Ka\v c--Moody symmetry, realised on fields by the chiral currents
\begin{eqnarray}\nonumber
J_L(\sigma)=\tfrac{{\mathsf k}}{2\pi}\,g(\sigma)\,\p_+g(\sigma)^{-1}\,,\qquad
\qquad J_R(\sigma)=\tfrac{{\mathsf k}}{2\pi}\,g(\sigma)^{-1}\,\p_-g(\sigma)
\end{eqnarray}
that generate the centrally extended current algebra $\,\widehat{\gt{g}}_\sfk^L
\oplus\widehat{\gt{g}}_\sfk^R$.\ Note that the currents become functions of the
respective light-cone coordinates $\,\sigma^\pm\,$ upon using the field
equations of the $\sigma$-model. Their (infinitesimal) action
integrates to
\begin{eqnarray}\nonumber
g(\sigma)\mapsto h_L(\sigma^+)\cdot g(\sigma)\cdot h_R(\sigma^-)^{-1}\,,
\end{eqnarray}
with independent chiral transformation maps $\,h_L,h_R\in{\mathsf L}{\rm G}\,$
from the loop group $\,{\mathsf L}{\rm G}\equiv C^\infty({\mathbb{S}}^1,{\rm G})$.\medskip
\noindent \emph{\textbf{The boundary $\mathcal{G}_\sfk$-bi-brane.}} We shall
first consider defects describing \emph{boundary} maximally
symmetric $\mathcal{G}_\sfk$-bi-branes of the WZW model, or maximally symmetric
$\mathcal{G}_\sfk$-branes for short. To this end, we focus on (the vicinity of)
a connected component $\,\ell\,$ of the edge set $\,\gt{E}_\Gamma\,$ of
$\,\Gamma$,\ which we take to be an oriented circle embedded in
$\,\Sigma\,$ at $\,t=0\,$ in a local coordinate system
$\,(t,\varphi)\,$ described on p.\,\pageref{page:adapt}. As argued
in \Rxcite{p.\ 12}{Runkel:2008gr}, the corresponding string
background $\,\gt{B}_{\mathsf k}^\p=(\mathcal{M}_{\mathsf k}^\p,\mathcal{B}^{\mathsf k}_\p,\cdot)\,$
consists of
\begin{itemize}
\item[(TD)] the target $\,\mathcal{M}_{\mathsf k}^\p=\mathcal{M}_{\mathsf k}\sqcup\{\bullet\}\,$
given by the disjoint union of the bulk target $\,\mathcal{M}_{\mathsf k}=({\rm G},
{\rm g}_{\mathsf k},\mathcal{G}_\sfk)\,$ and a single point $\,\{\bullet\}\,$ with no
structure over it;
\item[(D)] the $\mathcal{G}_\sfk$-bi-brane $\,\mathcal{B}_{\mathsf k}^\p=(Q_{\mathsf k}^\p,
\iota_{Q_{\mathsf k}^\p},\bullet,\omega_{\mathsf k}^\p,\Phi_{\mathsf k}^\p)$,\ with
\begin{itemize}
\item[(D.i)] the world-volume
\begin{eqnarray}\nonumber
Q^\p_{\mathsf k}=\bigsqcup_{\lambda\in\faff{\gt{g}}}\,\mathscr{C}_\lambda\,,\qquad\qquad
\mathscr{C}_\lambda=\bigl\{\ {\rm Ad}_x{\rm e}_\lambda \quad\vert\quad x\in{\rm G}\ \bigr\}\,,
\end{eqnarray}
given by the disjoint sum of the conjugacy classes of Cartan
elements $\,{\rm e}_\lambda={\rm e}^{\frac{2\pi{\mathsf i}\,\lambda}{{\mathsf k}}}\in{\rm G}\,$
labelled by weights $\,\lambda\,$ from the fundamental affine Weyl
alcove\footnote{In what follows, we shall always identify $\,\gt{g}\,$
with its dual $\,\gt{g}^*\,$ using the Cartan--Killing metric.} at
level $\,{\mathsf k}$,
\begin{eqnarray}\label{eq:faffggt}
P^{\mathsf k}_+(\gt{g})={\mathsf k}\,\xcA_W(\gt{g})\cap P(\gt{g})\,,
\end{eqnarray}
the latter being the intersection of the weight lattice $\,P(\gt{g}
)\,$ of $\,\gt{g}\,$ with its ${\mathsf k}$-inflated Weyl alcove $\,{\mathsf k}\,
\xcA_{\rm W}(\gt{g})$,\ i.e.\ a subset
\begin{eqnarray}\nonumber
\xcA_{\rm W}(\gt{g})=\bigl\{\ \lambda\in\gt{t} \quad\big\vert\quad {\rm tr}_\gt{g}(
\lambda\cdot\theta)\leq 1\quad\land\quad{\rm tr}_\gt{g}(\lambda\cdot\a_i)\geq 0\,,\ i
\in\ovl{1,{\rm rank}\,\gt{g}} \ \bigr\}
\end{eqnarray}
of the Cartan subalgebra $\,\gt{t}\subset\gt{g}$,\ defined in terms of
the simple roots $\,\a_i,\ i\in\ovl{1,{\rm rank} \,\gt{g}}\,$ of
$\,\gt{g}\,$ and its longest root $\,\theta$;
\item[(D.ii)] the $\mathcal{G}_\sfk$-bi-brane maps, defined component-wise by
the embedding $\,\iota_{Q^\p_{\mathsf k}}\vert_{\mathscr{C}_\lambda}\equiv\imath_\lambda:
\mathscr{C}_\lambda\emb{\rm G}\,$ of the conjugacy class $\,\mathscr{C}_\lambda\,$ in the
group manifold, and the constant map $\,\bullet:Q_{\mathsf k}^\p\to\{
\bullet\}$;
\item[(D.iii)] the curvature, also defined component-wise as
\begin{eqnarray}\label{eq:WZW-brane-curv}
\omega_{\mathsf k}^\p\vert_{\mathscr{C}_\lambda}=\tfrac{{\mathsf k}}{8\pi}\,\imath_\lambda^*
{\rm tr}_\gt{g}\left(\theta_L\wedge\tfrac{{\rm id}_\gt{g}+{\rm Ad}_\cdot}{{\rm id}_\gt{g}-
{\rm Ad}_\cdot}\,\theta_L\right)=:\omega^\p_{{\mathsf k},\lambda}\,;
\end{eqnarray}
\item[(D.iv)] the $\mathcal{G}_\sfk$-bi-brane 1-isomorphism given on each
component $\,\mathscr{C}_\lambda\,$ of $\,Q^\p_{\mathsf k}\,$ by the corresponding
trivialisation
\begin{eqnarray}\nonumber
\Phi^\p_{\mathsf k}\vert_{\mathscr{C}_\lambda}=:\Phi^\p_{{\mathsf k},\lambda}\ :\ \imath_\lambda^*
\mathcal{G}_\sfk\xrightarrow{\cong}I_{\omega^\p_{{\mathsf k},\lambda}}
\end{eqnarray}
of the restricted gerbe $\,\mathcal{G}_\sfk$.
\end{itemize}
\end{itemize}
The DGC obtained by plugging the above data into \Reqref{eq:DGC} is
the familiar statement
\begin{eqnarray}\nonumber
(J_L-J_R)\vert_\ell=0
\end{eqnarray}
of maximal symmetry of the boundary defect, the symmetry being
determined by a single copy of the level-${\mathsf k}$ Ka\v c--Moody
algebra $\,\widehat{\gt{g}}_\sfk\,$ embedded diagonally in the current-symmetry
algebra $\,\widehat{\gt{g}}_\sfk^L\oplus\widehat{\gt{g}}_\sfk^R\,$ of the bulk theory.
It is vital to note that -- as was argued for $\,{\rm G}={\rm SU}(N)\,$
in \Rxcite{Sec.\,8.1}{Gawedzki:2002se} and for an arbitrary compact
simple 1-connected Lie group $\,{\rm G}\,$ in
\Rxcite{Sec.\,5.1}{Gawedzki:2004tu} -- the stable isomorphisms
$\,\Phi^\p_{{\mathsf k},\lambda}\,$ exist for $\,\lambda\in\faff{\gt{g}}\,$
exclusively, and so they single out a subset of conjugacy classes in
$\,{\rm G}\,$ which coincides with the set of world-volumes of stable
(untwisted) maximally symmetric D-branes of the WZW model at level
$\,{\mathsf k}$,\ cf., e.g., \Rcite{Felder:1999ka}, which -- in turn -- are
in a one-to-one correspondence with the (untwisted) maximally
symmetric boundary states of the associated BCFT (ib.). \void{\begin{Rem}
The geometric data $\,\mathcal{B}_{\mathsf k}^\p\,$ endow each $\,\mathscr{C}_\lambda\,$ with
the structure of a model quasi-hamiltonian ${\rm G}$-space, as
introduced in \Rcite{Alekseev:1997}. Indeed, consider the component
curvature 2-form $\,\omega^\p_{{\mathsf k},\lambda}\,$ taken with the minus sign,
$\,\varpi_{{\mathsf k},\lambda}=-\omega^\p_{{\mathsf k},\lambda}$.\ The 2-form
$\,\varpi_{{\mathsf k},\lambda}\,$ is manifestly ${\rm Ad}_{\rm G}$-invariant,
\begin{eqnarray}\nonumber
\varpi_{{\mathsf k},\lambda}({\rm Ad}_x(g))=\varpi_{{\mathsf k},\lambda}(g)\,,\qquad\qquad x\in
{\rm G}\,,
\end{eqnarray}
and, together with the ${\rm G}$-equivariant embedding map $\,\mu_\lambda
\equiv\iota_\lambda\,$ as the ${\rm G}$-valued moment map, it satisfies
\begin{itemize}
\item the triviality relation:
\begin{eqnarray}\nonumber
{\rm d}\varpi_{{\mathsf k},\lambda}=-\mu_\lambda^*{\rm H}_{\mathsf k}\,,
\end{eqnarray}
\item the generalised moment-map condition:
\begin{eqnarray}\nonumber
\xi^A\,(R_A-L_A)\righthalfcup\varpi_{{\mathsf k},\lambda}=\tfrac{{\mathsf k}}{4\pi}\,\mu_\lambda^*
{\rm tr}_\gt{g}\bigl(\xi\,(\theta_L+\theta_R)\bigr)\,,\qquad\qquad\xi=\xi^A\,t_A
\in\gt{g}\,,
\end{eqnarray}
written in terms of the right-invariant Maurer--Cartan 1-form
$\,\theta_R(g)={\rm d} g\,g^{-1}=\theta_R^A\otimes t_A\,$ and in terms of the
right-invariant (resp.\ left-invariant) vector fields $\,R_A\,$
(resp.\ $\,L_A\,$) dual to $\,\theta_R\,$ (resp.\ to $\,\theta_L$),
\begin{eqnarray}\nonumber
R_A\righthalfcup\theta_R^B=\delta_A^{\ B}\,,\qquad\qquad L_A\righthalfcup\theta_L^B=\delta_A^{\
B}\,,
\end{eqnarray}
\item the kernel condition:
\begin{eqnarray}\nonumber
{\rm ker}\,\varpi_{{\mathsf k},\lambda}(h_\lambda)=\bigl\{\ \xi^A\,(R_A-L_A)(h_\lambda)
\quad \vert \quad \xi\in{\rm ker}\,\bigl({\rm Ad}_{\mu_\lambda(h_\lambda)}+{\rm id}_\gt{g}
\bigr) \ \bigr\}\,,
\end{eqnarray}
cf.\ \Rcite{Alekseev:1997} for a simple proof.
\end{itemize}
Thus, $\,(\mathscr{C}_\lambda,{\rm Ad}_{\rm G};\varpi_{{\mathsf k},\lambda},\mu_\lambda)\,$ composes a
quasi-hamiltonian ${\rm G}$-space (with respect to the adjoint action of the
group) in the sense of Alekseev--Malkin--Meinrenken. This structure
is central to the (pre)quantisation of the conjugacy class, as
discussed at some length in \Rcite{Krepski:2007}. It will also
reappear naturally in our analysis of the splitting-joining
interaction of the string in Section \ref{sec:fusion}.\end{Rem}\begin{center}*\hspace{2cm}*\hspace{2cm}*\end{center}}\medskip
\noindent \emph{\textbf{The non-boundary $\mathcal{G}_\sfk$-bi-brane.}} The next
type of maximally symmetric WZW defects that we want to discuss are
those implementing jumps by elements of the target Lie group in the
sense that the limiting values attained at a point $\,p\,$ on the
defect circle $\,\ell\,$ by the one-sided local extensions
$\,(g_{|1},g_{|2})\,$ of the embedding map $\,g:\Sigma\setminus\Gamma\to
{\rm G}\,$ to the defect line $\,\ell$,\ described in Definition
\ref{def:net-field}, are two generically distinct points in the
group manifold. A special class of such defects -- the central-jump
defects at which $\,g_{|2}=z\cdot g_{|1}\,$ for $\,z\,$ from the
centre $\,Z({\rm G})\,$ of $\,{\rm G}\,$ -- were considered at length in
\Rcite{Runkel:2008gr}. The more general jump defects, with -- as
above -- the jump given by $\,g_{|1}^{-1}\cdot g_{|2}\in{\rm G}$,\ were
first studied in \Rcite{Fuchs:2007fw}, where the notion of a
bi-brane was introduced. They shall be expanded upon in
\Rcite{Runkel:2010}. In the conventions of the latter paper, the
string background $\,\gt{B}_{\mathsf k}= (\mathcal{M}_{\mathsf k},\mathcal{B}_{\mathsf k},\cdot)\,$ for
these jump defects consists of
\begin{itemize}
\item[(TB)] the target $\,\mathcal{M}_{\mathsf k}=({\rm G},{\rm g}_{\mathsf k},\mathcal{G}_\sfk)\,$ of the
defect-free model;
\item[(B)] the $\mathcal{G}_\sfk$-bi-brane $\,\mathcal{B}_{\mathsf k}=(Q_{\mathsf k},d_1,d_0,\omega_{\mathsf k}
,\Phi_{\mathsf k})$,\ with
\begin{itemize}
\item[(B.i)] the world-volume
\begin{eqnarray}\nonumber
Q_{\mathsf k}={\rm G}\x Q^\p_{\mathsf k}\,;
\end{eqnarray}
\item[(B.ii)] the $\mathcal{G}_\sfk$-bi-brane maps, defined explicitly as
\begin{eqnarray}\nonumber
d_0(g,h_\lambda)=g\cdot h_\lambda\,,\qquad\qquad d_1(g,h_\lambda)=g\,;
\end{eqnarray}
\item[(B.iii)] the curvature, defined component-wise as
\begin{eqnarray}\nonumber
\omega_{\mathsf k}\vert_{{\rm G}\x\mathscr{C}_\lambda}=-{\rm pr}_2^*\omega^\p_{{\mathsf k},\lambda}+\rho_{\mathsf k}=:
\omega_{{\mathsf k},\lambda}\,,\qquad\qquad\rho_{\mathsf k}=\tfrac{{\mathsf k}}{4\pi}\,{\rm tr}_\gt{g}
\bigl({\rm pr}_1^*\theta_L\wedge{\rm pr}_2^*\theta_R\bigr)
\end{eqnarray}
in terms of the canonical projections $\,{\rm pr}_\a:{\rm G}\x{\rm G}\to{\rm G},\
\a\in\{1,2\}$;
\item[(B.iv)] the $\mathcal{G}_\sfk$-bi-brane 1-isomorphism with a
component-wise definition
\begin{eqnarray}\nonumber
\Phi_{\mathsf k}\vert_{{\rm G}\x\mathscr{C}_\lambda}=:\Phi_{{\mathsf k},\lambda}\ &:&\ d_1^*\mathcal{G}_\sfk
\equiv d_1^*\mathcal{G}_\sfk\otimes I_{{\rm pr}_2^*\omega^\p_{{\mathsf k},\lambda}}\otimes I_{-{\rm pr}_2^*
\omega^\p_{{\mathsf k},\lambda}}\xrightarrow{\quad{\rm id}_{d_1^*\mathcal{G}_\sfk}\otimes\Phi^{\p\,
\vee}_{{\mathsf k},\lambda}\otimes{\rm id}_{ I_{-{\rm pr}_2^*\omega^\p_{{\mathsf k},\lambda}}}\quad}
{\rm pr}_1^*\mathcal{G}_\sfk\otimes{\rm pr}_2^*\mathcal{G}_\sfk\otimes I_{-{\rm pr}_2^*\omega^\p_{{\mathsf k},\lambda}}\cr\cr
&&\hspace{5.2cm}\xrightarrow{\quad\mathcal{M}_{\mathsf k}\otimes{\rm id}_{ I_{-{\rm pr}_2^*
\omega^\p_{{\mathsf k},\lambda}}}\quad}d_0^*\mathcal{G}_\sfk\otimes I_{\omega_{{\mathsf k},\lambda}}\,,
\end{eqnarray}
invoking the 1-isomorphism
\begin{eqnarray}\nonumber
\mathcal{M}_{\mathsf k}\ :\ {\rm pr}_1^*\mathcal{G}_\sfk\otimes{\rm pr}_2^*\mathcal{G}_\sfk\xrightarrow{\cong}{\rm m}^*
\mathcal{G}_\sfk\otimes I_{\rho_{\mathsf k}}\,,\qquad\qquad{\rm m}\ :\ {\rm G}\x{\rm G}\to{\rm G}\ :\
(g,h)\mapsto g\cdot h
\end{eqnarray}
of the multiplicative structure on $\,\mathcal{G}_\sfk$,\ as introduced in
\Rcite{Carey:2004xt} and developed in
Refs.\,\cite{Waldorf:2008mult,Gawedzki:2009jj}.
\end{itemize}
\end{itemize}
The DGC is, once more, the statement of maximal symmetry,
\begin{eqnarray}\nonumber
(J^1_L-J^2_L)\vert_\ell=0=(J^1_R-J^2_R)\vert_\ell\,,
\end{eqnarray}
the symmetry being determined by the full bi-chiral level-${\mathsf k}$
Ka\v c--Moody algebra $\,\widehat{\gt{g}}_\sfk^L\oplus\widehat{\gt{g}}_\sfk^R\,$ of the bulk theory.
The chiral currents $\,J^\a_L,J^\a_R\,$ are defined as previously
but using the respective one-sided local extensions $\,g_{|\a}\,$ of
the patch component of the $\sigma$-model field. Owing to the form of
the energy-momentum tensor, given by a sum of terms quadratic in the
chiral currents, the continuity of the latter across the defect line
ensures the topologicality of the defect associated to
$\,\mathcal{B}_{\mathsf k}$,\ as defined in \Rcite{Runkel:2008gr}.
\begin{Rem}\label{rem:mult-str} It ought to be emphasised that the
existence and uniqueness (up to a 2-isomorphism) of the
1-isomorphism $\,\mathcal{M}_{\mathsf k}\,$ on a compact simple 1-connected Lie
group $\,{\rm G}\,$ implies, via a simple topological argument (cf.\
\Rcite{Runkel:2010}), that the maximally symmetric WZW
$\mathcal{G}_\sfk$-bi-brane $\,\mathcal{B}_{\mathsf k}\,$ has precisely as many connected
components (labelled by weights $\,\lambda\in\faff{\gt{g}}$) as its
boundary analogon $\,\mathcal{B}_{\mathsf k}^\p$. \end{Rem}\begin{center}*\hspace{2cm}*\hspace{2cm}*\end{center} \begin{Rem} The world-volume
$\,Q_{\mathsf k}\,$ of $\,\mathcal{B}_{\mathsf k}\,$ is ${\rm G}\x{\rm G}$-equivariantly
isomorphic to the disjoint union of the bi-conjugacy classes
\begin{eqnarray}
\mathscr{B}_{(t_\lambda,e)}=\bigl\{\ \bigl(x\cdot t_\lambda\cdot y^{-1},x\cdot y^{-
1}\bigr) \quad\big\vert\quad x,y\in{\rm G} \ \}\,,
\label{eq:biconj-def}
\end{eqnarray}
of \Rcite{Fuchs:2007fw} for the pairs of group elements $\,(t_\lambda,e)
\in\mathscr{C}_\lambda\x\{e\}\subset{\rm G}\x{\rm G}$. \end{Rem}\begin{center}*\hspace{2cm}*\hspace{2cm}*\end{center}
\end{Eg}
\section{The canonical structure and pre-quantisation of the
$\sigma$-model} \label{sec:can}
Having written out the $\sigma$-model action functional of interest in
\Reqref{eq:sigma}, we may now analyse the symplectic structure on
its state space, a task best completed within the framework of
covariant classical field theory (or first-order formalism) of
Refs.\,\cite{Gawedzki:1972ms,Kijowski:1973gi,Kijowski:1974mp,Kijowski:1976ze,Szczyrba:1976,Kijowski:1979dj},
cf.\ also \Rcite{Gotay:2006cft} for an exhaustive exposition of the
modern approach and a comprehensive list of references. This
formalism enables us to interpret the (inter-)bi-brane data in terms
of the canonical structure of the underlying two-dimensional field
theory, whereupon a clear-cut field-theoretic statement can be made
in regard to the relation between defects and dualities of the
$\sigma$-model, in the spirit of, e.g., \Rcite{Frohlich:2006ch}. We
begin by briefly reviewing those elements of the general formalism
that are instrumental in the subsequent analysis of the physical
system of interest.
\subsection{Elements of the covariant formalism}\label{sub:can-gen}
Let us first introduce some basic notions.
\begin{Def}\label{def:Cartan}
Let $\,\pi_\mathcal{F}:\mathcal{F}\to\mathscr{M}\,$ be a fibre bundle over a
(pseudo-)riemannian base $\,(\mathscr{M},{\rm g})\,$ of dimension
$\,\dim\,\mathscr{M}=:d$,\ and let $\,{\mathsf J}^1\mathcal{F}\to\mathscr{M}\,$ be the first-jet
bundle of $\,\mathcal{F}$,\ with local coordinates
$\,(x^\mu,\phi^A,\xi^B_\nu),\ \mu,\nu\in\ovl{1,d} ,\
A,B\in\ovl{1,N}$,\ where $\,N\,$ is the dimension of the typical
fibre of $\,\mathcal{F}$.\ Consider an $\mathcal{F}$-field theory $\,\mathscr{F}\,$ on
$\,\mathscr{M}$,\ i.e.\ a theory of continuously differentiable sections
$\,(\phi^A)^{A\in\ovl{1,N}}\,$ of the bundle $\,\mathcal{F}\,$ (termed the
\textbf{covariant configuration bundle of $\mathcal{F}$-field theory
$\,\mathscr{F}\,$} in this context), determined by the principle of least
action applied to the action functional
\begin{eqnarray}\label{eq:F-theory}
S_\mathscr{F}[\phi^A]=\int_\mathscr{M}\,\mathcal{L}_\mathscr{F}(x^\mu,\phi^A,\xi^B_\nu)
\vert_{\xi^B_\nu=\p_\nu\phi^B}\,{\mathsf d}^d x\,,
\end{eqnarray}
in which $\,{\mathsf d}^d x={\mathsf d} x^1\wedge{\mathsf d} x^2\wedge\ldots\wedge{\mathsf d}
x^d\in\Gamma(\wedge^d{\mathsf T}^*\mathscr{M})\,$ is the volume form in local
coordinates $\,x^\mu\,$ on $\,\mathscr{M}$,\ $\,\p_\mu=\frac{\p\ }{\p
x^\mu}\,$ are the associated partial derivatives, and the map
$\,\mathcal{L}_\mathscr{F}\,$ on $\,{\mathsf J}^1\mathcal{F}\,$ with values in the space of
scalar densities (of weight $0$) on $\,\mathscr{M}\,$ is termed the
\textbf{lagrangean (density) of $\mathcal{F}$-field theory $\,\mathscr{F}\,$} and
considered \textbf{regular} iff the matrix functional
$\,\frac{\delta^2\mathcal{L}_\mathscr{F}}{\delta\xi^A_\mu\delta \xi^B_\nu}\,$ (with a
multi-index $\,{}^\mu_A$) is invertible on sections of $\,\mathcal{F}\,$
that extremise $\,S_\mathscr{F}$.\ The \textbf{Cartan form $\,\Theta_\mathscr{F}\,$
of $\mathcal{F}$-field theory $\,\mathscr{F}\,$} is the $d$-form on $\,{\mathsf J}^1\mathcal{F}\,$
given by the formula
\begin{eqnarray}\nonumber
\Theta_\mathscr{F}(x^\mu,\phi^A,\xi^B_\nu)=\left(\mathcal{L}-\xi^C_\lambda\,\tfrac{\delta
\mathcal{L}}{\delta\xi^C_\lambda}\right)(x^\mu,\phi^A,\xi^B_\nu)\,{\mathsf d}^d x+
\tfrac{\delta\mathcal{L}}{\delta\xi^C_\lambda}(x^\mu,\phi^A,\xi^B_\nu)\,\delta\phi^C
\wedge(\p_\lambda\righthalfcup{\mathsf d}^d x)\,.
\end{eqnarray}
Above, and in what follows, we use the symbol $\,\delta\,$ to
distinguish differentiation in the direction of the fibre of
$\,{\mathsf J}^1\mathcal{F}$,\ termed \textbf{$\mathcal{F}$-vertical},\ from that along the
base $\,\mathscr{M}$,\ e.g., $\,\delta\phi^A\,$ and $\,\frac{\delta\ \ }{\delta
\xi^A_\mu}\,$ vs.\ $\,{\mathsf d} x^\mu\,$ and $\,\frac{\p\ \ }{\p x^\mu}$.
\begin{flushright}$\checkmark$\end{flushright}\end{Def} \noindent The significance of the Cartan form rests on the
following
\begin{Prop}\cite{Gawedzki:1990jc}\label{prop:Cartan}
Let $\,\mathscr{F}\,$ be an $\mathcal{F}$-field theory on a (pseudo-)riemannian
manifold $\,(\mathscr{M},{\rm g})$,\ determined by a regular lagrangean
density $\,\mathcal{L}_\mathscr{F}$,\ with a covariant configuration bundle
$\,\pi_\mathcal{F}:\mathcal{F} \to\mathscr{M}$,\ the attendant first-jet bundle
$\,{\mathsf J}^1\mathcal{F}\to\mathscr{M}$,\ and the Cartan form $\,\Theta_\mathscr{F}\,$ on the
latter. Then,
\begin{itemize}
\item[i)] the principle of least action applied to the functional
\begin{eqnarray}\nonumber
S_{\Theta_\mathscr{F}}[\Psi]=\int_\mathscr{M}\,\Psi^*\Theta_\mathscr{F}\,,\qquad\qquad\Psi\in
\Gamma({\mathsf J}^1\mathcal{F})
\end{eqnarray}
yields, as the Euler--Lagrange equations, the field equations of
$\,\mathscr{F}$,\ that is the Euler--Lagrange equations of the action
functional \eqref{eq:F-theory}, and -- for $\,\mathscr{M}\,$ with a
non-empty boundary -- also the boundary conditions of $\,\mathscr{F}$;\ the
equations follow from the condition
\begin{eqnarray}\label{eq:EuLagAl}
0=\mathscr{V}\righthalfcup\delta S_{\Theta_\mathscr{F}}[\Psi_{\rm cl}]\,,
\end{eqnarray}
to be satisfied by the extremal (or \emph{classical}) sections
$\,\Psi_{\rm cl}\,$ of $\,{\mathsf J}^1\mathcal{F}\,$ for an arbitrary
$\mathcal{F}$-vertical vector field $\,\mathscr{V}$,\ i.e.\ for $\,\mathscr{V}\in\Gamma({\mathsf T}
{\mathsf J}^1\mathcal{F})\cap{\rm ker}\,\pi_{\mathcal{F}\,*}=:\Gamma\bigl(({\mathsf T} J^1\mathcal{F})^{\perp_\mathcal{F}}
\bigr)\,$ (in particular, the boundary conditions of $\,\mathscr{F}\,$ are
implied by the vanishing of the boundary term on the right-hand side
of \Reqref{eq:EuLagAl});
\item[ii)] $\Theta_\mathscr{F}\,$ canonically determines a closed 2-form
$\,\Omega_\mathscr{F}\,$ on the space $\,{\mathsf P}_\mathscr{F}\,$ of extremal sections of
$\,{\mathsf J}^1\mathcal{F}$.
\end{itemize}
\end{Prop}
\noindent A complete proof can be extracted from the original paper.
However, the proof being constructive in nature, it appears useful
to give at least an idea thereof -- after \Rcite{Gawedzki:2007fo} --
to prepare the reader for the subsequent considerations.
\begin{itemize}
\item[Ad i)] The key point is to note that the Euler--Lagrange
equations obtained for the distinguished choice $\,\mathscr{V}=\tfrac{\delta\
}{\delta\xi^A_\mu}\,$ of a $\mathcal{F}$-vertical vector field on
$\,{\mathsf J}^1\mathcal{F}\,$ read
\begin{eqnarray}\label{eq:eom-xipar}
\tfrac{\delta^2\mathcal{L}_\mathscr{F}}{\delta\xi^C_\kappa\delta\xi^D_\lambda}(x^\mu,\phi^A,
\xi^B_\nu)\,(\xi^D_\lambda-\p_\lambda\phi^D)=0\,,
\end{eqnarray}
and so, assuming regularity of $\,\mathcal{L}_\mathscr{F}$,\ we conclude that the
equality
\begin{eqnarray}\label{eq:xi-as-pX}
\xi^A_\mu=\p_\mu\phi^A
\end{eqnarray}
holds true on $\,{\mathsf P}_\mathscr{F}$.\ The Euler--Lagrange equations of the
action functional \eqref{eq:F-theory} then follow straightforwardly.
\item[Ad ii)] Assume $\,\p\mathscr{M}=\emptyset$.\ Pick up a pair
$\,C_\a,\ \a\in\{1,2\}\,$ of Cauchy hypersurfaces in $\,\mathscr{M}\,$ and
cut out a region $\,\mathscr{M}_{1,2}\subset\mathscr{M}\,$ such that $\,\p
\mathscr{M}_{1,2}=C_1\sqcup(-C_2)$,\ where the minus in front of $\,C_2\,$
represents the reversal of the orientation on $\,C_2\,$ induced from
that on $\,\mathscr{M}$,\ and such that the two hypersurfaces can be
homotopically transformed into one another across $\,\mathscr{M}_{1,2}$.\
Write
\begin{eqnarray}\nonumber
S_{1,2}[\Psi_{\rm cl}]:=\int_{\mathscr{M}_{1,2}}\,\bigl(\Psi_{\rm cl}
\vert_{\mathscr{M}_{1,2}}\bigr)^*\Theta_\mathscr{F}\,.
\end{eqnarray}
Using \Reqref{eq:EuLagAl}, we find
\begin{eqnarray}\nonumber
\delta S_{1,2}[\Psi_{\rm cl}]=\int_{C_1}\,(\Psi_{\rm cl}
\vert_{C_1})^*\Theta_\mathscr{F}-\int_{C_2}\,(\Psi_{\rm cl}
\vert_{C_2})^*\Theta_\mathscr{F}\,,
\end{eqnarray}
and therefore conclude that the 2-form
\begin{eqnarray}\label{eq:presympF}
\Omega_\mathscr{F}[\Psi_{\rm cl}]:=\int_\mathscr{C}\,(\Psi_{\rm cl}\vert_\mathscr{C})^*
\delta\Theta_\mathscr{F}\,,
\end{eqnarray}
written for an \emph{arbitrary} Cauchy hypersurface $\,\mathscr{C}$,\ is
manifestly closed (and independent of the choice of $\,\mathscr{C}$) and
hence defines a presymplectic form on $\,{\mathsf P}_\mathscr{F}$.\ The proof
proceeds analogously for $\,\p\mathscr{M}\neq\emptyset$,\ and the analysis
below (for $\,\mathscr{M}=\Sigma\,$ with domain walls) is readily seen to
cover that case. The sole difference is the appearance of a more
complicated expression
\begin{eqnarray}\nonumber
\delta S_{1,2}[\Psi_{\rm cl}]=\Xi_{C_1}[\Psi_{\rm cl}\vert_{C_1}]-
\Xi_{C_2}[\Psi_{\rm cl}\vert_{C_2}]\,,
\end{eqnarray}
with the two functional 1-form contributions once more localised on
the two Cauchy hypersurfaces.
\end{itemize}
\begin{Rem}\label{rem:Mars-Wein} Vector fields that span the kernel of the
presymplectic form $\,\Omega_\mathscr{F}\,$ are identified with generators of
infinitesimal gauge transformations of $\,\mathscr{F}$.\ Upon performing
the standard symplectic reduction on the state space $\,{\mathsf P}_\mathscr{F}\,$
with respect to the characteristic distribution $\,K_\mathscr{F}\,$ of $\,\Omega_\mathscr{F}\,$
(assumed reducible) and subsequently restricting $\,\Omega_\mathscr{F}\,$ to
the space $\,\ovl{\ovl{\mathsf P}}_\mathscr{F}={\mathsf P}_\mathscr{F}// K_\mathscr{F}\,$ of leaves of
this distribution, we ultimately obtain a canonical form
$\,\ovl{\ovl\Omega}_\mathscr{F}\,$ on the physical (reduced) state space
$\,\ovl{\ovl{\mathsf P}}_\mathscr{F}$,\ alongside a Poisson bracket
\begin{eqnarray}
\{\mathscr{O}_1,\mathscr{O}_2\}_\mathscr{F}[\Psi_{\rm cl}]=\ovl{\ovl\Omega}_\mathscr{F}[\Psi_{\rm
cl}]( X_{\mathscr{O}_1},X_{\mathscr{O}_2})
\end{eqnarray}
of \textbf{hamiltonian functions $\,\mathscr{O}_i$},\ i.e.\ functionals on
the reduced state space which generate \textbf{hamiltonian vector
fields} associated with $\,\mathscr{O}_i\,$ as per
\begin{eqnarray}
\delta\mathscr{O}_i=-\mathscr{X}_{\mathscr{O}_i}\righthalfcup\,\ovl{\ovl\Omega}_\mathscr{F}[ \Psi_{\rm cl}]\,.
\end{eqnarray}
Here, $\,X_{\mathscr{O}_i}\,$ are vectors tangent to $\,{\mathsf P}_\mathscr{F} \,$ at
the state $\,\Psi_{\rm cl}$.\ They are defined by the corresponding
$\mathcal{F}$-vertical vector fields $\,\mathscr{X}_{\mathscr{O}_i}\,$ tangent to
$\,{\mathsf J}^1\mathcal{F}\,$ and satisfying the linearised variant of the field
equations of $\,\mathscr{F}$.\ In the case of a space-time with a non-empty
boundary (or domain walls), the $\,\mathscr{X}_{\mathscr{O}_i}\,$ are additionally
required to obey the linearised version of the boundary (resp.\
domain-wall gluing) conditions of $\,\mathscr{F}$.\end{Rem}\begin{center}*\hspace{2cm}*\hspace{2cm}*\end{center}
The reconstruction of the symplectic form on the state space of the
$\mathcal{F}$-field theory $\,\mathscr{F}\,$ is the first step towards a geometric
quantisation of the latter as it induces, iff $\,\frac{1}{2\pi}\,
\Omega_\mathscr{F}\,$ has integral periods over 2-cycles of $\,{\mathsf P}_\mathscr{F}$,\ a
circle bundle over $\,{\mathsf P}_\mathscr{F}\,$ whose space of sections, when
suitably polarised, can be identified with the Hilbert space of
$\,\mathscr{F}$,\ cf., e.g., \Rcite{Woodhouse:1992de}. In the present
paper, we do not address the question of the choice of the
polarisation, and so we content ourselves with the following
\begin{Def}\label{def:prequantise}
Let $\,\mathscr{F}\,$ be an $\mathcal{F}$-field theory on a (pseudo-)riemannian
manifold $\,(\mathscr{M},{\rm g})\,$ with a covariant configuration bundle
$\,\pi_\mathcal{F}: \mathcal{F}\to\mathscr{M}$,\ and let $\,({\mathsf P}_\mathscr{F},\Omega_\mathscr{F})\,$ be the
symplectic space of extremal sections of the first-jet bundle
$\,{\mathsf J}^1\mathcal{F}\,$ of $\,\mathcal{F}$,\ equipped with a symplectic form
$\,\Omega_\mathscr{F}\,$ of Proposition \ref{prop:Cartan}. The
\textbf{pre-quantum bundle $\,\pi_{\mathcal{L}_\mathscr{F}}:\mathcal{L}_\mathscr{F}\to
{\mathsf P}_\mathscr{F}\,$ of $\mathcal{F}$-field theory $\,\mathscr{F}\,$} is a circle bundle
over $\,{\mathsf P}_\mathscr{F}\,$ with connection $\,\nabla_{\mathcal{L}_\mathscr{F}}\,$ of
curvature
\begin{eqnarray}\nonumber
\curv(\nabla_{\mathcal{L}_\mathscr{F}})=\pi_{\mathcal{L}_\mathscr{F}}^*\Omega_\mathscr{F}\,.
\end{eqnarray}
Fix a choice $\,\mathcal{O}_{{\mathsf P}_\mathscr{F}}=\{\mathcal{O}^{{\mathsf P}_\mathscr{F}}_i\}_{i\in
\mathscr{I}_{{\mathsf P}_\mathscr{F}}}\,$ of an open cover of $\,{\mathsf P}_\mathscr{F}\,$ and a local
presentation, in the sense of Definition \ref{def:loco}, of the
pre-quantum bundle in terms of its \v Cech--Deligne data $\,
\mathcal{L}_\mathscr{F}\xrightarrow{\rm loc.}(\theta_{\mathscr{F}\,i},\gamma_{\mathscr{F}\,ij})\in
\cA^{2,1}(\mathcal{O}_{{\mathsf P}_\mathscr{F}})\,$ associated with $\,\mathcal{O}_{{\mathsf P}_\mathscr{F}}\,$
and subject to the cohomological constraints
\begin{eqnarray}\nonumber
D_{(1)}(\theta_{\mathscr{F}\,i},\gamma_{\mathscr{F}\,ij})=(\Omega_\mathscr{F}
\vert_{\mathcal{O}^{{\mathsf P}_\mathscr{F}}_i},0,1)\,.
\end{eqnarray}
A \textbf{pre-quantisation of $\mathcal{F}$-field theory $\,\mathscr{F}$},
understood in the sense of, e.g., \Rcite{Woodhouse:1992de}, is an
assignment, to every smooth function $\,h\in C^\infty({\mathsf P}_\mathscr{F},{\mathbb{R}}
)\,$ and to the associated \textbf{(global) hamiltonian vector field
$\,\mathscr{X}_h\,$} on $\,{\mathsf P}_\mathscr{F}$,\ determined by the relation
\begin{eqnarray}\nonumber
\mathscr{X}_h\righthalfcup\Omega_\mathscr{F}=-\delta h\,,
\end{eqnarray}
of a collection $\,\widehat\mathcal{O}_h:=(\widehat h_i)_{i\in\mathscr{I}_\mathscr{F}}\,$
of local linear operators
\begin{eqnarray}\label{eq:pre-ham-gen}
\widehat h_i:=-{\mathsf i}\,\pLie{\mathscr{X}_h}-\mathscr{X}_h\righthalfcup\theta_{\mathscr{F}\,i}+h
\vert_{\mathcal{O}^{{\mathsf P}_\mathscr{F}}_i}
\end{eqnarray}
on the space $\,\Gamma(\mathcal{L}_\mathscr{F})\,$ of sections of the pre-quantum
bundle, the latter being regarded as the \textbf{pre-quantisation
Hilbert space}. The collection $\,\widehat\mathcal{O}_h\,$ shall be termed
the \textbf{pre-quantum hamiltonian for $\,h$}.\ By the very
construction, the commutator of a pair $\,\widehat\mathcal{O}_{h_\a},\ \a\in
\{1,2\}\,$ of pre-quantum hamiltonians takes the canonical form
\begin{eqnarray}\label{eq:comm-preq-ham}
[\,\widehat\mathcal{O}_{h_1}\,,\,\widehat\mathcal{O}_{h_2}\,]=-{\mathsf i}\,\widehat
\mathcal{O}_{\{\,h_1\,,\,h_2\,\}_{\Omega_\mathscr{F}}}\,.
\end{eqnarray}
\begin{flushright}$\checkmark$\end{flushright}\end{Def}
\subsection{The covariant formalism for the $\sigma$-model}
Specialisation of the above general discussion to the non-linear
$\sigma$-model of \Reqref{eq:sigma} prerequires a number of
modifications, which -- while preserving the basic conceptual
framework -- serve to adapt the tools introduced to the setting in
hand, in which forms on the space-time $\,\Sigma\,$ are replaced by
locally smooth forms associated with a given triangulation
$\,\triangle(\Sigma)$,\ and in which the space-time itself is split
into domains, supporting the respective phases of the
two-dimensional field theory. As for the latter point, the reader is
advised to acquaint herself or himself, by way of a warm-up, with
the treatment of the world-sheet with a non-empty boundary in
\Rcite{Gawedzki:2001rm}.
The first modification consists in replacing the covariant
configuration bundle $\,\mathcal{F}\,$ with
\begin{Def}\label{def:cov-conf-bdle-si}
The \textbf{covariant configuration bundles $\,\pi_{\mathcal{F}_\sigma}:\mathcal{F}_\sigma
\to\Sigma\,$ of the non-linear $\sigma$-model for network-field
configurations $\,(X\,\vert\,\Gamma)\,$ in string background $\,\gt{B}\,$
on world-sheet $\,(\Sigma,\gamma)\,$ with defect quiver $\,\Gamma\,$} are given
by a disjoint sum of fibre bundles over the respective components of
the disjoint union of elements of $\,\gt{P}_\Sigma,\gt{E}_\Gamma\,$ and
$\,\gt{V}_\Gamma\,$ with restrictions
\begin{eqnarray}\nonumber
\mathcal{F}_\sigma\vert_{\wp\in\gt{P}_\Sigma}:=\wp\x M\to\wp\,,\qquad\qquad\mathcal{F}_\sigma
\vert_{\ell\in\gt{E}_\Gamma}:=\ell\x Q\to\ell\,,\qquad\qquad\mathcal{F}_\sigma
\vert_{\jmath\in\gt{V}_\Gamma}:=\jmath\x T_{n_\jmath}\to\jmath\,.
\end{eqnarray}
The associated first-jet bundles, $\,{\mathsf J}^1\mathcal{F}_\sigma\to\Sigma$,\ admit
local coordinates
\begin{itemize}
\item $\bigl(\sigma^a,X^\mu,\xi^\nu_b)\,$ over $\,\wp\in\gt{P}_\Sigma$,\
where $\,\sigma^a\,$ are local coordinates on the patch $\,\wp$,\ and
$\,X^\mu\,$ are local coordinates on $\,M$;
\item $\,(\varphi,X^A,\xi^B_\varphi)\,$ over $\,\ell\in\gt{E}_\Gamma$,\
where $\,\varphi\,$ is a local coordinate on the defect line
$\,\ell$,\ and $\,X^A\,$ are local coordinates on $\,Q$;
\item $(\sigma_\jmath,X^i)\,$ over $\,\jmath\in\gt{V}_\Gamma$,\ where
$\,\sigma_\jmath\,$ are the coordinates of the defect junction
$\,\jmath\,$ within $\,\Sigma$,\ and $\,X^i\,$ are local coordinates on
$\,T_{n_\jmath}\,$ (the first-jet extension is trivial over the
point $\,\jmath$).
\end{itemize}
\begin{flushright}$\checkmark$\end{flushright}\end{Def} \noindent The action functional \eqref{eq:sigma} being
defined in terms of local expressions sourced by plaquettes of a
triangulation $\,\triangle(\Sigma)\,$ of the world-sheet (and their
lower-dimensional submanifolds), the Cartan form naturally splits
into a sum over terms supported by the particular plaquettes. Below,
we define the Cartan form as an object glued up from these
contribution. Our point of departure is the local term $\,\mathcal{L}_p(\sigma
,X,\p X)\,$ of the lagrangean density for $\,S_\sigma\,$ coming from
the plaquette $\,p\in\triangle(\Sigma)$.\ In the minkowskian gauge
$\,\gamma=\eta\,$ of the intrinsic world-sheet metric, it is given by
the formula\footnote{In the formula, we employ a shorthand notation
$\,e\subset p\cap\Gamma\,$ to denote those edges of the plaquette
$\,p\,$ which lie on a defect line (and hence belong to the
triangulation $\,\triangle(\Gamma\setminus\gt{V}_\Gamma)$), and similarly for
$\,\jmath\in p\cap\gt{V}_\Gamma$,\ the latter denoting the defect
junctions among the vertices of the triangulation of the plaquette
$\,p$.}
\begin{eqnarray}\nonumber
\mathcal{L}_p(\sigma,X,\xi)\,{\mathsf d}^2\sigma&=&\tfrac{1}{2}\,{\rm g}_{\mu\nu}(X)\,
\star_\eta\bigl(\xi^\mu\wedge\star_\eta\xi^\nu\bigr)\,{\mathsf d}^2\sigma-
B_{i_p,\mu\nu}(X)\,\star_\eta\bigl(\xi^\mu\wedge\xi^\nu\bigr)\,
{\mathsf d}^2\sigma\cr\cr
&&+\sum_{e\subset p}\,\left(A_{i_p i_e,\mu}(X)\,\xi^\mu\wedge\delta_e-
{\mathsf i}\,\sum_{v\in e}\,\delta_v\,\log g_{i_p i_e i_v}^{\varepsilon_{pev}}(X)\,
{\mathsf d}^2\sigma\right)\cr\cr
&&+\sum_{e\subset p\cap\Gamma}\,\left(P_{i_e,A}(X)\,\xi^A\wedge\delta_e-
{\mathsf i}\,\sum_{v\in e}\,\delta_v\,\log K_{i_e i_v}^{-\varepsilon_{ev}}(X)\,{\mathsf d}^2
\sigma\right)\cr\cr
&&-{\mathsf i}\,\sum_{\jmath\in p\cap\gt{V}_\Gamma}\,\delta_\jmath\,\log f_{n_\jmath
,i_\jmath}(X)\,{\mathsf d}^2\sigma\,,
\end{eqnarray}
with
\begin{eqnarray}\nonumber
\tfrac{\delta^2\mathcal{L}_p}{\delta\xi^\mu_a\delta\xi^\nu_b}(\sigma,X,\xi)=-\bigl(
{\rm g}_{\mu\nu}\,\eta^{ab}-2B_{i_p,\mu\nu}\,\varepsilon^{ab}\bigr)(X)=:-L^{a
b}_{i_p,\mu\nu}(X)\,,
\end{eqnarray}
all written in terms of local coordinates $\,\sigma^a,\ a\in\{1,2\}\,$
on $\,p$,\ with $\,{\mathsf d}^2\sigma={\mathsf d}\sigma^1\wedge{\mathsf d}\sigma^2$,\ alongside
objects $\,\xi=\xi_a\,{\rm d}\sigma^a$,\ and the Dirac distributions
$\,\delta_x\equiv\delta^{(2)}(\sigma-\sigma_x)\,$ on $\,\Sigma$,\ as well as the
singular (Dirac-type) currents $\,\delta_e\,$ supported over $\,e\subset
p$,\ with the defining property
\begin{eqnarray}\nonumber
\forall_{\theta\in\Omega^1(p)}\ :\ \int_p\,\theta\wedge\delta_e=\int_e\,
\iota_e^*\theta\,,
\end{eqnarray}
where $\,\iota_e:e\emb p\,$ is the embedding map. In the minkowskian
gauge, we have
\begin{eqnarray}\nonumber
\star_\eta
1={\mathsf d}^2\sigma\,,\qquad\star_\eta{\mathsf d}\sigma^1=-{\mathsf d}\sigma^2\,,\qquad
\star_\eta{\mathsf d}\sigma^2=-{\mathsf d}\sigma^1\,,\qquad\star_\eta{\mathsf d}^2\sigma=-1\,.
\end{eqnarray}
This yields
\begin{Def}\label{def:Cart-form-si}
Let $\,\gt{B}=(\mathcal{M},\mathcal{B},\mathcal{J})\,$ be a string background of Definition
\ref{def:bckgrnd}. The \textbf{Cartan form of the non-linear
$\sigma$-model for network-field configurations $\,(X\,\vert\,\Gamma)\,$ in
string background $\,\gt{B}\,$ on world-sheet $\,(\Sigma,\gamma)\,$ with
defect quiver $\,\Gamma\,$} is a 2-form $\,\Theta_\sigma\,$ on the first-jet
bundles $\,{\mathsf J}^1\mathcal{F}_\sigma\,$ of the covariant configuration bundles
$\,\mathcal{F}_\sigma\,$ of the $\sigma$-model, given in terms of its restrictions
$\,\Theta_\sigma\vert_p=:\Theta_p\,$ to patches $\,p\in\triangle(\Sigma)\,$ of a
triangulation $\,\triangle(\Sigma)\,$ of $\,\Sigma\,$ subordinate to
$\,\mathcal{O}_\mathscr{M},\ \mathscr{M}\in\{M,Q,T\}\,$ with respect to $\,(X\,\vert\,\Gamma)\,$ that
take the form
\begin{eqnarray}\nonumber
\Theta_p(\sigma,X,\xi)&=&\tfrac{1}{2}\,{\rm g}_{\mu\nu}(X)\,\xi^\mu\wedge
\star_\eta\xi^\nu-{\rm g}_{\mu \nu}(X)\,\delta
X^\mu\wedge\star_\eta\xi^\nu\cr\cr
&&-B_{i_p,\mu\nu}(X)\,\xi^\mu\wedge\xi^\nu+2B_{i_p,\mu\nu}(X)\,\delta
X^\mu\wedge\xi^\nu\cr\cr
&&+\sum_{e\subset p}\,\bigl(A_{i_p i_e}(X)\wedge\delta_e-{\mathsf i}\,\sum_{v
\in e}\,\delta_v\,\log g_{i_p i_e i_v}^{\varepsilon_{pev}}(X)\,{\mathsf d}^2\sigma\bigr)
\cr\cr
&&+\sum_{e\subset p\cap_\Gamma}\,\bigl(P_{i_e}(X)\,\wedge\delta_e-{\mathsf i}\,
\sum_{v\in e}\,\delta_v\,\log K_{i_e i_v}^{-\varepsilon_{ev}}(X)\,{\mathsf d}^2\sigma
\bigr)\cr\cr
&&-{\mathsf i}\,\sum_{\jmath\in p\cap\gt{V}_\Gamma}\,\delta_\jmath\,\log f_{n_\jmath,
i_\jmath}(X)\,{\mathsf d}^2\sigma\,.
\end{eqnarray}
\begin{flushright}$\checkmark$\end{flushright}\end{Def} \begin{Rem} Consider a generic $\mathcal{F}_\sigma$-vertical vector field
$\,\mathscr{V}\,$ on $\,{\mathsf J}^1\mathcal{F}_\sigma\,$ with restrictions
\begin{eqnarray}\nonumber
\mathscr{V}\vert_{\gt{P}}=V^\mu\,\tfrac{\delta\ \ }{\delta X^\mu}+V^\mu_a\,\tfrac{\delta\
}{\delta \xi^\mu_a}\,,\qquad\qquad\mathscr{V}\vert_{\gt{E}_\Gamma}=V^A\,\tfrac{\delta\ \
}{\delta X^A}+V_\varphi^A\,\tfrac{\delta\ }{\delta\xi^A_\varphi}\,,\qquad
\qquad\mathscr{V}\vert_{\gt{V}_\Gamma}=V^i\,\tfrac{\delta\ \ }{\delta X^i}\,,
\end{eqnarray}
where the various components are constrained as per
\begin{eqnarray}\label{eq:tangojet-constr}
V^A\,\tfrac{\p\iota_\a^\mu}{\p X^A}=V^\mu\circ\iota_\a\,,\qquad
\qquad V^i\,\tfrac{\p\pi_n^{k,k+1\ A}}{\p X^i}=V^A\circ\pi_n^{k,k+
1}\,.
\end{eqnarray}
The requirement that $\,\mathscr{V}\,$ obey the linearised version of
\Reqref{eq:eom-xipar} is tantamount to the imposition of the
relation
\begin{eqnarray}\nonumber
V^\mu_a=\p_a V^\mu\,.
\end{eqnarray}
Hence, a vector field tangent to the space of extremal sections at a
section $\,\Psi_{\sigma,{\rm cl}}\,$ is necessarily of the form
\begin{eqnarray}\label{eq:tan-phasp}
\mathscr{V}\vert_{\gt{P}}=V^\mu\,\tfrac{\delta\ \ }{\delta X^\mu}+\p_a
V^\mu\,\tfrac{\delta\ }{\delta \xi^\mu_a}\,,
\end{eqnarray}
where the various components are related as in
\Reqref{eq:tangojet-constr}, and where the $\,V^\mu\,$ satisfy the
linearised version of \Reqref{eq:field-eqs}.\end{Rem}\begin{center}*\hspace{2cm}*\hspace{2cm}*\end{center}
We have
\begin{Prop}\label{prop:Cart-si-def}
Let $\,\Theta_\sigma\,$ be the Cartan form of the non-linear $\sigma$-model
for network-field configurations $\,(X\,\vert\,\Gamma)\,$ in string
background $\,\gt{B}\,$ on world-sheet $\,(\Sigma,\gamma)\,$ with defect
quiver $\,\Gamma$,\ explicited in Definition \ref{def:Cart-form-si}.
Given a section $\,\Psi_\sigma\in\Gamma({\mathsf J}^1\mathcal{F}_\sigma)\,$ of the first-jet
bundles $\,{\mathsf J}^1 \mathcal{F}_\sigma\to\Sigma\,$ of the covariant configuration
bundles for the $\sigma$-model, write
\begin{eqnarray}\nonumber
S_{\Theta_\sigma}[\Psi_\sigma]:=\int_\Sigma\,\Psi^*\Theta_\sigma\,.
\end{eqnarray}
The principle of least action applied to the functional
$\,S_{\Theta_\sigma}\,$ as per
\begin{eqnarray}\label{eq:var-STHsi}
\mathscr{V}\righthalfcup\delta S_{\Theta_\sigma}[\Psi_{\sigma,{\rm cl}}]=0\,,
\end{eqnarray}
with $\,\mathscr{V}\in\Gamma({\mathsf T}{\mathsf J}^1\mathcal{F}_\sigma)^{\perp_{\mathcal{F}_\sigma}}\,$ an
arbitrary $\mathcal{F}_\sigma$-vertical vector field on $\,{\mathsf J}^1\mathcal{F}_\sigma$,\
yields the field equations \eqref{eq:field-eqs} alongside the Defect
Gluing Condition \eqref{eq:DGC} for classical sections $\,\Psi_{\sigma
,{\rm cl}}\,$ of the $\sigma$-model.
\end{Prop}
\noindent As the proof of the proposition is rather technical, it
has been relegated to Appendix \ref{app:Cart-form-si}.\medskip
The Cartan form $\,\Theta_\sigma\,$ enables us to study the canonical
structure of the classical $\sigma$-model and provides non-trivial
insights into its quantum r\'egime, all that through the definition
of a (pre-)symplectic form on the space of states $\,\Psi_{\sigma,{\rm
cl}}(\sigma)=\bigl(\sigma^a,X^I(\sigma),\p_b X^J(\sigma)\bigr)\,$ of the model
(here, $\,I\,$ and $\,J\,$ are multi-indices taking values in the
index sets associated with coordinates on $\,M\sqcup Q\sqcup T$).
The latter space admits a natural parameterisation in terms of
initial data of an extremal section $\,\Psi_{\sigma,{\rm cl}}\,$
localised on a Cauchy hypersurface in $\,\Sigma\,$ -- a space-like
contour $\,\mathscr{C}\,$ in the case in hand. As argued in Remark
\ref{rem:states}, there are two qualitatively different species of a
classical state in the presence of a defect quiver in the
world-sheet: the untwisted state and the twisted state. Accordingly,
we have
\begin{Def}\label{def:untw-phspace}
Let $\,\gt{B}\,$ be a string background with target space $\,M$.\ The
\textbf{untwisted state space $\,{\mathsf P}_{\sigma,\emptyset}\,$ of the
non-linear $\sigma$-model for network-field configurations
$\,(X\,\vert\,\Gamma)\,$ in string background $\,\gt{B}\,$ on world-sheet
$\,(\Sigma,\gamma)\,$ with defect quiver $\,\Gamma\,$} is given by the
cotangent bundle over the free-loop space $\,{\mathsf L} M=
C^\infty({\mathbb{S}}^1,M)\,$ of the target space $\,M\,$ of the $\sigma$-model,
\begin{eqnarray}\nonumber
{\mathsf P}_{\sigma,\emptyset}={\mathsf T}^*{\mathsf L} M\,.
\end{eqnarray}
It has local coordinates $\,(X^\mu,{\mathsf p}_\nu)$,\ where $\,X:{\mathbb{S}}^1\to
M\,$ is a smooth loop in $\,M\,$ and $\,{\mathsf p}={\mathsf p}_\mu\,\delta X^\mu\,$
is a normal covector field on $\,X^\mu$.\begin{flushright}$\checkmark$\end{flushright}\end{Def} \noindent The twisted
counterpart is introduced in
\begin{Def}\label{def:tw-phspace}
Let $\,\gt{B}\,$ be a string background with target $\,\mathcal{M}=(M,{\rm g},\mathcal{G}
)\,$ and bi-brane $\,\mathcal{B}=\bigl(Q,\iota_\a,\omega,\Phi\ \vert\ \a\in\{1,
2\}\bigr)$,\ and let $\,\Gamma\,$ be a defect quiver embedded in a
world-sheet $\,(\Sigma,\gamma)\,$ in such a manner that there exists a
closed space-like curve $\,\mathscr{C}\cong{\mathbb{S}}^1\subset\Sigma\,$ that
intersects $\,I\in{\mathbb{N}}_{>0}\,$ defect lines $\,\ell_k\in\gt{E}_\Gamma,\
k\in\ovl{1, I}\,$ of $\,\Gamma\,$ at the respective points $\,\sigma_k\,$
so that the tangent vectors $\,\widehat t_k\,$ of the defect lines
$\,\ell_k\,$ are all time-like or anti-time-like at the $\,\sigma_k$,\
with $\,X_* \widehat t_k=:V_k\in{\mathsf T}_{q_k}Q$.\ Write $\,\varepsilon_k=+
1\,$ if $\,\widehat t_k\,$ is time-like, and $\,\varepsilon_k=-1\,$ if
$\,\widehat t_k\,$ is anti-time-like at $\,\sigma_k$.\ Fix a collection
of points $\,\{P_k\}_{k\in\ovl{1,I}}\in{\mathbb{S}}^1$,\ write
$\,{\mathbb{S}}^1_{\{P_k\}}:= {\mathbb{S}}^1\setminus\{P_k\}_{k\in\ovl{1,I}}\,$ and
define the space of smooth maps
\begin{eqnarray}\nonumber
{\mathsf L}_{Q|\{(P_k,\varepsilon_k)\}}M=\{\ (X,q_k\ \vert\ k\in\ovl{1,I})\in
C^\infty({\mathbb{S}}^1_{\{P_k\}},M)\x Q^{\x I} \quad\vert\quad \lim_{\epsilon\to
0^+}\,X\bigl(P_k+(-1)^{\a+1}\,\varepsilon_k\,\epsilon\bigr)= \iota_\a(q_k) \
\}\,.
\end{eqnarray}
Denote as $\,\widehat\tau_\a(P_k):=-\varepsilon_k\,\lim_{\epsilon\to 0^+}\,X_*
\widehat t\bigl(P_k+(-1)^{\a+1}\,\varepsilon_k\,\epsilon\bigr)\,$ the
(one-sided) limiting values of the pushforward of the tangent vector
field $\,\widehat t(\cdot)\,$ on $\,{\mathbb{S}}^1_{\{P_k\}}\,$ along $\,X$,\
and write $\,(\iota_1^{+1},\iota_2^{+1}):=(\iota_1,\iota_2)\,$ and
$\,( \iota_1^{-1},\iota_2^{-1}):=(\iota_2,\iota_1)$.\ The
\textbf{$k$-twisted state space $\,{\mathsf P}_{\sigma,\mathcal{B}|\{(P_k,\varepsilon_k)
\}}\,$ of the non-linear $\sigma$-model for network-field
configurations $\,(X\,\vert\,\Gamma)\,$ in string background $\,\gt{B}\,$
on world-sheet $\,(\Sigma,\gamma)\,$ with defect quiver $\,\Gamma\,$} is
naturally identified with the space
\begin{eqnarray}\nonumber
{\mathsf P}_{\sigma,\mathcal{B}|\{(P_k,\varepsilon_k)\}}=\Bigg\{\ (X,{\mathsf p}={\mathsf p}_\mu\,\delta X^\mu
,q_k,V_k\ \vert\ k\in\ovl{1,I})\in{\mathsf T}^*C^\infty({\mathbb{S}}^1_{\{P_k\}},M)
\x{\mathsf T} Q^{\x I} \quad\bigg\vert\quad \cr\cr\cr \land\quad \left\{
\begin{array}{l} \lim_{\epsilon\to 0^+}{\mathsf p}\bigl(P_k+(-1)^{\a+1}\,\epsilon\bigr)={\rm g}
\bigl(\iota_\a^{\varepsilon_k}(q_k)\bigr)\bigl(\varepsilon_k\,\iota_{\a\,*}^{\varepsilon_k}
V_k,\cdot\bigr)\cr\cr
{\rm g}\bigl(\iota_1(q_k)\bigr)\bigl(\widehat\tau_1(P_k),\iota_{1 \,*}(
\cdot)\bigr)-{\rm g}\bigl(\iota_2(q_k)\bigr)\bigl(\widehat\tau_2(P_k),
\iota_{2\,*}(\cdot)\bigr)=V_k\righthalfcup\omega(q_k)\end{array} \right. \ \Bigg\}\,.
\end{eqnarray}
The space $\,{\mathsf P}_{\sigma,\mathcal{B}|\{(P_k,\varepsilon_k)\}}\,$ shall be described
in terms of its local coordinates $\,(X^\mu,{\mathsf p}_\nu,q_k,V_k\ \vert\
k\in\ovl{1,I})$. \begin{flushright}$\checkmark$\end{flushright}\end{Def} \noindent We may now formulate the following
fundamental statements:
\begin{Prop}\label{prop:sympl-form-si-untw}
Let $\,\gt{B}\,$ be a string background with target $\,\mathcal{M}=(M,{\rm g},\mathcal{G}
)$,\ and let $\,{\mathsf P}_{\sigma,\emptyset}\,$ be the untwisted state space
of the non-linear $\sigma$-model for network-field configurations
$\,(X\,\vert\,\Gamma)\,$ in string background $\,\gt{B}\,$ on world-sheet
$\,(\Sigma,\gamma)\,$ with defect quiver $\,\Gamma$.\ Denote by
\begin{eqnarray}\nonumber
{\rm ev}_M\ :\ {\mathsf L} M\x{\mathbb{S}}^1\to M
\end{eqnarray}
the canonical evaluation map. The Cartan form $\,\Theta_\sigma\,$ of the
$\sigma$-model from Definition \ref{def:Cart-form-si} canonically
defines a closed 2-form on $\,{\mathsf P}_{\sigma,\emptyset}$,\ given by the
formula
\begin{eqnarray}\label{eq:Omsi-untw}
\Omega_{\sigma,\emptyset}=\delta\theta_{{\mathsf T}^*{\mathsf L} M}+\pi_{{\mathsf T}^*{\mathsf L} M}^*
\int_{{\mathbb{S}}^1}\,{\rm ev}_M^* {\rm H}\,,
\end{eqnarray}
in which
\begin{eqnarray}\label{eq:can-1-cot}
\theta_{{\mathsf T}^*{\mathsf L} M}[(X,{\mathsf p})]=\int_{{\mathbb{S}}^1}\,\Vol({\mathbb{S}}^1)\wedge{\mathsf p}
\,,\qquad\qquad{\mathsf p}={\mathsf p}_\mu\,\delta X^\mu
\end{eqnarray}
is the canonical 1-form on the total space of the cotangent bundle
$\,\pi_{{\mathsf T}^*{\mathsf L} M}:{\mathsf T}^*{\mathsf L} M\to{\mathsf L} M$,\ written using the
volume form $\,\Vol({\mathbb{S}}^1)\,$ on $\,{\mathbb{S}}^1$,\ and $\,{\rm H}=\curv(\mathcal{G}
)$.\ The 2-form is to be evaluated on an arbitrary classical section
$\,\Psi_{\sigma,{\rm cl}}\in\Gamma( {\mathsf J}^1\mathcal{F}_\sigma)\,$ of the first-jet
bundles of the covariant configuration bundles $\,\mathcal{F}_\sigma\,$ of the
$\sigma$-model. The section (state) is represented by its Cauchy data
$\,\bigl(X^\mu,{\mathsf p}_\nu\bigr)\in {\mathsf P}_{\sigma,\emptyset}\,$ localised
on an arbitrary untwisted Cauchy contour $\,\mathscr{C}\cong{\mathbb{S}}^1$.
\end{Prop}
\noindent A proof of the proposition is given in Appendix
\ref{app:sympl-form-si-untw}.
\begin{Prop}\label{prop:sympl-form-si-tw}
Let $\,\gt{B}\,$ be a string background with target $\,\mathcal{M}=(M,{\rm g},\mathcal{G}
)\,$ and $\mathcal{G}$-bi-brane $\,\mathcal{B}=(Q,\iota_\a,\omega,\Phi\ \vert\
\a\in\{1,2\})$,\ and let $\,{\mathsf P}_{\sigma,\mathcal{B}|\{(P_k,\varepsilon_k)\}}\,$ be
the $k$-twisted state space of the non-linear $\sigma$-model for
network-field configurations $\,( X\,\vert\,\Gamma)\,$ in string
background $\,\gt{B}\,$ on world-sheet $\,(\Sigma,\gamma)\,$ with defect
quiver $\,\Gamma$.\ Write $\,{\mathbb{S}}^1_{\{P_k\}}={\mathbb{S}}^1\setminus\{P_k\}_{k
\in\ovl{1,I}}\,$ and denote by
\begin{eqnarray}\nonumber
{\rm ev}_{M,\{P_k\}}\ :\ C^\infty({\mathbb{S}}^1_{\{P_k\}},M)\x{\mathbb{S}}^1_{\{P_k\}}\to
M
\end{eqnarray}
the canonical evaluation map, and by $\,{\rm pr}_{{\mathsf T}^*C^\infty
({\mathbb{S}}^1_{\{P_k\}},M)}:{\mathsf P}_{\sigma,\mathcal{B}|\{(P_k,\varepsilon_k)\}}\to{\mathsf T}^*
C^\infty({\mathbb{S}}^1_{\{P_k\}},M)\,$ and $\,{\rm pr}_{Q,k}:{\mathsf P}_{\sigma,\mathcal{B}|\{(
P_k,\varepsilon_k)\}}\to Q\,$ the canonical projections, the latter having
the $k$-th cartesian factor as the codomain. The Cartan form
$\,\Theta_\sigma\,$ of the $\sigma$-model from Definition
\ref{def:Cart-form-si} canonically defines a closed 2-form on
$\,{\mathsf P}_{\sigma,\mathcal{B}|\{(P_k,\varepsilon_k)\}}$,\ given by the formula
\begin{eqnarray}
\Omega_{\sigma,\mathcal{B}|\{(P_k,\varepsilon_k)\}}={\rm pr}_{{\mathsf T}^*C^\infty({\mathbb{S}}^1_{\{P_k\}},
M)}^*\bigl(\delta\theta_{{\mathsf T}^*C^\infty({\mathbb{S}}^1_{\{P_k\}},M)}+
\pi_{{\mathsf T}^*C^\infty({\mathbb{S}}^1_{\{P_k\}},
M)}^*\int_{{\mathbb{S}}^1_{\{P_k\}}}\,{\rm ev}_{M,\{P_k\}}^*{\rm H}\bigr)+\sum_{k=1}^I\,
\varepsilon_k\,{\rm pr}_{Q,k}^*\omega\,,\cr\cr\label{eq:Omsi-tw}
\end{eqnarray}
in which
\begin{eqnarray}\label{eq:can-1-cot-tw}
\theta_{{\mathsf T}^*C^\infty({\mathbb{S}}^1_{\{P_k\}},M)}[(X,{\mathsf p})]=
\int_{{\mathbb{S}}^1_{\{P_k\}}}\,\Vol({\mathbb{S}}^1_{\{P_k\}})\wedge{\mathsf p}\,,\qquad
\qquad{\mathsf p}={\mathsf p}_\mu\,\delta X^\mu
\end{eqnarray}
is the canonical 1-form on the total space of the cotangent bundle
$\,\pi_{{\mathsf T}^*C^\infty({\mathbb{S}}^1_{\{P_k\}},
M)}:{\mathsf T}^*C^\infty({\mathbb{S}}^1_{\{P_k\}},M)\to
C^\infty({\mathbb{S}}^1_{\{P_k\}},M)$,\ written using the volume form
$\,\Vol({\mathbb{S}}^1_{\{P_k\}})\,$ on $\,{\mathbb{S}}^1_{\{P_k\}}$.\ The 2-form is
to be evaluated on an arbitrary classical section $\,\Psi_{\sigma,{\rm
cl}}\in\Gamma({\mathsf J}^1\mathcal{F}_\sigma)\,$ of the first-jet bundles of the
covariant configuration bundles $\,\mathcal{F}_\sigma\,$ of the $\sigma$-model.
The section (state) is represented by its Cauchy data
$\,(X^\mu,{\mathsf p}_\nu,q_k,V_k\ \vert\ k\in\ovl{1,I})
\in{\mathsf P}_{\sigma,\mathcal{B}|\{(P_k,\varepsilon_k)\}}\,$ localised on an arbitrary
twisted Cauchy contour $\,\mathscr{C}\cong{\mathbb{S}}^1_{\{P_k\}}$,\ as in
Definition \ref{def:tw-phspace}.
\end{Prop}
\noindent A proof of the proposition is given in Appendix
\ref{app:sympl-form-si-tw}.\medskip
In order to give an explicit description of the pre-quantum bundles
for the two types of the state space of the $\sigma$-model, we should
first recall the necessary facts about the (Fr\'echet) manifold
$\,{\mathsf L} M$,\ as defined in \Rcite{Hamilton:1982}. We use, after
\Rcite{Gawedzki:1987ak}, the straightforward
\begin{Prop}\label{prop:cover-untw}
Let $\,{\mathsf L} M=C^\infty({\mathbb{S}}^1,M)\,$ be the free-loop space of a
manifold $\,M$,\ the latter coming with a choice
$\,\mathcal{O}_M=\{\mathcal{O}^M_i\}_{i\in\mathscr{I}_\mathscr{M}}\,$ of an open cover. Consider
the non-empty open sets\footnote{The free-loop space $\,{\mathsf L} M\,$ is
equipped with the compact-open topology.}
\begin{eqnarray}\nonumber
\mathcal{O}_\gt{i}=\{\ X\in{\mathsf L} M \quad\vert\quad \forall_{e,v\in\triangle(
{\mathbb{S}}^1)}\ :\ X(e)\subset\mathcal{O}^M_{i_e}\quad\land\quad X(v)\in
\mathcal{O}^M_{i_v} \ \}\,,
\end{eqnarray}
with the index $\,\gt{i}\,$ given by a pair $\,\bigl(\triangle({\mathbb{S}}^1)
,\phi\bigr)\,$ consisting of a choice $\,\triangle({\mathbb{S}}^1)\,$ of the
triangulation of the unit circle, with its edges $\,e\,$ and
vertices $\,v$,\ and a choice $\,\phi:\triangle({\mathbb{S}}^1)\to\mathscr{I}_M:f
\mapsto i_f\,$ of the assignment of indices of $\,\mathcal{O}_M\,$ to
elements of $\,\triangle({\mathbb{S}}^1)$.\ By varying these two choices
arbitrarily, whereby an index set $\,\mathscr{I}_{\mathcal{O}_{{\mathsf L} M}}\,$ is
formed, all of $\,{\mathsf L} M\,$ is covered, thus yielding an
\textbf{open cover $\,\mathcal{O}_{{\mathsf L} M}=\{\mathcal{O}_\gt{i} \}_{\mathscr{I}_{{\mathsf L} M}}\,$
of free-loop space $\,{\mathsf L} M$}.
\end{Prop}
\noindent Similarly,
\begin{Prop}\label{prop:cover-tw}
Let $\,\gt{B}\,$ be a string background with target $\,\mathcal{M}=(M,{\rm g},\mathcal{G}
)\,$ and $\mathcal{G}$-bi-brane $\,\mathcal{B}=(Q,\iota_\a,\omega,\Phi\ \vert\
\a\in\{1,2 \})$.\ Given a collection $\,\{P_k\}_{k\in \ovl{1,I}}\,$
of $\,I\in {\mathbb{N}}_{>0}\,$ points on the unit circle $\,{\mathbb{S}}^1$,\ and a
collection $\,\{\varepsilon_k\}_{k\in\ovl{1,I}}\,$ of $I$ elements of
$\,\{-1,+1\}$,\ let $\,{\mathsf L}_{Q|\{(P_k,\varepsilon_k)\}}M\,$ be the space
introduced in Definition \ref{def:tw-phspace}. Fix a choice
$\,\mathcal{O}_M=\{\mathcal{O}^M_i \}_{i\in\mathscr{I}_\mathscr{M}}\,$ of an open cover of $\,M$,\
and a choice $\,\mathcal{O}_Q=\{\mathcal{O}^Q_i\}_{i \in\mathscr{I}_Q}\,$ of an open cover
of $\,Q$,\ for which there exist \v Cech-extended $\mathcal{G}$-bi-brane
maps $\,(\iota_\a,\phi_\a),\ \a\in\{1,2\}$,\ as described in
Definition \ref{def:loco}. Consider the non-empty open sets
\begin{eqnarray}\nonumber
\mathcal{O}_{\{(P_k,\varepsilon_k)\}\,\gt{i}}=\left\{\ (X,q_k\ \vert\ k\in\ovl{1,I})
\in{\mathsf L}_{Q|\{P_k,\varepsilon_k\}}M \quad\bigg\vert\quad X(e)\subset
\mathcal{O}^M_{i_e}\ \land\ X(v)\in\mathcal{O}^M_{i_v}\ \land\ \left\{ \begin{array}{l} q_k
\in\mathcal{O}^Q_{i^{1,2}_{P_k}}\cr \iota_\a(q_k)\in\mathcal{O}^M_{\phi_\a(i^{1,
2}_{P_k})} \end{array} \right. \ \right\}
\end{eqnarray}
with the index $\,\gt{i}\,$ given by a triple $\,\bigl(\triangle_{\{
P_k\}}({\mathbb{S}}^1),\phi,\phi^{1,2}\bigr)\,$ consisting of a choice
$\,\triangle_{\{P_k\}}({\mathbb{S}}^1)\,$ of the triangulation of the unit
circle, with its edges $\,e\,$ and vertices $\,v\,$ of which $I$ are
fixed at the $\,P_k,\ k\in\ovl{1,I}$,\ and choices $\,\phi:\bigl(
\triangle_{\{P_k\}}({\mathbb{S}}^1)\setminus\{P_k\}_{k\in\ovl{1,I}}\bigr)
\to\mathscr{I}_M:f\mapsto i_f\,$ and $\,\phi^{1,2}:\{P_k\}_{k\in\ovl{1,I}}
\to\mathscr{I}_Q:P_k\mapsto i^{1,2}_{P_k}\,$ of the assignment of indices
of the open covers $\,\mathcal{O}_M\,$ and $\,\mathcal{O}_Q\,$ to elements of
$\,\triangle_{\{P_k\}}({\mathbb{S}}^1)$.\ By varying these choices
arbitrarily, whereby an index set $\,\mathscr{I}_{\mathcal{O}_{{\mathsf L}_{Q|\{P_k
,\varepsilon_k\}}M}}\,$ is formed, all of $\,{\mathsf L}_{Q|\{(P_k,\varepsilon_k)\}} M\,$
is covered, thus yielding an \textbf{open cover $\,\mathcal{O}_{{\mathsf L}_{Q|\{
P_k,\varepsilon_k\}}M}=\{\mathcal{O}_{\{(P_k,\varepsilon_k)\}\,\gt{i}}\}_{\gt{i}\in
\mathscr{I}_{{\mathsf L}_{Q|\{P_k,\varepsilon_k\}}M}}\,$ of space $\,{\mathsf L}_{Q|\{(
P_k,\varepsilon_k)\}}M$}.
\end{Prop}
\begin{Rem}\label{rem:overlaps} It is straightforward to describe
intersections of elements of the open cover $\,\mathcal{O}_{{\mathsf L} M}$.\ In so
doing, we follow \Rcite{Gawedzki:1987ak} once more. Given a pair
$\,\mathcal{O}_{\gt{i}^n},\ n\in\{1,2\}\,$ with the respective triangulations
$\,\triangle_n({\mathbb{S}}^1)\,$ (consisting of edges $\,e_n\,$ and vertices
$\,v_n$) and index assignments $\,(e_n,v_n)\mapsto(
i^n_{e_n},i^n_{v_n})$,\ we consider the triangulation $\,\ovl
\triangle({\mathbb{S}}^1)\,$ obtained by intersecting $\,\triangle_1({\mathbb{S}}^1
)\,$ with $\,\triangle_2({\mathbb{S}}^1)$,\ by which we mean that the edges
$\,\ovl e\,$ of $\,\ovl\triangle({\mathbb{S}}^1)\,$ are the non-empty
intersections of the edges of the $\,\triangle_n({\mathbb{S}}^1)$,\ and its
vertices $\,\ovl v\,$ are taken from the set-theoretic sum of the
two vertex sets. As previously, the incoming (resp.\ outgoing) edge
of $\,\ovl\triangle({\mathbb{S}}^1)\,$ at the vertex $\,\ovl v\,$ is denoted
by $\,\ovl e_+(\ovl v)\,$ (resp.\ $\,\ovl e_-(\ovl v)$). A non-empty
double intersection $\,\mathcal{O}_{\gt{i}^1}\cap\mathcal{O}_{\gt{i}^2}=:\mathcal{O}_{\gt{i}^1
\gt{i}^n}\,$ is then labelled by the triangulation $\,\ovl\triangle(
{\mathbb{S}}^1)$,\ taken together with the indexing convention such that
$\,i^n_{\ovl e}\,$ is the \v Cech index assigned -- via $\,\gt{i}^n\,$
-- to the edge of $\,\triangle_n ({\mathbb{S}}^1)\,$ containing $\,\ovl e
\in\ovl\triangle({\mathbb{S}}^1)$,\ and $\,i^n_{\ovl v}\,$ is the \v Cech
index assigned -- via $\,\gt{i}^n\,$ -- to $\,\ovl v\,$ if $\,\ovl
v\in\triangle_n({\mathbb{S}}^1)$,\ or the \v Cech index assigned -- via
$\,\gt{i}^n\,$ -- to the edge of $\,\triangle_n({\mathbb{S}}^1)\,$ containing
$\,\ovl v\,$ otherwise. Analogous remarks apply to
$\,\mathcal{O}_{{\mathsf L}_{Q|\{(P_k,\varepsilon_k)\}} M}\,$.\end{Rem}\begin{center}*\hspace{2cm}*\hspace{2cm}*\end{center}
\noindent We have the fundamental result:
\begin{Thm}\cite{Gawedzki:1987ak}\label{thm:trans-untw}
Let $\,\mathscr{M}\,$ be a manifold with a gerbe $\,\mathcal{G}\,$ of curvature
$\,\curv(\mathcal{G})=:{\rm H}\,$ over it, and denote by
\begin{eqnarray}\nonumber
{\rm ev}_\mathscr{M}\ :\ {\mathsf L}\mathscr{M}\x{\mathbb{S}}^1\to\mathscr{M}
\end{eqnarray}
the canonical evaluation map for the free-loop space $\,{\mathsf L}\mathscr{M}=
C^\infty({\mathbb{S}}^1,M)\,$ of $\,\mathscr{M}$.\ The gerbe $\,\mathcal{G}\,$ canonically
defines a circle bundle $\,\pi_{\mathcal{L}_\mathcal{G}}:\mathcal{L}_\mathcal{G}\to{\mathsf L} M\,$ with
connection $\,\nabla_{\mathcal{L}_\mathcal{G}}\,$ of curvature
\begin{eqnarray}\nonumber
\curv(\nabla_{\mathcal{L}_\mathcal{G}})=\int_{{\mathbb{S}}^1}\,{\rm ev}_\mathscr{M}^*{\rm H}\,,
\end{eqnarray}
to be termed the \textbf{transgression bundle}. The assignment
$\,\mathcal{G} \to\mathcal{L}_\mathcal{G}\,$ yields a cohomology map\linebreak
$\,{\mathbb{H}}^2\bigl(\mathscr{M},\mathcal{D}(2
)^\bullet\bigr)\to{\mathbb{H}}^1\bigl({\mathsf L}\mathscr{M},\mathcal{D}(1)^\bullet\bigr)$,\ termed
the \textbf{transgression map}. \end{Thm} \noindent A constructive proof
can be found in the original paper. As the underlying idea shall
subsequently be extended to $\,{\mathsf L}_{Q|\{(P_k,\varepsilon_k)\}}M$,\ we
review it below. The transgression bundle $\,\mathcal{L}_{\mathcal{G}}\to{\mathsf L}
\mathscr{M}\,$ can be defined in terms of its local data $\,(E_\gt{i},G_{\gt{i}
\gt{j}})\in\cA^{2,1}(\mathcal{O}_{{\mathsf L}\mathscr{M}})\,$ associated with the open cover
$\,\mathcal{O}_{{\mathsf L}\mathscr{M}}\,$ from Proposition \ref{prop:cover-untw} and
determined by the local data of $\,\mathcal{G}$,\ as written out in
Definition \ref{def:loco}. Here, the connection 1-forms $\,E_\gt{i}\,$
are the 1-forms on $\,\mathcal{O}_\gt{i}\ni X\,$ given by the formul\ae
\begin{eqnarray}\nonumber
E_\gt{i}[X]=-\sum_{e\in\triangle({\mathbb{S}}^1)}\,\int_e\,X_e^*B_{i_e}-
\sum_{v\in\triangle({\mathbb{S}}^1)}\,X^*A_{i_{e_+(v)}i_{e_-(v)}}(v)\,,
\end{eqnarray}
where $\,e_+(v)\,$ and $\,e_-(v)\,$ denote the incoming and the
outgoing edge meeting at $\,v$,\ respectively, and where $\,X_e=
X\vert_e$.\ The transition functions $\,G_{\gt{i}\gt{j}}\,$ are the
${\rm U}(1)$-valued functionals on $\,\mathcal{O}_{\gt{i}\gt{j}}\ni X\,$ defined as
\begin{eqnarray}\nonumber
G_{\gt{i}\gt{j}}[X]=\prod_{\ovl e\in\ovl\triangle({\mathbb{S}}^1)}\,{\rm e}^{-{\mathsf i}\,
\int_{\ovl e}\,X_{\ovl e}^*A_{i_{\ovl e}j_{\ovl e}}}\,\prod_{\ovl v
\in\ovl\triangle({\mathbb{S}}^1)}\,X^*\bigl(g_{i_{\ovl e_+(\ovl v)}i_{\ovl
e_-(\ovl v)}j_{\ovl e_+(\ovl v)}}\cdot g^{-1}_{j_{\ovl e_+(\ovl v)}
j_{\ovl e_-(\ovl v)}i_{\ovl e_-(\ovl v)}}\bigr)(\ovl v)
\end{eqnarray}
in terms of edges $\,\ovl e\,$ and vertices $\,\ovl v\,$ of the
triangulation $\,\ovl\triangle({\mathbb{S}}^1)\,$ from Remark
\ref{rem:overlaps}, and satisfying the standard cohomological
identities
\begin{eqnarray}\nonumber
E_\gt{j}-E_\gt{i}={\mathsf i}\,\delta\log G_{\gt{i}\gt{j}}\,,\qquad\qquad G_{\gt{j}\gt{k}}
\cdot G_{\gt{i}\gt{k}}^{-1}\cdot G_{\gt{i}\gt{j}}=1\,.
\end{eqnarray}
Under gauge transformations \eqref{eq:gauge-trans-gerbe} of the
local data of $\,\mathcal{G}$,\ the local symplectic potentials undergo
induced gauge transformations
\begin{eqnarray}\label{eq:E-gauge}
E_\gt{i}\mapsto E_\gt{i}-{\mathsf i}\,\delta\log H_\gt{i}\,,
\end{eqnarray}
where
\begin{eqnarray}\nonumber
H_\gt{i}[X]=\prod_{e\in\triangle({\mathbb{S}}^1)}\,{\rm e}^{{\mathsf i}\,\int_e\,X_e^*
\Pi_{i_e}}\,\prod_{v\in\triangle({\mathbb{S}}^1)}\,X^*\chi_{i_{e_+(v)}i_{e_-
(v)}}^{-1}(v)\,.
\end{eqnarray}
The physical significance of the last proposition can be phrased as
\begin{Cor}\label{cor:preqb-untw}
Let $\,\gt{B}\,$ be a string background with target $\,\mathcal{M}=(M,{\rm g},\mathcal{G}
)$,\ and let $\,({\mathsf P}_{\sigma,\emptyset},\Omega_{\sigma,\emptyset})\,$ be the
untwisted state space of the non-linear $\sigma$-model for
network-field configurations $\,(X\,\vert\,\Gamma)\,$ in string
background $\,\gt{B}\,$ on world-sheet $\,(\Sigma,\gamma)\,$ with defect
quiver $\,\Gamma$.\ Denote by $\,\pi_{{\mathsf T}^*{\mathsf L} M}:{\mathsf T}^*{\mathsf L} M\to{\mathsf L}
M\,$ the canonical map from the total space of the cotangent bundle
over the free-loop space $\,{\mathsf L} M=C^\infty({\mathbb{S}}^1,M)\,$ of $\,M\,$
onto its base. The pre-quantum bundle $\,\pi_{\mathcal{L}_{\sigma,\emptyset}}:
\mathcal{L}_{\sigma,\emptyset}\to{\mathsf P}_{\sigma,\emptyset}\,$ for the untwisted
sector of the $\sigma$-model is the circle bundle
\begin{eqnarray}\nonumber
\mathcal{L}_{\sigma,\emptyset}:=\pi_{{\mathsf T}^*{\mathsf L} M}^*\mathcal{L}_\mathcal{G}\otimes\bigl({\mathsf T}^*
{\mathsf L} M\x{\mathbb{S}}^1\bigr)\to{\mathsf T}^*{\mathsf L} M\cong{\mathsf P}_{\sigma,\emptyset}
\end{eqnarray}
given by the tensor product of the pullback, along $\,\pi_{{\mathsf T}^*
{\mathsf L} M}$,\ of the transgression bundle $\,\mathcal{L}_\mathcal{G}\,$ of Theorem
\ref{thm:trans-untw} and of the trivial circle bundle $\,{\mathsf T}^*{\mathsf L}
M\x{\mathbb{S}}^1\to{\mathsf T}^*{\mathsf L} M\,$ with a global connection 1-form equal to
the canonical 1-form $\,\theta_{{\mathsf T}^*{\mathsf L} M}\,$ on $\,{\mathsf T}^*{\mathsf L}
M$,\ explicited in Proposition \ref{prop:sympl-form-si-untw}. In
particular, given the open cover $\,\mathcal{O}_{{\mathsf L} M}=\{\mathcal{O}_\gt{i}\}_{\gt{i}
\in\mathscr{I}_{{\mathsf L} M}}\,$ of $\,{\mathsf L} M\,$ defined in Proposition
\ref{prop:cover-untw}, local data of $\,\mathcal{L}_{\sigma,\emptyset}\,$
associated with the induced open cover $\,\mathcal{O}_{{\mathsf P}_{\sigma,
\emptyset}}=\{\mathcal{O}^*_\gt{i}\}_{\gt{i}\in \mathscr{I}_{{\mathsf L} M}},\ \mathcal{O}^*_\gt{i}:=
\pi_{{\mathsf T}^*{\mathsf L} M}^{-1}(\mathcal{O}_\gt{i})\,$ of $\,{\mathsf P}_{\sigma,\emptyset}\,$
can be expressed in terms of the local data
$\,(E_\gt{i},G_{\gt{i}\gt{j}})\,$ of the bundle $\,\mathcal{L}_\mathcal{G}\,$ from
Theorem \ref{thm:trans-untw} as
\begin{eqnarray}\label{eq:loc-dat-preq-untw}
\theta_{\sigma,\emptyset\,\gt{i}}=\theta_{{\mathsf T}^*{\mathsf L}
M}\vert_{\mathcal{O}^*_\gt{i}}+\pi_{{\mathsf T}^*{\mathsf L}
M}^*E_\gt{i}\,,\qquad\qquad\gamma_{\sigma,\emptyset\,\gt{i}\gt{j}}=\pi_{{\mathsf T}^*
{\mathsf L} M}^*G_{\gt{i}\gt{j}}\,.
\end{eqnarray}
\end{Cor}
\medskip
\noindent Gaw\c{e}dzki's construction is readily verified to
generalise to the twisted case.
\begin{Thm}\label{thm:trans-tw}
Let $\,\gt{B}\,$ be a string background with target $\,\mathcal{M}=(M,{\rm g},\mathcal{G}
)\,$ and $\mathcal{G}$-bi-brane $\,\mathcal{B}=(Q,\iota_\a,\omega,\Phi\ \vert\ \a\in\{1
,2\})$,\ and let $\,{\mathsf L}_{Q|\{(P_k,\varepsilon_k)\}}M\,$ be the space
introduced in Definition \ref{def:tw-phspace}. Write $\,{\mathbb{S}}^1_{\{P_k
\}}={\mathbb{S}}^1\setminus\{P_k\}_{k\in\ovl{1,I}}\,$ and denote by
\begin{eqnarray}\nonumber
{\rm ev}_{M,\{P_k\}}\ :\ C^\infty({\mathbb{S}}^1_{\{P_k\}},M)\x{\mathbb{S}}^1_{\{P_k\}}\to
M
\end{eqnarray}
the canonical evaluation map. The pair $\,(\mathcal{G},\mathcal{B})\,$ canonically
defines a circle bundle $\,\pi_{\mathcal{L}_{(\mathcal{G},\mathcal{B})|\{(P_k,\varepsilon_k)\}}}:
\mathcal{L}_{(\mathcal{G},\mathcal{B})|\{(P_k,\varepsilon_k)\}}\to{\mathsf L}_{Q|\{(P_k,\varepsilon_k)\}}M\,$
with connection $\,\nabla_{\mathcal{L}_{(\mathcal{G},\mathcal{B})|\{(P_k,\varepsilon_k)\}}}\,$ of
curvature
\begin{eqnarray}\nonumber
\curv(\nabla_{\mathcal{L}_{(\mathcal{G},\mathcal{B})|\{(P_k,\varepsilon_k)\}}})=\pi_{\mathcal{L}_{(\mathcal{G},
\mathcal{B})|\{(P_k,\varepsilon_k)\}}}^*\bigl({\rm pr}_{C^\infty({\mathbb{S}}^1_{\{P_k\}},M)}^*
\int_{{\mathbb{S}}^1_{\{P_k\}}}\,{\rm ev}_{M,\{P_k\}}^*{\rm H}+\sum_{k=1}^I\,\varepsilon_k
\,{\rm pr}_{Q,k}^*\omega\bigr)\,,
\end{eqnarray}
written in terms of the canonical projections $\,{\rm pr}_{C^\infty(
{\mathbb{S}}^1_{\{P_k\}},M)}:{\mathsf L}_{Q|\{(P_k, \varepsilon_k)\}}M\to C^\infty(
{\mathbb{S}}^1_{\{P_k\}},M)\,$ and $\,{\rm pr}_{Q,k}:{\mathsf L}_{Q|\{(P_k,\varepsilon_k)\}}M
\to Q$,\ the latter having the $k$-th cartesian factor as the
codomain. Under the assignment
$\,(\mathcal{G},\mathcal{B})\to\mathcal{L}_{(\mathcal{G},\mathcal{B})|\{(P_k,\varepsilon_k)\}}$,\
(gauge-)equivalence classes of pairs $\,(\mathcal{G},\mathcal{B})$,\ as described in
Definition \ref{def:loco}, are mapped to isomorphism classes of
bundles with connection. \end{Thm}
\noindent\begin{proof}
We define $\,\mathcal{L}_{(\mathcal{G},\mathcal{B})|\{(P_k,\varepsilon_k)\}}\,$ explicitly in
terms of its local data $\,(E_{\{(P_k,\varepsilon_k)\}\,\gt{i}},G_{\{(P_k,
\varepsilon_k)\}\,\gt{i}\gt{j}})\in\cA^{2,1}(\mathcal{O}_{{\mathsf L}_{Q|\{(P_k,\varepsilon_k)\}}M})
\,$ associated with the open cover $\,\mathcal{O}_{{\mathsf L}_{Q|\{(P_k,\varepsilon_k)\}}
M}\,$ from Proposition \ref{prop:cover-tw} and determined by local
data of $\,(\mathcal{G},\mathcal{B})$,\ as written out in Definition \ref{def:loco}.
Write $\,(X^\mu,q_k\ \vert\ k\in\ovl{1,I})\equiv(X,\{q_k\})$.\ It is
easy to check that the objects
\begin{eqnarray}\nonumber
E_{\{(P_k,\varepsilon_k)\}\,\gt{i}}[(X,\{q_k\})]&=&-\sum_{e\in\triangle_{\{
P_k\}}({\mathbb{S}}^1)}\,\int_e\,X_e^*B_{i_e}-\sum_{v\in\triangle_{\{P_k\}}(
{\mathbb{S}}^1)\setminus\{P_k\}_{k\in\ovl{1,I}}}\,X^*A_{i_{e_+(v)}i_{e_-(v
)}}(v)\cr\cr
&&+\sum_{k=1}^I\,\varepsilon_k\,\bigl(\iota_1^*A_{i_{e_{\widetilde\sigma_k}(
P_k)}\phi_1(i^{1,2}_{P_k})}-\iota_2^*A_{i_{e_{\sigma_k}(P_k)}\phi_2(
i^{1,2}_{P_k})}+P_{i^{1,2}_{P_k}}\bigr)(q_k)\,,\cr\cr\cr
G_{\{(P_k,\varepsilon_k)\}\,\gt{i}\gt{j}}[(X,\{q_k\})]&=&\prod_{\ovl e\in\ovl
\triangle_{\{P_k\}}({\mathbb{S}}^1)}\,{\rm e}^{-{\mathsf i}\,\int_{\ovl e}\,X_{\ovl e}^*
A_{i_{\ovl e}j_{\ovl e}}}\cr\cr
&&\cdot\prod_{\ovl v\in\ovl\triangle_{\{P_k\}}({\mathbb{S}}^1)\setminus\{P_k
\}_{k\in\ovl{1,I}}}\,X^*\bigl(g_{i_{\ovl e_+(\ovl v)}i_{\ovl e_-(
\ovl v)}j_{\ovl e_+(\ovl v)}}\cdot g^{-1}_{j_{\ovl e_+(\ovl v)}
j_{\ovl e_-(\ovl v)}i_{\ovl e_-(\ovl v)}}\bigr)(\ovl v)\cr\cr
&&\cdot\prod_{k=1}^I\,\bigl[\iota_1^*\bigl(g_{i_{\ovl e_{\widetilde
\sigma_k}(P_k)}j_{\ovl e_{\widetilde\sigma_k}(P_k)}\phi_1(i^{1,2}_{P_k})}
\cdot g^{-1}_{j_{\ovl e_{\widetilde\sigma_k}(P_k)}\phi_1(i^{1,2}_{P_k})
\phi_1(j^{1,2}_{P_k})}\bigr)\cr\cr
&&\hspace{.7cm}\cdot\iota_2^*\bigl(g^{-1}_{i_{\ovl e_{\sigma_k}(P_k)}
j_{\ovl e_{\sigma_k}(P_k)}\phi_2(i^{1,2}_{P_k})}\cdot g_{j_{\ovl
e_{\sigma_k}(P_k)}\phi_2(i^{1,2}_{P_k})\phi_2(j^{1,2}_{P_k})}\bigr)
\cdot K_{i^{1,2}_{P_k}j^{1,2}_{P_k}}\bigr]^{\varepsilon_k}(q_k)\,,
\end{eqnarray}
written in terms of $\,(\sigma_k,\widetilde\sigma_k)=(+,-)\,$ if $\,\varepsilon_k
=+1$,\ and $\,(\sigma_k,\widetilde\sigma_k)=(-,+)\,$ otherwise, obey the
required cohomological constraints. Similarly, one verifies through
inspection that under a gauge transformation
\eqref{eq:gauge-trans-gerbe} of the local data of $\,\mathcal{G}$,\
accompanied by the $\mathcal{G}$-twisted gauge transformation
\eqref{eq:gauge-trans-bi} of the local data of the $\mathcal{G}$-bi-brane
1-isomorphism, the pair $\,(E_{\{(P_k,\varepsilon_k)\}\,\gt{i}},G_{\{(P_k,
\varepsilon_k)\}\,\gt{i}\gt{j}})\,$ undergoes induced gauge transformation
\begin{eqnarray}\label{eq:tw-preq-si-gauge}
(E_{\{(P_k,\varepsilon_k)\}\,\gt{i}},G_{\{(P_k,\varepsilon_k)\}\,\gt{i}\gt{j}})\mapsto
\bigr(E_{\{(P_k,\varepsilon_k)\}\,\gt{i}},G_{\{(P_k,\varepsilon_k)\}\,\gt{i}\gt{j}})+
D_{(0)}(H_{\{(P_k,\varepsilon_k)\}\,\gt{i}})\,,
\end{eqnarray}
with
\begin{eqnarray}\nonumber
H_{\{(P_k,\varepsilon_k)\}\,\gt{i}}[(X,\{q_k\})]&=&\prod_{e\in\triangle_{\{
P_k\}}({\mathbb{S}}^1)}\,{\rm e}^{{\mathsf i}\,\int_e\,X_e^*\Pi_{i_e}}\,\prod_{v\in
\triangle_{\{P_k\}}({\mathbb{S}}^1)\setminus\{P_k\}_{k\in\ovl{1,I}}}\,X^*
\chi_{i_{e_+(v)}i_{e_-(v)}}^{-1}(v)\cr\cr
&&\cdot\prod_{k=1}^I\,\bigl(\iota_1^*\chi_{i_{e_{\widetilde\sigma_k}(
P_k)}\phi_1(i^{1,2}_{P_k})}\cdot\iota_2^*\chi^{-1}_{i_{e_{\sigma_k}(
P_k)}\phi_2(i^{1,2}_{P_k})}\cdot W^{-1}_{i^{1,2}_{P_k}}
\bigr)^{\varepsilon_k}(q_k)\,.
\end{eqnarray}
\qed\end{proof} \noindent The physical content of the above result is
summarised in
\begin{Cor}\label{cor:preqb-tw}
Let $\,\gt{B}\,$ be a string background with target $\,\mathcal{M}=(M,{\rm g},\mathcal{G}
)\,$ and $\mathcal{G}$-bi-brane $\,\mathcal{B}=(Q,\iota_\a,\omega,\Phi\ \vert\ \a\in\{1
,2\})$,\ and let $\,({\mathsf P}_{\sigma,\mathcal{B}|\{(P_k,\varepsilon_k)\}},\Omega_{\sigma,\mathcal{B}|\{
(P_k,\varepsilon_k)\}})\,$ be the $k$-twisted state space of the non-linear
$\sigma$-model for network-field configurations $\,(X\,\vert\,\Gamma)\,$ in
string background $\,\gt{B}\,$ on world-sheet $\,(\Sigma,\gamma)\,$ with
defect quiver $\,\Gamma$.\ Denote by $\,{\rm pr}_{{\mathsf L}_{Q|\{(P_k,\varepsilon_k)\}}
M}:{\mathsf P}_{\sigma,\mathcal{B}|\{(P_k,\varepsilon_k)\}}\to{\mathsf L}_{Q|\{(P_k,\varepsilon_k)\}}M\,$
the canonical projection from the $k$-twisted state space to the
space $\,{\mathsf L}_{Q|\{(P_k,\varepsilon_k)\}}M\,$ from Definition
\ref{def:tw-phspace}. The pre-quantum bundle $\,\pi_{\mathcal{L}_{\sigma,\mathcal{B}|
\{(P_k,\varepsilon_k)\}}}:\mathcal{L}_{\sigma,\mathcal{B}|\{(P_k,\varepsilon_k)\}}\to{\mathsf P}_{\sigma,\mathcal{B}|
\{(P_k,\varepsilon_k)\}}\,$ for the twisted sector of the $\sigma$-model is
the circle bundle
\begin{eqnarray}\nonumber
\mathcal{L}_{\sigma,\mathcal{B}|\{(P_k,\varepsilon_k)\}}:={\rm pr}_{{\mathsf L}_{Q|\{(P_k,\varepsilon_k)\}}M}^*
\mathcal{L}_{(\mathcal{G},\mathcal{B})|\{(P_k,\varepsilon_k)\}}\otimes\bigl({\mathsf P}_{\sigma,\mathcal{B}|\{(P_k,
\varepsilon_k)\}}\x{\mathbb{S}}^1\bigr)\to {\mathsf P}_{\sigma,\mathcal{B}|\{(P_k,\varepsilon_k)\}}
\end{eqnarray}
given by the tensor product of the pullback, along $\,{\rm pr}_{{\mathsf L}_{Q|
\{(P_k,\varepsilon_k)\}}M}$,\ of the bundle $\,\mathcal{L}_{(\mathcal{G},\mathcal{B})|\{(P_k,
\varepsilon_k)\}}\,$ of Theorem \ref{thm:trans-tw}, and of the trivial
circle bundle $\,{\mathsf P}_{\sigma,\mathcal{B}|\{(P_k,\varepsilon_k)\}}\x{\mathbb{S}}^1\to
{\mathsf P}_{\sigma,\mathcal{B}|\{(P_k,\varepsilon_k)\}}\,$ with a global connection 1-form
equal to the pullback, along the canonical projection $\,{\rm pr}_{{\mathsf T}^*
C^\infty({\mathbb{S}}^1_{\{P_k\}},M)}:{\mathsf P}_{\sigma,\mathcal{B}|\{(P_k,\varepsilon_k)\}}\to
{\mathsf T}^*C^\infty({\mathbb{S}}^1_{\{P_k\}},M)$,\ of the canonical 1-form
$\,\theta_{{\mathsf T}^*C^\infty({\mathbb{S}}^1_{\{P_k\}},M)}\,$ on $\,{\mathsf T}^*
C^\infty({\mathbb{S}}^1_{\{P_k\}},M)$,\ explicited in Proposition
\ref{prop:sympl-form-si-tw}. In particular, given the open cover
$\,\mathcal{O}_{{\mathsf L}_{Q\vert\{(P_k,\varepsilon_k)\}}M}=\{\mathcal{O}_{\{(P_k,\varepsilon_k)\}\,
\gt{i}}\}_{\gt{i}\in\mathscr{I}_{{\mathsf L}_{Q\vert\{(P_k,\varepsilon_k)\}}M}}\,$ of
\linebreak $\,{\mathsf L}_{Q\vert\{(P_k,\varepsilon_k)\}}M\,$ defined in
Proposition \ref{prop:cover-tw}, local data of
$\,\mathcal{L}_{\sigma,\mathcal{B}|\{(P_k,\varepsilon_k) \}}\,$ associated with the induced
open cover $\,\mathcal{O}_{{\mathsf P}_{\sigma,\mathcal{B}|
\{(P_k,\varepsilon_k)\}}}=\{{\rm pr}_{{\mathsf L}_{Q|\{(P_k,\varepsilon_k)\}}M}^{-1}(\mathcal{O}_{\{(
P_k,\varepsilon_k)\}\,\gt{i}})\}_{\gt{i}\in\mathscr{I}_{{\mathsf L}_{Q\vert\{(P_k,\varepsilon_k)\}}
M}}\,$ of $\,{\mathsf P}_{\sigma,\mathcal{B}|\{(P_k,\varepsilon_k)\}}\,$ can be expressed in
terms of the local data $\,(E_{\{(P_k,\varepsilon_k)\}\,\gt{i}},G_{\{(P_k,
\varepsilon_k)\}\,\gt{i}\gt{j}})\,$ of the bundle $\,\mathcal{L}_{(\mathcal{G},\mathcal{B})\vert\{(P_k
,\varepsilon_k)\}}\,$ from Theorem \ref{thm:trans-tw} as
\begin{eqnarray}
\theta_{\sigma,\mathcal{B}\vert\{(P_k,\varepsilon_k)\}\,\gt{i}}&=&{\rm pr}_{{\mathsf T}^*C^\infty(
{\mathbb{S}}^1_{\{P_k\}},M)}^*\theta_{{\mathsf T}^*C^\infty({\mathbb{S}}^1_{\{P_k\}},M)}+
{\rm pr}_{{\mathsf L}_{Q|\{(P_k,\varepsilon_k)\}}M}^*E_\gt{i}\,,\cr
\label{eq:loc-dat-preq-tw}\\
\gamma_{\sigma,\mathcal{B}\vert\{(p_k,\varepsilon_k)\}\,\gt{i}\gt{j}}&=&{\rm pr}_{{\mathsf L}_{Q|\{(P_k,
\varepsilon_k)\}}M}^*G_{\gt{i}\gt{j}}\,.\nonumber
\end{eqnarray}
\end{Cor}
\bigskip
Prior to passing to the subsequent sections, in which we exploit the
knowledge, gained heretofore, of the canonical and pre-quantum
structure on the space of states of the $\sigma$-model in the presence
of defects, we pause to briefly discuss a simple application of our
results, of particular relevance to the study of the concept of
`emergent geometry of string theory'. \begin{Rem}\label{rem:NCG}
\emph{\textbf{The non-commutative geometry of the bi-brane
world-volume.}} The presence of the defect-line contributions
\begin{eqnarray}\nonumber
\omega(q_k)=\omega_{AB}(q_k)\,\delta X^A\wedge\delta X^B\,,\qquad q_k\in Q
\end{eqnarray}
in \Reqref{eq:Omsi-tw} is a clear-cut indication that the
quantisation of the defect $\sigma$-model yields a non-commu\-ta\-tive
deformation of the algebra of functions on the bi-brane
world-volume, the latter being generated by the coordinate functions
$\,X^A$.\ This is a bi-brane variant of the long-known phenomenon of
the (gerbe-induced) non-commutativity of the D-brane geometry in the
so-called `stringy r\'egime', first discussed\footnote{For an
earlier account of the phenomenon, exhibiting its rich mathematical
structure in the \emph{closed}-string sector, cf.\
\Rcite{Frohlich:1993es} and Refs.\
\cite{Frohlich:1995mr,Frohlich:1996zc,Frohlich:1998zm}. In the more
recent \Rcite{Recknagel:2006hp}, the gerbe-related description of
the open string in the WZW model was worked out along the lines of
these earlier papers.} in \Rcite{Douglas:1997fm}. The actual form of
the non-commutativity of the quantum position operators depends
strongly on the choice of the quantisation scheme, as indicated in
\Rcite{Seiberg:1999vs}. However, under certain circumstances, one
can get some insight into the matter already on the (semi)classical
level. Indeed, assume that the string background $\,\mathcal{M}\,$ comes
with a small dimensionless parameter $\,\epsilon\,$ (derived, e.g., from
a common length scale for $\,M\,$ and $\,Q$), and that there exists
a geometric r\'egime, to be referred to as a \textbf{decoupling
r\'egime} (e.g., a vicinity of a distinguished point in the large
target space)
\begin{eqnarray}\nonumber
X^\mu,X^A=O(\epsilon^{d_X})\,,\qquad d_X\in{\mathbb{N}}
\end{eqnarray}
in which the target-space metric behaves as
\begin{eqnarray}\nonumber
{\rm g}=O(\epsilon^{d_{\rm g}})\,,\qquad d_{\rm g}\in{\mathbb{N}}\,.
\end{eqnarray}
The condition of a vanishing Weyl anomaly, mentioned in Remark
\ref{rem:Weyl-anom}, then fixes the scaling behaviour of the gerbe
curvature to be
\begin{eqnarray}\nonumber
{\rm H}=O(\epsilon^{d_{\rm H}})\,,\qquad d_{\rm H}\geq d_{\rm g}
\end{eqnarray}
(consistently with the field equations \eqref{eq:field-eqs}), and so
the sum of the first two terms in $\,\Omega_{\sigma,\mathcal{B}|\{(P_k,\varepsilon_k
)\}}\,$ scales as $\,O(\epsilon^{d_{\rm g}})$.\ The bi-brane curvature, on
the other hand, need not decrease at the same rate since the
relation
\begin{eqnarray}\label{eq:curv-constr}
\iota_1^*{\rm H}-\iota_2^*{\rm H}={\mathsf d}\omega
\end{eqnarray}
that follows from Eqs.\,\eqref{eq:DG-is-H}-\eqref{eq:DPhi-is} and
determines the scaling behaviour of the defect-line terms in the
symplectic structure contains the difference of the pullbacks along
the two bi-brane maps, which may affect the value of the critical
exponent $\,d_\omega\,$ in
\begin{eqnarray}\nonumber
\omega=O(\epsilon^{d_\omega})\,.
\end{eqnarray}
Thus, whenever $\,d_{\rm g}-d_\omega>0$,\ we may, in the decoupling
r\'egime described, approximate the symplectic structure as
\begin{eqnarray}\nonumber
\Omega_{\sigma,\mathcal{B}|\{(q_k,\varepsilon_k)\}}=\sum_{k=1}^I\,\varepsilon_k\,{\rm pr}_{Q,k}^*\omega
\bigl(1+O\bigl(\epsilon^{d_{\rm g}-d_\omega}\bigr)\bigr)\,.
\end{eqnarray}
From now onwards, we restrict our attention to the case of $\,k=1$,\
with $\,P_1=P,\ q_1=q\,$ and $\,\varepsilon_1=+1$.
If the bi-brane curvature is invertible in the decoupling r\'egime,
with the inverse defining a\linebreak Poisson bivector
\begin{eqnarray}\nonumber
\Pi=\Pi^{AB}\,\p_A\wedge\p_B\,,\qquad\qquad\Pi^{AB}=-\tfrac{1}{4}\,
\bigl(\omega^{-1}\bigr)^{AB}\,,
\end{eqnarray}
we find natural defect observables $\,X^A(P)\equiv X^A\,$
represented by the hamiltonian vector fields
\begin{eqnarray}\nonumber
\mathscr{X}_{X^A}=2\Pi^{AB}(X)\,\tfrac{\delta \ \ }{\delta X^B} \,.
\end{eqnarray}
Canonical quantisation of their Poisson bracket
\begin{eqnarray}\nonumber
\{X^A,X^B\}_{\Omega_{\sigma,\mathcal{B}|\{(P,+1)\}}}=2\Pi^{AB}(X)
\end{eqnarray}
gives a non-commutative algebra of stringy coordinates on $\,Q$,\ as
claimed. By the usual argument, the closedness of $\,\omega\,$ ensures
the vanishing of the jacobiator of the Poisson bracket thus defined,
necessary for the associativity of the non-commutative deformation
of the algebra of functions on the bi-brane.\medskip
By way of illustration of the general phenomenon, we treat in some
detail the case of the maximally symmetric WZW $\mathcal{G}_\sfk$-bi-branes from
Example \ref{eg:WZW-def}. The WZW model for a compact Lie group
$\,{\rm G}\,$ comes with a natural parameter $\,\epsilon_\sfk=\frac{1}{{\mathsf k}}\,$
that sets -- through the Cartan--Killing metric $\,{\rm g}_{\mathsf k}\,$ (and
in conjunction with the string tension, which we suppressed in the
present notation) -- the characteristic length scale of the string
background and plays the r\^ole of Planck's constant $\,\hbar\,$ and
a parameter of the non-commutative deformation of the algebra of
functions on $\,{\rm G}\,$ (or a submanifold thereof) determined by the
operator content of the quantised $\sigma$-model, cf., e.g.,
Refs.\,\cite{Frohlich:1993es} and \cite{Recknagel:2006hp}. Here,
very large but finite values of $\,{\mathsf k}\,$ give access to a
semiclassical approximation of the quantum geometry of the WZW
string\footnote{For a proposal of an algebraic description of that
geometry in the quantum r\'egime, consistent with the quantum-group
symmetries of the rational conformal field theory of the WZW model,
cf.\
Refs.\,\cite{Pawelczyk:2001zi,Pawelczyk:2002kd,Pawelczyk:2003nb,Pawelczyk:2005ye}.}.
More specifically, let us write elements of $\,{\rm G}$,\ and hence
also fields of the WZW model, in terms of the canonical (Riemann
normal) coordinates $\,X^A=\epsilon_\sfk\,\widetilde X_A\,$ on the group
manifold, suitably rescaled, to wit,
\begin{eqnarray}\nonumber
g={\rm e}^{\epsilon_\sfk\,\widetilde X^A\,t_A}\,,
\end{eqnarray}
and subsequently pass to the decoupling r\'egime
\begin{eqnarray}\nonumber
\widetilde X^A=O(1)\,,\qquad\qquad\epsilon_\sfk\ll 1
\end{eqnarray}
of \Rcite{Alekseev:1999bs}. Having thus restricted our analysis to
world-sheets embedded in an immediate vicinity of the group unit in
a large group manifold, we readily establish the equalities
\begin{eqnarray}\nonumber
d_X=1\,,\qquad\qquad d_{{\rm g}_{\mathsf k}}=1\,,\quad\qquad d_{{\rm H}_{\mathsf k}}=2
\,,
\end{eqnarray}
and -- for $\,\lambda\in\faff{\gt{g}}\,$ small and for values of the
$\mathcal{G}_\sfk$-bi-brane field $\,(g,h_\lambda)\,$ restricted to a small
neighbourhood of $\,(e,e)\in{\rm G}\x{\rm G}\,$ --
\begin{eqnarray}\nonumber
\omega^\p_{{\mathsf k},\lambda}=O(1)\,,\qquad\qquad\omega_{{\mathsf k},\lambda}=-{\rm pr}_2^*
\omega^\p_{{\mathsf k},\lambda}+O(\epsilon_\sfk)\,.
\end{eqnarray}
This follows straightforwardly from the relations
\begin{eqnarray}\nonumber
\theta_L=O(\epsilon_\sfk)\,,\qquad\qquad{\rm id}_\gt{g}-{\rm Ad}_g=-\epsilon_\sfk\,
{\rm ad}_{\widetilde X}+O(\epsilon_\sfk^2)\,.
\end{eqnarray}
Consequently, the deformation of the commutative algebra of
functions on both the boundary and the non-boundary maximally
symmetric WZW $\mathcal{G}_\sfk$-bi-brane, as encoded by the decoupling limit of
the respective symplectic forms, is determined by the properties of
the 2-form
\begin{eqnarray}\nonumber
\omega_{{\mathsf k},\lambda}^\p(\widetilde X)=-\tfrac{1}{8\pi}\,{\rm tr}_\gt{g}\bigl(
t_A\,{\rm ad}^{-1}_{\widetilde X}t_B\bigr)\,\delta\widetilde X^A\wedge\delta
\widetilde X^B+O(\epsilon_\sfk)\,,\qquad\widetilde X=\widetilde X^A\,t_A
\end{eqnarray}
which coincides with the Kirillov--Kostant--Souriau symplectic form
on the coadjoint orbit $\,\mathscr{O}_\lambda\cong{\rm G}/{\rm G}_\lambda\,$ (for
$\,{\rm G}_\lambda\,$ the ${\rm Ad}_\cdot$-stabiliser of $\,\lambda\,$ in $\,{\rm G}$)
of Refs.\,\cite{Kostant:1970,Souriau:1970,Kirillov:1975}.
Equivalently, the deformation is characterised by the associated
Poisson bivector
\begin{eqnarray}\nonumber
\Pi^\p_{{\mathsf k},\lambda}(\widetilde X)=4\pi\,f_{ABC}\,\widetilde X^A\,
\tfrac{\delta\ \ }{\delta\widetilde X^B}\wedge\tfrac{\delta\ \ }{\delta\widetilde
X^C}+O(\epsilon_\sfk)\,.
\end{eqnarray}
Taking the latter as the germ of a deformation quantisation of the
smooth geometry of conjugacy classes $\,\mathscr{C}_\lambda\,$ close to the
group unit leads to the emergence of the so-called fuzzy conjugacy
classes, analogous to the fuzzy sphere of
Refs.\,\cite{Hoppe:1982PhD,Madore:1992}. These are precisely the
non-commutative geometries emerging from perturbative calculations
of the quantised WZW model. They were first explored in the present
context in
Refs.\,\cite{Alekseev:1999bs,Alekseev:2000fd,Alekseev:2000wg}, cf.\
also \Rcite{Recknagel:2006hp} for a gerbe-related discussion based
on the spectral data of the supersymmetric extension of the boundary
WZW model. \end{Rem}\begin{center}*\hspace{2cm}*\hspace{2cm}*\end{center}
\section{State-space isotropics from defects, and $\sigma$-model
dualities}\label{sec:def-as-iso}
Now that we have developed a symplectic formalism for the
description of the two-dimensional field theory in hand, we may
apply it to study defects. Thus, motivated by the discussion,
presented in
Refs.\,\cite{Frohlich:2004ef,Frohlich:2006ch,Bachas:2008jd}, of the
r\^ole that defects play in mediating dualities of the underlying
two-dimensional field theory, and also by the study of chosen
examples of dualities in the gerbe-theoretic context in
Refs.\,\cite{Schweigert:2007wd,Runkel:2008gr,Sarkissian:2008dq}, we
seek to establish an appropriate rigorous result within the
canonical framework, to wit, we want to describe the symplectic
relations within the untwisted state space $\,({\mathsf P}_{\sigma,\emptyset}=
{\mathsf T}^*{\mathsf L} M,\Omega_{\sigma,\emptyset})\,$ of the $\sigma$-model inhabiting
two adjacent patches $\,\wp_\a,\ \a\in\{1,2\}\,$ separated by a
space-like connected component $\,\ell\cong{\mathbb{S}}^1\,$ of the defect
quiver $\,\Gamma\,$ that are induced by the intermediary $\mathcal{G}$-bi-brane
structure $\,\mathcal{B}$.\ To this end, using independence of $\,\Omega_{\sigma,
\emptyset}\,$ over each of the two patches of the choice of the
Cauchy contour used to define it, we push the respective Cauchy
contours, $\,C_1\,$ and $\,C_2$,\ to $\,\ell$.\ Following this
simple prescription, we find
\begin{Prop}\label{prop:DGC-as-iso}
Let $\,\gt{B}\,$ be a string background with target $\,\mathcal{M}=(M,{\rm g},\mathcal{G}
)\,$ and $\mathcal{G}$-bi-brane $\,\mathcal{B}=(Q,\iota_\a,\omega,\Phi\ \vert\ \a\in\{
1,2\})$,\ and let $\,({\mathsf P}_{\sigma,\emptyset}={\mathsf T}^*{\mathsf L}
M,\Omega_{\sigma,\emptyset})\,$ be the untwisted state space of the
non-linear $\sigma$-model for network-field configurations
$\,(X\,\vert\,\Gamma)\,$ in string background $\,\gt{B}\,$ on world-sheet
$\,(\Sigma,\gamma)\,$ with defect quiver $\,\Gamma$.\ Consider the symplectic
structure on the product space $\,{\mathsf P}_{\sigma,\emptyset}\x{\mathsf P}_{\sigma,
\emptyset}\equiv{\mathsf P}_{\sigma,\emptyset}^{\x 2}\,$ determined by the
`difference' symplectic form
\begin{eqnarray}\nonumber
\Omega_{\sigma,\emptyset}^-={\rm pr}_1^*\Omega_{\sigma,\emptyset}-{\rm pr}_2^*\Omega_{\sigma,
\emptyset}\,.
\end{eqnarray}
The $\mathcal{G}$-bi-brane $\,\mathcal{B}\,$ together with the Defect Gluing
Condition \eqref{eq:DGC} canonically defines an isotropic
submanifold in $\,({\mathsf P}_{\sigma,\emptyset}^{\x 2},\Omega_{\sigma,
\emptyset}^-)$,\ given by
\begin{eqnarray}\nonumber
\gt{I}_\sigma(\mathcal{B})&=&\{\ (\psi_1,\psi_2)\in{\mathsf P}_\sigma^{\x 2}\,,\quad
\psi_\a=( X_\a,{\mathsf p}_\a)\,,\ \a\in\{1,2\} \quad\vert\quad (X_1,X_2)
\in(\iota_1 \x\iota_2)({\mathsf L} Q)\cr\cr
&&\hspace{.5cm}\land\quad\exists_{X\in(\iota_1\x\iota_2)^{-1}\{(X_1,
X_2)\}}\ :\ {\rm DGC}_\mathcal{B} (\psi_1,\psi_2,X)=0 \ \}
\end{eqnarray}
in terms of Cauchy data $\,\psi_\a$.\ The latter subspace is a
fibration over the free-loop space $\,{\mathsf L} Q$,\ and we shall
identify it with the corresponding subspace in $\,{\mathsf P}_{\sigma,
\emptyset}^{\x 2}\x{\mathsf L} Q\,$ in what follows.
\end{Prop}
\noindent\begin{proof}
Take a pair of states $\,(\psi_1,\psi_2)\in\gt{I}_\sigma(\mathcal{B})\,$ with
$\,(X_1,X_2)=\bigl(\iota_1(X),\iota_2(X)\bigr)$,\ satisfying the DGC
\eqref{eq:DGC}, i.e.
\begin{eqnarray}\label{eq:DGC-coord}
{\mathsf p}_{2\,\mu}\,\mathscr{W}^\mu_2-{\mathsf p}_{1\,\mu}\,\mathscr{W}^\mu_1=-2(X_*\widehat t
)^A\,\omega_{AB}\,\mathscr{W}^B\,,
\end{eqnarray}
for $\,\mathscr{W}=\mathscr{W}^A\,\tfrac{\delta\ \ }{\delta X^A}\,$ an arbitrary vector
field from $\,\Gamma({\mathsf T} Q)\,$ restricted to $\,X$,\ the latter being
modelled on $\,{\mathbb{S}}^1\,$ with a normalised tangent vector field
$\,\widehat t$,\ and for $\,\mathscr{W}_\a=\iota_{\a\,*}\mathscr{W}$.\ We want to
consider the distinguished subspace $\,\gt{T}\gt{I}_\sigma(\mathcal{B})\,$ of
$\,\Gamma({\mathsf T}{\mathsf P}_\sigma^{\x 2})\,$ over $\,\gt{I}_\sigma(\mathcal{B})\,$ spanned by
vector fields
\begin{eqnarray}\nonumber
\widetilde\mathscr{V}=\widetilde\mathscr{V}_1\oplus\widetilde\mathscr{V}_2\,,\qquad
\qquad\widetilde\mathscr{V}_\a=\mathscr{V}_\a^\mu\,\tfrac{\delta\ \ }{\delta X_\a^\mu}+
\mathscr{P}_{\a\,\mu}\,\tfrac{\delta\ \ }{\delta{\mathsf p}_{\a\,\mu}}
\end{eqnarray}
describing tangential deformations $\,\mathscr{V}_\a\equiv\mathscr{V}_\a^\mu\,
\frac{\delta\ \ }{\delta X_\a^\mu}=\iota_{\a\,*}(\mathscr{V}^A\,\frac{\delta\ \ }{\delta
X^A})\,$ of the $\,X_\a \,$ induced by deformations $\,\mathscr{V}=\mathscr{V}^A\,
\frac{\delta\ \ }{\delta X^A}\,$ of the defect loop $\,X$,\ and augmented by
deformations $\,\mathscr{P}_{\a\, \mu}\,\frac{\delta\ \ }{\delta{\mathsf p}_{\a\,\mu}}\,$
of the normal covector fields satisfying a linearised version of
\Reqref{eq:DGC-coord},
\begin{eqnarray}
&&\sum_{\a=1,2}\,(-1)^\a\,\bigl[\mathscr{P}_{\a\,\mu}\,\mathscr{W}_\a^\mu+
{\mathsf p}_{\a\,\mu}\,\mathscr{V}^A\,\bigl(\mathscr{W}^B\,\p_A\p_B\iota_\a^\mu+\p_A
\mathscr{W}^B\,\p_B\iota_\a^\mu\bigr)\bigr]\cr\cr
&=&-2(X_*\widehat t)^A\,\bigl(\mathscr{V}^B\,\bigl(\p_B\omega_{AC}\,\mathscr{W}^C+
\omega_{AC}\,\p_B\mathscr{W}^C\bigr)+\omega_{BC}\,\mathscr{W}^C\,\p_A\mathscr{V}^B\bigr)\,,
\label{eq:lin-DGC}
\end{eqnarray}
where $\,\p_A=\frac{\p\ \ }{\p X^A}\,$ and where all fields are
implicitly functions on $\,{\mathbb{S}}^1$.\ Clearly, $\,\mathscr{V}\,$ is a vector
field on $\,Q\,$ and the pair $\,\mathscr{W}_\a,\ \a\in\{1,2\}\,$ can be
completed to another admissible deformation of the $\,X_\a$,\
\begin{eqnarray}\nonumber
\widetilde\mathscr{W}=\widetilde\mathscr{W}_1\oplus\widetilde\mathscr{W}_2\,,\qquad
\qquad\widetilde\mathscr{W}_\a=\mathscr{W}_\a^\mu\,\tfrac{\delta\ \ }{\delta X_\a^\mu}+
\mathscr{Q}_{\a\,\mu}\,\tfrac{\delta\ \ }{\delta{\mathsf p}_{\a\,\mu}}\,,
\end{eqnarray}
by the addition of deformations $\,\mathscr{Q}_{\a\,\mu}\,\frac{\delta\ \ }{\delta
{\mathsf p}_{\a\,\mu}}\,$ of the normal covector fields satisfying an
analogon of \Reqref{eq:lin-DGC}. Upon subtracting the resulting
equation from \Reqref{eq:lin-DGC} and using \Reqref{eq:DGC} in
conjunction with the closedness of $\,\Gamma({\mathsf T} Q)\,$ under the Lie
bracket, we obtain the relation
\begin{eqnarray}\label{eq:lin-DGC-asym}\qquad\qquad
\sum_{\a=1,2}\,(-1)^\a\,\bigl(\mathscr{Q}_{\a\,\mu}\,\mathscr{V}_\a^\mu-\mathscr{P}_{\a
\,\mu}\,\mathscr{W}_\a^\mu\bigr)=-\mathscr{W}\righthalfcup\mathscr{V}\righthalfcup X_*\widehat t\righthalfcup{\mathsf d}
\omega+X_*\widehat t\righthalfcup{\mathsf d}\bigl(\mathscr{W}\righthalfcup\mathscr{V}\righthalfcup\omega\bigr)\,.
\end{eqnarray}
We may now evaluate $\,\Omega_{\sigma,\emptyset}^-$,\ taken at an extremal
section $\,(\Psi_\sigma^1,\Psi_\sigma^2)\,$ determined by the Cauchy data
$\,(\psi_1,\psi_2)\in\gt{I}_\sigma(\mathcal{B};X)$,\ on a pair $\,(\widetilde V,
\widetilde W)\,$ of vectors obtained by evaluating the pair $\,(
\widetilde\mathscr{V},\widetilde\mathscr{W})\,$ of vector fields from $\,\gt{T}
\gt{I}_\sigma(\mathcal{B})\,$ at $\,(\psi_1,\psi_2)$.\ Putting
Eqs.\,\eqref{eq:Omsi-untw} and \eqref{eq:lin-DGC-asym} together and
employing \Reqref{eq:curv-constr}, we readily verify the desired
identity
\begin{eqnarray}\nonumber
\Omega_{\sigma,\emptyset}^-[(\psi_1,\psi_2)](\widetilde V,\widetilde
W)=0\,.
\end{eqnarray}
It leads us to conclude that the defect defines an isotropic
subspace $\,\gt{T}\gt{I}_\sigma(\mathcal{B})\,$ within $\,\Gamma({\mathsf T} {\mathsf P}_\sigma^{\x
2}\vert_{\gt{I}_\sigma(\mathcal{B})})\,$ over the distinguished submanifold
$\,\gt{I}_\sigma(\mathcal{B})$. \qed\end{proof}\medskip
Whenever the subspace $\,\gt{I}_\sigma(\mathcal{B})\,$ with an isotropic tangent
$\,\gt{T}\gt{I}_\sigma(\mathcal{B})\,$ is actually a graph, the $\mathcal{G}$-bi-brane
defines a symplectomorphism of the untwisted state space\footnote{A
symplectomorphism can be viewed as a maximal isotropic in $\,{\mathsf T}
{\mathsf P}_\sigma^{\x 2}$, cf., e.g., \Rcite{Woodhouse:1992de}.}, which can
be understood as an identification between states, chosen
arbitrarily, incident on the defect carrying the data of the
$\mathcal{G}$-bi-brane $\,\mathcal{B}\,$ from one side of the defect line and those
emerging from it on the other side. However, it is to be kept in
mind that states on either side of the defect line carry charges of
the symmetries of the untwisted $\sigma$-model, notably, the energy
and, in the case of extended (internal) symmetry, additional charges
to which the symmetry currents couple. Thus, for the defect to
describe a duality\footnote{Note that the introduction of the
bi-brane, whose world-volume is a priori not related to the target
space, allows for a unified treatment of \emph{symmetries} which do
not leave a connected component of the target space (and solely put
in correspondence extremal sections that map the world-sheet into
different regions thereof) and proper \emph{dualities} which act
between different connected components of the target space,
oftentimes of inequivalent topology. A notable example of the latter
type is the T-duality between principal torus bundles, cf.\ Example
\ref{ex:Tdual}.} of the untwisted theory, we should demand that the
charges of the two states identified with one another at the defect
match, or, equivalently, that the corresponding symmetry currents be
continuous at the defect. Whereas, in concrete examples, one may
wish to impose weaker correspondence constraints, allowing for a
partial breakdown of some internal symmetries, the gauge symmetry of
the $\sigma$-model, that is the conformal symmetry, should always be
preserved. As was shown in \Rcite{Runkel:2008gr}, one linear
combination of the conformal currents, to wit, the one that
generates diffeomorphisms preserving the defect line,
$\,T_{++}-T_{--},\ T_{\pm\pm}={\rm g}_{\mu \nu}(X)\,\p_\pm
X^\mu\,\p_\pm X^\nu\,$ (in the adapted coordinates), is
automatically preserved at the defect by virtue of the DGC -- this
is the content of Theorem \ref{thm:conf-def}. The last property
identifies the defects considered as \textbf{conformal} in the sense
of \Rcite{Oshikawa:1996dj}. For a space-like defect line, which is
what we have been considering in the present section, it is the
other linear combination,
\begin{eqnarray}\label{eq:ham-dens-si}
T_{++}+T_{--}\equiv\tfrac{1}{2}\,\bigl(\bigl({\rm g}^{-1}\bigr)^{\mu
\nu}(X)\,{\mathsf p}_\mu(X)\,{\mathsf p}_\nu(X)+{\rm g}_{\mu\nu}(X)\,(X_*\widehat
t)^\mu\,(X_*\widehat t)^\nu\bigr)=\mathcal{H}_\sigma\,,
\end{eqnarray}
that gives the hamiltonian density $\,\mathcal{H}_\sigma\,$ of the
$\sigma$-model. It generates conformal transformations on the
world-sheet which deform the defect line, and hence the continuity
of $\,\mathcal{H}_\sigma\,$ at the latter is related to the extendibility of
the defect, as made precise in
\begin{Def}
Let $\,\gt{B}\,$ be a string background with target $\,\mathcal{M}=(M,{\rm g},\mathcal{G}
)\,$ and $\mathcal{G}$-bi-brane $\,\mathcal{B}=(Q,\iota_\a,\omega,\Phi\ \vert\ \a\in\{1
,2\})$,\ and let $\,(X\,\vert\,\Gamma)\,$ be a network-field
configuration in string background $\,\gt{B}\,$ on world-sheet
$\,(\Sigma,\gamma)\,$ with defect quiver $\,\Gamma$.\ Denote by $\,U\,$ a
tubular neighbourhood of an edge $\,\ell\,$ of $\,\Gamma\,$ within
$\,\Sigma$,\ with the property $\,U\cap\Gamma=\ell$.\ The neighbourhood
$\,U\,$ is split by the oriented line $\,\ell\,$ into subsets
$\,U_\a,\ \a\in\{1,2\}\,$ as in Definition \ref{def:net-field}. An
\textbf{extension $\,\widehat X\,$ of network-field configuration
$\,(X\,\vert\,\Gamma)\,$ on neighbourhood $\,U\,$ of defect line
$\,\ell\,$} is a map $\,\widehat X:U\to Q$,\ such that
\begin{eqnarray}\nonumber
\widehat X\vert_\ell=X\,,\qquad\qquad\iota_\a\circ\widehat
X\vert_{U_\a}=X\vert_{U_\a}\,,\qquad\qquad
\end{eqnarray}
and such that the relation
\begin{eqnarray}\nonumber
\iota_1^*{\rm g}\bigl(\widehat X(p)\bigr)\bigl(\widehat X_*\widehat
u^\perp,\cdot\bigr)-\iota_2^*{\rm g}\bigl(\widehat X(p)\bigr)\bigl(
\widehat X_*\widehat u^\perp,\cdot\bigr)-\widehat X_*\widehat u\righthalfcup
\omega\bigl(\widehat X(p)\bigr)=0\,,\qquad\qquad\widehat u^\perp=
\gamma^{-1}\bigl(\widehat u\righthalfcup\Vol(\Sigma,\gamma),\cdot\bigr)
\end{eqnarray}
is satisfied at any point $\,p\in U\,$ and for an arbitrary vector
$\,\widehat u\in{\mathsf T}_p\Sigma$.\ A defect $\,\ell\,$ of a network field
configuration that admits an extension on a neighbourhood of
$\,\ell\,$ shall be termed \textbf{extendible}.\begin{flushright}$\checkmark$\end{flushright}\end{Def} \noindent The
less restrictive condition of continuity of the conformal current
gives rise to
\begin{Def}\label{def:def-top}
Let $\,\gt{B}\,$ be a string background with target $\,\mathcal{M}=(M,{\rm g},\mathcal{G}
)$,\ and let $\,\Gamma\,$ be a defect quiver embedded in a world-sheet
$\,\Sigma$.\ Consider the non-linear $\sigma$-model for network-field
configurations $\,(X\,\vert\,\Gamma)\,$ in string background $\,\gt{B}\,$
on world-sheet $\,(\Sigma,\gamma)\,$ with defect quiver $\,\Gamma$.\ Choose a
local coordinate system $\,\{\sigma^a \}^{a\in\{1,2\}}\,$ in the
neighbourhood of a point in $\,\Gamma$.\ The defect $\,\Gamma\,$ is called
\textbf{topological} iff the conformal current $\,T$,\ with local
components
\begin{eqnarray}\nonumber
T^{ab}=\tfrac{2}{\sqrt{\det\,\gamma}}\,\tfrac{\delta S_\sigma}{\delta\gamma_{ab}}\,,
\end{eqnarray}
is continuous across $\,\Gamma$. \begin{flushright}$\checkmark$\end{flushright}\end{Def} \begin{Rem} The notion of
topologicality can be regarded as a classical counterpart of the
quantum concept introduced in \Rcite{Petkova:2000ip}. \end{Rem}\begin{center}*\hspace{2cm}*\hspace{2cm}*\end{center}
Extendible defects have the desired property of topologicality, as
stated in
\begin{Thm}\cite[Sec.\,2.9]{Runkel:2008gr}
The non-linear $\sigma$-model of Definition \ref{def:sigmod} for
network-field configurations $\,(\Gamma ,X)\,$ in string background
$\,\gt{B}\,$ on world-sheet $\,(\Sigma,\gamma)\,$ with defect quiver $\,\Gamma\,$
composed of extendible defects is invariant with respect to
arbitrary (gauge) transformations
\begin{eqnarray}\nonumber
&X\mapsto X\circ D\,,\qquad\qquad\gamma\mapsto D^*\gamma\,,\qquad D\in
{\xcD iff}^+(\Sigma)\,,&\cr\cr
&\gamma\mapsto{\rm e}^{2w}\cdot\gamma\,,\qquad{\rm e}^{2w}\in{\rm Weyl}(\gamma)&
\end{eqnarray}
from the semidirect product $\,{\xcD iff}^+(\Sigma){\hspace{-0.04cm}\ltimes\hspace{-0.05cm}}{\rm Weyl}(\gamma)\,$ of
the group $\,{\xcD iff}^+(\Sigma)\,$ of (orientation-preserving)
diffeomorphisms of $\,\Sigma\,$ with the group $\,{\rm Weyl}(\gamma)\,$ of
Weyl rescalings of the metric $\,\gamma$.\ All components of the
conformal current are continuous across the defect lines of $\,\Gamma$,\
that is the defect is \textbf{topological} in the sense of
Definition \ref{def:def-top}. \end{Thm} \begin{Rem} From the point of view of
the categorial quantisation of the $\sigma$-model in the presence of
defects, as discussed, e.g., in \Rcite{Runkel:2008gr}, it is natural
to expect a topological defect to be deformable, and hence
necessarily extendible. However, the statement in the quantum theory
is usually formulated in terms of correlation functions assigned to
(decorated) world-sheets with embedded defect quivers, and so -- in
the present setting -- the issue of finding its proper classical
counterpart is obscured by aspects of the quantisation procedure
such as the choice of the renormalisation scheme that affects the
field-theoretic functionals entering the DGC, cf., e.g.,
Refs.\,\cite{Bachas:2004sy,Alekseev:2007in,Bachas:2009mc} for an
illustration. As we are not addressing here the issue of
quantisation beyond the construction of the pre-quantum bundle, we
shall restrict ourselves to topological rather than extendible
defects in what follows. \end{Rem}\begin{center}*\hspace{2cm}*\hspace{2cm}*\end{center} \noindent Thus, it is amidst
topological defects that we should look for those that describe
dualities of the untwisted $\sigma$-model. Before we do that, however,
let us make the very notion of duality precise, using the various
field-theoretic constructs introduced hitherto.
\begin{Def}\label{def:pqsymm}
Let $\,\gt{B}\,$ be a string background with target $\,\mathcal{M}=(M,{\rm g},\mathcal{G}
)$,\ and let $\,({\mathsf P}_{\sigma,\emptyset}={\mathsf T}^*{\mathsf L} M,\Omega_{\sigma,
\emptyset})\,$ be the untwisted state space of the non-linear
$\sigma$-model for network-field configurations $\,(X\,\vert\,\Gamma )\,$
in string background $\,\gt{B}\,$ on world-sheet $\,(\Sigma,\gamma)\,$ with
defect quiver $\,\Gamma$.\ Furthermore, let $\,\pi_{\mathcal{L}_{\sigma,
\emptyset}}:\mathcal{L}_{\sigma,\emptyset}\to{\mathsf P}_{\sigma,\emptyset}\,$ be the
pre-quantum bundle for the untwisted sector of the $\sigma$-model,
constructed in Corollary \ref{cor:preqb-untw}. A \textbf{pre-quantum
duality of the untwisted sector of the non-linear $\sigma$-model for
network-field configurations $\,(X\,\vert\,\Gamma)\,$ in string
background $\,\gt{B}\,$ on world-sheet $\,(\Sigma,\gamma)\,$ with defect
quiver $\,\Gamma\,$} is a pair $\,(\gt{I}_\sigma, \gt{D}_\sigma)\,$ which consists
of
\begin{itemize}
\item a graph $\,\gt{I}_\sigma\subset{\mathsf P}_{\sigma,\emptyset}^{\x 2}$,\
isotropic with respect to the `difference' symplectic form $\,\Omega_{\sigma,
\emptyset}^-\,$ of Proposition \ref{prop:DGC-as-iso}, and having the
property that the difference
\begin{eqnarray}\label{eq:Hsi-}
\mathcal{H}_\sigma^-={\rm pr}_1^*\mathcal{H}_\sigma-{\rm pr}_2^*\mathcal{H}_\sigma
\end{eqnarray}
of the pullbacks, along the canonical projections $\,{\rm pr}_\a:
{\mathsf P}_{\sigma,\emptyset}\x{\mathsf P}_{\sigma,\emptyset}\to {\mathsf P}_{\sigma,\emptyset},\
\a\in\{1,2\}\,$,\ of the hamiltonian density $\,\mathcal{H}_\sigma\,$ of the
$\sigma$-model, as given in \Reqref{eq:ham-dens-si}, vanishes
identically on restriction to $\,\gt{I}_\sigma$;
\item a bundle isomorphism
\begin{eqnarray}\label{eq:Dsi}
\gt{D}_\sigma\ :\ {\rm pr}_1^*\mathcal{L}_{\sigma,\emptyset}\vert_{\gt{I}_\sigma}
\xrightarrow{\cong}{\rm pr}_2^*\mathcal{L}_{\sigma,\emptyset}\vert_{\gt{I}_\sigma}\,,
\end{eqnarray}
between the restrictions to $\,\gt{I}_\sigma\,$ of the pullbacks of
$\,\mathcal{L}_{\sigma,\emptyset}\,$ along the canonical projections
$\,{\rm pr}_\a$.
\end{itemize}\begin{flushright}$\checkmark$\end{flushright}\end{Def}
\begin{Rem} Pre-quantum dualities of the $\sigma$-model which are consistent
with a given choice of the polarisation of the pre-quantum bundle
defining the Hilbert space of the theory give rise to bona fide
dualities of the quantised $\sigma$-model. \end{Rem}\begin{center}*\hspace{2cm}*\hspace{2cm}*\end{center} \noindent The first
relation between conformal defects and $\sigma$-model dualities is
established in the following
\begin{Thm}\label{thm:def-dual}
Let $\,\gt{B}\,$ be a string background with target $\,\mathcal{M}=(M,{\rm g},\mathcal{G}
)\,$ and $\mathcal{G}$-bi-brane $\,\mathcal{B}=\bigl(Q,\iota_\a,\omega,\Phi\ \vert\ \a
\in\{ 1,2\}\bigr)$,\ and consider the non-linear $\sigma$-model for
network-field configurations $\,(X\,\vert\,\Gamma)\,$ in string
background $\,\gt{B}\,$ on world-sheet $\,(\Sigma,\gamma)\,$ with defect
quiver $\,\Gamma$.\ The $\mathcal{G}$-bi-brane $\,\mathcal{B}\,$ together with the
Defect Gluing Condition \eqref{eq:DGC} canonically defines a
pre-quantum duality of the untwisted sector of the $\sigma$-model iff
the following conditions are satisfied:
\begin{itemize}
\item[i)] both loop-space maps
\begin{eqnarray}\nonumber
\widetilde\iota_\a\ :\ {\mathsf L} Q\to{\mathsf L} M\ :\ X\mapsto\iota_\a\circ X
\,,\quad\a\in\{1,2\}\,,
\end{eqnarray}
induced by the $\mathcal{G}$-bi-brane maps $\,\iota_\a\,$ (and hence also
the latter), are surjective submersions onto connected components of
$\,{\mathsf L} M$;
\item[ii)] let $\,s_{\gt{i}_\a,\a}:\mathcal{O}_{\gt{i}_\a}\to{\mathsf L} Q\,$ denote
smooth local sections of $\,\widetilde\iota_\a\,$ over elements
$\,\mathcal{O}_{\gt{i}_\a}\,$ of an open cover $\,\mathcal{O}_{{\mathsf L} M}=\{\mathcal{O}_\gt{i}
\}_{\gt{i}\in\mathscr{I}_{{\mathsf L} M}}\,$ of $\,{\mathsf L} M$,\ e.g., the one from
Proposition \ref{prop:cover-untw} induced from a sufficiently fine
open cover of $\,M$;\ given a pair $\,s_{\gt{i}_\a,\a}^n,\ n\in\{1,
2\}\,$ of such sections satisfying $\,\widetilde\iota_\a\circ
s_{\gt{i}_\a,\a}^1=\widetilde\iota_\a\circ s_{\gt{i}_\a,\a}^2$,\ the
relations
\begin{eqnarray}\label{eq:iotasI-iotasJ-tan}
\widetilde\iota_{3-\a}\circ s_{\gt{i}_\a,\a}^1=\widetilde\iota_{3-\a}
\circ s_{\gt{i}_\a,\a}^2\,,\qquad\qquad\widetilde\iota_{3-\a\,*}\circ
s_{\gt{i}_\a,\a\,*}^1=\widetilde\iota_{3-\a\,*}\circ s_{\gt{i}_\a,
\a\,*}^2
\end{eqnarray}
hold true;
\item[iii)] let $\,s_{i_\a,\a}:\mathcal{O}_{i_\a}\to Q\,$ denote smooth
local sections of $\,\iota_\a\,$ over elements $\,\mathcal{O}_{i_\a}\,$ of
an open cover $\,\mathcal{O}_M=\{\mathcal{O}_i\}_{i\in\mathscr{I}_M}\,$ of $\,M\,$
compatible, in an obvious manner, with the $\,s_{\gt{i}_\a,\a}\,$
introduced previously; given a pair $\,s_{i_\a,\a}^n,\ n\in\{1,2
\}\,$ of such sections associated with a pair $\,s_{\gt{i}_\a,\a}^n,\
n\in\{1,2\}\,$ as above, the relations
\begin{eqnarray}\label{eq:sIom-sJom}
s^{1\,*}_{i_\a,\a}\omega=s^{2\,*}_{i_\a,\a}\omega
\end{eqnarray}
obtain;
\item[iv)] for arbitrary $\,((X_1,{\mathsf p}_1),(X_2,{\mathsf p}_2))\in
\mathcal{O}_{\gt{i}_1}\x\mathcal{O}_{\gt{i}_2}\subset\gt{I}_\sigma\,$ and, in the notation of
the preceding points, for any $\,(s_{\gt{i}_1,1},s_{\gt{i}_2,2})\,$ such
that $\,s_{\gt{i}_1,1}(X_1)=s_{\gt{i}_2,2}(X_2)$,\ the following
identity is satisfied:
\begin{eqnarray}\nonumber
\bigl({\rm g}^{-1}\bigr)^{\mu\nu}\bigl(\widetilde\iota_2\circ s_{i_1,
1}(X_1)\bigr)\,\tfrac{\p s_{\gt{i}_2,2}^A}{\p X_1^\mu}\,\tfrac{\p
s_{\gt{i}_2,2}^B}{\p X_1^\nu}\,\bigl[{\mathsf p}_{1\,\rho}\,\tfrac{\p
\iota_1^\rho}{\p X^A}+2\omega_{AC}\bigl(s_{\gt{i}_1,1}(X_1)\bigr)\,
\tfrac{\p s_{\gt{i}_1,1}^C}{\p X_1^\rho}\,(X_{1\,*}\widehat t)^\rho
\bigr]\cr\cr \cdot\bigl[{\mathsf p}_{1\,\sigma}\,\tfrac{\p\iota_1^\sigma}{\p X^B}
+2\omega_{BD}\bigl(s_{\gt{i}_1,1}(X_1)\bigr)\,\tfrac{\p s_{\gt{i}_1,
1}^D}{\p X_1^\sigma}\,(X_{1\,*}\widehat t)^\sigma\bigr]+{\rm g}_{\mu\nu}
\bigl(\widetilde\iota_2\circ s_{i_1,1}(X_1)\bigr)\,\tfrac{\p
\iota_2^\mu}{\p X^A}\,\tfrac{\p\iota_2^\nu}{\p X^B}\,\tfrac{\p
s_{\gt{i}_1,1}^A}{\p X_1^\rho}\,\tfrac{\p s_{\gt{i}_1,1}^B}{\p X_1^\sigma}
\,(X_{1\,*}\widehat t)^\rho\,(X_{1\,*}\widehat t)^\sigma\cr\cr
=\bigl({\rm g}^{-1}\bigr)^{\mu\nu}(X_1)\,{\mathsf p}_{1\,\mu}\,{\mathsf p}_{1\,\nu}+
{\rm g}_{\mu\nu}(X_1)\,(X_{1\,*}\widehat t)^\mu\,(X_{1\,*}\widehat t
)^\nu\,.
\end{eqnarray}
\end{itemize}
\end{Thm}
\noindent\begin{proof}
Recall that, in virtue of Proposition \ref{prop:DGC-as-iso},
$\,\mathcal{B}\,$ canonically defines an isotropic submanifold $\,\gt{I}_\sigma(
\mathcal{B})\subset{\mathsf P}_{\sigma,\emptyset}^{\x 2}$.\ Choose good open covers
$\,\mathcal{O}_M=\{\mathcal{O}^M_i\}_{i\in\mathscr{I}_M}\,$ and $\,\mathcal{O}_Q=\{\mathcal{O}^Q_i\}_{i\in
\mathscr{I}_Q}\,$ such that there exist \v Cech extensions $\,(\iota_\a,
\phi_\a)\,$ of the $\mathcal{G}$-bi-brane maps and an open cover of
$\,\gt{I}_\sigma(\mathcal{B})\,$ is induced, as in Proposition
\ref{prop:cover-untw}, with triangulations of both loops in $\,(
\psi_1,\psi_2)\in\gt{I}_\sigma(\mathcal{B})\,$ coming from a triangulation of the
parent loop $\,X\in{\mathsf L} Q$.\ Thus, in particular, at each point
$\,(\psi_1,\psi_2)\in\gt{I}_\sigma(\mathcal{B})$,\ we have a common triangulation
$\,\triangle({\mathbb{S}}^1)$,\ with edges $\,e\,$ and vertices $\,v$,\ of
the unit circle parameterising $\,X\,$ and $\,X_\a=\iota_\a\circ
X$,\ and, for each element $\,f \in\triangle({\mathbb{S}}^1)$,\ a triple of
indices $\,(i_f^1,i_f^2,i_f^{1,2})\in\mathscr{I}_M\x\mathscr{I}_M\x\mathscr{I}_Q$,\
related as per
\begin{eqnarray}\label{eq:comm-triang-index}
i_f^\a=\phi_\a(i_f^{1,2})\,.
\end{eqnarray}
Next, fix a local presentation of $\,\gt{B}\,$ associated with this
choice of covers as in Definition \ref{def:loco}. It is then a
matter of a simple calculation to verify that the local data
$\,(\theta_{\sigma,\emptyset\,\gt{i}},\gamma_{\sigma,\emptyset\,\gt{i}\gt{j}})\,$ of
the pre-quantum bundle $\,\mathcal{L}_{\sigma,\emptyset}\,$ associated -- as
in Corollary \ref{cor:preqb-untw} -- with the open cover of a
cartesian factor in $\,\gt{I}_\sigma(\mathcal{B})\,$ (induced as in Proposition
\ref{prop:cover-untw}) satisfy the identities
\begin{eqnarray}
{\rm pr}_2^*\theta_{\sigma,\emptyset\,\gt{i}^2}-{\rm pr}_1^*\theta_{\sigma,\emptyset
\,\gt{i}^1}&=&-{\mathsf i}\,{\mathsf d}\log f_{\sigma,\mathcal{B}\,(\gt{i}^1,\gt{i}^2)}\,,
\label{eq:f12-as-iso}\\\cr
{\rm pr}_2^*\gamma_{\sigma,\emptyset\,\gt{i}^2\gt{j}^2}&=&f_{\sigma,\mathcal{B}\,(\gt{i}^1,
\gt{i}^2)}\cdot{\rm pr}_1^*\gamma_{\sigma,\emptyset\,\gt{i}^1\gt{j}^1}\cdot f_{\sigma,
\mathcal{B}\,(\gt{j}^1,\gt{j}^2)}^{-1}\,,
\end{eqnarray}
written in terms of the canonical projections $\,{\rm pr}_\a:\gt{I}_\sigma(\mathcal{B}
)\to{\mathsf P}_{\sigma,\emptyset},\ \a\in\{1,2\}\,$ and of the ${\rm U}(1)$-valued
functionals
\begin{eqnarray}\label{eq:duality-iso-bib}\qquad\qquad
f_{\sigma,\mathcal{B}\,(\gt{i}^1,\gt{i}^2)}[(\psi_1,\psi_2)]=\prod_{e\in\triangle(
{\mathbb{S}}^1)}\,{\rm e}^{{\mathsf i}\,\int_e\,X_e^*P_{i^{1,2}_e}}\cdot\prod_{v\in
\triangle({\mathbb{S}}^1)}\,X^*K^{-1}_{i_{e_+(v)}^{1,2}i_{e_-(v)}^{1,2}}(v)
\end{eqnarray}
on $\,\mathcal{O}^*_{\gt{i}^1}\x\mathcal{O}^*_{\gt{i}^2}\subset\gt{I}_\sigma(\mathcal{B})$,\ where
$\,\mathcal{O}^*_{\gt{i}^\a}=\pi_{{\mathsf T}^*{\mathsf L} M}^{-1}\bigl(\mathcal{O}_{\gt{i}^\a}
\bigr)$.\ Hence, the $\,f_{\sigma,\mathcal{B}\,\gt{i}^{1,2}}\,$ can be identified
with local data of an isomorphism $\,\gt{D}_\sigma(\mathcal{B})\,$ from
Definition \ref{def:pqsymm}. It remains to establish the conditions
under which the isotropic submanifold $\,\gt{I}_\sigma(\mathcal{B})\subset
{\mathsf P}_{\sigma,\emptyset}^{\x 2}\,$ becomes a graph.
For this to be the case, it is necessary that the two maps
$\,\widetilde\iota_\a\,$ be surjective so that any loop in $\,M\,$
can be descended from a loop in $\,Q$.\void{More specifically, it
may happen that the target space $\,M\,$ is a disjoint sum of
several manifolds, with the patch embedding map $\,X_{|\a}\,$
sending $\,\wp_\a\,$ into a separate connected component $\,M_\a\,$
of $\,M\,$ by continuity. In this case, we should demand that
$\,\iota_\a\,$ covers all of $\,{\mathsf L} M_\a\,$ as we vary the parent
loop $\,X\vert_\ell$.} Having thus established a correspondence,
fibred over $\,Q$,\ between loops in either cartesian factor of
$\,{\mathsf P}_{\sigma,\emptyset}^{\x 2}$,\ or -- in the world-sheet picture
-- on either side of the defect line, we still have to require that
upon choosing a specific parent loop $\,X\in{\mathsf L} Q\,$ and thus
picking up a pair $\,(X_1,X_2)\,$ of loops from $\,{\mathsf L} M\,$ and
putting them in correspondence, and upon determining either of the
loop momenta, $\,{\mathsf p}_1\,$ or $\,{\mathsf p}_2$,\ the other loop momentum
is already fixed uniquely by the DGC. Inspection of the latter,
\begin{eqnarray}\nonumber
{\mathsf p}_1\circ\widetilde\iota_{1\,*}-{\mathsf p}_2\circ\widetilde\iota_{2\,*}
-X_*\widehat t\righthalfcup\omega=0\,,
\end{eqnarray}
reveals that for this to hold, also the tangent maps $\,\widetilde
\iota_{\a\,*}\,$ must admit local right inverses. That is,
altogether, the $\,\widetilde\iota_\a\,$ should be surjective
submersions, with smooth local sections $\,s_{\gt{i}_\a,\a}:
\mathcal{O}_{\gt{i}_\a}\to{\mathsf L} Q\,$ satisfying the identities
\begin{eqnarray}\nonumber
\widetilde\iota_\a\circ s_{\gt{i}_\a,\a}={\rm id}_{\mathcal{O}_{\gt{i}_\a}}\,,\qquad
\qquad\widetilde\iota_{\a\,*}\circ s_{\gt{i}_\a,\a\,*}={\rm id}_{\Gamma({\mathsf T}{\mathsf L}
M\vert_{\mathcal{O}_{\gt{i}_\a}})}\,.
\end{eqnarray}
Indeed, for $\,X_\a\in\mathcal{O}_{\gt{i}_\a},\ X\in(\iota_1\x\iota_2)^{-1}\{(
X_1,X_2)\}\,$ and a pair of sections $\,(s_{\gt{i}_1,1},s_{\gt{i}_2,2}
)\,$ such that
\begin{eqnarray}\label{eq:X-as-sec}
X=s_{\gt{i}_\a,\a}(X_\a)\,,
\end{eqnarray}
there arise functional relations
\begin{eqnarray}\label{eq:Xal-of-Xal}
X_2(X_1)=\widetilde\iota_2\circ s_{\gt{i}_1,1}(X_1)\,,\qquad\qquad
X_1(X_2)=\widetilde\iota_1\circ s_{\gt{i}_2,2}(X_2)
\end{eqnarray}
between the loop coordinates, alongside the functional relations
\begin{eqnarray}
{\mathsf p}_2({\mathsf p}_1,X_1)&=&{\mathsf p}_1\circ\widetilde\iota_{1\,*}\circ
s_{\gt{i}_2,2\,*}- \bigl(X_*\widehat t\righthalfcup\omega(X)\bigr)\circ s_{\gt{i}_2,
2\,*}\,,\cr\label{eq:pal-of-pal}&&\\
{\mathsf p}_1({\mathsf p}_2,X_2)&=&{\mathsf p}_2\circ\widetilde\iota_{2\,*}\circ
s_{\gt{i}_1,1\,*}+\bigl(X_*\widehat t\righthalfcup\omega(X)\bigr)\circ s_{\gt{i}_1,
1\,*}\nonumber
\end{eqnarray}
between the loop momentum coordinates on $\,\gt{I}_\sigma(\mathcal{B})$.\ The
dependence of the loop momentum $\,{\mathsf p}_\a\,$ on the loop coordinate
$\,X_{3-\a}\,$ is given by \Reqref{eq:X-as-sec}. The above are
statements valid at every point along the loop, and the pushforward
operators $\,s_{\gt{i}_\a,\a\,*}\,$ are to be understood as
characterising the local sections of the $\,\widetilde\iota_\a\,$
that enter the definition of the $\,s_{\gt{i}_\a,\a}$.\ The relations
define a graph iff they agree for any two choices $\,s_{\gt{i}_\a,
\a}^n,\ n\in\{1,2\}\,$ of sections (i.e.\ for any two choices
$\,X^n\in{\mathsf L} Q,\ n\in\{1,2\}\,$ of the parent loop) corresponding
to a given pair $\,X_\a\,$ of loops in $\,M$.\ This is tantamount to
imposing conditions \eqref{eq:iotasI-iotasJ-tan} (which ensure that
arbitrary \emph{curves} of loops in $\,{\mathsf L} M\,$ are mapped into one
another in a unique manner), together with
\begin{eqnarray}\nonumber
\bigl(X^1_*\widehat t\righthalfcup\omega(X^1)\bigr)\circ s^1_{\gt{i}_\a,\a\,*}=
\bigl(X^2_*\widehat t\righthalfcup\omega(X^2)\bigr)\circ s^2_{\gt{i}_\a,\a\,*}\,,
\end{eqnarray}
or -- equivalently --
\begin{eqnarray}\nonumber
X_{\a\,*}\widehat t\righthalfcup\bigl(s^{1\,*}_{\gt{i}_\a,\a}\omega-s^{2\,
*}_{\gt{i}_\a,\a}\omega\bigr)(X_\a)=0\,.
\end{eqnarray}
The corresponding local statement, at a given point $\,X_\a(\varphi)
\in\mathcal{O}^M_{i_\a}\,$ along the loop in $\,M$,\ yields
\Reqref{eq:sIom-sJom} by virtue of the arbitrariness of the vector
$\,X_{\a\,*}\widehat t$.
Finally, the identity from point iv) is a simple rewrite of the
condition $\,\mathcal{H}_\sigma^-\vert_{\gt{I}_\sigma}\equiv 0\,$ taking into
account the relations \eqref{eq:X-as-sec}-\eqref{eq:pal-of-pal}.
\qed\end{proof} \begin{Rem} It deserves to be noted that the requirement that the
state correspondence engendered by the defect be independent of the
choice of the local section of the surjective submersion
$\,\widetilde\iota_\a:{\mathsf L} Q\to{\mathsf L} M\,$ is automatically satisfied
in the (physically) most natural setting, which is that of $\,Q\,$
being a submanifold within $\,M\x M\,$ projecting surjectively on
both cartesian factors. \end{Rem}\begin{center}*\hspace{2cm}*\hspace{2cm}*\end{center} \begin{Rem} Our discussion indicates that
surjective submersions play a prominent r\^ole in the canonical
description of dualities of the $\sigma$-model on world-sheets with
defect quivers. This is to be compared with the categorial treatment
of gerbes and gerbe bi-modules in \Rcite{Fuchs:2009si} which also
appears to distinguish maps of this kind, albeit in a more formal
manner. \end{Rem}\begin{center}*\hspace{2cm}*\hspace{2cm}*\end{center}\medskip
Once the conditions for the defect to describe a duality of the
untwisted sector of the $\sigma$-model have been established, it is
tempting to reverse the question and enquire as to the necessary
conditions for a duality to define a bi-brane that can subsequently
be put over a defect line. General as it stands, the question falls
beyond the compass of the present paper. We may, nonetheless, try to
draw useful insights from the study of a wide class of dualities for
which there exists a concise explicit description in terms of
canonical transformations on the state space of the untwisted sector
of the $\sigma$-model, determined by generating functionals of a
restricted `linear' form, to be described below. Dualities of this
type, including abelian and non-abelian dualities, as well as the
Poisson--Lie T-duality of the WZW model, were examined in a series
of papers by Alvarez, Refs.\,\cite{Alvarez:2000bh,Alvarez:2000bi},
from which we borrow some of our conventions and a number of
observations.
The chief idea of the approach advertised above consists in
explicitly enforcing the isotropy of the space $\,\gt{T}\gt{I}_\sigma\,$ of
sections of the tangent bundle of a graph $\,\gt{I}_\sigma\,$ in
$\,{\mathsf P}_\sigma^{\x 2}\,$ representing the duality by trivialising the
symplectic potential of $\,\Omega^-_{\sigma,\emptyset}\,$ with the help of
a generating functional of a canonical transformation $\,\psi_1
\mapsto\psi_2\,$ determined by the graph $\,\gt{I}_\sigma\ni(\psi_1,
\psi_2),\ \psi_\a=(X_\a,{\mathsf p}_\a),\ \a\in\{1,2\}$.\ In so doing, the
generating functional is chosen such that the transformation between
the two sets of variables: $\,(X_{1 \,*} \widehat t,{\mathsf p}_1)\,$ and
$\,(X_{2\,*}\widehat t,{\mathsf p}_2)\,$ induced by the canonical
transformation is invertible and preserves the hamiltonian density
\eqref{eq:ham-dens-si}. The latter condition, in conjunction with
the distinguished form of the hamiltonian density (a sum of terms
quadratic in $\,{\mathsf p}\,$ and $\,X_*\widehat t$,\ respectively), was
used in \Rcite{Alvarez:2000bh} to restrict the choice of the
generating functional, for a specific trivialisation of
$\,\Omega^-_{\sigma, \emptyset}$,\ to those depending linearly on the
$\,X_{\a*}\widehat t\,$ and further constrained by the requirement
of orthogonality with respect to the metric
$\,\bigl({\rm g},{\rm g}^{-1}\bigr)\,$ entering the definition of the
hamiltonian density.
An obvious problem with the above description of a canonical
transformation lies with the lack of a global definition of the
symplectic potential of $\,\Omega^-_{\sigma,\emptyset}\,$ in general, a
simple variation on the theme of the lack of a global definition of
the topological term in the $\sigma$-model action functional, only
transferred one degree lower in cohomology and -- simultaneously --
from the target space to its free-loop space. Below, we resolve this
problem by considering the full structure of the pre-quantum bundle
over the state space of the untwisted sector of the $\sigma$-model, in
a local presentation suggested by Corollary \ref{cor:preqb-untw} in
conjunction with Proposition \ref{prop:cover-untw}. Moreover, we
extend the analysis to a larger class of trivialisations of
$\,\Omega^-_{\sigma,\emptyset}$,\ thereby gaining access to a canonical
description of geometric symmetries of the $\sigma$-model.
With view towards organising the discussion of our results, we begin
by providing a precise description of the class of dualities to be
considered in the sequel.
\begin{Def}\label{def:dualiTN}
Let $\,\gt{B}\,$ be a string background with target $\,\mathcal{M}=(M,{\rm g},\mathcal{G}
)$,\ and let $\,(\gt{I}_\sigma,\gt{D}_\sigma)\,$ be a pre-quantum duality of
the untwisted sector of the non-linear $\sigma$-model for network-field
configurations $\,(X\,\vert\,\Gamma)\,$ in string background $\,\gt{B}\,$
on world-sheet $\,(\Sigma,\gamma)\,$ with defect quiver $\,\Gamma$.\ Assume
that $\,(\gt{I}_\sigma,\gt{D}_\sigma )\,$ is determined by a generating
functional $\,\mathscr{F}_\sigma\,$ of a canonical transformation $\,\mathscr{D}_\sigma:
{\mathsf P}_{\sigma,\emptyset}\to{\mathsf P}_{\sigma,\emptyset}$,\ given as a collection
of smooth real-valued functionals $\,\Phi_{\sigma\,\gt{i}}\,$ on elements
of an open cover $\,\mathcal{O}_{\gt{I}_\sigma}=\{\mathcal{O}^{\gt{I}_\sigma}_\gt{i}\}_{\gt{i}\in
\mathscr{I}_{\gt{I}_\sigma}}\,$ of $\,\gt{I}_\sigma$,\ i.e.\ $\,\gt{I}_\sigma\,$ is the
graph of $\,\mathscr{D}_\sigma\,$ and the local data of $\,\mathscr{F}_\sigma\,$ yield a
local presentation of the bundle isomorphism $\,\gt{D}_\sigma$.\ Assume
further that the $\,\Phi_{\sigma\,\gt{i}}\,$ depend \emph{at most
linearly} on the variables $\,(X_{\a\,*}\widehat t,{\mathsf p}_\a),\ \a\in
\{1,2\}$.\ Fix $\,\mathcal{O}_{\gt{I}_\sigma}\,$ to be the open cover induced
from the open covers $\,\mathcal{O}_{{\mathsf L} M}\,$ of the free-loop space
$\,{\mathsf L} M=C^\infty({\mathbb{S}}^1, M)\,$ of the target space $\,M\,$ from
Proposition \ref{prop:cover-untw} on the cartesian factors of
$\,\gt{I}_\sigma$,\ coming from a sufficiently fine good open cover
$\,\mathcal{O}_M\,$ of $\,M$,\ so that elements of $\,\mathcal{O}_{\gt{I}_\sigma}\,$ are
of the form
\begin{eqnarray}\nonumber
\mathcal{O}^{\gt{I}_\sigma}_{(\gt{i}^1,\gt{i}^2)}=\mathcal{O}^*_{\gt{i}^1}\x\mathcal{O}^*_{\gt{i}^2}
\end{eqnarray}
for $\,\mathcal{O}^*_{\gt{i}^\a}\,$ as defined in Corollary
\ref{cor:preqb-untw}. We call $\,(\gt{I}_\sigma,\gt{D}_\sigma)\,$ a
\textbf{pre-quantum duality of type $T$ of the untwisted sector of
the non-linear $\sigma$-model for network-field configurations
$\,(X\,\vert\,\Gamma)\,$ in string background $\,\gt{B}\,$ on world-sheet
$\,(\Sigma,\gamma)\,$ with defect quiver $\,\Gamma\,$} iff its local data can
be put in the form
\begin{eqnarray}
\Phi_{\sigma\,(\gt{i}_1,\gt{i}_2)}[(\psi_1,\psi_2)]=\sum_{e\in\triangle
({\mathbb{S}}^1)}\,\int_e\,(X_{1\,e},X_{2\,e})^*P_{(i^1_e,i^2_e)}+{\mathsf i}\,
\sum_{v\in\triangle({\mathbb{S}}^1)}\,(X_1,X_2)^*\log K_{(i^1_{e_+(v)},
i^2_{e_+(v)})(i^1_{e_-(v)},i^2_{e_-(v)})}(v)\cr\cr
\label{eq:Phi-om-loc}
\end{eqnarray}
for $\,(\psi_1,\psi_2)\in\mathcal{O}^{\gt{I}_\sigma}_{(\gt{i}^1,\gt{i}^2)}\,$ with
$\,\psi_\a=(X_\a,{\mathsf p}_\a)$,\ some smooth 1-forms $\,P_{(i^1,i^2
)}\,$ on $\,\mathcal{O}^M_{i^1}\x\mathcal{O}^M_{i^2}\,$ and some smooth ${\rm U}(1)$-valued
maps $\,K_{(i^1,i^2)(j^1,j^2)}=K^{-1}_{(j^1,j^2)(i^1,i^2)}\,$ on
$\,\mathcal{O}^M_{i^1 j^1}\x\mathcal{O}^M_{i^2 j^2}$.\ The data are required to
satisfy the identities
\begin{eqnarray}
{\rm pr}_1^*\bigl(\theta_{{\mathsf T}^*{\mathsf L} M}+\pi_{{\mathsf T}^*{\mathsf L} M}^*E_{\gt{i}^1}
\bigr)-{\rm pr}_2^*\bigl(\theta_{{\mathsf T}^*{\mathsf L} M}+\pi_{{\mathsf T}^*{\mathsf L} M}^*
E_{\gt{i}^2}\bigr)&=&-{\mathsf i}\,\delta\log f_{\sigma\,(\gt{i}_1,\gt{i}_2)}\,,\cr
\label{eq:dual-preq-triv}&&\\ {\rm pr}_1^* \pi_{{\mathsf T}^*{\mathsf L} M}^*G_{\gt{i}^1
\gt{j}^1}\cdot{\rm pr}_2^*\pi_{{\mathsf T}^*{\mathsf L} M}^*G_{\gt{i}^2\gt{j}^2}^{-1}&=&
f_{\sigma\,(\gt{i}_1,\gt{i}_2)}\cdot f_{\sigma\,(\gt{j}_1,\gt{j}_2)}^{-1}\,,
\nonumber
\end{eqnarray}
written in terms of the smooth ${\rm U}(1)$-valued functionals
\begin{eqnarray}\label{eq:Phifa}
f_{\sigma\,(\gt{i}_1,\gt{i}_2)}={\rm e}^{-{\mathsf i}\,\Phi_{\sigma\,(\gt{i}_1,\gt{i}_2)}}
\end{eqnarray}
and of the canonical projections $\,{\rm pr}_\a:{\mathsf P}_{\sigma,\emptyset}\x
{\mathsf P}_{\sigma,\emptyset}\to{\mathsf P}_{\sigma,\emptyset},\ \a\in\{1,2\}$,\ the
canonical 1-form $\,\theta_{{\mathsf T}^*{\mathsf L} M}\,$ on the total space of
the cotangent bundle $\,\pi_{{\mathsf T}^*{\mathsf L} M}:{\mathsf T}^*{\mathsf L} M\to{\mathsf L} M\,$
from Proposition \ref{prop:sympl-form-si-untw} and the local data
$\,(E_\gt{i},G_{\gt{i}\gt{j}})\,$ of the transgression bundle
$\,\mathcal{L}_\mathcal{G}\to{\mathsf L} M\,$ from Theorem \ref{thm:trans-untw}.
Analogously, we call $\,(\gt{I}_\sigma,\gt{D}_\sigma)\,$ a \textbf{pre-quantum
duality of type $N$ of the untwisted sector of the non-linear
$\sigma$-model for network-field configurations $\,(X\,\vert\,\Gamma)\,$ in
string background $\,\gt{B}\,$ on world-sheet $\,(\Sigma,\gamma )\,$ with
defect quiver $\,\Gamma\,$} iff, in the same notation, its local data
can be put in the form
\begin{eqnarray}\label{eq:Phi-unipot}
\Phi_{\sigma\,(\gt{i}_1,\gt{i}_2)}[(\psi_1,\psi_2)]=-\int_{{\mathbb{S}}^1}\,\Vol(
{\mathbb{S}}^1)\,{\mathsf p}_{2\,\mu}\,F^\mu(X_1)+W_{(\gt{i}^1,\gt{i}^2)}[(\psi_1,
\psi_2)]
\end{eqnarray}
for an arbitrary smooth map
\begin{eqnarray}\nonumber
F\ :\ M\to M\,,
\end{eqnarray}
and for smooth real-valued functionals $\,W_{(\gt{i}^1,\gt{i}^2 )}\,$ on
the $\,\mathcal{O}^{\gt{I}_\sigma}_{(\gt{i}^1,\gt{i}^2)}$,\ of the form
\begin{eqnarray}
W_{(\gt{i}_1,\gt{i}_2)}[(\psi_1,\psi_2)]=\sum_{e\in\triangle({\mathbb{S}}^1)}\,
\int_e\,(X_{1\,e},X_{2\,e})^*P_{(i^1_e,i^2_e)}+{\mathsf i}\,\sum_{v\in
\triangle({\mathbb{S}}^1)}\,(X_1,X_2)^*\log K_{(i^1_{e_+(v)},i^2_{e_+(v)})(
i^1_{e_-(v)},i^2_{e_-(v)})}(v)\,,\cr\cr\label{eq:Wi1i2}
\end{eqnarray}
with $\,P_{(i^1,i^2)}\in\Omega^1(\mathcal{O}^M_{i^1}\x\mathcal{O}^M_{i^2})\,$ and
$\,K_{(i^1,i^2)(j^1,j^2)}=K^{-1}_{(j^1,j^2)(i^1,i^2)}\in
{\rm U}(1)_{\mathcal{O}^M_{i^1 j^1}\x\mathcal{O}^M_{i^2 j^2}}$.\ (It is understood that
there is no dependence on the $\,{\mathsf p}_\a\,$ in $\,W_{(\gt{i}^1,\gt{i}^2
)}$.) Here, the identities to be satisfied by $\,f_{\sigma\,(\gt{i}_1,
\gt{i}_2)}\,$ as in \Reqref{eq:Phifa} read
\begin{eqnarray}
{\rm pr}_1^*\bigl(\theta_{{\mathsf T}^*{\mathsf L} M}+\pi_{{\mathsf T}^*{\mathsf L} M}^*E_{\gt{i}^1}
\bigr)-{\rm pr}_2^*\bigl(\theta_{{\mathsf T}^*{\mathsf L} M}^*+\pi_{{\mathsf T}^*{\mathsf L} M}^*
E_{\gt{i}^2}\bigr)&=&-{\mathsf i}\,\delta\log f_{\sigma\,(\gt{i}_1,\gt{i}_2)}\,,\cr
\label{eq:dual-preq-triv-bis}&&\\
{\rm pr}_1^*\pi_{{\mathsf T}^*{\mathsf L} M}^*G_{\gt{i}^1\gt{j}^1}\cdot{\rm pr}_2^*\pi_{{\mathsf T}^*
{\mathsf L} M}^*G_{\gt{i}^2\gt{j}^2}^{-1}&=&f_{\sigma \,(\gt{i}_1,\gt{i}_2)}\cdot
f_{\sigma\,(\gt{j}_1,\gt{j}_2)}^{-1}\,,\nonumber
\end{eqnarray}
where
\begin{eqnarray}\nonumber
\theta_{{\mathsf T}^*{\mathsf L} M}^*[(X,{\mathsf p})]=-\int_{{\mathbb{S}}^1}\,\Vol({\mathbb{S}}^1)\wedge
X^\mu\,\delta{\mathsf p}_\mu\,.
\end{eqnarray}
\begin{flushright}$\checkmark$\end{flushright}\end{Def} \begin{Rem} The form of the edge terms in the definition
\eqref{eq:Phi-om-loc} of the local data of the generating functional
of the duality of type $T$ is dictated by the requirement that the
ensuing canonical transformation induce a linear map $\,(X_{1\,
*}\widehat t,{\mathsf p}_1)\mapsto(X_{2\,*}\widehat t,{\mathsf p}_2)$,\ as
discussed earlier in this section and in Alvarez's papers, and the
vertex corrections are perfectly consistent with this requirement in
the local description of the generating functional.
The expression \eqref{eq:Phi-unipot} defining the generating
functional of the duality of type $N$, on the other hand, should be
regarded as a natural local deformation of the global generating
functional
\begin{eqnarray}\nonumber
\Phi_{\rm id}[(\psi_1,\psi_2)]=-\int_{{\mathbb{S}}^1}\,\Vol({\mathbb{S}}^1)\,{\mathsf p}_{2\,\mu}
\,X_1^\mu\,,
\end{eqnarray}
readily verified to yield the identity canonical transformation on
$\,{\mathsf P}_{\sigma,\emptyset}$.\ Thus, unlike dualities of type $T$,
dualities of type $N$ are continuously deformable to the trivial
duality (i.e.\ to the identity symplectomorphism).\end{Rem}\begin{center}*\hspace{2cm}*\hspace{2cm}*\end{center}
We are now ready to present our findings which can be summarised as
follows
\begin{Thm}\label{thm:duali-T-bib} Let $\,\gt{B}\,$ be a string background
with target $\,\mathcal{M}=(M,{\rm g},\mathcal{G})$.\ Consider the non-linear
$\sigma$-model for network-field configurations $\,(X\,\vert\,\Gamma)\,$ in
string background $\,\gt{B}\,$ on world-sheet $\,(\Sigma,\gamma)\,$ with
defect quiver $\,\Gamma$.\ For every duality $\,(\gt{I}_\sigma,\gt{D}_\sigma)\,$
of type $T$ of the $\sigma$-model, there exists a topological defect
with a $\mathcal{G}$-bi-brane $\,\mathcal{B}_{\gt{D}_\sigma}=\bigl(Q,\iota_\a,\omega,\Phi\
\vert\ \a\in\{1,2\} \bigr)\,$ over it with the following properties:
\begin{itemize}
\item[i)] the world-volume $\,Q\,$ is a submanifold of the cartesian
square $\,M\x M\,$ of the target space $\,M$;
\item[ii)] the $\mathcal{G}$-bi-brane maps are given by the canonical
projections $\,\iota_\a={\rm pr}_\a:Q\to M,\ \a\in\{1,2\}$;
\item[iii)] $Q\,$ carries a symplectic form
\begin{eqnarray}\nonumber
\Omega_{\gt{D}_\sigma}(X_1,X_2):=\omega^{1\wedge 2}_{\mu\nu}(X_1,X_2)\,{\mathsf d}
X_1^\mu\wedge{\mathsf d} X_2^\nu\,,\qquad\qquad(X_1,X_2)\in Q
\end{eqnarray}
defined by the curvature (no summation over the repeated indices)
\begin{eqnarray}\label{eq:curv-dualB-wedge}
\omega(X_1,X_2)=\sum_{0<i\leq j<3}\,(-1)^{i+1}\,\omega_{i\wedge j}\,,
\qquad\qquad \omega_{i\wedge j}=\omega^{i\wedge j}_{\mu\nu}(X_1,X_2)\,{\mathsf d}
X_i^\mu\wedge{\mathsf d} X_j^\nu
\end{eqnarray}
of $\,\mathcal{B}_{\gt{D}_\sigma}$,\ the latter being given in terms of globally
smooth maps $\,\omega^{i\wedge j}_{\mu\nu}\in C^\infty(Q,{\mathbb{R}})$;
\item[iv)] the \textbf{duality background}
\begin{eqnarray}\nonumber
\gt{B}_{\gt{D}_\sigma}=(\mathcal{M},\mathcal{B}_{\gt{D}_\sigma},\cdot)
\end{eqnarray}
satisfies the \textbf{duality-background constraints}
\begin{eqnarray}\label{eq:dualiT-back-constr}
{\rm E}_2=-\omega_{1\wedge 2}\circ{\rm E}_1^{-1}\circ\omega_{1\wedge 2}\,,
\end{eqnarray}
written in terms of the \textbf{background operators}
\begin{eqnarray}\nonumber
{\rm E}_\a=:{\rm g}_\a+\omega_{\a\wedge\a}\ :\ \Gamma({\mathsf T} Q)\to\Gamma({\mathsf T}^*Q)\ :\
\mathscr{V}\mapsto{\rm E}_\a(\mathscr{V},\cdot)\,,\qquad\qquad{\rm g}_\a={\rm pr}_\a^*{\rm g}
\,,\qquad\a\in\{1,2\}\,.
\end{eqnarray}
\end{itemize}
\end{Thm}
\noindent\begin{proof}
Let us adopt the notation of Definition \ref{def:dualiTN}. Using the
identity
\begin{eqnarray}\nonumber
\delta\int_e\,X_e^*\eta=-\int_e\,X_\ell^*\delta\eta+X^*\eta\vert_{\p e}\,,
\end{eqnarray}
valid for an arbitrary edge $\,e\in\triangle({\mathbb{S}}^1)\,$ and for any
$\,\eta\in\Omega^1\bigl(X(e)\bigr)$,\ we readily extract from the first
of identities \eqref{eq:dual-preq-triv} the relations
\begin{eqnarray}
{\mathsf p}_1-{\mathsf p}_2=(X_{1\,*}\widehat t,X_{2\,*}\widehat t)\righthalfcup\bigl(
{\rm pr}_1^*B_{i^1}-{\rm pr}_2^*B_{i^2}+{\mathsf d} P_{(i^1,i^2)}\bigr)(X_1,X_2)\,,
\label{eq:p-diff-symm}\\\cr {\rm pr}_1^*A_{i^1 j^1}-{\rm pr}_2^*A_{i^2 j^2}+
P_{(j^1,j^2)}-P_{(i^1,i^2)}-{\mathsf i}\,{\mathsf d}\log K_{(i^1,i^2)(j^1,j^2)}=0
\,, \label{eq:can-trans-cond-loc}
\end{eqnarray}
implied by the requirement that both the edge term and the vertex
term of the identity vanish independently. The relations are to be
satisfied on the submanifold $\,Q\subset M\x M\,$ obtained by taking
the set of all pairs of points in $\,M\,$ intersected by pairs of
loops from $\,(\pi_{{\mathsf T}^*{\mathsf L} M},\pi_{{\mathsf T}^*{\mathsf L} M})(\gt{I}_\sigma)$.\
The manifold $\,Q\,$ canonically projects \emph{onto} $\,M$.\ Taking
the exterior derivative of both sides of
\Reqref{eq:can-trans-cond-loc} and, subsequently, using
\Reqref{eq:DG-is-H}, we obtain the equality
\begin{eqnarray}\nonumber
{\rm pr}_1^*B_{j^1}-{\rm pr}_2^*B_{j^2}+{\mathsf d} P_{(j^1,j^2)}={\rm pr}_1^*B_{i^1}-
{\rm pr}_2^*B_{i^2}+{\mathsf d} P_{(i^1,i^2)}\,,
\end{eqnarray}
from which we infer the existence of a globally defined 2-form
\begin{eqnarray}\label{eq:om-def}
\omega:={\rm pr}_1^*B_{i^1}-{\rm pr}_2^*B_{i^2}+{\mathsf d} P_{(i^1,i^2)}\in\Gamma(\wedge^2
{\mathsf T}^*Q)\,.
\end{eqnarray}
This is in keeping with \Reqref{eq:p-diff-symm} as the latter
requires that the expression on the right-hand side be a smooth
1-form. We also note that it yields a relation
\begin{eqnarray}
K_{(j^1,j^2)(k^1,k^2)}\cdot K_{(i^1,i^2)(k^1,k^2)}^{-1}\cdot K_{(i^1
,i^2)(j^1,j^2)}\cdot{\rm pr}_2^*g_{i^2 j^2 k^2}\cdot{\rm pr}_1^*g_{i^1 j^1
k^1}^{-1}=:C_{(i^1,i^2)(j^1,j^2)(k^1,k^2)}\,,\cr\cr\label{eq:KC-gg-C}
\end{eqnarray}
in which $\,(C_{(i^1,i^2)(j^1,j^2)(k^1,k^2)})\,$ is a locally
constant ${\rm U}(1)$-valued \v Cech 2-cochain on $\,Q$.\ Clearly,
\begin{eqnarray}\label{eq:Cobstr-cech}
\bigl(\check{\d}^{(2)}C\bigr)_{(i^1,i^2)(j^1,j^2)(k^1,k^2)(l^1,l^2)}=1\,,
\end{eqnarray}
and the class $\,[(C_{(i^1,i^2 )(j^1,j^2)(k^1,k^2)})]\in\check{H}^2\bigl(Q
,{\rm U}(1)\bigr)\,$ is readily seen to define the obstruction to the
existence of a $\mathcal{G}$-bi-brane $\,(Q,\omega,{\rm pr}_1,{\rm pr}_2,\Phi)\,$ with
1-isomorphism $\,\Phi:{\rm pr}_1^*\mathcal{G}\xrightarrow{\cong}{\rm pr}_2^*\mathcal{G}\otimes
I_\omega\,$ with local data $\,(P_{(i^1,i^2)},K_{(i^1,i^2)(j^1,j^2)}
)$.\ Indeed, Eqs.\,\eqref{eq:can-trans-cond-loc}, \eqref{eq:om-def}
and \eqref{eq:KC-gg-C} can be rewritten concisely in the familiar
form
\begin{eqnarray}\nonumber
{\rm pr}_1^*(B_{i^1},A_{i^1 j^1},g_{i^1 j^1 k^1})+D_{(1)}(P_{(i^1,i^2)},
K_{(i^1,i^2)(j^1,j^2)})+(0,0,C_{(i^1,i^2)(j^1,j^2)(k^1,k^2)})\cr\cr
={\rm pr}_2^*(B_{i^2},A_{i^2 j^2},g_{i^2 j^2 k^2})+(\omega\vert_{\mathcal{O}^Q_{(i_1
,i^2)}},0,0)\,,
\end{eqnarray}
and it is immediately clear that a pair of 2-cochains $\,(C_{(i^1,
i^2)(j^1,j^2)(k^1,k^2)})\,$ and $\,(C_{(i^1,i^2)(j^1,j^2)(k^1,k^2)}
')\,$ cohomologous as per $\,(C_{(i^1,i^2)(j^1,j^2)(k^1,k^2)}')=(
C_{(i^1, i^2)(j^1,j^2)(k^1,k^2)})\cdot\check{\d}^{(1)}c\,$ for some
(locally constant) 1-cochain $\,c\,$ corresponds to a pair of
1-cochains $\,( K_{(i^1,i^2)(j^1,j^2)})\,$ and
$\,(K_{(i^1,i^2)(j^1,j^2)}')\,$ related by the shift
$\,(K_{(i^1,i^2)(j^1,j^2)}')=(K_{(i^1,i^2)(j^1 ,j^2)})\cdot
c^{-1}$.\
Finally, \Reqref{eq:p-diff-symm} rewrites as
\begin{eqnarray}\label{eq:DGC-from-dualiT}
{\mathsf p}_1-{\mathsf p}_2-(X_{1\,*}\widehat t,X_{2\,*}\widehat t)\righthalfcup\omega(X_1,X_2
)=0\,,
\end{eqnarray}
and so we recover the complete description of a conformal defect up
to the obstruction\linebreak
$\,[(C_{(i^1,i^2)(j^1,j^2)(k^1,k^2)})]$.\ The latter is removed on
taking into account the second of identities
\eqref{eq:dual-preq-triv}. Indeed, using
Eqs.\,\eqref{eq:can-trans-cond-loc} and \eqref{eq:KC-gg-C}, we
readily cast the above relation in the compact form
\begin{eqnarray}\nonumber
\prod_{\ovl v\in\triangle({\mathbb{S}}^1)}\,(X_1,X_2)^*\left(\tfrac{C_{(
i^1_{\ovl e_+(\ovl v)},i^2_{\ovl e_+(\ovl v)})(i^1_{\ovl e_-(\ovl v
)},i^2_{\ovl e_-(\ovl v)})(j^1_{\ovl e_+(\ovl v)},j^2_{\ovl e_+(\ovl
v )})}}{C_{(j^1_{\ovl e_+(\ovl v)},j^2_{\ovl e_+(\ovl v)})(j^1_{\ovl
e_-(\ovl v)},j^2_{\ovl e_-(\ovl v)})(i^1_{\ovl e_-(\ovl v)},
i^2_{\ovl e_-(\ovl v)})}}\right)(\ovl v)=1\,,
\end{eqnarray}
which -- in view of the arbitrariness of $\,(X_1,X_2)(\ovl v)\,$ and
of the triangulation used -- requires
\begin{eqnarray}\label{eq:C-is-C}
C_{(i^1,i^2)(j^1,j^2)(k^1,k^2)}=C_{(i^1,i^2)(j^1,j^2)(l^1,l^2)}
\end{eqnarray}
for any pair of quadruples $\,(i^\a,j^\a,k^\a,l^\a)\in\mathscr{I}_M^4,\ \a
\in\{1,2\}\,$ such that $\,\mathcal{O}^M_{i^\a j^\a k^\a l^\a}\neq
\emptyset$.\ Hence,
\begin{eqnarray}\nonumber
C_{(i^1,i^2)(j^1,j^2)(k^1,k^2)}=C_{(i^1,i^2)(j^1,j^2)(j^1,j^2)}=:
\widetilde C_{(i^1,i^2)(j^1,j^2)}\,,
\end{eqnarray}
and the newly defined maps $\,\widetilde C_{(i^1,i^2)(j^1,j^2)}$,\
with values in the set $\,\{-1,1\}$,\ form a locally constant
2-cochain -- in particular,
\begin{eqnarray}\nonumber
\widetilde C_{(j^1,j^2)(i^1,i^2)}=\widetilde C_{(i^1,i^2)(j^1,j^2
)}^{-1}\,.
\end{eqnarray}
Using Eqs.\,\eqref{eq:Cobstr-cech} and \eqref{eq:C-is-C}, we then
find
\begin{eqnarray}\nonumber
C_{(i^1,i^2)(j^1,j^2)(k^1,k^2)}&=&C_{(j^1,j^2)(k^1,k^2)(l^1,l^2)}
\cdot C_{(i^1,i^2)(k^1,k^2)(l^1,l^2)}^{-1}\cdot C_{(i^1,i^2)(j^1,j^2
)(l^1,l^2)}\cr\cr
&=&\widetilde C_{(j^1,j^2)(k^1,k^2)}\cdot\widetilde C_{(i^1 ,i^2)(
k^1,k^2)}^{-1}\cdot\widetilde C_{(i^1,i^2)(j^1,j^2)}\,.
\end{eqnarray}
Clearly, the \v Cech cohomology class of this 2-cochain is trivial
and it can be absorbed into a redefinition of the local data of the
1-isomorphism $\,\Phi$,\ cf.\ \Reqref{eq:KC-gg-C}. This leaves us
with statements iii) and iv) of the theorem to demonstrate.
The remainder of the proof uses solely elementary analysis of
canonical transformations defined in terms of generating functions,
cf., e.g., \Rxcite{Sec.\,6.5}{Marsden:1994}. Thus, upon recalling
that the space $\,\gt{I}_\sigma\,$ is -- by assumption -- diffeomorphic
to $\,{\mathsf T}^*{\mathsf L} M$,\ we can choose the loop variables $\,(X_1,X_2
)\,$ as independent local coordinates on $\,\gt{I}_\sigma$,\ which has
the following two consequences: First of all, \Reqref{eq:om-def}
yields three independent relations:
\begin{eqnarray}
\omega_{1\wedge 1}&=&{\rm pr}_1^*B_{i^1}+[{\mathsf d} P_{(i^1,i^2)}]_{1\wedge 1}\,,
\label{eq:om-1wedge1}\\\cr \omega_{2\wedge 2}&=&{\rm pr}_2^*B_{i^2}-[{\mathsf d}
P_{(i^1,i^2)}]_{2\wedge 2}\,,\label{eq:om-2wedge2}\\\cr \omega_{1\wedge
2}&=&[{\mathsf d} P_{(i^1,i^2)}]_{1\wedge 2}\,, \label{eq:om-1wedge2}
\end{eqnarray}
written in terms of the components $\,\omega_{i\wedge j}\,$ of
$\,\omega\,$ from \Reqref{eq:curv-dualB-wedge} and those of $\,{\mathsf d}
P_{(i^1,i^2)}$,\ defined analogously. Secondly, we may extract from
\Reqref{eq:DGC-from-dualiT} a pair of coupled equations
\begin{eqnarray}\nonumber
\left(\begin{array}{cc} -\omega_{1\wedge 1} & {\rm id}_{\Gamma({\mathsf T}^*{\mathsf L} M)}\cr\cr
-\omega_{1\wedge 2} & 0 \end{array}\right)\, \left(\begin{array}{c} X_{1\,*}\widehat
t\cr\cr {\mathsf p}_1 \end{array}\right)=\left(\begin{array}{cc} \omega_{1\wedge 2} & 0
\cr\cr -\omega_{2\wedge 2} & {\rm id}_{\Gamma({\mathsf T}^*{\mathsf L} M)} \end{array}\right)\,
\left(\begin{array}{c} X_{2\,*}\widehat t \cr\cr {\mathsf p}_2 \end{array}\right)\,,
\end{eqnarray}
to be understood as representing the action of linear operators on
sections of $\,{\mathsf T}{\mathsf L} Q\oplus{\mathsf T}^*{\mathsf L} Q$,\ with 2-form fields
acting on vector fields through contraction, i.e.\ as per $\,\omega_{i
\wedge j}\vartriangleright X_{\a\,*}\widehat t:=X_{\a\,*}\widehat t\righthalfcup\omega_{i
\wedge j}$.\ Clearly, for the transformation between the two pairs
$\,\bigl(X_{\a \,*}\widehat t,{\mathsf p}_\a\bigr),\ \a\in\{1,2\}\,$ thus
defined to be invertible, we have to demand that $\,\omega_{1 \wedge
2}$,\ regarded as a map from $\,\Gamma({\mathsf T} Q)\,$ to $\,\Gamma({\mathsf T}^* Q)$,\
possess an inverse,
\begin{eqnarray}\nonumber
\omega_{1\wedge 2}^{-1}=\tfrac{1}{4}\,\bigl(\bigl(\omega^{1\wedge 2}
\bigr)^{-1}\bigr)^{\mu\nu}\,\p_\mu\wedge\p_\nu\,,\qquad\qquad\bigl(
\bigl(\omega^{1\wedge 2}\bigr)^{-1}\bigr)^{\lambda\mu}\,\omega^{1\wedge
2}_{\mu\nu}=\delta^\lambda_{\ \nu}\,,
\end{eqnarray}
acting on 1-forms as $\,\omega_{1\wedge 2}^{-1}\vartriangleright(\eta_\mu\,{\mathsf d}
X^\mu):=\frac{1}{2}\,\bigl(\bigl(\omega^{1\wedge 2}\bigr)^{-1}
\bigr)^{\mu\nu}\,\eta_\mu\,\p_\nu$.\ This proves statement iii) of
the theorem. Having ensured the invertibility of $\,\omega_{1\wedge
2}$,\ we may express $\,\bigl(X_{\a \,*}\widehat t,{\mathsf p}_\a\bigr)\,$
through $\,\bigl(X_{3-\a\,*}\widehat t,{\mathsf p}_{3-\a}\bigr)$.\
Demanding that $\,\mathcal{H}^-_\sigma\,$ of \Reqref{eq:Hsi-} vanish
identically on $\,\gt{I}_\sigma\,$ then produces a relation
\begin{eqnarray}\label{eq:MgMg-om}
{\rm M}_\omega^{\rm T}\circ\widehat{\rm g}_1\circ{\rm M}_\omega=\widehat{\rm g}_2\,,
\end{eqnarray}
written in terms of the operators
\begin{eqnarray}\nonumber
\widehat{\rm g}_\a=\left(\begin{array}{cc} {\rm g}_\a & 0 \cr\cr 0 & {\rm g}^{-1}_\a
\end{array}\right)\,,\qquad\a\in\{1,2\}
\end{eqnarray}
and
\begin{eqnarray}\nonumber
{\rm M}_\omega=\left(\begin{array}{cc} \omega_{1\wedge 2}^{-1}\circ\omega_{2\wedge 2} &
-\omega_{1\wedge 2}^{-1}\cr\cr \omega_{1\wedge 2}+\omega_{1\wedge 1}\circ
\omega_{1\wedge 2}^{-1}\circ\omega_{2\wedge 2} & -\omega_{1\wedge 1}\circ
\omega_{1\wedge 2}^{-1} \end{array}\right)\,,
\end{eqnarray}
and of the transpose of the latter,
\begin{eqnarray}\nonumber
{\rm M}_\omega^{\rm T}=\left(\begin{array}{cc} \omega_{2\wedge 2}\circ\omega_{1\wedge
2}^{-1} & -\omega_{1\wedge 2}-\omega_{2\wedge 2}\circ\omega_{1\wedge 2}^{-1}
\circ\omega_{1\wedge 1}\cr\cr \omega_{1\wedge 2}^{-1} & -\omega_{1\wedge
2}^{-1}\circ\omega_{1\wedge 1} \end{array}\right)\,.
\end{eqnarray}
We shall next demonstrate that \Reqref{eq:MgMg-om} is equivalent to
the duality-background constraints \eqref{eq:dualiT-back-constr}. To
this end, we first note that the latter actually encodes a pair of
independent relations for the symmetric and antisymmetric component
of the background operator $\,{\rm E}_2$,\ respectively. Explicitly,
\begin{eqnarray}\nonumber
{\rm g}_2&=&-\tfrac{1}{2}\,\omega_{1\wedge 2}\circ\bigl({\rm E}_1^{-1}+
{\rm E}_1^{-1\,{\rm T}}\bigr)\circ\omega_{1\wedge 2}\equiv-\tfrac{1}{2}\,
\omega_{1\wedge 2}\circ\bigl[({\rm g}_1+\omega_{1\wedge 1})^{-1}\circ({\rm g}_1
-\omega_{1\wedge 1})\circ({\rm g}_1-\omega_{1\wedge 1})^{-1}\cr\cr
&&+({\rm g}_1+\omega_{1\wedge 1})^{-1}\circ({\rm g}_1+\omega_{1\wedge 1})\circ(
{\rm g}_1-\omega_{1\wedge 1})^{-1}\bigr]\circ\omega_{1\wedge 2}=-\omega_{1
\wedge 2}\circ({\rm g}_1+\omega_{1\wedge 1})^{-1}\circ{\rm g}_1\circ({\rm g}_1-
\omega_{1\wedge 1})^{-1}\circ\omega_{1\wedge 2}\cr\cr
&=&-\omega_{1\wedge 2}\circ\bigl[({\rm g}_1-\omega_{1\wedge 1})\circ
{\rm g}_1^{-1}\circ({\rm g}_1+\omega_{1\wedge 1})\bigr]^{-1}\circ\omega_{1
\wedge 2}\equiv-\omega_{1\wedge 2}\circ({\rm g}_1-\omega_{1\wedge 1}\circ
{\rm g}_1^{-1}\circ\omega_{1\wedge 1})^{-1}\circ\omega_{1\wedge 2}\,,
\end{eqnarray}
and, analogously,
\begin{eqnarray}\nonumber
\omega_{2\wedge 2}=\omega_{1\wedge 2}\circ{\rm g}_1^{-1}\circ\omega_{1\wedge 1}
\circ({\rm g}_1-\omega_{1\wedge 1}\circ {\rm g}_1^{-1}\circ\omega_{1\wedge 1}
)^{-1}\circ\omega_{1\wedge 2}\,.
\end{eqnarray}
The above are to be compared with the independent relations
determined by the continuity constraint \eqref{eq:MgMg-om}. These
are easily found to be
\begin{eqnarray}\nonumber
{\rm g}_2^{-1}&=&\omega_{1\wedge 2}^{-1}\circ(\omega_{1\wedge 1}\circ
{\rm g}_1^{-1}\circ\omega_{1\wedge 1}-{\rm g}_1)\circ\omega_{1\wedge 2}^{-1}\,,
\cr\cr
{\rm g}_2&=&\omega_{2\wedge 2}\circ\omega_{1\wedge 2}^{-1}\circ{\rm g}_1\circ
\omega_{1\wedge 2}^{-1}\circ\omega_{2\wedge 2}-\bigl(\omega_{2\wedge 2}\circ
\omega_{1\wedge 2}^{-1}\circ\omega_{1\wedge 1}+\omega_{1\wedge 2}\bigr)\circ
{\rm g}_1^{-1}\circ\bigl(\omega_{1\wedge 2}+\omega_{1\wedge 1}\circ\omega_{1
\wedge 2}^{-1}\circ\omega_{2\wedge 2}\bigr)\,,\cr\cr 0&=&\omega_{1\wedge
2}^{-1}\circ{\rm g}_1\circ\omega_{1\wedge 2}^{-1}\circ\omega_{2\wedge 2}-
\omega_{1\wedge 2}^{-1}\circ\omega_{1\wedge 1}\circ{\rm g}_1^{-1}\circ\bigl(
\omega_{1\wedge 2}+\omega_{1\wedge 1}\circ\omega_{1\wedge 2}^{-1}\circ\omega_{2
\wedge 2}\bigr)\,.
\end{eqnarray}
The remaining relation is a transpose of the bottom one. Evidently,
the top one is an inverse of the symmetric component of
\Reqref{eq:dualiT-back-constr}, and so we are left with the other
two to examine.
Upon using the bottom relation in the middle one, we reduce the
latter to the form
\begin{eqnarray}\label{eq:rem-g2}
{\rm g}_2=-\omega_{1\wedge 2}\circ{\rm g}_1^{-1}\circ\bigl(\omega_{1\wedge 2}+
\omega_{1\wedge 1}\circ\omega_{1\wedge 2}^{-1}\circ\omega_{2\wedge 2}\bigr)
\,,
\end{eqnarray}
which can be combined with (the inverse of) the top one and
subsequently substituted back into the bottom relation to yield
\begin{eqnarray}\nonumber
\omega_{2\wedge 2}&=&\omega_{1\wedge 2}\circ{\rm g}_1^{-1}\circ\omega_{1\wedge
1}\circ{\rm g}_1^{-1}\circ\bigl(\omega_{1\wedge 2}+\omega_{1\wedge 1}\circ
\omega_{1\wedge 2}^{-1}\circ\omega_{2\wedge 2}\bigr)=-\omega_{1\wedge 2}\circ
{\rm g}_1^{-1}\circ\omega_{1\wedge 1}\circ\omega_{1\wedge 2}^{-1}\circ{\rm g}_2
\cr\cr
&=&-\omega_{1\wedge 2}\circ{\rm g}_1^{-1}\circ\omega_{1\wedge 1}\circ\bigl(
\omega_{1\wedge 1}\circ{\rm g}_1^{-1}\circ\omega_{1\wedge 1}-{\rm g}_1\bigr)^{-
1}\circ\omega_{1\wedge 2}\,,
\end{eqnarray}
which is the desired form of the antisymmetric component of
\Reqref{eq:dualiT-back-constr}. At this stage, it remains to verify
that the two components found hitherto ensure that the remaining
relation \eqref{eq:rem-g2} is satisfied identically. With $\,\omega_{2
\wedge 2}\,$ as above, its right-hand side takes the form
\begin{eqnarray}\nonumber
{\rm g}_2=-\omega_{1\wedge 2} \circ{\rm g}_1^{-1}\circ\bigl[\omega_{1\wedge 2}-
\omega_{1\wedge 1}\circ{\rm g}_1^{-1}\circ\omega_{1\wedge 1}\circ\bigl(
\omega_{1\wedge 1}\circ{\rm g}_1^{-1}\circ\omega_{1\wedge 1}-{\rm g}_1\bigr)^{-
1}\circ\omega_{1\wedge 2}\bigr]\,,
\end{eqnarray}
and so we must show the identity
\begin{eqnarray}\nonumber
-\omega_{1\wedge 2}\circ {\rm g}_1^{-1}\circ\bigl[\omega_{1\wedge 2}-\omega_{1
\wedge 1}\circ{\rm g}_1^{-1}\circ\omega_{1\wedge 1}\circ\bigl(\omega_{1\wedge
1}\circ{\rm g}_1^{-1}\circ\omega_{1\wedge 1}-{\rm g}_1\bigr)^{-1}\circ\omega_{1
\wedge 2}\bigr]\cr\cr =\omega_{1\wedge 2}\circ(\omega_{1\wedge 1}\circ
{\rm g}_1^{-1}\circ\omega_{1\wedge 1}-{\rm g}_1)^{-1}\circ\omega_{1\wedge 2}\,,
\end{eqnarray}
which follows straightforwardly upon regrouping its terms.\qed\end{proof}
\medskip
A similar result can be established for dualities of type $N$,\
namely,
\begin{Thm}\label{thm:duali-N-bib}
Let $\,\gt{B}\,$ be a string background with target $\,\mathcal{M}=(M,{\rm g},\mathcal{G}
)$.\ Consider the non-linear $\sigma$-model for network-field
configurations $\,(X\,\vert\,\Gamma)\,$ in string background $\,\gt{B}\,$
on world-sheet $\,(\Sigma,\gamma)\,$ with defect quiver $\,\Gamma$.\ To every
duality $\,(\gt{I}_\sigma, \gt{D}_\sigma)\,$ of type $N$ of the $\sigma$-model,
there is associated a topological defect with a $\mathcal{G}$-bi-brane
$\,\mathcal{B}_{\gt{D}_\sigma}=\bigl(Q, \iota_\a,\omega,\Phi\ \vert \
\a\in\{1,2\}\bigr)\,$ over it with the following properties:
\begin{itemize}
\item[i)] the world-volume $\,Q\,$ is a submanifold $\,Q=({\rm id}_M\x F)
(M)\subset M\x M\,$ of the cartesian square $\,M\x M\,$ of the
target space $\,M$;
\item[ii)] $F\,$ is an isometry of the metric manifold $\,(M,{\rm g})$;
\item[iii)] the $\mathcal{G}$-bi-brane maps are given by the canonical
projections $\,\iota_\a={\rm pr}_\a:Q\to M,\ \a\in\{1,2\}$;
\item[iv)] the curvature $\,\omega\,$ vanishes identically;
\item[v)] the pullback of the 1-isomorphism $\,\Phi\,$ along
the isomorphism $\,{\rm id}_M\x F:M\xrightarrow{\cong}Q\,$ is of the form
\begin{eqnarray}\label{eq:iso-for-sympl-nom}
({\rm id}_M\x F)^*\Phi\ :\ \mathcal{G}\xrightarrow{\cong}F^*\mathcal{G}\,.
\end{eqnarray}
\end{itemize}
\end{Thm}
\noindent\begin{proof}
We use the notation of Definition \ref{def:dualiTN} and -- reasoning
as in the proof of Theorem \ref{thm:duali-T-bib} -- choose
$\,(X_1,{\mathsf p}_2)\,$ as independent local coordinates on
$\,\gt{I}_\sigma$,\ which leads to the relation
\begin{eqnarray}\label{eq:XFX}
X_2=F[X_1]\,,
\end{eqnarray}
extracted from the first of identities \eqref{eq:dual-preq-triv}.
Clearly, for the generating functional \eqref{eq:Phi-unipot} to
define a duality of the $\sigma$-model, $\,F\,$ has to induce an
invertible map on $\,M$.\ The remaining relation encoded by the
first of identities \eqref{eq:dual-preq-triv} reads
\begin{eqnarray}\label{eq:sep-sympl-id}
\int_{{\mathbb{S}}^1}\,\Vol({\mathbb{S}}^1)\wedge\bigl({\mathsf p}_1-\widehat F_*\vartriangleright{\mathsf p}_2
\bigr)=E_{\gt{i}_2}[X_2]-E_{\gt{i}_1}[X_1]-\delta W_{\gt{i}_1\gt{i}_2}[(\psi_1,
\psi_2)]\,,
\end{eqnarray}
where we introduced the operator $\,\widehat F_*=\frac{\delta F^\mu}{\delta
X_1^\nu}\,\frac{\delta \ }{\delta X_2^\mu}\otimes\delta X_1^\nu$,\ acting on
$\,{\mathsf p}_2={\mathsf p}_{2\,\mu}\,\delta X_2^\mu\,$ through contraction,
\begin{eqnarray}\nonumber
\widehat F_*\vartriangleright\delta X_2^\mu:=\frac{\delta F^\mu}{\delta X_1^\nu}\,\delta
X_1^\nu\,.
\end{eqnarray}
The left-hand side of \Reqref{eq:sep-sympl-id} being globally
defined, so must be its right-hand side, hence
\begin{eqnarray}\label{eq:bigO}
E_{\gt{i}_2}[X_2]-E_{\gt{i}_1}[X_1]-\delta W_{\gt{i}_1\gt{i}_2}[(\psi_1,\psi_2)
]=:O[X_1]\,,
\end{eqnarray}
for some $\,O\in\Gamma(\wedge^1{\mathsf T}^*{\mathsf L} M)\,$ induced by a global
2-form on $\,M\,$ as per
\begin{eqnarray}\nonumber
O=\int_{{\mathbb{S}}^1}\,{\rm ev}_M^*\omega\,,
\end{eqnarray}
where $\,{\rm ev}_M:{\mathsf L} M\x{\mathbb{S}}^1\to M\,$ is the canonical evaluation
map. Here, we made explicit use of relation \eqref{eq:XFX} to
express the combination of local objects on the left-hand side of
\Reqref{eq:bigO} as a functional of the independent variable
$\,X_1\,$ exclusively. Substituting formula \eqref{eq:bigO} back
into \Reqref{eq:sep-sympl-id}, we now establish a linear
transformation between the pairs $\,(X_{1\,*}\widehat t,{\mathsf p}_1)\,$
and $\,(X_{2\,*}\widehat t,{\mathsf p}_2)\,$ which reads
\begin{eqnarray}\nonumber
\left(\begin{array}{cc} \widehat F_*^{\rm T} & 0 \cr\cr -\omega & {\rm id}_{\Gamma(
{\mathsf T}^*{\mathsf L} M)} \end{array}\right)\,\left(\begin{array}{c} X_{1\,*}\widehat t
\cr\cr {\mathsf p}_1 \end{array}\right)=\left(\begin{array}{cc} {\rm id}_{\Gamma({\mathsf T}{\mathsf L} M)}
& 0 \cr\cr 0 & \widehat F_* \end{array}\right)\,\left(\begin{array}{c} X_{2\,*}
\widehat t \cr\cr {\mathsf p}_2 \end{array}\right)\,,
\end{eqnarray}
with $\,\widehat F_*^{\rm T}=\frac{\delta F^\mu}{\delta X_1^\nu}\,\delta
X_1^\nu\otimes\frac{\delta \ }{\delta X_2^\mu}\,$ acting on $\,X_{1\,*}\widehat
t\,$ via contraction,
\begin{eqnarray}\nonumber
\widehat F_*^{\rm T}\vartriangleright X_{1\,*}\widehat t=(X_{1\,*}\widehat
t)^\nu\,\frac{\delta F^\mu}{\delta X_1^\nu}\,\frac{\delta \ }{\delta X_2^\mu}\,.
\end{eqnarray}
The invertibility of the transformation thus defined necessitates
the existence of an inverse of the tangent map $\,F_*$,\ which
identifies $\,F\,$ as a ($C^1$-)diffeomorphism of $\,M$.\ Demanding,
furthermore, that the transformation preserve the hamiltonian
density yields the constraints
\begin{eqnarray}\label{eq:Ndual-constr}
\omega=0\,,\qquad\qquad F^*{\rm g}={\rm g}\,,
\end{eqnarray}
and so $\,F\,$ is a ($C^1$-)isometry of $\,(M,{\rm g})$.
Finally, taking into account the assumed form of the functionals
$\,W_{\gt{i}_1\gt{i}_2}$,\ we readily establish -- reasoning along the
same lines as in the proof of Theorem \ref{thm:duali-T-bib} -- that
local data $\,(P_{(i_1,i_2)},K_{(i_1,i_2)(j_1,j_2)})\,$ define a
1-isomorphism
\begin{eqnarray}\nonumber
\Phi\ :\ {\rm pr}_1^*\mathcal{G}\xrightarrow{\cong}{\rm pr}_2^*\mathcal{G}
\end{eqnarray}
over the manifold $\,({\rm id}_M\x F)(M)\cong M$,\ with a pullback along
$\,{\rm id}_M\x F\,$ as claimed in the thesis of the theorem.\qed\end{proof}
\medskip
Prior to passing to the discussion of the canonical interpretation
of defect junctions, we pause to present a couple of examples that
give some flesh to the abstract constructions of the present
section.\bigskip
\begin{Eg}\textbf{Duality of type $T$ from the T-duality defect.}
\label{ex:Tdual}\\[-8pt]
\noindent An important example of a proper duality that
(generically) involves a non-trivial change of the topology of the
connected component of the target space is provided by T-duality,
generalising the duality between the $\sigma$-model with target space
$\,{\mathbb{S}}^1_R$,\ i.e.\ a circle of radius $\,R$,\ and that with target
space $\,{\mathbb{S}}^1_{\frac{1}{R}}$,\ i.e.\ a circle of the (T-)dual
radius $\,\frac{1}{R}\,$ (in certain natural units). In the latter
case, translational charges of the string are interchanged with the
winding charges under the duality. A local description of the
algebraic relations between the various components of the background
established by the duality was first worked out in
Refs.\,\cite{Buscher:1987qj,Buscher:1987sk}, whence they are called
\textbf{the Buscher rules}, cf.\ also \Rcite{Giveon:1994fu} for a
review of the early studies of the subject, and
Refs.\,\cite{Alvarez:2000bh,Alvarez:2000bi} for an analysis carried
out in the canonical framework. Global issues were attacked in
\Rcite{Alvarez:1993qi} and, more recently, in
Refs.\,\cite{Bouwknegt:2003vb}, where the important concept of a
correspondence space was introduced and the topological transitions
accompanying T-dualisation in the presence of non-trivial
backgrounds were studied in a systematic manner (cf.\ also
\Rcite{Belov:2007qj} for an attempt at a full-fledged
gerbe-theoretic formulation). The duality was also studied in the
context of the lagrangean description of the string in the presence
of world-sheet defects in \Rcite{Sarkissian:2008dq}.
The string background $\,\gt{B}_T=(\mathcal{M}_T,\mathcal{B}_T,\cdot)\,$ for the
T-duality defect that we want to consider here consists of
\begin{itemize}
\item[(TT)] the target $\,\mathcal{M}_T=(M_T,{\rm g}_T,\mathcal{G}_T)\,$ with the
target space
\begin{eqnarray}\nonumber
M_T={\mathbb{T}}^n_1\sqcup{\mathbb{T}}^n_2
\end{eqnarray}
given by the disjoint union of a pair of $n$-dimensional tori, with
the metric $\,{\rm g}_T\,$ of constant restrictions
\begin{eqnarray}\nonumber
{\rm g}_T\vert_{{\mathbb{T}}^n_\a}={\rm g}_\a\,,
\end{eqnarray}
and the gerbe $\,\mathcal{G}_T\,$ of trivial restrictions
\begin{eqnarray}\nonumber
\mathcal{G}_T\vert_{{\mathbb{T}}^n_\a}= I_{{\rm B}_\a}
\end{eqnarray}
with constant curvings $\,{\rm B}_\a\in\Gamma(\wedge^2{\mathsf T}^*{\mathbb{T}}^n_\a)$;
\item[(T)] the $\mathcal{G}_T$-bi-brane $\,\mathcal{B}_T=(Q_T,{\rm pr}_1,{\rm pr}_2,\omega_T,\Phi_T
)$,\ with
\begin{itemize}
\item[(T.i)] the world-volume
\begin{eqnarray}\nonumber
Q_T={\mathbb{T}}^n_1\x{\mathbb{T}}^n_2\subset M_T\x M_T\,;
\end{eqnarray}
\item[(T.ii)] the $\mathcal{G}_T$-bi-brane maps, given by the canonical
projections
\begin{eqnarray}\nonumber
\iota_\a={\rm pr}_\a\ :\ {\mathbb{T}}^n_1\x{\mathbb{T}}^n_2\to{\mathbb{T}}^n_\a\subset M_T\,;
\end{eqnarray}
\item[(T.iii)] the closed curvature $\,\omega_T$,\ with components
\begin{eqnarray}\nonumber
\omega_{T\,\a\wedge\a}={\rm pr}_\a^*{\rm B}_\a\,,\qquad\qquad\omega_{T\,1\wedge 2}
={\rm F}_{\rm P}\,,
\end{eqnarray}
given in terms of the curvings $\,{\rm B}_\a\,$ and of the curvature
2-form $\,\pi_{P_{{\mathbb{T}}^n_1\x {\mathbb{T}}^n_2}}^*{\rm F}_{\rm P}=\curv(
\nabla_{P_{{\mathbb{T}}^n_1\x{\mathbb{T}}^n_2}})\,$ of a connection
$\,\nabla_{P_{{\mathbb{T}}^n_1\x{\mathbb{T}}^n_2}}\,$ on the Poincar\'e bundle
$\,\pi_{P_{{\mathbb{T}}^n_1\x{\mathbb{T}}^n_2}}:P_{{\mathbb{T}}^n_1\x{\mathbb{T}}^n_2}\to{\mathbb{T}}^n_1\x
{\mathbb{T}}^n_2\,$ over the double torus $\,{\mathbb{T}}^n_1\x{\mathbb{T}}^n_2$;
\item[(T.iv)] the $\mathcal{G}_T$-bi-brane 1-isomorphism
\begin{eqnarray}\nonumber
\Phi_T\ :\ I_{{\rm pr}_1^*{\rm B}_1}\xrightarrow{\cong} I_{{\rm pr}_2^*{\rm B}_2
+\omega_T}\,,
\end{eqnarray}
induced (e.g., on the level of the local data) by the Poincar\'e
bundle $\,P_{{\mathbb{T}}^n_1\x{\mathbb{T}}^n_2}$.
\end{itemize}
\end{itemize}
Given these background data, the DGC \eqref{eq:DGC} produces the
compact formul\ae
\begin{eqnarray}\nonumber
\pi_2=-X_{1\,*}\widehat t\righthalfcup{\rm F}_{\rm P}\,,\qquad\qquad\pi_1=X_{2
\,*}\widehat t\righthalfcup{\rm F}_{\rm P}
\end{eqnarray}
defining an isotropic graph $\,\gt{I}_T\subset{\mathsf P}_{\sigma,
\emptyset}^{\x 2}\,$ and written here in terms of the
\emph{canonical} momentum fields
\begin{eqnarray}\nonumber
\pi_\a={\rm p}_\a-X_{\a\,*}\widehat t\righthalfcup{\rm B}_\a\,.
\end{eqnarray}
In local angle coordinates $\,\theta^\mu_\a,\ \mu\in\ovl{1,n}\,$ on
$\,{\mathbb{T}}^n_\a$,\ we have a simple expression for the curvature 2-form
of the Poincar\'e bundle:
\begin{eqnarray}\nonumber
{\rm F}_{\rm P}=\tfrac{1}{2\pi}\,\delta_{\mu\nu}\,{\mathsf d}\theta_1^\mu\wedge
{\mathsf d}\theta_2^\nu\,.
\end{eqnarray}
This form ensures the required symplecticity of $\,(Q,{\rm F}_{\rm P}
)$.\ The duality-background constraints
\eqref{eq:dualiT-back-constr}, on the other hand, are identical with
the Buscher rules of Refs.\,\cite{Buscher:1987qj,Buscher:1987sk},
relating components of the T-dual pairs $\,({\rm g}_\a,{\rm B}_\a)\,$ as
per
\begin{eqnarray}\nonumber
{\rm g}_2=-{\rm F}_{\rm P}\circ({\rm g}_1-{\rm B}_1\circ{\rm g}_1^{-1}\circ{\rm B}_1
)^{-1}\circ{\rm F}_{\rm P}\,,\qquad\qquad{\rm B}_2=-{\rm F}_{\rm P}\circ
{\rm g}_1^{-1}\circ{\rm B}_1\circ{\rm F}_{\rm P}^{-1}\circ{\rm g}_2\,.
\end{eqnarray}
\end{Eg}\medskip
\begin{Eg}\textbf{Duality of type $N$ from the central-jump WZW defect.}
\label{ex:ZGjump}\\[-8pt]
\noindent An example of a geometric duality engendered by an
extendible defect associated with an isometry of the target is
provided by the central-jump WZW defect -- a subdefect of the
non-boundary maximally symmetric WZW defect at which the
discontinuity of the ${\rm G}$-valued lagrangean field $\,g:\Sigma\to
{\rm G}\,$ of the $\sigma$-model is constrained to take values in the
disjoint union of the distinguished point-like conjugacy classes
$\,\mathcal{C}_{\lambda_z}=\{z\}\,$ of elements $\,z\in Z({\rm G})\,$ of the centre
$\,Z({\rm G})\,$ of the target Lie group $\,{\rm G}$,
\begin{eqnarray}\nonumber
g_{|2}=z\cdot g_{|1}\,.
\end{eqnarray}
The world-volume of the associated $\mathcal{G}_\sfk$-bi-brane, equipped with a
$Z({\rm G})$-invariant (Cartan--Killing) metric and of a vanishing
curvature, all in conformity with \Reqref{eq:Ndual-constr}, is
identified with $\,{\rm G}\x Z({\rm G})$,\ and the (pullback)
$\mathcal{G}_\sfk$-bi-brane 1-isomorphisms of \Reqref{eq:iso-for-sympl-nom},
\begin{eqnarray}\nonumber
\cA_{{\mathsf k},z}\ :\ \mathcal{G}_\sfk\xrightarrow{\cong}\bigl(z^{-1}\bigr)^*\mathcal{G}_\sfk\,,
\end{eqnarray}
one for each element of $\,Z({\rm G})$,\ form part of the data of the
$Z({\rm G})$-equivariant structure on $\,\mathcal{G}_\sfk\,$ constructed explicitly
in \Rcite{Gawedzki:2003pm}. The extendibility of the defect was
verified in \Rcite{Runkel:2008gr}, where the defect data were
subsequently shown to encode a piece of the Moore--Seiberg data of
the WZW model, to wit, the fusing matrix restricted to the
simple-current sector of the quantised CFT.
\end{Eg}
\section{Fusion of states through defect junctions}\label{sec:fusion}
Hereunder, we continue to unravel, in the canonical framework
adopted in the present paper, the physical contents of the gluing
conditions satisfied by the $\sigma$-model field and components of the
string background at the defect quiver, this time focusing on the
DJI
\begin{eqnarray}\label{eq:DJI}
\Delta_{T_{n_\jmath}}\omega=0\,,
\end{eqnarray}
to be imposed at any defect junction $\,\jmath\in\gt{V}_\Gamma\,$ of
valence $\,n_\jmath$.\ From the point of view of the underlying
gerbe theory, the identity expresses a consistency condition for the
trivialising 2-isomorphism $\,\varphi_{n_\jmath}\,$ assigned to
$\,\jmath$,\ cf.\ \Reqref{eq:Dphin-is}. Much in the same fashion as
the DGC \eqref{eq:DGC} constrains propagation of states in the
world-sheet with an embedded defect quiver by determining which
states of the untwisted sector of the theory are transmitted through
the defect line, the DJI turns out to be associated intimately with
the natural geometric splitting-joining interactions of the string
in that it restricts the spectrum of states emerging from a
collision taking place at the defect quiver with defect
junctions\footnote{Throughout the present section, one ought to keep
in mind the contents of the clarifying footnote
\footref{foot:mink-vs-eukl}.}. Thus, in particular, it will be
shown, in the companion paper \cite{Suszek:2010b}, to define an
intertwiner for a representation of the symmetry algebra of the
$\sigma$-model on the space of multi-string states associated with an
interaction vertex decorated with a defect junction, cf.\ the recent
findings of Refs.\,\cite{Runkel:2009su,Runkel:2010} to this effect.
This result can be regarded as a straightforward completion of the
chain of results: the old one, reported in \Rcite{Gawedzki:1987ak},
which shows that the $\sigma$-model gerbe transgresses to a circle
bundle over the configuration space of the untwisted sector of the
theory and thus defines a pre-quantum bundle of the theory, and the
novel one, presented in the previous section, which demonstrates
that the bi-brane, considered together with the attendant DGC, on
one hand transgresses to an isomorphism of the pre-quantum bundle
over an isotropic submanifold in the space of two-string states, and
on the other hand canonically defines a pre-quantum bundle of the
twisted sector of the theory.
In order to illustrate our point and -- in so doing -- introduce
convenient means of description, let us consider the following
(simplest possible)\bigskip
\begin{Eg}\textbf{The splitting-joining interaction in the absence of
defects.}\label{ex:fusion-triv}\\[-8pt]
\noindent Let $\,{\mathsf I}=[0,\pi]\,$ denote the closed $\pi$-unit
interval, and write
\begin{eqnarray}\label{eq:prev-shift-id}\qquad\qquad
\varsigma_1={\rm id}_{{\mathbb{S}}^1}\,,\qquad\qquad\varsigma_2\ :\ \varphi\mapsto 2\pi-
\varphi\,,\qquad\qquad\tau\ :\ \varphi\mapsto\varphi+\pi\,,\qquad
\varphi\in{\mathbb{S}}^1
\end{eqnarray}
for the identity map, the standard parity-reversal map and the
$\pi$-shift map on the unit circle, respectively. We shall think of
$\,{\mathsf I}\,$ as a submanifold of the unit circle $\,{\mathbb{S}}^1$,\ and so,
in particular, $\,\varsigma_2({\mathsf I})=-[\pi,2\pi]\,$ (the minus denotes the
orientation reversal) and $\,\tau({\mathsf I})=[\pi,2\pi]$.\ We then take
the cartesian product $\,{\mathsf P}_{\sigma,\emptyset}^{\x 2}\,$ of two
copies of the untwisted state space $\,{\mathsf P}_{\sigma,\emptyset}={\mathsf T}^*
{\mathsf L} M$,\ and, for an arbitrarily chosen free open path $\,Y_{1,2}
\in C^\infty({\mathsf I},M)\equiv{\mathsf I} M\,$ in $\,M$,\ define a subspace
\begin{eqnarray}\label{eq:fusion-sub-comp-triv}\qquad\qquad
{\mathsf P}_{\sigma,\emptyset}^{\circledast(\mathcal{B}_{\rm triv};Y_{1,2})}=\left\{\
(\psi_1,\psi_2)\in{\mathsf P}_{\sigma,\emptyset}^{\x 2}\,,\quad \psi_\a=(X_\a,
{\mathsf p}_\a)\,,\ \a\in\{1,2\} \quad\middle\vert \quad \left\{ \begin{array}{l}
X_\a\vert_{\varsigma_\a({\mathsf I})}=Y_{1,2} \cr
{\mathsf p}_1\vert_{\mathsf I}={\mathsf p}_2\vert_{\varsigma_2({\mathsf I})} \end{array} \right. \
\right\}\,,
\end{eqnarray}
where $\,\mathcal{B}_{\rm triv}\,$ stands for the trivial $\mathcal{G}$-bi-brane
from Example \ref{eg:triv-def}. The gluing condition for the loop
momenta of the two states can be thought of as a trivial instance of
the DGC \eqref{eq:DGC} imposed along the half-loop
interval\footnote{The direction of $\,{\mathsf p}_2\,$ is determined,
according to our original conventions, by the orientation of the
reversed half-loop $\,\varsigma_2({\mathsf I})$.}. Clearly, elements of
$\,{\mathsf P}_{\sigma,\emptyset}^{\circledast(\mathcal{B}_{\rm triv};Y_{1,2})}\,$ are
generic states assigned to the two incoming legs of the standard
stringy `pair-of-pants' diagram with the contour $\,\ell\cong{\mathsf I}$,\
which carries no extra string-background data, fixed (arbitrarily)
within the world-sheet $\,\Sigma\,$ as in \Rfig{fig:pants}. Upon
varying the half-loop $\,Y_{1,2}$,\ we obtain a subspace
\begin{eqnarray}\label{eq:fusion-sub-triv}
{\mathsf P}_{\sigma,\emptyset}^{\circledast\mathcal{B}_{\rm triv}}=\bigcup_{Y_{1,2}
\in{\mathsf I} M}\,{\mathsf P}_{\sigma,\emptyset}^{\circledast(\mathcal{B}_{\rm triv};Y_{1,2}
)}\subset{\mathsf P}_{\sigma,\emptyset}^{\x 2}\,.
\end{eqnarray}
in the space of untwisted two-string states, which, for the reason
just named and also for other reasons that shall become clear
shortly when we come to discuss less trivial examples, we choose to
call the \textbf{$\mathcal{B}_{\rm triv}$-fusion subspace of the untwisted
string}.
We may next consider a mapping
\begin{eqnarray}\label{eq:int-map-triv}
\gt{i}_{\sigma,(\circledast\mathcal{B}_{\rm triv}:\mathcal{J}_{\rm triv}:\mathcal{B}_{\rm triv})}
\ :\ {\mathsf P}_{\sigma,\emptyset}^{\circledast\mathcal{B}_{\rm triv}}\to{\mathsf P}_{\sigma,
\emptyset}\,,
\end{eqnarray}
labelled by the trivial inter-bi-brane of Example \ref{eg:triv-def},
which assigns to a pair of states $\,(\psi_1,\psi_2)\,$ a third
state $\,\psi_3\,$ with the loop embedding field satisfying a pair
of `half-loop' gluing conditions
\begin{eqnarray}\label{eq:X-glue-int-triv}
X_2\vert_{\mathsf I}=X_3\vert_{\mathsf I}\,,\qquad\qquad X_1\vert_{\tau({\mathsf I})}=
X_3\vert_{\tau({\mathsf I})}\,,
\end{eqnarray}
and with the loop momentum field constrained analogously as per
\begin{eqnarray}\label{eq:p-glue-int-triv}
{\mathsf p}_2\vert_{\mathsf I}={\mathsf p}_3\vert_{\mathsf I}\,,\qquad\qquad{\mathsf p}_1\vert_{\tau(
{\mathsf I})}={\mathsf p}_3\vert_{\tau({\mathsf I})}\,,
\end{eqnarray}
across -- in the simple case in hand -- the distinguished defect
$\,(\mathcal{B}_{\rm triv};X_3)$.\ The conditions identify $\,\psi_3\,$ as a
generic state to be placed around the `waist' in the `pair-of-pants'
diagram of \Rfig{fig:pants}. Accordingly, we call
$\,\gt{i}_{\sigma,(\circledast\mathcal{B}_{\rm triv}:\mathcal{J}_{\rm triv}: \mathcal{B}_{\rm
triv})}\,$ the \textbf{$2\to 1$ cross-$(\mathcal{B}_{\rm triv}, \mathcal{J}_{\rm
triv})$ interaction of the untwisted string}.
\begin{figure}[hbt]~\\[5pt]
$$
\raisebox{-50pt}{\begin{picture}(50,50)
\put(-79,-4){\scalebox{0.25}{\includegraphics{pants-intuit.pdf}}}
\end{picture}
\put(0,0){
\setlength{\unitlength}{.60pt}\put(-28,-16){
\put(-30,0) { (a) }
\put(-35,40) { $\ell$ }
\put(-76,105) { $\psi_1$ }
\put(-17,105) { $\psi_2$ }
\put(-100,170) { $\Sigma$ }
}\setlength{\unitlength}{1pt}}}
\hspace{6cm}
\raisebox{-50pt}{\begin{picture}(50,50)
\put(-79,-4){\scalebox{0.25}{\includegraphics{pants-fusion.pdf}}}
\end{picture}
\put(0,0){
\setlength{\unitlength}{.60pt}\put(-28,-16){
\put(-30,0) { (b) }
\put(-35,40) { $\ell$ }
\put(-76,105) { $\psi_1$ }
\put(-17,105) { $\psi_2$ }
\put(8,215) { $\psi_3$ }
\put(-100,170) { $\Sigma$ }
}\setlength{\unitlength}{1pt}}}
$$
\caption{A canonical description of the splitting-joining
interaction. (a) Fusion of the states $\,\psi_1\,$ and $\,\psi_2\,$
along a half-loop $\,\ell\subset\Sigma,\ \ell\cong{\mathsf I}$.\ (b) The
interaction $\,\gt{i}_{\sigma,(\circledast\mathcal{B}_{\rm triv}:\mathcal{J}_{\rm triv}:
\mathcal{B}_{\rm triv})}\,$ sends a two-string state $\,(\psi_1,\psi_2)\,$
from the $\mathcal{B}_{\rm triv}$-fusion subspace into an emergent state
$\,\psi_3\,$ across the loose half-loops.} \label{fig:pants}
\end{figure}
Due to the trivial character of the gluing conditions, the
interaction $\,\gt{i}_{\sigma,(\circledast\mathcal{B}_{\rm triv}:\mathcal{J}_{\rm triv}:
\mathcal{B}_{\rm triv})}\,$ is manifestly surjective and many-to-one (pairs
of loops differing by the choice of the differentiable extension of
the given `loose' half-loop embedding field and an extension of the
attendant momentum field to the fused half-loop all map to the same
loop in $\,{\mathsf P}_{\sigma,\emptyset}$). It leads us to
\begin{Def}\label{def:cross-int-sub-triv}
Let $\,\gt{B}\,$ be a string background with target $\,\mathcal{M}=(M,{\rm g},\mathcal{G}
)$,\ and let $\,({\mathsf P}_{\sigma,\emptyset},\Omega_{\sigma,\emptyset})\,$ be the
untwisted state space of the non-linear $\sigma$-model for
network-field configurations $\,(X\,\vert\,\Gamma)\,$ in string
background $\,\gt{B}\,$ on world-sheet $\,(\Sigma,\gamma)\,$ with defect
quiver $\,\Gamma$.\ Furthermore, let $\,{\mathsf P}_{\sigma,
\emptyset}^{\circledast\mathcal{B}_{\rm triv}}\,$ be the $\mathcal{B}_{\rm
triv}$-fusion subspace in $\,{\mathsf P}_{\sigma,\emptyset}^{\x 2}\equiv
{\mathsf P}_{\sigma,\emptyset}\x{\mathsf P}_{\sigma,\emptyset}\,$ given in
Eqs.\,\eqref{eq:fusion-sub-comp-triv}-\eqref{eq:fusion-sub-triv},
and $\,\gt{i}_{\sigma,(\circledast\mathcal{B}_{\rm triv}:\mathcal{J}_{\rm triv}:\mathcal{B}_{\rm
triv})}\,$ the $2\to 1$ cross-$(\mathcal{B}_{\rm triv},\mathcal{J}_{\rm triv})$
interaction defined by
Eqs.\,\eqref{eq:int-map-triv}-\eqref{eq:p-glue-int-triv}. The
\textbf{$2\to 1$ cross-$(\mathcal{B}_{\rm triv},\mathcal{J}_{\rm triv})$ interaction
subspace of the untwisted string} is the space
\begin{eqnarray}\nonumber
\gt{I}_\sigma(\circledast\mathcal{B}_{\rm triv}:\mathcal{J}_{\rm triv}:\mathcal{B}_{\rm triv})&=&
\bigl\{\ (\psi_1,\psi_2,\psi_3)\in{\mathsf P}_{\sigma,\emptyset}^{\circledast
\mathcal{B}_{\rm triv}}\x{\mathsf P}_{\sigma,\emptyset} \quad\vert\quad \psi_3=
\gt{i}_{\sigma,(\circledast\mathcal{B}_{\rm triv}:\mathcal{J}_{\rm triv}:\mathcal{B}_{\rm triv})}
(\psi_1,\psi_2) \ \bigr\}\,.
\end{eqnarray}
\begin{flushright}$\checkmark$\end{flushright}\end{Def} It is physically pertinent to enquire as to the distinctive
features of $\,\gt{I}_\sigma(\circledast\mathcal{B}_{\rm triv}:\mathcal{J}_{\rm triv}:
\mathcal{B}_{\rm triv})$,\ the latter viewed as a subspace in a symplectic
space. These could then be interpreted as a canonical manifestation
of the basic interaction process in the string theory in hand. The
answer is contained in the following
\begin{Prop}\label{prop:int-sub-triv}
Let $\,\gt{B}\,$ be a string background with target $\,\mathcal{M}=(M,{\rm g},\mathcal{G}
)$,\ and let $\,({\mathsf P}_{\sigma,\emptyset},\Omega_{\sigma,\emptyset})\,$ be the
untwisted state space of the non-linear $\sigma$-model for
network-field configurations $\,(X\,\vert\,\Gamma)\,$ in string
background $\,\gt{B}\,$ on world-sheet $\,(\Sigma,\gamma)\,$ with defect
quiver $\,\Gamma$.\ Consider the symplectic manifold $\,({\mathsf P}_{\sigma,
\emptyset}^{\x 3},\Omega_{\sigma,\emptyset}^{+-})\,$ defined as
\begin{eqnarray}\nonumber
{\mathsf P}_{\sigma,\emptyset}^{\x 3}:={\mathsf P}_{\sigma,\emptyset}\x{\mathsf P}_{\sigma,
\emptyset}\x{\mathsf P}_{\sigma,\emptyset}\,,\qquad\qquad\Omega_{\sigma,
\emptyset}^{+-}:={\rm pr}_1^*\Omega_{\sigma,\emptyset}+{\rm pr}_2^*\Omega_{\sigma,
\emptyset}-{\rm pr}_3^*\Omega_{\sigma,\emptyset}
\end{eqnarray}
in terms of the canonical projections $\,{\rm pr}_\a:{\mathsf P}_{\sigma,
\emptyset}^{\x 3}\to{\mathsf P}_{\sigma,\emptyset}$.\ Furthermore, let
$\,\pi_{\mathcal{L}_{\sigma,\emptyset}}:\mathcal{L}_{\sigma,\emptyset}\to{\mathsf P}_{\sigma,
\emptyset}\,$ be the pre-quantum bundle for the untwisted sector of
the $\sigma$-model. Then, the $2\to 1$ cross-$(\mathcal{B}_{\rm triv},\mathcal{J}_{\rm
triv})$ interaction subspace $\,\gt{I}_\sigma(\circledast\mathcal{B}_{\rm triv}:
\mathcal{J}_{\rm triv}:\mathcal{B}_{\rm triv} )\,$ is an isotropic submanifold in
$\,({\mathsf P}_{\sigma,\emptyset}^{\x 3},\Omega_{\sigma, \emptyset}^{+-})\,$ and
there exists a canonical bundle isomorphism
\begin{eqnarray}\nonumber
\gt{J}_{\sigma,(\circledast\mathcal{B}_{\rm triv}:\mathcal{J}_{\rm triv}:\mathcal{B}_{\rm triv}
)}\ :\ \bigl({\rm pr}_1^*\mathcal{L}_{\sigma,\emptyset}\otimes{\rm pr}_2^*\mathcal{L}_{\sigma,
\emptyset}\bigr)\vert_{\gt{I}_\sigma(\circledast\mathcal{B}_{\rm triv}: \mathcal{J}_{\rm
triv}:\mathcal{B}_{\rm triv})}\xrightarrow{\cong}{\rm pr}_3^*\mathcal{L}_{\sigma,\emptyset}
\vert_{\gt{I}_\sigma(\circledast\mathcal{B}_{\rm triv}: \mathcal{J}_{\rm triv}:\mathcal{B}_{\rm
triv})}
\end{eqnarray}
between the restrictions to $\,\gt{I}_\sigma(\circledast\mathcal{B}_{\rm triv}:
\mathcal{J}_{\rm triv}:\mathcal{B}_{\rm triv})\,$ of the (tensor) pullback bundles.
\end{Prop}
\noindent A proof of the proposition can be obtained through
specialisation of the proof of Theorem \ref{thm:cross-def-int-untw}
upon setting $\,\mathcal{B}\equiv\mathcal{B}_{\rm triv}\,$ and $\,\mathcal{J}\equiv\mathcal{J}_{\rm
triv}$,\ the latter two being as in Example \ref{eg:triv-def}.
It is owing to the purely geometric nature of the field theory under
consideration that we obtain a simple yet structured representation
of the interaction in the canonical description, which -- as is
obvious from the hitherto discussion -- generalises
straightforwardly to higher-rank fusion and interaction subspaces.
\end{Eg}\medskip
We are now ready to go directly to the main point of interest of
this section, which is a canonical interpretation of the DJI for
world-sheets decorated with non-trivial defect quivers. In order to
describe these in a fashion suggested by the above example, we shall
have to modify our construction of the fusion subspace and that of
the cross-defect interaction suitably.
\subsection{Interactions in the untwisted sector}
We begin with the canonical analysis of the splitting-joining
interaction of untwisted states across a non-trivial defect
(sub-)quiver. While reading the formal definitions and mathematical
expressions ~appearing in this part of the section, it is good to
keep in mind the physical situation depicted in \Rfig{fig:fusion}
that is being modelled by them.
\begin{figure}[hbt]~\\[5pt]
$$
\raisebox{-50pt}{\begin{picture}(50,50)
\put(-79,-4){\scalebox{0.25}{\includegraphics{fusion.pdf}}}
\end{picture}
\put(0,0){
\setlength{\unitlength}{.60pt}\put(-28,-16){
\put(-35,40) { $\ell_{1,2}$ }
\put(-78,103) { $\psi_1$ }
\put(-2,148) { $\psi_2$ }
\put(1,215) { $\psi_3$ }
\put(63,177) { $\ell_{2,3}$ }
\put(-65,192) { $\ell_{3,1}$ }
\put(-120,160) { $\wp_1$ }
\put(105,133) { $\wp_2$ }
\put(70,240) { $\wp_3$ }
\put(10,180) { $\jmath$ }
\put(8,73) { $\jmath^\vee$ }
}\setlength{\unitlength}{1pt}}}
$$
\caption{The splitting-joining interaction mediated by defects,
crossing at a pair of defect junctions $\,\jmath\,$ and
$\,\jmath^\vee$.\ Fusion of the states $\,\psi_1\,$ and $\,\psi_2\,$
along the defect $\,(\mathcal{B};Y_{1,2})\,$ produces an emergent state
$\,\psi_3\,$ via the $2\to 1$ cross-$(\mathcal{B},\mathcal{J})$ interaction.}
\label{fig:fusion}
\end{figure}
\begin{Def}\label{def:int-sub-untw}
Let $\,\gt{B}\,$ be a string background with target $\,\mathcal{M}=(M,{\rm g},\mathcal{G}
)$,\ $\mathcal{G}$-bi-brane $\,\mathcal{B}=\bigl(Q,\iota_\a,\omega,\Phi\ \vert\ \a\in
\{1,2\}\bigr)\,$ and $(\mathcal{G},\mathcal{B})$-inter-bi-brane $\,\mathcal{J}=\bigl(T_n,
\bigl(\varepsilon^{k,k+1}_n,\pi^{k,k+1}_n \ \vert\ k\in\ovl{1,n}\bigr),
\varphi_n\ \vert\ n\in{\mathbb{N}}_{\geq 3}\bigr)$,\ and let $\,({\mathsf P}_{\sigma,
\emptyset},\Omega_{\sigma,\emptyset})\,$ be the untwisted state space of
the non-linear $\sigma$-model for network-field configurations
$\,(X\,\vert\,\Gamma)\,$ in string background $\,\gt{B}\,$ on world-sheet
$\,(\Sigma,\gamma)\,$ with defect quiver $\,\Gamma$.\ For $\,{\mathsf I}=[0,\pi]$,\
denote by $\,{\mathsf I} Q=C^\infty({\mathsf I},Q)\,$ the free open-path space of
$\,Q$.\ The \textbf{$\mathcal{B}$-fusion subspace of the untwisted string}
is the subset of $\,{\mathsf P}_{\sigma,\emptyset}^{\x
2}={\mathsf P}_{\sigma,\emptyset}\x {\mathsf P}_{\sigma,\emptyset}\,$ given by the
formula
\begin{eqnarray}
{\mathsf P}_{\sigma,\emptyset}^{\circledast\mathcal{B}}&=&\{\ (\psi_1,\psi_2)\in
{\mathsf P}_\sigma^{\x 2}\,,\quad \psi_\a=(X_\a,{\mathsf p}_\a)\,,\ \a\in\{1,2\}
\quad\vert\quad (X_1\vert_{\mathsf I},X_2\vert_{\varsigma_2({\mathsf I})})\in(\iota_1
\x\iota_2)({\mathsf I} Q)\cr\cr
&&\hspace{.5cm}\land\quad\exists_{Y_{1,2}\in(\iota_1\x\iota_2)^{-1}
\{(X_1,X_2)\}}\ :\ \tx{DGC}_\mathcal{B}(\psi_1\vert_{\mathsf I},\psi_2
\vert_{\varsigma_2({\mathsf I})},Y_{1,2})=0 \ \}\,.\label{eq:fus-prod-untw}
\end{eqnarray}
It is a fibration over the free-path space $\,{\mathsf I} Q$,\ and we shall
identify it with the corresponding subspace in $\,{\mathsf P}_{\sigma,
\emptyset}^{\x 2}\x {\mathsf I} Q\,$ in what follows. Consider a map
$\,\gt{i}^{2\to 1}_{\sigma,(\circledast\mathcal{B}:\mathcal{J}:\mathcal{B})}$,\ to be termed the
\textbf{$2\to 1$ cross-$(\mathcal{B},\mathcal{J})$ interaction of the untwisted
string}, which assigns to pairs of states from
$\,{\mathsf P}_{\sigma,\emptyset}^{\circledast\mathcal{B}}\,$ subsets of $\,{\mathsf P}_{\sigma,
\emptyset}\,$ such that a pair $\,(\psi_1, \psi_2)\,$ fused along a
free open path $\,Y_{1,2}\in{\mathsf I} Q\,$ is mapped to the set of all
those states $\,\psi_3=(X_3,{\mathsf p}_3)\,$ that satisfy the relations
\begin{eqnarray}\qquad\qquad
X_2\vert_{\mathsf I}=\iota_1\circ Y_{2,3}\,,\qquad X_3\vert_{\mathsf I}=\iota_2
\circ Y_{2,3}\,,\qquad X_1\vert_{\tau({\mathsf I})}=\iota_1\circ Y_{1,3}
\,,\qquad X_3\vert_{\tau({\mathsf I})}=\iota_2\circ Y_{1,3}\,,
\label{eq:nfix-half}\\\cr Y_{I,J}\vert_{\p{\mathsf I}}=\pi_3^{I,J}\circ Z
\,,\qquad(I,J)\in\{(1,2),(2,3),(1,3)\}\,,\\\cr
\tx{DGC}_\mathcal{B}(\psi_2\vert_{\mathsf I},\psi_3\vert_{\mathsf I},Y_{2,3})=0\,,\qquad
\qquad\tx{DGC}_\mathcal{B}(\psi_1\vert_{\tau({\mathsf I})},\psi_2\vert_{\tau({\mathsf I}
)},Y_{1,3})=0\label{eq:loop-nfix-half}
\end{eqnarray}
for $\,\pi_3^{1,3}\equiv\pi_3^{3,1}$,\ some free open paths $\,Y_{1,
3},Y_{2,3}\in{\mathsf I} Q\,$ and a map $\,Z:\p{\mathsf I}\to T_{3,++-}\cup T_{3,-
-+}\,$ from the set $\,\{0,\pi\}\,$ into the components $\,T_{3,++
-}\,$ and $\,T_{3,--+}\,$ of $\,T_3\subset T\,$ corresponding to the
values $\,\varepsilon_3^{1,2}=\pm 1= \varepsilon_3^{2,3}=-\varepsilon_3^{3,1}\,$ of the
orientation maps, with $\,Z(0)\in T_{3,--+}\,$ and $\,Z(\pi)\in T_{3
,++-}$.\ The \textbf{$2\to 1$ cross-$(\mathcal{B},\mathcal{J})$ interaction subspace
of the untwisted string} is then the subset of
$\,{\mathsf P}_{\sigma,\emptyset}^{\x 3}=
{\mathsf P}_{\sigma,\emptyset}\x{\mathsf P}_{\sigma,\emptyset}\x{\mathsf P}_{\sigma,\emptyset}\,$
given by the formula
\begin{eqnarray}\nonumber
\gt{I}_\sigma(\circledast\mathcal{B}:\mathcal{J}:\mathcal{B})=\{\ (\psi_1,\psi_2,\psi_3)\in
{\mathsf P}_{\sigma,\emptyset}^{\circledast\mathcal{B}}\x{\mathsf P}_{\sigma,\emptyset}
\quad\vert\quad \psi_3\in\gt{i}^{2\to 1}_{\sigma,(\circledast\mathcal{B}:\mathcal{J}:\mathcal{B}
)}(\psi_1,\psi_2) \ \}\,.
\end{eqnarray}
Once again, the latter subspace is a fibration over the cartesian
cube $\,{\mathsf I} Q^3$,\ and we shall identify it with the corresponding
subspace in $\,{\mathsf P}_\sigma^{\x 3}\x{\mathsf I} Q^{\x 3}\,$ in what follows.
\begin{flushright}$\checkmark$\end{flushright}\end{Def} \noindent We have
\begin{Thm}\label{thm:cross-def-int-untw}
Let $\,\gt{B}=(\mathcal{M},\mathcal{B},\mathcal{J})\,$ be a string background, and let
$\,({\mathsf P}_{\sigma,\emptyset},\Omega_{\sigma,\emptyset})\,$ be the untwisted
state space of the non-linear $\sigma$-model for network-field
configurations $\,(X\,\vert\,\Gamma)\,$ in string background $\,\gt{B}\,$
on world-sheet $\,(\Sigma,\gamma)\,$ with defect quiver $\,\Gamma$,\ with the
pre-quantum bundle for the untwisted sector of the $\sigma$-model over
it, $\,\pi_{\mathcal{L}_{\sigma,\emptyset}}:\mathcal{L}_{\sigma,\emptyset}\to{\mathsf P}_{\sigma,
\emptyset}$.\ Furthermore, let $\,({\mathsf P}_{\sigma,\emptyset}^{\x 3},
\Omega_{\sigma,\emptyset}^{+-})\,$ be the symplectic manifold defined in
Proposition \ref{prop:int-sub-triv}. Then, the following statements
hold true:
\begin{itemize}
\item[i)] the $2\to 1$ cross-$\mathcal{B}$ interaction subspace of the
untwisted string, $\,\gt{I}_\sigma(\circledast\mathcal{B}:\mathcal{J}:\mathcal{B})$,\ constructed
in Definition \ref{def:int-sub-untw}, is an isotropic submanifold of
$\,({\mathsf P}_{\sigma,\emptyset}^{\x 3},\Omega_{\sigma, \emptyset}^{+-})$;
\item[ii)] the background $\,\gt{B}\,$ canonically
induces a bundle isomorphism
\begin{eqnarray}\nonumber
\gt{J}_{\sigma,(\circledast\mathcal{B}:\mathcal{J}:\mathcal{B})}\ :\ \bigl({\rm pr}_1^*\mathcal{L}_{\sigma,
\emptyset}\otimes{\rm pr}_2^*\mathcal{L}_{\sigma,\emptyset}\bigr)\vert_{\gt{I}_\sigma(
\circledast\mathcal{B}:\mathcal{J}:\mathcal{B})}\xrightarrow{\cong}{\rm pr}_3^*\mathcal{L}_{\sigma,
\emptyset}\vert_{\gt{I}_\sigma(\circledast\mathcal{B}:\mathcal{J}:\mathcal{B})}
\end{eqnarray}
between the restrictions to $\,\gt{I}_\sigma(\circledast\mathcal{B}:\mathcal{J}:\mathcal{B})\,$
of the (tensor) pullback bundles.
\end{itemize}
\end{Thm} \noindent A proof of the theorem is given in Appendix
\ref{app:cross-def-int-untw}. \begin{Rem}\label{rem:duality-scheme} The
relation between defects and dualities of the $\sigma$-model worked out
in the previous section distinguished those bi-branes whose maps
$\,\iota_\a:Q\to M\,$ are \emph{surjective submersions}. The
canonical analysis of the splitting-joining interaction of the
untwisted string immediately leads to similar conclusions for the
inter-bi-brane. Indeed, it is clear that for a given interaction
vertex of \Rfig{fig:fusion} to allow the appearance of arbitrary
outgoing and incoming states, i.e.\ for the interaction subspace to
project onto each cartesian component
$\,{\mathsf P}_{\sigma,\emptyset}\subset{\mathsf P}_{\sigma, \emptyset}^{\x n}$,\ the
inter-bi-brane maps $\,\pi_n^{k,k+1}:T_n \to Q\,$ should all be
surjective. Taking into account the additional requirement of
topologicality of the defect, we note that -- at least in the case
of extendible defects -- the inter-bi-brane maps should, moreover,
be submersions, so that, once more, surjective submersions become
singled out. These are particularly interesting in the case of
inter-bi-branes admitting \textbf{induction}, as introduced in
\Rxcite{Sec.\,2.8}{Runkel:2008gr}. The latter is motivated by the
physical observation that a defect junction $\,\jmath\,$ of valence
$\,n_\jmath>3$,\ represented by a defect-field insertion in the
underlying CFT, can be regarded as a product of a stepwise limiting
procedure in which a collection of `elementary' 3-valent vertices
are merged by sending the lengths of the interconnecting defect
lines to zero, whereby the associated defect fields of the CFT may
have to be renormalised in order to remove the ensuing divergencies.
In what follows, we briefly recall the idea of induction in
restriction to the component of the background obtained by fixing
the values of the orientation maps $\,\varepsilon_n^{k,k+1}\,$ to be all
$\,+1\,$ except for $\,\varepsilon_n^{n,1}=-1\,$ for all $\,n\in {\mathbb{N}}_{\geq
3}\,$ and taking the inter-bi-brane 2-isomorphisms $\,\varphi_n\,$
restricted to the corresponding submanifolds
$\,T_{n,++\cdots+-}\subset T_n\,$ (the remaining components of the
background are left unrestricted). As the very construction of the
$\sigma$-model in string background $\,\gt{B}\,$ clearly indicates, there
are no additional geometric insights to gain from considering the
more general case. Indeed, as long as we are concerned with a
\emph{single} defect junction (which is, in particular, all we need
to determine the associated field-space data), we are at liberty to
choose an arbitrary relative-orientation pattern for the defect
lines converging at that junction.
Denote the inter-bi-brane maps $\,\pi_3^{k,k+1},\ k\in\{1,2,3\}\,$
as
\begin{eqnarray}\nonumber
\pi_3^{1,2}=d^{(2)}_0\,,\qquad\qquad\pi_3^{2,3}=d^{(2)}_2\,,\qquad
\qquad\pi_3^{3,1}=d^{(2)}_1\,.
\end{eqnarray}
The background $\,\gt{B}\,$ shall be termed a \textbf{string
background with induction} iff, for each $\,n\geq 3$,\ there exist
smooth maps
\begin{eqnarray}\nonumber
d^{(n)}_i\ :\ T_{n+1,++\cdots+-}\to T_{n,++\cdots+-}\,,\qquad
i\in\ovl{0,n}
\end{eqnarray}
satisfying the identities
\begin{eqnarray}\label{eq:simpl-id-dd}
d^{(n-1)}_i\circ d^{(n)}_j=d^{(n-1)}_{j-1}\circ d^{(n)}_i\qquad
\tx{for}\quad i<j\,,
\end{eqnarray}
and such that the inter-bi-brane 2-isomorphisms $\,\varphi_n\,$ are
induced from $\,\varphi_3\,$ in a natural manner illustrated in
\Rxcite{Sec.\,2.8}{Runkel:2008gr} on an explicit example amenable to
a straightforward generalisation (which we leave out here for the
sake of conciseness). Identities \eqref{eq:simpl-id-dd} arise as
simple consistency conditions to be imposed on the limiting values
attained by the $\sigma$-model field at the defect junctions of an
embedded defect quiver as we decompose the defect quiver at these
junctions into clusters of defect junctions of lower valence bridged
by intermediate defect lines, prior to passing to the limit of the
vanishing length of the intermediate defect lines, cf.\
\Rfig{fig:induction}.
\begin{figure}[hbt]~\\[5pt]
$$
\raisebox{-50pt}{\begin{picture}(50,50)
\put(-70,-4){\scalebox{0.25}{\includegraphics{simpl-mov.pdf}}}
\end{picture}
\put(0,0){
\setlength{\unitlength}{.60pt}\put(-28,-16){
\put(-80,235) { $d^{(4)}_4$ }
\put(-80,137) { $d^{(4)}_3$ }
\put(17,100) { $d^{(4)}_2$ }
\put(55,137) { $d^{(4)}_1$ }
\put(55,235) { $d^{(4)}_0$ }
}\setlength{\unitlength}{1pt}}}
$$
\caption{Simplicial moves on a defect junction of valence 5.}
\label{fig:induction}
\end{figure}
On the other hand, it is tempting to view them as the simplicial
identities obeyed by the face maps of a simplicial space composed by
the family of manifolds $\,\{M,Q,T_{3,++-},T_{4,+++-},\ldots\}$.\
That this is the proper manner of thinking of the string background
with induction can be seen as follows: First of all, the bi-brane
maps provide a natural completion of the family $\,(d^{(n)}_i\
\vert\ i\in\ovl{0,n}\,,\ n\in{\mathbb{N}}_{\geq 2})\,$ of smooth maps
interrelated as per \Reqref{eq:simpl-id-dd}, which can readily be
seen upon setting
\begin{eqnarray}\nonumber
d^{(1)}_0:=\iota_1\,,\qquad\qquad d^{(1)}_1:=\iota_2
\end{eqnarray}
and recalling relations \eqref{eq:proto-simpl}. Furthermore, it is
natural, from the point of view of the associated $\sigma$-model, to
incorporate the trivial defect into the formal definition of the
string background by allowing degenerate defect quivers in which
some defect lines carry the (trivial) data of the trivial defect.
Indeed, we should always be able to insert a circular trivial defect
into the world-sheet, or attach a trivial-defect line to a given
defect junction, between any two of its defect lines, whereby the
valence of the defect junction is increased by $1$. It is easy to
see, going through similar consistency checks of the limiting values
of the $\sigma$-model fields as those used in the derivation of
\Reqref{eq:simpl-id-dd}, that this can be formalised as a
requirement of the existence of distinguished sections
\begin{eqnarray}
s^{(n-1)}_i\ :\ T_{n,++\cdots+-}\to T_{n+1,++\cdots+-}\,,\qquad i\in
\ovl{0,n-1}\,,\cr\label{eq:deg-sec}\\ s^{(1)}_j\ :\ Q\to
T_{3,++-}\,,\qquad j\in\{0,1 \}\,,\qquad\qquad\qquad s^{(0)}_0\ :\
M\to Q\nonumber
\end{eqnarray}
of the respective surjective submersions, satisfying the identities
\begin{eqnarray}
s_i^{(n+1)}\circ s_j^{(n)}=s_{j+1}^{(n+1)}\circ s_i^{(n)}\qquad
\tx{if}\quad i\leq j\,,\label{eq:simpl-id-ss}\\\cr d_i^{(n+1)}\circ
s_j^{(n)}=\left\{\begin{array}{lcl} s_{j-1}^{(n-1)}\circ d_i^{(n)}\quad &
\tx{if} & i<j\cr\cr {\rm id}_{T_{n+1,++\cdots+-}}\quad & \tx{if} & i=j
\quad\lor\quad i=j+1\cr\cr s_j^{(n-1)}\circ d_{i-1}^{(n)}\quad &
\tx{if} & i>j+1 \end{array}\right..\label{eq:simpl-id-ds}
\end{eqnarray}
Here, $\,s^{(0)}_0\,$ puts the image, with respect to $\,X:\Sigma\setminus\Gamma\to
M$,\ of an arbitrary point from the interior of a world-sheet patch
on the world-volume of the trivial defect. Similarly, $\,s^{(1
)}_j\,$ expresses the possibility of viewing the image, with respect to $\,X:
\Gamma\setminus\gt{E}_\Gamma\to Q$,\ of a point from the interior of a defect
line embedded in $\,\Sigma\,$ as the image of the degenerate 3-valent
defect junction, with the trivial-defect line joining the original
one from the side of $\,U_1\,$ (for $\,j=1$) or $\,U_2\,$ (for $\,j=
0$) in the notation of Definition \ref{def:net-field}. Finally, the
maps $\,s^{(n-1)}_i\,$ represent the process of increasing the
valence of a given defect junction through attachment of a
trivial-defect line between the neighbouring defect lines $\,\ell_{n
-i-1,n-i}\,$ and $\,\ell_{n-i,n-i+1}\,$ that converge at this
junction (with the usual convention $\,\ell_{0,1}\equiv\ell_{n,1}
\equiv\ell_{n,n+1}$).
Altogether, Eqs.\,\eqref{eq:simpl-id-dd}-\eqref{eq:simpl-id-ds}
reproduce the full set of simplicial identities for the face maps
$\,d^{(n)}_i\,$ and degeneracy maps $\,s^{(n)}_i\,$ of a simplicial
space
\begin{eqnarray}\nonumber
\alxydim{@C=1.cm@R=.05cm}{\cdots \ar@<1ex>[r]^{d^{(4)}_i}
\ar@<.5ex>[r] \ar@<0ex>[r] \ar@<-.5ex>[r] \ar@<-1ex>[r] & T_4
\ar@<.75ex>[r]^{d^{(3)}_i} \ar@<.25ex>[r] \ar@<-.25ex>[r]
\ar@<-.75ex>[r] & T_3 \ar@<.5ex>[r]^{d^{(2)}_i} \ar@<0.ex>[r]
\ar@<-.5ex>[r] & Q \ar@<.5ex>[r]^{d^{(1)}_i} \ar@<-.5ex>[r]
& M}
\end{eqnarray}
A \textbf{simplicial string background $\,\gt{B}=(\mathcal{M},\mathcal{B},\mathcal{J})$},\
i.e.\ a string background with induction, equipped with the family
\eqref{eq:deg-sec} of sections, can be regarded as a straightforward
generalisation of a simplicial background describing a proper
(geometric) symmetry of the $\sigma$-model with target $\,\mathcal{M}=(M,{\rm g},
\mathcal{G})\,$ endowed with structure of a ${\rm K}$-space for some group
$\,{\rm K}$,\ acting on $\,(M,{\rm g})\,$ by isometries
\begin{eqnarray}\nonumber
\ell\ :\ {\rm K}\x M\to M\ :\ (g,m)\mapsto\ell_g(m)=:g.m\,,\qquad\qquad
\ell_g^*{\rm g}={\rm g}
\end{eqnarray}
that lift to the gerbe $\,\mathcal{G}\,$ in the sense made precise in
\Rcite{Gawedzki:2010rn}. The relevant simplicial space is given by
the nerve
\begin{eqnarray}\nonumber
\alxydim{@C=1.cm@R=.05cm}{{\mathsf N}({\rm K}{\hspace{-0.04cm}\ltimes\hspace{-0.05cm}} M)^\bullet\quad :
\hspace{-.5cm} & \cdots \ar@<1ex>[r]^{{}^{\tx{\tiny $\txG$}}\hspace{-2pt} d^{(4)}_i} \ar@<.5ex>[r]
\ar@<0ex>[r] \ar@<-.5ex>[r] \ar@<-1ex>[r] & {\rm K}^3\x M
\ar@<.75ex>[r]^{{}^{\tx{\tiny $\txG$}}\hspace{-2pt} d^{(3)}_i} \ar@<.25ex>[r] \ar@<-.25ex>[r]
\ar@<-.75ex>[r] & {\rm K}^2\x M \ar@<.5ex>[r]^{{}^{\tx{\tiny $\txG$}}\hspace{-2pt} d^{(2)}_i}
\ar@<0.ex>[r] \ar@<-.5ex>[r] & {\rm K}\x M \ar@<.5ex>[r]^{{}^{\tx{\tiny $\txG$}}\hspace{-2pt}
d^{(1)}_i} \ar@<-.5ex>[r] & M}\,,
\end{eqnarray}
of the action groupoid
\begin{eqnarray}\nonumber
\alxydim{@C=2.cm}{{\rm K}{\hspace{-0.04cm}\ltimes\hspace{-0.05cm}} M\quad : \hspace{-2cm} & {\rm K}\x M
\ar@<.5ex>[r]^{{\rm pr}_2=:{}^{\tx{\tiny $\txG$}}\hspace{-2pt} d^{(1)}_0} \ar@<-.5ex>[r]_{\ell=:{}^{\tx{\tiny $\txG$}}\hspace{-2pt}
d^{(1)}_1} & M}\,,
\end{eqnarray}
written in terms of the action $\,\ell\,$ and of the canonical
projection $\,{\rm pr}_2$.\ The action groupoid is understood as the
small category with object and morphism sets
\begin{eqnarray}\nonumber
\obj\,({\rm K}{\hspace{-0.04cm}\ltimes\hspace{-0.05cm}} M)=M\,,\qquad\qquad{\rm Mor}\,({\rm K}{\hspace{-0.04cm}\ltimes\hspace{-0.05cm}} M)={\rm K}\x M\,,
\end{eqnarray}
with the identity morphism ($e\,$ is the group unit)
\begin{eqnarray}\nonumber
{\rm id}_m=(e,m)\,,
\end{eqnarray}
and with source and target maps
\begin{eqnarray}\nonumber
s(g,m)=m\,,\qquad\qquad t(g,m)=g.m\,,
\end{eqnarray}
which, altogether, lead to a natural identification of the spaces of
$n$-tuples of composable morphisms (i.e.\ the remaining members of
the family $\,{\mathsf N}({\rm K}{\hspace{-0.04cm}\ltimes\hspace{-0.05cm}} M)^\bullet\,$ of spaces) with the
respective product spaces $\,{\rm K}^n\x M$.\ The (inter-)bi-brane data
compose a ${\rm K}$-equivariant structure on $\,\mathcal{G}\,$ as in
\Rcite{Gawedzki:2010rn}. This structure was shown to be a
prerequisite of the gauging of the internal (rigid) ${\rm K}$-symmetry
of the $\sigma$-model (the latter being obtained as a lift of the
geometric action $\,\ell\,$ to the phase space of the $\sigma$-model)
and it is readily proven necessary for the gauged $\sigma$-model to
descend to the coset $\,M/{\rm K}$.\ Reasoning by analogy, we
conjecture that the existence of a simplicial string background
associated with a given $\sigma$-model duality is a necessary
ingredient of a consistent formulation of string theory on a
`quotient' background of a $\sigma$-model descended from the original
one upon `gauging' the duality group, whenever such a group and the
attendant `quotient' can be defined. Here, the submersive
surjectivity of the face maps is necessary to establish duality
equivalences on the entire phase space and to ensure that all states
are transmitted by any defect quiver carrying the duality data. More
specifically, the bi-brane face maps $\,d^{(1)}_i:Q\to M\,$ encode
an element-wise presentation of the set of duality transformations
on the space of states, the 3-valent inter-bi-brane face maps
$\,d^{(2)}_i:T_3\to Q\,$ render the presentation distributive with
respect to the group operation on the set of dualities, and the
requirement that the 4-valent inter-bi-brane structure induced from
the 3-valent one by means of the face maps $\,d^{(3)}_i:T_4\to
T_3\,$ be independent of the choice of the defining simplicial move
enforce the associativity of the presentation. Finally, the
existence of an induced inter-bi-brane structure on the component
world-volumes $\,T_n\,$ of valence $\,n\geq 5$,\ independent of the
choice of defining simplicial moves, guarantees that the associative
presentation of the duality group carries over to arbitrary
interaction schemes (i.e.\ to arbitrary defect quivers). As the
treatment of coset $\sigma$-models in \Rcite{Gawedzki:2010rn} suggests,
there are, generically, extra constraints to be imposed on the thus
obtained `duality-equivariant' structure for the $\sigma$-model to
descend to the duality quotient. A motivating explicit instantiation
of this idea (if also far from being understood rigorously to date)
is the notion of a T-fold, advanced in \Rcite{Hull:2004in}, in which
the target is described in terms of local charts (carrying local
metric and gerbe data) patched together using T-duality
transformations. Let us also point out that the above
\textbf{duality scheme} bears a deep affinity with the
\textbf{categorial descent scheme} discussed in
\Rcite{Fuchs:2009si}. We hope to return to these issues in the near
future.\end{Rem}\begin{center}*\hspace{2cm}*\hspace{2cm}*\end{center}
\subsection{Interactions in the twisted sector}
Another class of string processes in which the inter-bi-brane and
the associated DJI are naturally expected to transgress to the
canonical description is the splitting-joining interaction of
twisted states. When assembling the necessary formal ingredients, we
are guided by the depiction of the corresponding network-field
configuration on the world-sheet, the simplest of its kind,
presented in \Rfig{fig:tw-fusion}. Thus, we consider three defect
lines, $\,\ell_{1,2},\ell_{2,3}\,$ and $\,\ell_{3,1}$,\ converging
at a defect junction $\,\jmath$.\ To each defect line, we attach the
corresponding Cauchy contour $\,C_{I,J},\
(I,J)\in\{(1,2),(2,3),(3,1)\}$,\ oriented as the ones drawn in the
figure and crossing the respective defect lines transversally, each
at a single point. The contours are next pushed towards one another
in such a manner that they overlap pairwise along open arcs, all
crossing at a pair of points in $\,\Sigma$,\ the defect junction
$\,\jmath\,$ being one of them.
\begin{figure}[hbt]~\\[5pt]
$$
\raisebox{-50pt}{\begin{picture}(50,50)
\put(-79,-4){\scalebox{0.25}{\includegraphics{tw-fusion.pdf}}}
\end{picture}
\put(0,0){
\setlength{\unitlength}{.60pt}\put(-28,-16){
\put(-35,40) { $\ell$ }
\put(-90,105) { $\psi_{\ell_{1,2}}$ }
\put(-5,145) { $\psi_{\ell_{2,3}}$ }
\put(23,213) { $\psi_{\ell_{3,1}}$ }
\put(-95,140) { $\ell_{1,2}$ }
\put(55,112) { $\ell_{2,3}$ }
\put(-10,220) { $\ell_{3,1}$ }
\put(63,177) { $\ell_R$ }
\put(-65,192) { $\ell_L$ }
\put(10,180) { $\jmath$ }
}\setlength{\unitlength}{1pt}}}
$$
\caption{The splitting-joining interaction of a triple of 1-twisted
states. The Cauchy contours representing the states are drawn in
black.} \label{fig:tw-fusion}
\end{figure}
Our first task consists in identifying the various subspaces within
the cartesian square and the cartesian cube of the 1-twisted state
space of the $\sigma$-model, of relevance to the problem in hand.
\begin{Def}\label{def:int-sub-tw}
Let $\,\gt{B}=(\mathcal{M},\mathcal{B},\mathcal{J})\,$ be a string background as in Definition
\ref{def:bckgrnd}. Fix $\,P\in{\mathbb{S}}^1\,$ and $\,\varepsilon\in\{-1,+1\}$,\
and let $\,({\mathsf P}_{\sigma,\mathcal{B}|(P,\varepsilon)},\Omega_{\sigma,\mathcal{B}|(P,\varepsilon)})\,$ be
the 1-twisted state space of the non-linear $\sigma$-model for
network-field configurations $\,(X\,\vert\,\Gamma)\,$ in string
background $\,\gt{B}\,$ on world-sheet $\,(\Sigma,\gamma)\,$ with defect
quiver $\,\Gamma$.\ The \textbf{$\mathcal{B}_{\rm triv}$-fusion subspace of the
1-twisted string} is a subspace in
\begin{eqnarray}\label{eq:Ptw2}
{\mathsf P}_{\sigma,\mathcal{B}|(\varepsilon_1,\varepsilon_2)}^{\x 2}:={\mathsf P}_{\sigma,\mathcal{B}|(\pi,\varepsilon_1)}\x
{\mathsf P}_{\sigma,\mathcal{B}|(\pi,\varepsilon_2)}
\end{eqnarray}
given by the formula
\begin{eqnarray}
{\mathsf P}_{\sigma,\mathcal{B}|(\varepsilon_1,\varepsilon_2)}^{\circledast\mathcal{B}_{\rm triv}}&=&\{\ (
\psi_1,\psi_2)\in{\mathsf P}_{\sigma,\mathcal{B}|(\varepsilon_1,\varepsilon_2 )}^{\x 2}\,,\quad
\psi_\a=(X,{\mathsf p},q_\a,V_\a)\,,\ \a\in\{1,2\}\cr\cr
&&\hspace{.5cm}\vert\quad X_1\vert_{\mathsf I}=X_2\vert_{\varsigma_2({\mathsf I})}\quad
\land\quad{\mathsf p}_1\vert_{\mathsf I}={\mathsf p}_2\vert_{\varsigma_2({\mathsf I})} \ \}\,.
\label{eq:fus-prod-tw}
\end{eqnarray}
Clearly, the specific choice $\,P_1=\pi=P_2\,$ of the intersection
point is immaterial to the outcome of our analysis. Consider a map
$\,\gt{i}^{2\to 1}_{\sigma,(\circledast\mathcal{B}_{\rm triv}:\mathcal{J}:\mathcal{B}_{\rm triv}
)}$,\ to be termed the \textbf{$2\to 1$ cross-$(\mathcal{B}_{\rm triv},\mathcal{J})$
interaction of the untwisted string}, which assigns to pairs of
states from $\,{\mathsf P}_{\sigma,\mathcal{B}|(\varepsilon_1,\varepsilon_2)}^{\circledast\mathcal{B}_{\rm
triv}}\,$ subsets of $\,{\mathsf P}_{\sigma,\mathcal{B}|(\pi,\varepsilon_3)}\,$ such that a
pair $\,(\psi_1,\psi_2)\,$ is mapped to the set of all those states
$\,\psi_3=(X,{\mathsf p},q_3,V_3)\,$ that satisfy the relations
\begin{eqnarray}\label{eq:glue-inter-tw}
\begin{array}{cc} X_2\vert_{\mathsf I}=X_3\vert_{\mathsf I}\,,\qquad & X_1\vert_{\tau({\mathsf I}
)}=X_3\vert_{\tau({\mathsf I})}\,,\cr\cr
{\mathsf p}_2\vert_{\mathsf I}={\mathsf p}_3\vert_{\mathsf I}\,,\qquad & {\mathsf p}_1\vert_{\tau({\mathsf I}
)}={\mathsf p}_3\vert_{\tau({\mathsf I} )}\,, \end{array}
\end{eqnarray}
augmented with the constraints
\begin{eqnarray}\label{eq:glue-inter-junct}
q_k=\pi_3^{k,k+1}(t_3)\,,\qquad k\in\{1,2,3\}\,,
\end{eqnarray}
the latter being expressed in terms of a fixed point $\,t_3\in T_{3,
++-}\subset T\,$ from the component $\,T_{3,++-}\,$ of $\,T_3\subset
T\,$ corresponding to the values $\,\varepsilon_3^{1,2}=+1=\varepsilon_3^{2,3}=-
\varepsilon_3^{3,1}\,$ of the orientation maps. The \textbf{$2\to 1$
cross-$(\mathcal{B}_{\rm triv},\mathcal{J} )$ interaction subspace of the 1-twisted
string} is then the subset of the space
\begin{eqnarray}\nonumber
{\mathsf P}_{\sigma,\mathcal{B}|(\varepsilon_1,\varepsilon_2,\varepsilon_3)}^{+-}:={\mathsf P}_{\sigma,\mathcal{B}|(\pi,
\varepsilon_1)}\x{\mathsf P}_{\sigma,\mathcal{B}|(\pi,\varepsilon_2)}\x{\mathsf P}_{\sigma ,\mathcal{B}|(\pi,\varepsilon_3)}
\end{eqnarray}
given by the formula
\begin{eqnarray}\nonumber
\gt{I}_\sigma(\circledast\mathcal{B}_{\rm triv}:\mathcal{J}:\mathcal{B}_{\rm triv})^{\mathcal{B}|(\varepsilon_1
,\varepsilon_2,\varepsilon_3)}=\{\ (\psi_1,\psi_2,\psi_3)\in{\mathsf P}_{\sigma,\mathcal{B}|(
\varepsilon_1,\varepsilon_2)}^{\circledast\mathcal{B}_{\rm triv}}\x{\mathsf P}_{\sigma,\mathcal{B}|(\pi,
\varepsilon_3)}\quad\vert\quad \psi_3\in\gt{i}^{2\to 1}_{\sigma,(\circledast
\mathcal{B}_{\rm triv}:\mathcal{J}:\mathcal{B}_{\rm triv})}(\psi_1,\psi_2)\ \}\,.
\end{eqnarray}
\begin{flushright}$\checkmark$\end{flushright}\end{Def} \noindent We are now ready to give
\begin{Thm}\label{thm:cross-def-int-tw}
Let $\,\gt{B}=(\mathcal{M},\mathcal{B},\mathcal{J})\,$ be a string background as in Definition
\ref{def:bckgrnd}, and let
$\,({\mathsf P}_{\sigma,\mathcal{B}|(\pi,\varepsilon)},\linebreak\Omega_{\sigma,\mathcal{B}|(\pi,\varepsilon)}),\
\pi\in{\mathbb{S}}^1\,$ be the 1-twisted state space of the non-linear
$\sigma$-model for network-field configurations $\,(X\,\vert\,\Gamma)\,$ in
string background $\,\gt{B}\,$ on world-sheet $\,(\Sigma,\gamma)\,$ with
defect quiver $\,\Gamma$,\ with the pre-quantum bundle for the 1-twisted
sector of the $\sigma$-model over it,
$\,\pi_{\mathcal{L}_{\sigma,\mathcal{B}|(\pi,\varepsilon)}}:\mathcal{L}_{\sigma,\mathcal{B}|(\pi,\varepsilon)}\to
{\mathsf P}_{\sigma,\mathcal{B}|(\pi,\varepsilon)}$.\ Endow the space
\begin{eqnarray}\nonumber
{\mathsf P}_{\sigma,\mathcal{B}|(\varepsilon_1,\varepsilon_2,\varepsilon_3)}^{\x 3}:={\mathsf P}_{\sigma,\mathcal{B}|(\pi,
\varepsilon_1)}\x{\mathsf P}_{\sigma,\mathcal{B}|(\pi,\varepsilon_2)}\x{\mathsf P}_{\sigma,\mathcal{B}|(\pi,\varepsilon_3)}
\end{eqnarray}
with the symplectic structure defined by the 2-form
\begin{eqnarray}\nonumber
\Omega^{+-}_{\sigma,\mathcal{B}|(\varepsilon_1,\varepsilon_2,\varepsilon_3)}:={\rm pr}_1^*\Omega_{\sigma,\mathcal{B}|(\pi,
\varepsilon_1)}+{\rm pr}_2^*\Omega_{\sigma,\mathcal{B}|(\pi,\varepsilon_2)}-{\rm pr}_3^*\Omega_{\sigma,\mathcal{B}|(\pi,
\varepsilon_3)}
\end{eqnarray}
in terms of the canonical projections $\,{\rm pr}_k:{\mathsf P}_{\sigma,\mathcal{B}|(\varepsilon_1
,\varepsilon_2,\varepsilon_3)}^{\x 3}\to{\mathsf P}_{\sigma,\mathcal{B}|(\pi,\varepsilon_k)},\ k\in\{1,2,3
\}$.\ and consider the pullback circle bundle
\begin{eqnarray}\nonumber
\mathcal{L}_{\sigma,\mathcal{B}|(\varepsilon_1,\varepsilon_2,\varepsilon_3)}^{+-}:=\to{\mathsf P}_{\sigma,\mathcal{B}|(\varepsilon_1,\varepsilon_2,
\varepsilon_3)}^{\x 3}\,.
\end{eqnarray}
Then, the following statements hold true:
\begin{itemize}
\item[i)] the $2\to 1$ cross-$(\mathcal{B}_{\rm triv},\mathcal{J})$ interaction
subspace $\,\gt{I}_\sigma(\circledast\mathcal{B}_{\rm triv}:\mathcal{J}:\mathcal{B}_{\rm
triv})^{\mathcal{B}|(\varepsilon_1,\varepsilon_2,\varepsilon_3)}$ of the 1-twisted string is an
isotropic submanifold in the symplectic manifold $\,({\mathsf P}_{\sigma,\mathcal{B}|(
\varepsilon_1,\varepsilon_2,\varepsilon_3)}^{\x 3},\Omega^{+-}_{\sigma,\mathcal{B}|(\varepsilon_1,\varepsilon_2,
\varepsilon_3)})$;
\item[ii)] the background $\,\gt{B}\,$ canonically
induces a trivialisation
\begin{eqnarray}\nonumber
\gt{J}_{\sigma,(\circledast\mathcal{B}_{\rm triv}:\mathcal{J}:\mathcal{B}_{\rm triv})}^{\mathcal{B}|(
\varepsilon_1,\varepsilon_2,\varepsilon_3)}\ :\ \bigl({\rm pr}_1^*\mathcal{L}_{\sigma,\mathcal{B}|(\pi,\varepsilon_1)}
\otimes{\rm pr}_2^*\mathcal{L}_{\sigma,\mathcal{B}|(\pi,\varepsilon_2)}\bigr)\vert_{{\gt{I}_\sigma(
\circledast\mathcal{B}_{\rm triv}:\mathcal{J}:\mathcal{B}_{\rm triv})^{\mathcal{B}|(\varepsilon_1,\varepsilon_2,
\varepsilon_3)}}}\xrightarrow{\cong}{\rm pr}_3^*\mathcal{L}_{\sigma,\mathcal{B}|(\pi,\varepsilon_3)}
\vert_{\gt{I}_\sigma(\circledast\mathcal{B}_{\rm triv}:\mathcal{J}:\mathcal{B}_{\rm triv})^{\mathcal{B}
|(\varepsilon_1,\varepsilon_2,\varepsilon_3)}}
\end{eqnarray}
between the restrictions to $\,\gt{I}_\sigma( \circledast\mathcal{B}_{\rm triv}:
\mathcal{J}:\mathcal{B}_{\rm triv})^{\mathcal{B}|(\varepsilon_1,\varepsilon_2,\varepsilon_3)}$ of the (tensor)
pullback bundles.
\end{itemize}
\end{Thm} \noindent A proof of the theorem is given in Appendix
\ref{app:cross-def-int-tw}. \begin{Rem} A generic interaction process on a
world-sheet with an arbitrary embedded defect quiver is a
combination of the two `pure' types considered in detail above, with
higher-rank fusion and interaction subspaces involved. Our
conclusions are readily seen to generalise to arbitrary such
processes. \end{Rem}\begin{center}*\hspace{2cm}*\hspace{2cm}*\end{center}\bigskip
\begin{Eg}\textbf{The WZW fusion ring and the maximally symmetric
inter-bi-brane.}\label{ex:Verlinde}\\[-8pt]
\noindent The correspondence between inter-bi-brane 2-isomorphisms
and interaction subspaces in the space of states of the untwisted
string may, in fact, carry over to the quantised theory, as
demonstrated in Refs.\,\cite{Runkel:2009su,Runkel:2010}, where the
inter-bi-brane for the non-boundary maximally symmetric WZW bi-brane
of Example \ref{eg:WZW-def} was reconstructed following the general
scheme laid out in \Rcite{Runkel:2008gr}.
The point of departure of the reconstruction scheme proposed in
\Rcite{Runkel:2010} is the simplicial ${\rm G}\x{\rm G}$-space given by
the nerve
\begin{eqnarray}\nonumber
\alxydim{@C=1.cm@R=.05cm}{{\mathsf N}({\rm G}{\hspace{-0.04cm}\ltimes\hspace{-0.05cm}}{\rm G})^\bullet\quad :
\hspace{-.5cm} & \cdots \ar@<1ex>[r]^{{}^{\tx{\tiny $\txG$}}\hspace{-2pt} d^{(4)}_i} \ar@<.5ex>[r]
\ar@<0ex>[r] \ar@<-.5ex>[r] \ar@<-1ex>[r] & {\rm G}^4
\ar@<.75ex>[r]^{{}^{\tx{\tiny $\txG$}}\hspace{-2pt} d^{(3)}_i} \ar@<.25ex>[r] \ar@<-.25ex>[r]
\ar@<-.75ex>[r] & {\rm G}^3 \ar@<.5ex>[r]^{{}^{\tx{\tiny $\txG$}}\hspace{-2pt} d^{(2)}_i}
\ar@<0.ex>[r] \ar@<-.5ex>[r] & {\rm G}^2 \ar@<.5ex>[r]^{{}^{\tx{\tiny $\txG$}}\hspace{-2pt} d^{(1)}_i}
\ar@<-.5ex>[r] & {\rm G}}\,,
\end{eqnarray}
of the action groupoid
\begin{eqnarray}\nonumber
\alxydim{@C=2.cm}{{\rm G}{\hspace{-0.04cm}\ltimes\hspace{-0.05cm}}{\rm G}\quad : \hspace{-2cm} & {\rm G}^2
\ar@<.5ex>[r]^{{\rm pr}_2=:{}^{\tx{\tiny $\txG$}}\hspace{-2pt} d^{(1)}_0} \ar@<-.5ex>[r]_{\varrho=:{}^{\tx{\tiny $\txG$}}\hspace{-2pt}
d^{(1)}_1} & {\rm G}}\,,
\end{eqnarray}
defined in terms of the \emph{right} regular action
\begin{eqnarray}\nonumber
\varrho\ :\ {\rm G}\x{\rm G}\to{\rm G}\ :\ (g,h)\mapsto g\cdot h=:\varrho_h(g
)\,.
\end{eqnarray}
Note that it is the first factor in each component $\,{\rm G}^n\,$ of
the nerve that plays the r\^ole of the ${\rm G}$-space from Remark
\ref{rem:duality-scheme}. Apart from that, the construction follows
that of the nerve of the group (viewed as a small category) first
presented in \Rcite{Segal:1968}. The component inter-bi-brane
world-volumes $\,T_n,\ n\in{\mathbb{N}}_{\geq 3}\,$ were assumed to be
composed of full orbits under the ${\rm G}\x{\rm G}$-action on
$\,{\rm G}^n\,$ intertwined, by the manifestly
${\rm G}\x{\rm G}$-equivariant face maps of $\,{\mathsf N}({\rm G}{\hspace{-0.04cm}\ltimes\hspace{-0.05cm}}{\rm G}
)^\bullet$,\ with the standard action of $\,{\rm G}\x{\rm G}\,$ on
$\,{\rm G}\equiv{\mathsf N}({\rm G}{\hspace{-0.04cm}\ltimes\hspace{-0.05cm}}{\rm G})^{(0)}\,$ by left and right regular
translations. They were then shown to split into disjoint unions of
such orbits,
\begin{eqnarray}\label{eq:IBB-WZW}
T_{n,++\cdots-}={\rm G}\x\widetilde\bigsqcup_{\lambda_1,\lambda_2,\ldots,\lambda_n
\in\faff{\gt{g}}}\,\mathcal{T}_{\lambda_1,\lambda_2,\ldots,\lambda_{n-1}}^{\hspace{35pt}
\lambda_n}\,,
\end{eqnarray}
e.g.
\begin{eqnarray}\label{eq:IBB-WZW-3}
T_{n,++-}={\rm G}\x\widetilde\bigsqcup_{\lambda,\mu,\nu\in\faff{\gt{g}}}\,
\mathcal{T}_{\lambda,\mu}^{\ \ \nu}\,,
\end{eqnarray}
with
\begin{eqnarray}
\mathcal{T}_{\lambda,\mu}^{\ \ \nu}=\bigsqcup_{[w]\in\mathcal{S}_\lambda\backslash{\rm G}/
\mathcal{S}_\mu}\,\left\{\ \bigl({\rm Ad}_x{\rm e}_\lambda,{\rm Ad}_{x\cdot w}{\rm e}_\mu\bigr)
\quad\middle\vert\quad x\in{\rm G}\quad\land\quad{\rm e}_\lambda\cdot{\rm Ad}_w
{\rm e}_\mu\in\mathscr{C}_\nu\ \right\}\,,
\end{eqnarray}
where $\,\mathcal{S}_\lambda\,$ is the ${\rm Ad}_\bullet$-stabiliser of the Cartan
element $\,{\rm e}_\lambda\,$ (cf.\ \Rcite{Runkel:2010} for a general
definition). The proof of the splitting of the inter-bi-brane
world-volume into a disjoint union of diagonal ${\rm Ad}_\bullet$-orbits,
each multiplied with the `reference' factor $\,{\rm G}$,\ rests upon
the identification of the sign-weighted sum of pullbacks, along the
inter-bi-brane maps, of the bi-brane world-volume $\,\omega_{\mathsf k}\,$
appearing in the DJI \eqref{eq:DJI} (to be satisfied necessarily by
all vector fields tangent to the inter-bi-brane world-volume) as a
pre-symplectic form on partially symplectically reduced, in a manner
detailed in \Rcite{Alekseev:1993rj}, space of classical field
configurations of the level-${\mathsf k}$ Chern--Simons theory with gauge
group $\,{\rm G}_{{\mathbb{R}} \x{\mathbb{C}} P^1}\,$ on $\,{\mathbb{R}}\x{\mathbb{C}} P^1\,$ (with
$\,{\mathbb{R}}\,$ playing the r\^ole of the time axis) in the presence of
$n-1$ vertical time-like Wilson lines of holonomies fixed to lie in
the respective conjugacy classes $\,\mathscr{C}_{\lambda_i},\
i\in\ovl{1,n-1}$,\ and a single vertical anti-time-like Wilson line
of a holonomy constrained to lie in $\,\mathscr{C}_{\lambda_n}$.\ The
quantisation of the weight labels, all taken from the discrete set
$\,\faff{\gt{g}}\,$ of \Reqref{eq:faffggt}, expresses the requirement
that there exist component 1-isomorphisms $\,\Phi_{{\mathsf k},\lambda_i},\
i\in\ovl{1,n}\,$ entering the definition of the inter-bi-brane
2-isomorphism, cf.\ Diagram \eqref{diag:2iso}. Finally, the set of
admissible components of the inter-bi-brane world-volume is
restricted, as indicated by the tilde over the symbol of the
disjoint union over $\,\faff{\gt{g}}\,$ in Eqs.\,\eqref{eq:IBB-WZW}
and \eqref{eq:IBB-WZW-3}, to those which support a
\emph{non-vanishing} inter-bi-brane 2-isomorphism (conjectured to
correspond to the non-vanishing Verlinde fusion coefficients). The
existence of the latter is topologically obstructed on a generic
space $\,{\rm G}\x\mathcal{T}_{\lambda_1,\lambda_2,\ldots,\lambda_{n-1}}^{\hspace{1.2cm}
\lambda_n}\,$ owing to, in particular, the non-simple connectedness of
that space, cf.\ Proposition \ref{prop:2iso-class-1iso}.
In the presence of the multiplicative structure on $\,\mathcal{G}_\sfk$,\
mentioned in Example \ref{eg:WZW-def}, cf.\ also Remark
\ref{rem:mult-str}, the question of existence of the inter-bi-brane
2-isomorphism over $\,{\rm G}\x\mathcal{T}_{\lambda_1,\lambda_2,\ldots,\lambda_{n-
1}}^{\hspace{1.2cm}\lambda_n}\,$ reduces to the same question for a
so-called fusion 2-isomorphism over $\,\mathcal{T}_{\lambda_1,\lambda_2,\ldots,
\lambda_{n-1}}^{\hspace{35pt}\lambda_n}$.\ Thus, for instance, in the case
of the elementary inter-bi-brane $\,T_{3,++-}$,\ it boils down to
constructing a 2-isomorphism
\begin{eqnarray}\nonumber
\alxydim{@C=4em@R=3em}{\bigl({\rm pr}_1^*\mathcal{G}_\sfk\otimes{\rm pr}_2^*\mathcal{G}_\sfk\bigr)\big
\vert_{\mathcal{T}_{\lambda_1,\lambda_2}^{\quad\lambda_3}} \ar[d]_{{\rm pr}_1^*\Phi^\p_{{\mathsf k}
,\lambda_1}\otimes{\rm pr}_2^*\Phi^\p_{{\mathsf k},\lambda_2}} \ar[r]^{\mathcal{M}_{\mathsf k}
\vert_{\mathcal{T}_{\lambda_1,\lambda_2}^{\quad\lambda_3}}} & \bigl({\rm m}^*\mathcal{G}_\sfk\otimes
I_{\rho_{\mathsf k}}\bigr)\big\vert_{\mathcal{T}_{\lambda_1,\lambda_2}^{\quad\lambda_3}}
\ar[d]^{{\rm m}^*\Phi^\p_{{\mathsf k},\lambda_3}\otimes{\rm id}_{ I_{\rho_{\mathsf k}}}} \\
I_{{\rm pr}_1^*\omega^\p_{{\mathsf k},\lambda_1}+{\rm pr}_2^*\omega^\p_{{\mathsf k},\lambda_2}}\big
\vert_{\mathcal{T}_{\lambda_1,\lambda_2}^{\quad\lambda_3}} \ar@{=>}[ur]|{\varphi_{\lambda_1,
\lambda_2}^{\quad\lambda_3}} \ar@{=}[r] & I_{{\rm m}^*\omega^\p_{{\mathsf k},\lambda_3}+
\rho_{\mathsf k}}\big\vert_{\mathcal{T}_{\lambda_1,\lambda_2}^{\quad\lambda_3}}}
\end{eqnarray}
whose definition involves, beside the 1-isomorphism $\,\mathcal{M}_{\mathsf k}\,$
of the multiplicative structure, the component 1-isomorphisms
$\,\Phi^\p_{{\mathsf k},\lambda_i},\ i\in\{1,2,3\}\,$ of the \emph{boundary}
maximally symmetric WZW bi-brane described in Example
\ref{eg:WZW-def}. The task in hand was explicitly carried out for
$\,{\rm G}={\rm SU}(2)\,$ in \Rcite{Runkel:2009su}, whereby it was shown that
the inter-bi-brane 2-isomorphism exists iff the Verlinde fusion
coefficient $\,N_{\lambda_1,\lambda_2}^{\quad\lambda_3}\,$ for the triple of
chiral sectors of the quantised WZW model labelled by $\,(\lambda_1,
\lambda_2,\lambda_3)\,$ does not vanish, in accord with the prediction of
the categorial quantisation of the WZW model elaborated in
Refs.\,\cite{Fuchs:2004xi,Frohlich:2006ch}. This result constitutes
a quantum variant of the pre-quantum result discussed in the present
section.
\end{Eg}
\begin{Rem} Incidentally, the last example indicates the possibility of
adding further structure to the correspondence between the
inter-bi-brane and the interaction subspace, valid whenever the
corresponding defect perserves some of the symmetry of the untwisted
sector of the $\sigma$-model. The structure in question is that of an
intertwiner of the representations of the symmetry algebra
associated with the states undergoing the splitting-joining
interaction mediated by the defect quiver. Making this observation
rigorous calls for a detailed study of the issue of symmetry
transmission across the defect, which shall be addressed in the
framework of generalised geometry in the companion paper
\cite{Suszek:2010b}. \end{Rem}\begin{center}*\hspace{2cm}*\hspace{2cm}*\end{center}
\section{Conclusions and outlook}\label{sec:out}
At the focus of our interest in the present paper lay the
state-space interpretation of conformal defects in the classical and
(pre-)quantum formulation of the two-dimensional (bosonic)
non-linear $\sigma$-model on an oriented multi-phase world-sheet. The
latter theory is defined in terms of cohomological structures,
termed the gerbe $\,\mathcal{G}$,\ the $\mathcal{G}$-bi-brane $\,\mathcal{B}\,$ and the
$(\mathcal{G},\mathcal{B})$-inter-bi-brane $\,\mathcal{J}$,\ coming from the 2-category
$\,\gt{BGrb}^\nabla(M\sqcup Q\sqcup T)\,$ of bundle gerbes (with
connection) over the codomain $\,M\sqcup Q\sqcup T\,$ of the
$\sigma$-fields fields. The issue of interest was addressed in the
canonical framework of description of the state space of the theory,
reconstructed in the classical r\'egime using the techniques of
covariant classical field theory (Propositions
\ref{prop:sympl-form-si-untw} and \ref{prop:sympl-form-si-tw}) and
subsequently extended to the quantum r\'egime by means of
transgression maps (Theorems \ref{thm:trans-untw} and
\ref{thm:trans-tw}), derived along the lines of the long-known
explicit construction for the closed string with a mono-phase
world-sheet. The transgression maps were employed to induce a
pre-quantum bundle over the state space of the $\sigma$-model from
gerbe and bi-brane data, in both the untwisted sector and the
twisted sector of that space (Corollaries \ref{cor:preqb-untw} and
\ref{cor:preqb-tw}). In the presence of the pre-quantum bundles, the
notion of a pre-quantum duality of the $\sigma$-model was formalised,
whereupon a precise correspondence was established between
pre-quantum dualities and (non-intersecting) topological world-sheet
defects. The correspondence associates a duality to a topological
defect with bi-brane maps given by surjective submersions satisfying
some additional technical conditions (Theorem \ref{thm:def-dual}),
and -- conversely -- identifies the bi-brane encoded by the data of
a duality of one of the two distinguished types: type $T$ (Theorem
\ref{thm:duali-T-bib}) and type $N$ (Theorem \ref{thm:duali-N-bib}).
From a further extension of the canonical framework to the
interacting multi-string state space, both classical and
(pre-)quantum, an intuitive picture was shown to emerge of the
defect junctions and the attendant geometric data from
$\,\gt{BGrb}^\nabla(M\sqcup Q\sqcup T)\,$ acting as intertwiners between
representations of the symmetry algebra realised on the spaces of
states of the string in interaction, be it untwisted or twisted
(Theorems \ref{thm:cross-def-int-untw} and
\ref{thm:cross-def-int-tw}). In the minimal scenario, the symmetry
algebra in question is the Virasoro algebra of the conformal group
in two dimensions. The case in which this algebra is extended by
some Ka\v c--Moody algebra induced from distinguished isometries of
the (pseudo-)riemannian target space of the $\sigma$-model is
elaborated in \Rcite{Suszek:2010b}. As a by-product of our analysis
of conformal defects, the notion of a simplicial string background
was introduced (Remark \ref{rem:duality-scheme}) and a conjectural
statement was made with regard to its r\^ole in defining string
theory on (generically non-geometric) `duality quotients',
generalising the concept of a T-fold in a manner dictated by a
duality scheme. The scheme is largely motivated by the idea of
categorial descent for the 2-category of bundle gerbes and the
gerbe-theoretic construction of the gauged $\sigma$-model.\medskip
There are two main inferences that can be drawn from our findings
recapitulated above: The first is the manifest naturality, i.e.\
completeness and minimality of the full-blown 2-categorial structure
$\,\gt{BGrb}^\nabla(M\sqcup Q\sqcup T)\,$ viewed as a scheme of
description of the two-dimensional dynamics (including the purely
geometric interactions) of the field theory in hand, and of its
modifications obtained through gauging and orientifolding.
Additional evidence in favour of the said naturality can be
extracted from the analysis of the algebraic structure on the set of
continuous internal symmetries of the multi-phase $\sigma$-model. This
issue is examined at length in the companion paper
\cite{Suszek:2010b}, whereby the concept of generalised geometry
twisted by the background $\,\gt{B}\,$ is seen to arise, encompassing
the familiar construction of
Refs.\,\cite{Hitchin:2004ut,Gualtieri:2003dx} as a special case. The
second basic inference is the fundamental r\^ole of world-sheet
defects (and so also of the attendant cohomological structures over
$\,M\sqcup Q\sqcup T$) in probing the `topography' of the moduli
space of two-dimensional (bosonic) non-linear $\sigma$-models via the
associated dualities. Taken in conjunction, the two offer insights
into the very deep structure underlying the lagrangean formulation
of (critical) string theory, and that from the level of readily
tractable geometric constructs from the smooth category. This alone
provides strong motivation for further work in the directions
suggested by the hitherto results.
An outstanding problem from this last category is the precise
relation between the 2-category $\,\gt{BGrb}^\nabla(M\sqcup Q\sqcup
T)$,\ considered together with the transgressed structures over the
state space of the $\sigma$-model, and elements of the categorial
quantisation scheme thereof (in the sense of Segal, cp.\
\Rcite{Segal:1987sk}). There is ample and highly non-trivial
evidence indicating that certain topologically protected (and
quantised) results of the gerbe-theoretic approach, such as, e.g.,
the existence and uniqueness results for the $\sigma$-model in a given
string background, its quotients and orientifolds, as well as the
cohomological data describing the fusion of topological defects
(e.g., the conditions of existence of inter-bi-brane 2-isomorphisms,
admitting a straightforward interpretation in terms of spaces of
conformal blocks, cp.\ \Rcite{Runkel:2010}, and the recoupling
coefficients for simple associator moves on topological defect
quivers embedded in the world-sheet, related to the fusing matrices
of Moore and Seiberg, cp.\ \Rcite{Runkel:2008gr}), carry over
\emph{unaltered} to the quantum r\'egime, as defined rigorously by
the categorial quantisation (or by the operator-algebraic
quantisation, for that matter). It therefore seems apposite to
enquire whether the cohomological structure of the gerbe and its
2-categorial descendants is sufficiently rigid to encode (still
more) essential non-perturbative data of the quantised $\sigma$-model,
also for string backgrounds which -- unlike the previously examined
cases of canonical gerbes on compact Lie groups and their maximally
symmetric (inter-)bi-branes -- are devoid of a rich symmetry that
could independently constrain the quatisation procedure. It stands
to reason that our understanding in this matter can be furthered by
a search for a direct `holographic' relation between the 2-category
$\,\gt{BGrb}^\nabla(M\sqcup Q\sqcup T)\,$ for the (rational)
two-dimensional $\sigma$-model and the higher categorial structure
behind a three-dimensional Topological Field Theory (TFT) that
defines the categorial quantisation scheme of the $\sigma$-model in a
manner detailed in Refs.\,\cite{Felder:1999cv,Felder:1999mq}. A link
between the two structures was established in the largely tractable
WZW setting (in which the relevant TFT is the Chern--Simons theory
with the gauge group given by the target Lie group of the WZW model)
in \Rcite{Carey:2004xt} but even in this highly symmetric example an
exhaustive analysis of the relation between the two-dimensional CFT
on a multi-phase world-sheet and the corresponding three-dimensional
TFT coupled to a collection of intersecting Wilson lines is lacking
to date.
In an attempt at gauging the actual extent to which gerbe theory is
an intrinsic element of a CFT description of string theory, one
could be even more audacious and explore string backgrounds away
from criticality (at which the Weyl anomaly vanishes), implicitly
chosen as the basis (or completion) of the $\sigma$-model discussion.
One possible way of grappling with the issue in hand might be the
concept of a generalised Ricci flow, winning an ever increasing
popularity of late. Here, the hope would be that the ideas of
Refs.\,\cite{Streets:2007PhD,Young:2008PhD}, originally applied to
(principal) fibre bundles, could be successfully adapted to handle
gerbes over (pseudo-)riemannian bases. A novel alternative for this
line of development appears to be offered by the study of the
so-called String structures of \Rcite{Killingback:1986rd}, cp.\
\Rcite{Waldorf:2009uf} for a modern treatment in the
higher-categorial language.
Another important question merely touched upon by hitherto
gerbe-theoretic analyses carried out with reference to world-sheet
defects takes its origin in the concept of a non-geometric string
background. While the general notion of a simplicial string
background forwarded in the present paper seems to be perfectly
tailored to describe the latter, specific conditions for the
existence of a `duality quotient' and extra constraints prerequisite
for gauging a `duality group' should be worked out, and explicit
examples of those intricate stringy (non-)geometries should be
found. An obvious point of departure for these general
considerations is an in-depth understanding of the cohomological
duality between principal torus bundles with gerbes, or T-duality.
Already this outwardly (physically) well-studied subject offers
interesting conceptual challenges such as, e.g., the geometric
description of the procedure of descending the gauged $\sigma$-model
from the correspondence space (i.e.\ from the intermediary
bi-toroidal fibration linking the dual backgrounds) to the T-dual
toroidal fibration via elimination of the gauge field and symplectic
reduction. Its peculiarity consists in that it mixes various
tensorial objects defining the string background, as accounted for,
e.g., by the Buscher rules of Example \ref{ex:Tdual} (or, more
generally, by the duality-background constraints
\eqref{eq:dualiT-back-constr}). This prompts to conceive a
significant departure from the established mode of description of
geometric constructs such as the metric structure, with its global
tensorial representation by the metric field, and the gerbe, with
its local differential-geometric presentation. Such a departure,
capable of incorporating also the dilaton field, should lead to the
emergence of a unified geometric treatment of the various components
of the full multiplet of massless closed-string excitations. It is
worth pointing out that a possible first step towards such a unified
treatment is Hitchin's construction, advanced in
\Rcite{Hitchin:2005in}, of a generalised metric on the generalised
tangent bundle with a torsion-full metric connection, combining, in
a most natural manner, the metric tensor and local gerbe data.
Last but not least, given the prominent r\^ole played by
$\,\gt{BGrb}^\nabla(M\sqcup Q\sqcup T)\,$ in the geometric quantisation
of the $\sigma$-model, it is tempting to investigate the conditions of
compatibility of a choice of polarisation of the pre-quantised
theory with structures carried by the conformal defect, and the
ensuing constraints on the admissible \emph{quantum} dualities. The
passage from the classical to the quantum r\'egime is bound to
result in a renormalisation of the functional relations determining
the bahaviour of the $\sigma$-model fields at the defect (cp., e.g.,
Refs.\,\cite{Bachas:2004sy,Alekseev:2007in}), and it would be
desirable to attain a good understanding of these quantum effects.
\medskip
Thus, altogether, it seems fair to conclude the present paper with
the constatation that the hitherto incursions into the physics of
conformal defects of the two-dimensional non-linear $\sigma$-model,
aided substantially and organised neatly by gerbe theory, present us
with a panoply of string-theoretic and mathematical problems,
including those of a truly fundamental nature, of which but a small
proportion have been elucidated so far. We are hoping to return to
some of them in a future publication.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,859 |
Q: Ideavim Folding and Expanding Code Blocks What is the key binding for folding and expanding code blocks in ideavim plugin ?
(za works in vrapper for eclipse , but not in ideavim )
A: Adding these lines to your ~/.ideavimrc allows the zO and zC commands, which recursively open and close the folds under the cursor:
nnoremap zC :action CollapseRegionRecursively<CR>
nnoremap zO :action ExpandRegionRecursively<CR>
I find these to be super-helpful in vim, and was missing them when using PyCharm with IdeaVim.
(HT: https://github.com/JetBrains/ideavim/pull/97 )
A: source : ideavim help: fold
zo Open one fold under the cursor. When a count is given, that
many folds deep will be opened. In Visual mode one level of
folds is opened for all lines in the selected area.
zc Close one fold under the cursor. When a count is given, that
many folds deep are closed. In Visual mode one level of folds
is closed for all lines in the selected area.
'foldenable' will be set.
zM Close all folds: set 'foldlevel' to 0.
'foldenable' will be set.
zR Open all folds. This sets 'foldlevel' to highest fold level.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,828 |
Blue Ridge Mountain Towns
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The 11 Best Places for Snow Tubing in North Carolina (2022-2023)
October 27, 2022 October 2, 2022 by Maggie Watts
Disclaimer: This post may contain affiliate links. All hosted affiliate links follow our editorial & privacy policies.
Though the tourist season may have its peaks and valleys, the North Carolina mountains are truly a nature-lover's playground 365 days a year.
Spring brings an abundance of wonderful wildflowers to the region's best hiking trails, from striking orange flame azaleas and white mountain laurel to vivid pink rhododendrons.
Summer offers warmer weather, an explosion of verdant green foliage, and a chance to cool off in the countless waterfalls and lakes that dot the Blue Ridge Mountains of North Carolina.
Autumn is peak season, with the radiant fall colors attracting hundreds of thousands of visitors.
From driving to scenic Blue Ridge Parkway overlooks and hiking in Pisgah National Forest to camping in state parks and exploring the myriad mountain towns along the way, there are an endless array of things to do at this time of year.
But you could make an argument that winter is the most magical time to visit the mountains of North Carolina. The crowds are smaller, the rates are cheaper, and there's a decent chance the higher elevations may be dusted with snow.
There are twinkling Christmas lights in all the small towns, Christmas tree farms dotted across the NC High Country, and tons of resorts offering opportunities for snow-skiing, snowboarding, and snow tubing.
Snow tubing in the North Carolina mountains is extremely popular, especially with families. Most of the NC snow tubing resorts have snow-making capabilities, so the fun doesn't have to stop on sunny winter days.
Read on for our in-depth guide to the best snow tubing in NC, with ski resorts and tubing parks listed geographically, and details about the activities and attractions at each one.
READ MORE: The Best Places for Snow-Skiing in North Carolina
Snow Tubing in North Carolina Mountains Guide
Wolf Ridge Snow Tubing Park (Mars Hill)
Zip N Slip Snow Tubing Park (Mars Hill)
Frozen Falls Tube Park, a.k.a Sapphire Valley Snow Tubing (Sapphire Valley)
Hawksnest Snow Tubing and Zipline (Seven Devils)
Sugar Mountain Resort (Sugar Mountain)
Jonas Ridge Snow Tubing Park (Newland)
Beech Mountain Resort (Beech Mountain)
Moonshine Mountain Snow Tubing Park (Hendersonville)
Black Bear Snow Tubing (Hendersonville)
Cataloochee Ski Area (Maggie Valley)
Highlands Outpost (Scaly Mountain)
READ MORE: 20 Awesome Things to Do for Winter in North Carolina
Snow Tubing In Asheville NC
The largest of the North Carolina mountain towns (population around 95,000), Asheville NC is an artistic town filled with loads of history.
The city's thriving cultural heart is the River Arts District. But the entire metropolitan area is filled with delicious restaurants and breweries, and surrounded by excellent hiking trails, waterfalls and unique treehouse rentals.
There are also plenty of snow tubing parks and ski resorts near Asheville for family and friends to enjoy.
Read on for our Asheville snow tubing recommendations, and note that the parks are listed in order of their geographic proximity to Downtown Asheville.
READ MORE: The 30 Best Things to Do in Asheville NC
Snow Tubing in North Carolina, photo via Canva
1. Wolf Ridge Snow Tubing Park
578 Valley View Circle, Mars Hill NC • 828-689-4111
Located in Mars Hill NC, the Wolf Ridge Snow Tubing Park features a 350-foot-long slope that's 100 feet wide at the bottom and 60 feet wide at the top.
The park also features a "magic carpet" to carry tubers back to the top, shuttle, and lights for nighttime snow tubing. Wolf Ridge sits at an elevation of 4,700 feet, and offers stunning views of Big Bald Mountain.
Located 20 minutes from Asheville, the snow tubing park is part of Wolf Ridge Ski Resort, which offers skiing and snowboarding slopes, as well as lessons at the resort's Snow Sports School.
The park makes its own snow, so you never have to worry whether Mother Nature will provide her own fresh powder. There are no designated lanes here, which means that guests can link up their tubes and ride together.
The Wolf Ridge Tubing season lasts from the first week of December through mid-April, with holiday rates ending in late February. The season's start and end dates are tentative, as they rely on how much snow the area receives.
The Wolf Ridge Tube Run is open from 9AM to 10PM Mon-Sat, and 9AM to 4:30PM on Sunday (except holiday weekends). Rates are cheaper on weekdays– $20 per hour, or $30 for 2 hours– rising to $25 an hour, or $35 for two hours, Friday to Sunday.
While at the ski resort, guests can also explore their four nature trails and two diamond runs, or relax by the fire inside with hot chocolate.
Guests can stay in cabins through Wolf Mountain Realty. We recommend a 4BR Wolf Laurel Resort Home with Views & Hot Tub, or a cozy 2BR resort cabin. But there are other vacation rentals (some pet-friendly) available via VRBO.
While you're in the area, watch a show at the Southern Appalachian Repertory Theatre, take a cooking class at The Farmer's Hands, hike Big Bald Mountain, or pick up a Xmas tree at Frosty Mountain Christmas Tree Farm.
READ MORE: The Best Western NC Christmas Tree Farms in Asheville, Boone & Beyond
Photo courtesy ZipNSlip Snow Tubing Park
2. Zip N Slip Snow Tubing Park
10725 US-23, Mars Hill NC • 828-689-8444
The Zip N Slip Snow Tubing Park in Mars Hills NC features four tubing lanes, with increased snowmaking capabilities for the 2022 season (which begins in mid-December and ends in late February).
It is strongly encouraged that visitors make reservations in advance, as the available slots often sell out. Each snow tubing session lasts up to two hours, and the park offers nighttime snow tubing with well-lit slopes.
Over the summer, Zip N Slip doubled its snow-making capabilities and increased their tubing lanes from 4 to 10. They also upgraded their facilities to make the 2022-2023 season even better than last year.
Current rates for tubing are $30 for an hour, or $45 for two-hour sessions. Groups of more than 15 people get a $7.50 discount for each person ($22.50/$37.50), and kids ages 3 to 5 are free.
Zip N Slip does not offer on-site lodging, but there are many hotels and cabin rentals available nearby.
We recommend the Marshall House Inn, Bend of Ivy Lodge, and the Buck House Inn on Bald Mountain. Nearby cabin rentals can also be found through Wolf Mountain Realty.
While you're in the area, be sure to check out the animals at Farmory's Safari Edventure, explore the hiking trails at Wild Harmony Park, and go horseback riding with Sandy Bottom Trail Rides.
READ MORE: The 15 Best Things to Do in Bryson City NC & Swain County
Snow Tubing in NC, photo via Canva
3. Frozen Falls Tube Park (a.k.a. Sapphire Valley Snow Tubing)
127 Cherokee Trail, Sapphire Valley NC • 828-743-7663
Frozen Falls Tube Park is in Sapphire Valley NC, about 60 miles southwest of Asheville on NC-280 W and US-64 W. This exciting snow tubing park features a 500-foot-long run, with a 60-foot vertical drop!
There are no age, weight, or height restrictions at Frozen Falls. But guests will need to fill out a release form when they purchase tickets, and will be held responsible for any injuries at the park.
Tubing tickets are sold in advance for the entire weekend (Friday through Sunday) at the Sapphire Valley Community Center, starting at 9AM on Friday morning.
The public rate for one session is $30, while the amenity rate for Sapphire Valley guests is $22.50. Frozen Falls snow tubing sessions last up to 1.75 hours, with sessions starting at 10AM, noon, 2PM, and 4PM.
Guests can choose between on-site lodging or stay in a 4BR Sapphire Valley Resort Cabin w/Mountain Views, a 3BR Charming Home 4 miles from Sapphire Valley, a 5BR Luxury Cabin, or one of many other VRBO options.
While you're in the area, be sure to check out the waterfalls near Brevard and Cashiers, explore Gorges State Park, and visit the scenic Horsepasture River.
You can also relax and unwind at the Canyon Spa at Lonesome Valley, or pick up a Christmas tree at Tom Sawyer Christmas Tree Farm & Elf Village.
READ MORE: The 10 Best Christmas Towns in North Carolina
Snow Tubing In Boone NC
Boone NC is one of several scenic small towns in the North Carolina High Country, perhaps best known as home to Appalachian State University.
Located approximately two hours north of Asheville and Hendersonville, the town is filled with tasty restaurants, beautiful waterfalls, picturesque campgrounds, and excellent hiking trails.
An array of ski resorts and snow tubing options surround the town, which becomes a winter wonderland around the holidays. Decorations light up King Street, and snow-flecked Boone Christmas tree farms are in abundance.
Read on for our guide to snow tubing parks in and around Boone NC, which are listed in order of geographic proximity.
READ MORE: The 20 Best Things to Do in Boone NC
Photo via Hawksnest Tubing
4. Hawksnest Snow Tubing & Zipline
2058 Skyland Drive, Seven Devils NC • 828-963-6561
Located 14 miles southwest of Boone, Hawksnest Snow Tubing & Zipline is the largest snow tubing park on the East Coast. It offers 30+ lanes of snow tubing, spanning from 400 to 1,000 feet long.
The park won the title of "TripAdvisor Traveler's Choice," and has been ranked #1 among snow tubing parks in NC. Children must be at least 3 years old to go snow tubing, and sessions last up to an hour and 45 minutes.
Mon-Fri tubing sessions are $35, but on weekends and holidays the price goes up to $45. You can get two sessions for $45 on weekdays.
Available sessions start at 10AM daily, and stop at 4PM Mon-Thu, 6PM Fri-Sat, and 2PM on Sundays. The opening and closing dates at Hawksnest are dependent on weather, but the park typically opens during Thanksgiving week.
In addition to snow tubing, the park also offers ziplining and concessions on-site. Food is available in the first-floor café, including hot drinks and light snacks.
There's no on-site lodging, but there are many cabin rentals nearby. We recommend a 2BR Seven Devils Condo, a 2BR Banner Elk Cabin, or a larger 5BR Banner Elk Home. There are also an abundance of rental options on VRBO.
While you're in the area, check out the Wilderness Run Alpine Coaster, hike the scenic Otter Falls trail, visit the Sugar Creek Gem Mine, or get some Christmas cheer at the Hawk Mountain Christmas Tree Farm.
READ MORE: The 15 Best Things to Do in Banner Elk NC
Sugar Mountain Snow Tubing, photos via SkiSugar.com
5. Sugar Mountain Resort
1009 Sugar Mountain Drive, Sugar Mountain NC • 828-898-4521
Located about 15 miles southwest of Boone, the Sugar Mountain Resort is home to snow tubing lanes that range up to 700 feet in length. Thankfully, there's a "magic carpet" that transports tubers back up to the top.
Children under age 3 aren't allowed on the tubing slopes. Children ages 6 and under must ride with an adult, and kids ages 7-8 must be supervised by a parent. Tubing sessions last up to an hour and 45 minutes.
Sugar Mountain Resort's winter season typically begins in early November and runs through March, depending upon the weather. In 2021 they added a 2,000-foot-long Big Birch chairlift, plus five new snowmaking machines.
Daily tubing sessions start at 10AM, noon, 2PM, 4PM, 6PM, and 8PM each day, with tickets available on a first-come-first-serve basis. Sessions cost $33 on weekdays and $40 on weekends and holidays (Dec 15 to Jan 1).
During "March Madness", when the North Carolina tubing season has unofficially ended, they offer discounted rates, with weekday sessions at $25 and weekends $30.
This popular North Carolina ski resort also offers a variety of other activities, ski and snowboarding slopes, ice skating, golf, and snowshoeing. Visitors will also find shopping, numerous great restaurants, and a spa at the resort.
For lodging, we recommend the nearby Chetola Resort Lodge or condos in Blowing Rock NC, or the privately owned Sugar Mountain accommodations available on Booking.com or VRBO.
While you're in the area, be sure to visit Grandfather Mountain State Park (and the Mile High Swinging Bridge at the adjacent paid attraction), or pick up a tree at Evergreen Christmas Tree Farm.
READ MORE: The 10 Best Blue Ridge Parkway Hotels & Cabin Rentals in NC
Jonas Ridges Snow Tubing, photo via jonasridgesnowtube.com
6. Jonas Ridge Snow Tubing Park
9472 NC-181, Newland NC • 828-733-4155
Located 25 miles south of Boone near Linville Falls, Jonas Ridge Snow Tubing Park is a family-friendly place with 5 snow tubing lanes and a "magic carpet" conveyer belt to the top of the ski slope.
Jonas Ridge offers tubes specially made to intensify the speed of their rides, and uses snow guns to ensure fresh powder is always available. The park is also equipped with plenty of lights for nighttime snow tubing!
The park requires all children ages six and under to be accompanied by an adult, and a maximum of four people may ride together at once.
Full snow tubing sessions are $35, or $30 for children six or under. Jonas Ridge also offers 1-hour sessions for $25 and $20, plus discounted rates for active military and groups of 15 or more.
Note that Jonas Ridge Snow Tubing Park typically opens in mid-December, and the season lasts through March. But this strongly relies on weather conditions. Closure updates can be found on Jonas Ridge's social media channels.
There's no on-site lodging here, but there are a variety of cabins and hotels near the park. We recommend the 2BR Newland Cottage, 4BR Newland Escape (with hot tub!), or 5BR Riverfront Home. Additional accommodation options are available from VRBO.
While you're in the area, visit nearby attractions such as Linville Caverns, Bobby McLean Memorial Park, the hiking trails at Linville Gorge Wilderness Area, or wine tastings at local wineries.
READ MORE: The 15 Best NC Wineries to Visit
Beech Mountain Snow Tubing, photo via Facebook
7. Beech Mountain Resort
1007 Beech Mountain Pkwy, Beech Mountain NC• 828-387-2011
Towering at an elevation of 5,506 feet, Beech Mountain NC is the highest town east of the Rockies.
Located 24 miles west of Boone, Beech Mountain Resort offers snow tubing lanes ranging up to 700 feet in length. It's also known for its ski slopes, and offers shopping, restaurants, and even a brewery in the Resort Village.
Their early season goes from Nov 20 to Dec 9, while the late season lasts from March 13 to 27. Note that reservations are necessary, and early and late season tickets are only sold in person.
Sessions start every other hour, from 10AM to 6PM on weekends and from noon to 6PM on weekdays. Tubing North Carolina at Beech Mountain costs $32 during the week or $38 on weekends and holidays (Dec 10-Jan 1).
Beech Mountain has a very strict height policy. For snow tubing, all riders must be at least 42 inches tall, and children are not allowed to ride on their parent's lap. Tubes are also limited to one person each.
There are a number of cabins and hotels near the resort. We recommend the Little Main Street Inn, Beech Alpen Inn, and 4 Seasons at Beech Mountain. Nearby vacation home and cabin rentals are also available through VRBO.
While you're in the area, take some time to hike the trails in scenic Emerald Outback, meet the alpacas and other animals of Apple Hill Farm, or get into the Christmas spirit at Twin Oakes Christmas Tree Farm.
READ MORE: The 6 Best Boone NC Breweries & Brewpubs (with Map)
Snow Tubing In Hendersonville NC
Located approximately 20 miles east of Brevard and 25 miles south of Asheville, Hendersonville NC is a town rich with history that also features great restaurants, shopping, apple orchards, wineries, and other attractions.
Though located at a considerably lower elevation (2,152 feet) than the towns of the NC High Country, the area is home to several fun-filled snow tubing parks for everyone to enjoy.
Read on for our guide to snow tubing in and around Hendersonville, and note that the parks are listed in order of geographical proximity to the town.
READ MORE: The 15 Best Things to Do in Hendersonville NC
Moonshine Mountain Snow Tubing, photo via moonshinemountain.com
8. Moonshine Mountain Snow Tubing Park
5865 Willow Ridge Road, Hendersonville NC • 828-696-0333
Located about 6 miles from Downtown Hendersonville, the Moonshine Mountain Snow Tubing Park features 500 feet of tubing slopes. And since the park makes its own snow, there's always an abundance of powder to ride on!
All snow tubers must be at least 36 inches tall, and weigh no more than 300 pounds. Moonshine Mountain's tube lift has a 250-pound weight limit, but guests who exceed it may walk to the top of the slopes instead.
The price for snow tubing at Moonshine Mountain is $35 for all riders, and the sessions last two hours. They don't take reservations, and guests must pay in cash (but there is an ATM on site).
In addition to snow tubing, the park also features snacks from the Haus Heidelberg Food Truck, as well as hot chocolate by the fire inside the lodge. Guests will also find a gift shop to browse inside the lodge.
There are lots of lodging options near the park. We recommend staying at the Echo Mountain Inn, The Lodge at Flat Rock, or Highland Lake Inn & Resort. Vacation home rentals and cabin rentals are also available from VRBO.
Popular attractions in the area include the Elijah Mountain Gem Mining, Pulitzer Prize-winning poet Carl Sandburg's home, hiking in DuPont State Recreational Forest, and u-pick fun at local apple orchards.
READ MORE: The 10 Best Hendersonville NC Restaurants for Foodies
photo via Black Bear Snow Tubing
9. Black Bear Snow Tubing
373 Kerr Road, Hendersonville NC • 828-685-1155
Located 13 miles northeast of Downtown Hendersonville, the family-owned Black Bear Snow Tubing park features a snow tubing slope over 500 feet in length.
Black Bear park makes its own snow, and offers tubing on its summer slope during the hotter weeks in winter.
Because the park is located at a lower altitude, slope conditions here rely on Hendersonville weather. The park is typically open from October through early March, but it was closed until Dec 18 in 2021.
Children must be 4 years old and weigh at least 35 pounds to ride, and 4- to 6-year-olds must be accompanied by an adult. Group tubing is not allowed, and the maximum weight limit is 300 pounds.
Tickets must be purchased on-site, without reservations. A standard two-hour tubing session is $40, but they also offer one-hour sessions for $30. Groups of 20 or more can get $5 off their individual tickets.
The Black Bear Snow Tubing park includes a heated lodge where guests can relax with hot drinks, hot dogs, and other concessions. There's also an on-site playground for kids.
If you want to stay nearby, we recommend the Broad River Inn, a 3BR Hendersonville cabin with hot tub, or a 3BR Charming Hendersonville Cottage. Vacation homes and cabin rentals are also available through VRBO.
While you're in the area, take time to explore the town of Hendersonville, get breathtaking views at Jump Off Rock, or see a play at the Flat Rock Playhouse. If you have kids, take them to the Hands On Museum for a day of fun!
READ MORE: The 15 Best Treehouse Rentals in the North Carolina Mountains
NC Snow Tubing via Canva
Snow Tubing In Maggie Valley NC
With the tagline "nestled in nature's glory," Maggie Valley is a scenic small town located 18 miles east of Great Smoky Mountains National Park.
With a variety of colorful North Carolina wildflowers and many black bears, Maggie Valley is a nature lover's dream destination.
While you're in the area, be on the lookout for Cataloochee Valley Elk!
Many elk from the national park herd have begun migrating into Maggie Valley, and they can often be seen throughout the town at dawn and dusk.
READ MORE: The 10 Best Things to Do in Maggie Valley NC
Photo via Cataloochee Ski Area
10. Cataloochee Ski Area
4721 Soco Road, Maggie Valley NC • 828-926-0285
Located less than 5 miles from Maggie Valley (and a close drive from Waynesville), the Cataloochee Ski Area's Tube World features specially crafted tubes, a "magic carpet" lift, and a snow-covered run.
The park uses two snowmaking guns that can create snow at up to 80 gallons per minute, and over a distance of up to 50 yards.
Tubing NC at the Cataloochee Ski Area lasts up to an hour and 45 minutes, and costs $30 per session during the week or $35 per session on weekends and holidays.
There's no on-site lodging at the Cataloochee Ski Area, but they have lodging partners nearby.
We recommend staying at Cozy Creek Cottages, The Five Star Inn, a 3BR Luxury Log Cabin, or a 3BR Maggie Valley Retreat with hot tub. If you want to rent a cozy cabin or a larger vacation home in the area, check out VRBO.
While you're in the area, be sure to visit beautiful Soco Falls, stop by the Wheels Through Time Motorcycle Museum, or visit the Museum of the Cherokee Indian in nearby Cherokee NC.
Since Maggie Valley is only 30 minutes from the eastern entrance to Great Smoky Mountains National Park, you should also make some time to explore America's most visited national park!
READ MORE: The 20 Best Things to Do in Waynesville NC & Haywood County
Scaly Mountain Snow Tubing, photo via Facebook
11. Highlands Outpost
7420 Dillard Road, Scaly Mountain NC • 828-526-3737
Formerly known as the Scaly Mountain Outdoor Center, this snow tubing spot located near the North Georgia border changed its name to Highlands Outpost in 2021
Their tubing course features a "magic carpet" lift to the top of the slopes as well as a Kiddie Slope, which costs just $15 for two hours.
Highland Outpost's winter tubing season begins on Oct 30 and lasts through March. The park also reopens in May for summer tubing fun.
Guests must be at least 42 inches tall and at least four years old to go snow tubing at Scaly Mountain. However, the Outdoor Center also offers ice skating, gem mining, and trout fishing.
Rates for the snow tubing are $35 per person for a 2-hour session, or combo tickets for tubing and one hour of ice skating are $45.
While you're in the area, you can also explore Highlands Aerial Park, visit popular North Carolina waterfalls like Dry Falls and Glen Falls, or check out Black Rock Mountain State Park just across the state line in Clayton GA.
On-site lodging is not available, but there are plenty of options nearby.
We recommend the treehouse setting of the 2BR Sky Valley Casa Dolce Vita, the 4BR Wine Down vacation home, or the 4 bedroom Spacious Pet-Friendly Cabin in Dillard GA. Additional cabin rentals and vacation home rentals are available in the area through VRBO. –by Maggie Watts; lead image via Canva
Categories North Carolina Mountains
20 Awesome Things to Do for Winter in North Carolina
Visiting Lake Winfield Scott in Suches GA (Near Blairsville)
We encourage anyone who loves the Blue Ridge region to learn about the Leave No Trace principles of responsible environmental stewardship.
Stay on marked trails, take only pictures, pack out your trash, and be considerate of others who share the trails and parks you explore.
Remember that waterfalls and rocky summits can be dangerous. Never try to climb waterfalls or get close to a ledge to get a selfie.
When you're exploring the wilderness, it's better to be safe than to be a statistic!
Maggie Watts
10 Great Romantic Getaways in Virginia for Couples
The 20 Best Places to Visit in North Carolina for 2023
The 20 Best Lakes in North Carolina (Mountains, Coast & Beyond)
20 Things to Do in the Blue Ridge Mountains of Virginia
The 15 Best Hendersonville NC Restaurants for 2023
© 2023 Blue Ridge Mountains Travel Guide | Work With Us | Editorial Policy | Privacy Policy | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,825 |
Präexistenz (von lateinisch prä- "vor(her)" und exsistere "hervortreten") steht für:
die vorgeburtliche Existenz der Seele, siehe Präexistenzlehre
Präexistenz Christi, eine Lehre der christlichen Dogmatik | {
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{"url":"https:\/\/www.helpwithphysics.com\/2013\/05\/gravitational-time-dilation.html","text":"# Gravitational Time Dilation\n\nCalculate planet speed of Neptune and Mercury relative sun orbit. \u00a0Two people live on Mercury. \u00a0One stays on Mercury and one moves to Neptune for 40 years then returns to mercury. \u00a0Did the person on Mercury age the same as the person on Neptune during the 40 year interval? \u00a0Did he age more, less, or equal? Explain.\n\n$R(Neptune) =4 500 000 000 km =4.5*10^{12} m$\u00a0 radius of Neptune orbit\n\n$T(Neptune) =164.79 years = 60190 days =5.2*10^9 seconds$\n\n$V (Neptune) = \\omega*R = (2*\\pi\/T)*R = 5437.4 m\/s =5.43 km\/s$\n\n$R(Mercury) = 57 909 100 km =5.79*10^{10} m$\u00a0 radius of Mercury orbit\n\n$T(Mercury) =87.97 days = 7.6*10^6 seconds$\n\n$V(Mercury) =(2*\\pi\/T)*R =47867 m\/s =47.87 km\/s$\n\nIf you take one men from Mercury to Neptune, the orbital speeds of both planets are small comparing to the speed of light (300 000 km\/s). This means the the RELATIVISTIC time dilation WITH SPEED will be small enough and will not count.\u00a0 $(\\sqrt{1-v^2\/c^2} =1)$\n\nWhat counts however is the GRAVITATIONAL time dilation (explained also by relativity) that will be significantly different for each person. The DEFINITION of gravitational time dilation is the following: clocks far from massive bodies run faster and clocks near massive bodies run slower.\n\nIn a few words time is passing slower for a person in a bigger gravitational potential. because the gravity of Neptune is much bigger that the gravitation of Mercury, than the time will pass slower for the person on Neptune that for the person on Mercury.\u00a0 The person on mercury will age faster than the person on Neptune.\n\n( Since the gravitational attraction of sun on both planets is small enough as compared to the gravitational attraction of the planet itself, the above observation is correct. If you live at higher altitude on earth, time is passing slower for you than for people on plain).","date":"2018-11-12 22:44:01","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6150503754615784, \"perplexity\": 1067.708339379242}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-47\/segments\/1542039741151.56\/warc\/CC-MAIN-20181112215517-20181113001517-00305.warc.gz\"}"} | null | null |
\section{Introduction}
Analyzing gravitational phenomena Einstein used the following
postulate (which he called equivalence principle): what ever
measurements we perform inside some spacetime region we cannot
distinguish between the case when there is a homogeneous
gravitational field and the case when all bodies in this region
have constant acceleration with respect to some inertial frame.
(And since any field can be considered homogeneous in a small
enough region, this principle can be applied to a neighborhood of
any point).
Einstein concluded from this principle that the spacetime
metric is pseudo-Riemannian and in absence of all other fields but
gravity the test particles are traveling along geodesics of this
metric \cite{einstein}.
Yet V.A.Fock \cite{fock} noticed that this formulation is not
exact enough: according to general relativity , the presence of
gravitation means spacetime to be curved, i.e. curvature tensor is
nonzero, $R_{ijkl} \neq 0$. This is valid in any frame, in
particular in a uniformly accelerated one. Hence in presence of
gravity $R_{ijkl} \neq 0$ while in uniformly accelerated frame
$R_{ijkl} = 0$, and this can be distinguished experimentally by
emitting a "cloud" of a particles endowed with clocks to various
directions with various speeds. With a help of the clocks one can
determine the proper time $ds$ along every trajectory and then
calculate the metric. By numerical differentiation of the metric we
can obtain the values of $R_{ijkl}$ and then compare all them
with zero.
That is why most of authors do postulate the Riemannian metric
within the strict mathematical account of general relativity. We
would like to give here more profound and at the same time more
strict mathematical grounds for this fact.
The main drawback of traditional definition of Riemannian
geometry of spacetime is that it is formulated in terms of length
({\em viz.}\/ proper time) of idealized infinitely small intervals
rather than real ones of finite size. Besides that, this definition
demands length of space-like intervals to be determined which is not
desirable from the operationalistic point of view. Some authors
(see, {\em e.g.} \cite{c4}) give the equivalent definition
including only proper time along finite parts of time-like curves.
The postulate the metric to be Riemannian in the sense of this
definition. This is more operationalistic but yet not motivated
physically.
In our paper we show that one can reformulate the initial Einstein's
equivalence principle in such a way that both Riemannian metric of
spacetime and the geodesic motion of test particles will be obtained
from it.
Considering only the gravitational field this result is of few
interest, but the question becomes essential in presence of non
gravitational fields - it stipulates the choice of covariant analogue
of an equation. For example in \cite{c7} one asserts that
conformally invariant scalar field equations $\Box\phi + 1/6\,R
\phi=0$ come into contradiction with equivalence principle since
they contain scalar curvature $R$ (a more detailed analysis of
this issue can be found in \cite{c8}). Nevertheless, such
reasonings does not seem convincing: for example, usual Maxwell
equations in curved space contain explicitly the curvature, but
they undoubted by agree with equivalence principle. However if we
twice differentiate both parts of the equation and exchange
$F_{kl;ij}$ by $F_{kl;ji}+{{R_{ij}}^p}_l F_{kp} +
{{R_{ij}}^p}_k F_{pl}$ we obtain equations containing the
curvature explicitly. Thus the presence of curvature tensor in an
equation does not mean at all the violation of equivalence
principle. The formulation of equivalence principle proposed below
allows to solve this problem in a physically meaningful and
mathematically strict way.
The idea of our reformulation is the following. The Fock's
experiment with the cloud of particles described above is idealized
since all real measurements have their errors. Therefore all the
values calculated via these measurements, in particular, the
curvature, have their errors too. Thus if the error of so calculated
curvature tensor will be great enough (of the order of the curvature
itself) it would not be possible to determine whether the genuine
value of curvature tensor is equal to zero or not. In the meantime,
the Einstein's principle claims that any {\em real} (rather than
exact) measurement performed in small enough region will not allow us
to distinguish real (possibly curved) space from flat one.
\section{Mathematical formulation}\label{s2}
Begin with a formalization of basic notions.
\begin{definition}\label{d1} A spacetime region $M$ is called
$\epsilon${\sc -small} with respect to some fixed frame iff for any
$i$ and any $a,b\in M$ $$ \vert x^i(a) - x^i(b)\vert < \epsilon $$
\end{definition}
This definition depends on coordinate frame; however the
reformulation of equivalence principle based on this definition turns
out not to depend on frame.
The spacetime properties determine the
relative movement of uncharged particles. There are devices to
measure coordinates and other kinematic features of the motion: time,
velocity (e.g. using the Doppler effect), acceleration etc. However
one mostly measure time or length (e.g. Doppler measurement of
velocity contains determining frequency - the time interval between
neighbor maxima). So, further we shall consider only time and
length measurement. Clearly the measuring of small intervals of time
and length can be performed with smaller absolute error. Let us
denote by $\lambda(\epsilon)$ the error of our measurements in the
$\epsilon$-small region.
\begin{definition}\label{d2} A {\sc spacetime} is a triple
$(M,\Gamma,\tau)$ with $M$ -- a smooth manifold, $\Gamma$ -- a
family of smooth curves on $M$ (trajectories of test particles) and
for any $\gamma \in \Gamma$ a smooth function $\tau:\gamma \times
\gamma \rightarrow {\bf R}$ (the proper time along $\gamma$) is
determined such that
\[
\tau(a,c) = \tau(a,b) + \tau(b,c)
\]
\noindent whenever
\[
\gamma^{-1}(a) < \gamma^{-1}(b) < \gamma^{-1}(c)
\]
\end{definition}
The {\sc flat} spacetime is a triple $(M_0,\Gamma_0,\tau_0)$ where
$M_0 = {\bf R}^4$, $\Gamma_0$ is set of all time-like straight lines,
$\tau_0$ is Minkowski metric.
\begin{definition}\label{d3} A spacetime $(M,\Gamma,\tau)$
is called $\lambda$-flat if for any point $m\in M$ there exists
such a frame that for a sufficiently small $\epsilon > 0$ all
coordinate and time measurements in any $\epsilon$-small region of
$M$ coincide (up to an error $\le \lambda(\epsilon)$) with the
analogous result in the flat spacetime.
\end{definition}
The final formulation of the equivalence principle must not of course
depend on accessible devices ({\em i.e.}\/ the kind of the function
$\lambda$). Thus, instead of a single function we must operate with
a class of such functions $\Lambda = \{\lambda\}$. We shall assume
possible refinement of any measurement, namely $\Lambda$ together
with every $\lambda$ is assumed to contain also
the function $k\lambda$ for every $0 < k < 1$.
\begin{definition}\label{d4} A spacetime is siad to be
$\Lambda$-flat if it is $\lambda$-flat for all $\lambda \in
\Lambda$.
\end{definition}
The formulation of the equivalence principle we propose is the
following:
\[
\hbox{\em the spacetime is} \; \lambda\hbox{\em -flat}
\]
If we exclude degenerate cases ($\Lambda$ is too great and Fock's
reasoning is valid or $\Lambda$ is too small so that
$\Lambda$-flatness implies nothing) the proposed formulation yields
us a basis for Riemannian metrics and geodesic motion.
\section{Main results}\label{sres}
\begin{theorem}\label{th1} For any class of functions
$\Lambda:{\bf R}^+ \rightarrow {\bf R}^+$ such that $\lambda \in
\Lambda$ implies $k \lambda \in \Lambda$ for any positive $k\le 1$
one of the following statements is valid: \end{theorem}
\begin{itemize}
\item[A.] Any spacetime is $\Lambda$-flat.
\item[B.] Only pseudo-Riemannian spacetime are $\Lambda$-flat and
the class of curves $\Gamma$ is arbitrary.
\item[C.] Only pseudo-Riemannian spacetimes are $\Lambda$-flat
and $\Gamma$ is the set of timelike geodesics.
\item[D.] Only flat spacetime is $\Lambda$-flat.
\end{itemize}
The proof will be organized according to the following plan:
\begin{picture}(120,100)(0,-10)
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\put(20,0){\mbox{Lemma \ref{l5}.}}
\put(20,30){\mbox{Lemma \ref{l3}.}}
\put(20,60){\mbox{Lemma \ref{l1}.}}
\multiput(41,-1)(0,30){3}{\mbox{YES}}
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\multiput(71,16)(0,30){3}{\mbox{NO}}
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\lim\limits_{\overline{\epsilon \to
0}}\frac{\lambda(\epsilon)}{\epsilon^3} < +\infty$}}
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\lim\limits_{\overline{\epsilon \to
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\lim\limits_{\overline{\epsilon \to
0}}\frac{\lambda(\epsilon)}{\epsilon} < +\infty$}}
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\begin{lemma}\label{l1} If there exists $\lambda \in \Lambda$ for
which
\begin{equation}\label{e1}
\lim\limits_{\overline{\epsilon \to
0}}\frac{\lambda(\epsilon)}{\epsilon}\,=\,K\, < +\infty
\end{equation}
\noindent then any $\Lambda$-flat spacetime is pseudo-Riemannian.
\end{lemma}
\paragraph{Proof.} Consider a curve $\gamma \in \Gamma$ and a point
$a\in m$ in a coordinate frame $\{x^i\}$. Let $a$ have the
coordinates $\{x^i_0\}$ in this frame.
In accordance with the definition of $\underline{\lim}$ there exists
such a sequence $\epsilon_n$ that $\lambda(\epsilon_n)/\epsilon_n$
tends to $K$. Let all $\epsilon_n$ be small enough (it can be
assumed with no loss of generality) then for any $n$ all measurements
in the region
$$ \vert x^i - x^i_0 \vert \,<\, \frac{\epsilon_n}{2} $$
\noindent with the error not greater than $\lambda(\epsilon_n)$
coincide with same measurements in flat spacetime, in particular
\[
\vert \delta\tau -\delta\tau_0\vert \,\le\, \lambda(\epsilon_n)
\]
\noindent where
\[
\begin{array}{rcl}
\delta\tau &=& \tau(x^i_0+\delta x^i, x^i_0) \cr
\delta\tau_0 &=& \tau_0(x^i_0+\delta x^i, x^i_0)
\end{array}
\]
\noindent and $\tau,\tau_0$ are metrics along two geodesics both
passing through the region described above.
If $x_0+\delta x$ lies on the frontier of the region then $\vert
\delta x^i \ge C\epsilon_n$ for some $C = \hbox{const}$ thus
\[
\left\vert \frac{\delta\tau}{\delta x^i} -
\frac{\delta\tau_0}{\delta x^i} \right\vert \,\le\,
\frac{\lambda(\epsilon_n)}{C\epsilon_n}
\]
\noindent therefore
\[
\frac{\delta\tau_0}{\delta x^i} -
\frac{\lambda(\epsilon_n)}{C\epsilon_n} \,\le\,
\frac{\delta\tau}{\delta x^i} \,\le\,
\frac{\delta\tau}{\delta x^i} +
\frac{\lambda(\epsilon_n)}{C\epsilon_n}
\]
So when $n\to \infty$ (and $\epsilon_n\to 0$)
\[
\frac{\delta\tau_0}{\delta x^i} - \frac{K}{C} \,\le\,
\underline{\lim}\frac{\delta\tau}{\delta x^i} \,\le\,
\overline{\lim}\frac{\delta\tau}{\delta x^i} \,\le\,
\frac{d\tau_0}{dx^i} + \frac{K}{C}
\]
\noindent All the above reasonings are valid for any $k\lambda$ with
$k<1$ hence
\[
\frac{d\tau_0}{dx^i} + \frac{kK}{C} \,\le\,
\underline{\lim}\frac{\delta\tau}{\delta x^i} \,\le\,
\overline{\lim}\frac{\delta\tau}{\delta x^i} \,\le\,
\frac{d\tau_0}{dx^i} + \frac{kK}{C}
\]
Since $k$ can be taken arbitrary small, we have
\[
\lim\frac{\delta\tau}{\delta x^i} \,=\,
\frac{d\tau_0}{dx^i}
\]
\noindent {\em i.e.}\/ in any point in some coordinate frame the
metric of our spacetime coincides with Minkowskian one, that is why
it is pseudo-Riemannian.
\hspace*{\fill}$\Box$\medskip
\begin{lemma}\label{l2} If for any $\lambda \in \Lambda$
\begin{equation}\label{e2}
\underline{\lim}\frac{\lambda(\epsilon)}{\epsilon} \,= \,
+\infty
\end{equation}
\noindent then any spacetime is $\Lambda$-flat.
\end{lemma}
\paragraph{Proof.} (\ref{e2}) implies $\lim\limits{\epsilon\to 0}
\lambda(\epsilon)/\epsilon = +\infty$. Hence for arbitrary $N$ we
have $\lambda(\epsilon)>N\epsilon$ beginning from some $\epsilon$.
In particular, it is valid for $N>\sup\vert d\tau/dx^i\vert$, hence
$$ \delta\tau < N\delta x^i \le N\epsilon < \lambda(\epsilon) $$
\noindent thus $\vert \delta\tau_0 - \delta\tau\vert <
\lambda(\epsilon)$ for any $\lambda \in \Lambda$.
\hspace*{\fill}$\Box$\medskip
\begin{lemma}\label{l3} If there exists $\lambda \in \Lambda$ such that
\begin{equation}\label{e3}
\underline{\lim}\frac{\lambda(\epsilon)}{\epsilon^2} \,< \,
+\infty
\end{equation}
\noindent then in any $\Lambda$-flat spacetime the set $\Gamma$ is a
set of geodesics.
\end{lemma}
\paragraph{Proof} is similar to that of Lemma \ref{l1}, but uses
the second derivatives of $\tau$:
\[
\left\vert \frac{d^2x^i}{d\tau^2} -
\frac{d^2x^i}{d\tau^2_0} \right\vert \le
\frac{\lambda(\epsilon)}{\epsilon^2}
\]
\noindent hence is some coordinate frame $d^2x^i/d\tau^2 = 0$ thus
$D^2x^i/ds^2 = 0$.
\hspace*{\fill}$\Box$\medskip
\begin{lemma}\label{l4} If for any $\lambda\in\Lambda$
\begin{equation}\label{e4}
\underline{\lim}\frac{\lambda(\epsilon)}{\epsilon^2} \,= \,
+\infty
\end{equation}
\noindent then any pseudo-Riemannian space with any set of
trajectories $\Gamma$ is $\Lambda$-flat.
\end{lemma}
\paragraph{Proof.} Is similar to that of Lemma \ref{l2}. We obtain
that
\[
\lambda(\epsilon) >
\vert D^2x^i/ds^2 \vert =
\vert D^2x^i/ds^2 - 0 \vert =
\vert D^2x^i/ds^2 -D^2x^i_0/ds^2 \vert
\]
\hspace*{\fill}$\Box$\medskip
\begin{lemma}\label{l5} If for some $\lambda\in\Lambda$
\begin{equation}\label{e5}
\underline{\lim}\frac{\lambda(\epsilon)}{\epsilon^3} \,< \,
+\infty
\end{equation}
\noindent then any $\Lambda$-flat spacetime is flat.
\end{lemma}
\paragraph{Proof.} In this case the Fock's reasoning is valid: in
terms of $d^3\tau/dx^3_i$ we can define the curvature tensor with
arbitrary small error.
\hspace*{\fill}$\Box$\medskip
\begin{lemma}\label{l6} If for any $\lambda\in\Lambda$
\begin{equation}\label{e6}
\underline{\lim}\frac{\lambda(\epsilon)}{\epsilon^3} \,= \,
+\infty
\end{equation}
\noindent and for some $\lambda\in\Lambda$
\begin{equation}\label{e7}
\underline{\lim}\frac{\lambda(\epsilon)}{\epsilon^2} \,< \,
+\infty
\end{equation}
\noindent then any pseudo-Riemannian space with the set of
geodesics $\Gamma$ is $\Lambda$- flat.
\end{lemma}
\paragraph{Proof} is carried out likewise. This competes the proof
of the main theorem.
\hspace*{\fill}$\Box$\medskip
The physical meaning of the results obtained is the following:
$\lim\frac{\lambda(\epsilon)}{\epsilon} \,< \,
+\infty$ means the possibility of arbitrary exact measuring
of velocities, $\lim\frac{\lambda(\epsilon)}{\epsilon^2} \,< \,
+\infty$ means the possibility of arbitrary exact measuring
of accelerations, and $\lim\frac{\lambda(\epsilon)}{\epsilon^3} \,<
\, +\infty$ means the possibility of arbitrary exact measuring of
derivatives of accelerations. So the physical meaning of the
result we obtained is the following: the equivalence principle is
valid only for measuring velocities and accelerations (in any point
they can be turned to zero by corresponding choice of the
coordinate frame) but not valid for derivatives of accelerations
(which correspond to invariant tidal forces).
\section{Some remarks on other fields}\label{srem}
We require the results of all measurements (including trajectories
of particles traveling under other fields) performed in
$\epsilon$-small region (see Def.1) to coincide up to
$\lambda(\epsilon)$ with the results of analogous experiments in
flat spacetime. We also require it to be so for all functions
$\lambda$ of a class $\Lambda$ containing a function
$\underline{\lim}{\lambda(\epsilon)}/{\epsilon^2} \,< \, +\infty$
satisfying and containing no function $\lambda$ with
$\underline{\lim}{\lambda(\epsilon)}/{\epsilon^3} \,< \, +\infty$
Let us demonstrate that the equivalence principle in this
formulation holds for equation of radiating charged particle and
theories with conformally invariant scalar field and does not hold
for scalar-tensor Brans-Dicke theory.
The equation describing the motion of a {\em radiating charged
particle} contains second derivatives of its velocity $\ddot{u}_i =
d^2u_i/ds^2$ viz. third derivatives of coordinates measured up to
$\lambda(\epsilon)/\epsilon^3$. However
$\lambda(\epsilon)/\epsilon^3$ tends to infinity for any
$\lambda\in\Lambda$, hence the smaller is the region, the greater
is vagueness in determining $\ddot{u}_i$, thus any equation
containing $\ddot{u}_i$ does not contradict our equivalence
principle (including equations containing the curvature
explicitly).
On {\em conformally-invariant scalar field.\/} As we already
mentioned, the only way to measure the curvature is by exploring
its influence on trajectories of particles in accordance with the
formula $\ddot{x}^i = \varphi_{,i}$. Since $\ddot{x}^i$ is measured
up to $\lambda(\epsilon)/\epsilon^3$, the measurement of
$\varphi_{,i}$ has the same value, hence $\Box\varphi =
(\varphi_{,i}^{,i}$ is measured up to
$\lambda(\epsilon)/\epsilon^3$ and in accordance with the preceding
reasoning any equation containing $\Box\varphi$ do not contradict
the operational equivalence principle.
In {\em Brans-Dicke theory} the principle does not hold since
micro-black holes do not move along geodesics there.
\medskip
\section{Locally almost isotropic space is almost uniform: on the
physical meaning of Schur's theorem}\label{s5}
The Schur's theorem asserts that if a space is locally isotropic
(i.e. in any point the curvature tensor has no directions chosen by
some properties)
\begin{equation}\label{e8}
R_{ijkl} \,=\, K(x)(g_{ik}g_{jl} - g_{il}g_{jk})
\end{equation}
\noindent then the space is homogeneous.
\medskip
Since all the real measurements are not exact their results can
provide only local almost isotropicity. To what extent we can
consider it to be homogeneous? It was the problem arised in 1960-s
by Yu.A.Volkov.
The main difficulty in solving the problem is that the usual proof
of Schur's theorem is based on Bianchi identities applied to
(\ref{e8}) where after summing we obtain $K_{,i} = 0$ hence $K =
{\rm const}$. However the fact that curvatures along different
directions are almost equal does not provide $K_{;i}$ to be small
enough. So, if we consider a space with almost equal curvatures
along all the directions as locally isotropic one, we cannot obtain
any isotropy. The goal of this section is to answer this
question assuming the local almost isotropicity to be the closeness
of results of all the measurements along any direction.
\begin{definition}\label{d4a} A spacetime region $M$ is called
$\epsilon$-small if for any $a,b \in M$
\[
\begin{array}{c}
|x^i(a) - x^i(b)| \le \epsilon \cr
\rho(a,b) \le \epsilon
\end{array}
\]
\noindent where $\rho$ is a metric on $M$.
\end{definition}
\begin{definition}\label{d5} By an $(x^i)$-distance between points
$a,b\in M$ we shall call $\max_i\{|x^i(a) - x^i(b)|, \rho(a,b)\}$.
The $\epsilon$-neighborhood of a point $a$ is the set of
all points $m$ of $M$ such that the $(x^i)$-distance between $m$
and $a$ does not exceed $\epsilon$. All geometric and kinematic
measurements, as it was already mentioned in section \ref{s2} are
reduced to the measurement of the metric, the distances and proper
time interval along trajectories of particles.
\end{definition}
\begin{definition}\label{d6} A spacetime region $M$ is
$\epsilon$-locally isotropic if for any $a\in M$ one can
define the action of an apropriate rotation group so that $a$ is
invariant under this action and all results obtained in
the $\epsilon$-neighborhood of $a$ coincide up to
$\lambda(\epsilon)$ with the results obtained after the action of
any element of the group.
\end{definition}
\begin{definition}\label{d7} In an analogous way we shall call a
region $\delta$-uniform if one can define an action of a transition
group on this region so that the results of any measurement on a
system of $N$ particles coincide with the precision
$\lambda(\epsilon)$ with analogous measurement on the system,
obtained from the first one after applying to it any element of the
group.
A region is called $\epsilon$-locally $\delta$-uniform if all the
above is valid for the measurement in $\epsilon$-neighborhoods of
any two points of the region. Further we shall consider only
geodesically connected domains.
\end{definition}
The problem now is to find the least $\delta$ (over $\epsilon$,
$\delta$ and $L$) such that any $\epsilon$-locally
$\lambda$-isotropic region of the size $L$ is $\delta$-uniform.
\begin{theorem}\label{th2} Any $\epsilon$-locally
$\lambda$-isotropic region of the size $L$ is:
\begin{enumerate}
\item $C\epsilon$-locally $\frac{CL\lambda}{\epsilon}$-uniform
\item $C(L/\epsilon)^2\lambda$-uniform
\end{enumerate}
\noindent where $C = {\rm const}$, and there are spaces for which
this evaluations cannot be diminished.
\end{theorem}
\paragraph{Sketch of the Proof.} 1). Since the local metric of
Riemannian space is close to Euclidean there exists some $c$ of
order 1 such that into the intersection of two $\epsilon$-neighborhoods
of two points on the distance $\epsilon$ one can inscribe a
$C\epsilon$-neighborhood.
2). Consider two $C\epsilon$-neighborhoods of arbitrary points
$x,y\in M$. Let $x^1,y^1$ be points on their frontiers (in
pseudo-Riemannian case we choose these points so that $xx^1$ and
$yy^1$ would be of the same kind). Since $M$ is geodesically
connected, $x^1$ and $y^1$ can be connected with a geodesic whose
$(x^i)$-length does not exceed $L$ (see Definition \ref{d5}). It
can be divided into $L/\epsilon$ parts of $(x^i)$-length of order
$\epsilon$. Then we approximate the geodesic by a broken line so
that the intervals $x^1z^1, z^1z^2, \ldots , z^ny^1$ have the
$(x^i)$-length $\epsilon$.
Now let us rotate $C\epsilon$-neighborhood of $x$ in order to
make it appears all inside the intersection of
$\epsilon$-neighborhoods of $x^1$ and $z^1$.
Then we rotate it around $z^1$, so that it gets to the
$\epsilon$-neighborhood of $z_2$ and so on until the
$\epsilon$-neighborhood of the point $y$. The results of all
measurements do not differ more than $\delta y\lambda$, hence the
results in neighborhoods of $x^1$ and $y^1$ do not differ by more
than $\lambda L/\epsilon$.
Here is an example when the evaluation cannot be refined: when
all differences of measurements are of the same sign i.e.
curvature monotonically varies from $x^1$ to $y^1$.
3). In a similar way we obtain the result for the global uniformity.
An interval of curve of length $\sim L$ is composed of $\sim
CL/\epsilon$ intervals of length $\sim C\epsilon$. The results of
measurements along this intervals are indistinguishable up to
$L\lambda/\epsilon$, hence the error in time or length measurement
along all the curve does not exceed $(CL/\epsilon)\cdot
(L\lambda/\epsilon) = C(L/\epsilon)^2\lambda$. This completes the
proof of Theorem \ref{th2}.
\section{Physical interpretation of the results and possible
applications}
If we perform in an $\epsilon$-small (see Definition \ref{d4a})
region a measurement with error $\lambda$ we know the metric
$g_{ij}\sim \delta\tau/\delta x^j$ up to $\lambda/\epsilon$,
the Cristoffel symbols $\Gamma^i_{jk}\sim \partial
g_{ik}/\partial x^j \sim \delta^2\tau/\delta x^{j2}$ up to
$\lambda/\epsilon^2$, and the curvature up to $\lambda/\epsilon^3$.
In any point we can choose a coordinate frame making
$\Gamma^i_{jk}$ zero, therefore measurements with error
$\lambda\sim \epsilon^2$ do not allow us to distinguish
non-isotropic case from a locally isotropic (and even from flat
one, i.e. any Riemannian space is $\epsilon$-locally
$C\epsilon^2$-isotropic). For the sake of such distinction the
error of the curvature must not exceed its value $K$, so the
relation $\lambda/K\epsilon^2$ characterize the relative error of
measurement of local {\em isotropy}.
The relative error of measuring the local isotropy is the
same. Therefore local isotropy implies local uniformly with
$\epsilon/\lambda$ times greater error. Hence if $\lambda\sim
\epsilon^3$ we obtain (for small enough $\epsilon$) the
$\epsilon^2$-uniformity obtained above for any Riemannian metric.
To obtain non-trivial information on local uniformity one must have
$\lambda\le \epsilon^4$ which corresponds to the possibility of
exact enough measurement of curvature tensor and its derivatives,
viz. tidal forces and their spacetime gradients. Mathematically
it means that if both and $R_{ijkl}$ and $R_{ijkl;m}$ are almost
isotropic then the space is almost uniform.
Thus the physical result is the following: if accelerations
and tidal forces an locally isotropic then nothing can be said on
uniformity of the region. However if gradients of tidal forces are
also isotropic then the space is locally uniform.
Imagine we verify the isotropy in a few points (e.g. close to
the Earth) - in general the same points as other points of space.
If it happens that in these points the space is $\epsilon$-locally
$\lambda$-isotropic then it is naturally to assume $\epsilon$-local
$\lambda$-isotropy everywhere and the results obtained give us the
evaluation of its homogeneity.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 267 |
Q: Web API that uses a default connection string to authenticate and an external database to get data I have a Solution that has an MVC architectural structure and a WEB Api. The web page connects to the WEB Api to first login and get a token (by using a local DB). Once it has the token, the website connects to it again as to get a Product. The method on the Web API is displayed below:
public IEnumerable<string> Get(int id)
{
try
{
Commonlayer.Views.ProductView pv = new ProductRepository().GetProductV(id);
return new string[] { pv.Description, pv.Email, pv.ImageLink, pv.Name, pv.Price.ToString() };
}
catch(Exception ex)
{
return new string[] { "1", "2", "3"};
}
}
What this does is that it connects to my Data Access Layer which is responsibly for getting a Product from the SQL Database. However this keeps failing, probably because I cannot connect to the DB properly. I tried to add a connection string other than my current local DB string in the WEB API Config, however that results in an error when it tries to login (possibly because it is confused on which connection string to use?).
This is the default connection string:
<connectionStrings>
<add name="DefaultConnection" connectionString="Data Source=(LocalDb)\v11.0;AttachDbFilename=|DataDirectory|\aspnet-WebAPI.mdf;Initial Catalog=aspnet-WebAPI;Integrated Security=True" providerName="System.Data.SqlClient" />
</connectionStrings>
At this point I have no idea what to do, any pointers would be great.
A: I found out the answer. I needed two connection strings in my WEB API, One for the local db and one for the external. However I was writing the connection strings in different manners, solved now.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,259 |
\section{Introduction}
Our aim is to describe the dynamics of a soliton solution of
the Korteweg-de Vries equation in the presence of a random potential, depending both
on space and time and which is white in time. After the first paper \cite{Wa} showing ``superdiffusion"
of the soliton of the KdV equation in the presence of an external force which is a white noise in time
(see also \cite{BKKS}, \cite{He}),
the interest in such questions of soliton dynamics in
the presence of either deterministic or random perturbations has recently increased in the
mathematical community. In \cite{Ga}, e.g. the question is investigated with the help of
inverse scattering methods, for different types of time-white noise perturbations, still for
the KdV equation, while in
\cite{jfgs0}, \cite{jfgs1}, the case of a soliton of the NLS equation is studied, with the
presence of a slowly varying deterministic external potential. Random potential perturbations for NLS equations
have also been considered in \cite{Ga0} and \cite{dBF}. The diffusion of solitons of the KdV equation in the
presence of additive noise was numerically investigated in \cite{Pr}.
Also, in \cite{dBD1}, we studied the soliton dynamics for a KdV equation with an additive space-time noise.
Our aim here is to reproduce the analysis of \cite{dBD1} in the case of a random potential, which is stationary
in space : the solution of the stochastic equation starting from a soliton at initial time will then stay
close to a modulated soliton up to times small compared to ${\varepsilon}^{-2}$ where ${\varepsilon}$ is the amplitude of
the random perturbation (see below).
In the present case, where the noise is multiplicative (the random potential) we are then able to analyze more
precisely the modulation equations for the soliton parameters and the linearized equation for the remaining
(dispersive) part
of the solution, and especially its asymptotic behavior in time.
We consider a stochastic KdV equation which may be written in It\^o form as
\begin{equation}
\label{1}
du+({\partial}_x^3 u+\frac12 {\partial}_x(u^2)) dt = {\varepsilon} u dW
\end{equation}
where ${\varepsilon}>0$ is a small parameter, $u$ is a random process defined on
$(t,x) \in {\mathbb R}^+\times {\mathbb R}$,
$W$ is a Wiener process on $L^2({\mathbb R})$ whose covariance operator $\phi \phi^*$
is such that $\phi$ is a convolution
operator on $L^2({\mathbb R})$ defined by
$$ \phi f(x)= \displaystyle \int_{{\mathbb R}} k(x-y) f(y) dy, \; \mathrm{for} \; f \in L^2({\mathbb R}). $$
The convolution kernel $k$ satisfies
\begin{equation}
\label{h1}
\|k\|_1:= \int_{{\mathbb R}} (k^2+(k')^2) dx < +\infty.
\end{equation}
Considering a complete orthonormal system $(e_i)_{i \in {\mathbb N}}$ in $L^2({\mathbb R})$,
we may alternatively write $W$ as
\begin{equation}
\label{2}
W(t,x)= \displaystyle \sum_{i \in {\mathbb N}} \beta_i(t) \phi e_i(x),
\end{equation}
$(\beta_i)_{i \in {\mathbb N}}$ being an independent family of real valued Brownian
motions.
The correlation function of the process $ W$ is then given by
$$ {\mathbb E}(W(t,x) W(s,y))=c(x-y)(s\wedge t), \quad x,y \in {\mathbb R},
\quad s,t>0, $$
where
$$c(z)=\int_{{\mathbb R}} k(z+u)k(u) du. $$
The existence and uniqueness of solutions for stochastic KdV equations of the
type (\ref{1}) but with an additive noise have been studied in \cite{dBD0},
\cite{dBDT99}, \cite{dBDT04}. The multiplicative case with homogeneous noise as described
above was considered in \cite{dBD06}:
assuming, together with the above condition, that $k$ is an integrable function of $x \in {\mathbb R}$
allowed us to prove the global existence and
uniqueness of solutions to equation (\ref{1}) in the energy space $H^1({\mathbb R})$,
that is in the space where both the mass
\begin{equation}
\label{masse}
m(u)=\displaystyle \frac{1}{2} \int_{{\mathbb R}} u^2(x) dx
\end{equation}
and the energy
\begin{equation}
\label{hamiltonien}
H(u)=\displaystyle \frac{1}{2}\displaystyle \int_{{\mathbb R}} ({\partial}_x u)^2 dx -\displaystyle \frac{1}{6} \displaystyle \int_{{\mathbb R}} u^3 dx
\end{equation}
are well defined. Note that $m$ and $H$ are conserved for the
equation without noise, that is
\begin{equation}
\label{KdV}
{\partial}_t u + {\partial}_x^3 u + \frac12 {\partial}_x (u^2) =0.
\end{equation}
Under the above conditions on $k$, it was then proved in \cite{dBD06} that for any given initial data
$u_0 \in H^1({\mathbb R})$, there is a unique solution $u$ of (\ref{1}) with paths a.s. continuous for $t\in {\mathbb R}^+$
with values in $H^1({\mathbb R})$.
Our aim in this article is to analyze the qualitative influence of a noise on a
soliton solution of the deterministic equation.
More precisely, we study the qualitative behavior of solutions of (\ref{1}) in the limit
${\varepsilon}$ tends to zero, assuming that the initial state of the solution is a
soliton of
equation (\ref{KdV}). We recall indeed that equation (\ref{KdV}) possesses a
two-parameter family of solitary waves (or soliton) solutions, propagating with a constant velocity $c>0$, with the expression
$u_{c,x_0}(t,x)={\varphi}_c(x-ct+x_0)$, $x_0 \in {\mathbb R}$, where
\begin{equation}
\label{soliton}
{\varphi}_c(x)= \displaystyle \frac{3c}{2\cosh^2(\sqrt{c}\frac{x}{2})}
\end{equation}
satisfies the equation
\begin{equation}
\label{eqsoliton}
{\varphi}_c^{\prime \prime} -c{\varphi}_c +\frac12 {\varphi}_c^2 =0.
\end{equation}
We do not recall here the well-known results concerning the stability of the soliton solutions
$u_{c,x_0}$ in equation (\ref{KdV}), but we refer to \cite{Ben}, \cite{BSS}, \cite{MM2}
or \cite{PW} for a review of the stability questions using PDE methods, or to \cite{GGKM} and
\cite{Sch} for a review of the stability of the solitons with the help of the inverse scattering transform.
Let us consider as in \cite{dBD1} the solution ${u^{\vep}}(t,x)$ of
equation (\ref{1}) which is such that ${u^{\vep}}(0,x)={\varphi}_0(x)$ where
$c_0>0$ is fixed. Then, in Section 2, we show, as we did in \cite{dBD1} for the additive equation
that up to times $C{\varepsilon}^{-2}$, where $C$ is a constant, we may write the solution
${u^{\vep}}$ as
\begin{equation}
\label{modulation}
{u^{\vep}}(t,x)= {\varphi_{c^{\varepsilon}(t)}} (x-{x^{\varepsilon}(t)}) +{\varepsilon} {\eta^{\varepsilon}} (x-{x^{\varepsilon}(t)})
\end{equation}
where the modulation parameters ${c^{\varepsilon}(t)}$ and ${x^{\varepsilon}(t)}$ satisfy a system of stochastic differential
equations and the remaining term ${\varepsilon} {\eta^{\varepsilon}}$ is small in $H^1({\mathbb R})$.
We then prove in Section 3 that the process ${\eta^{\varepsilon}}$
converges as ${\varepsilon}$ goes to zero, in quadratic mean, to a centered Gaussian
process $\eta$ which satisfies an additively driven linear equation, with a
conservative deterministic part;
we also investigate the behavior of the process $\eta$ as $t$ goes to
infinity and prove that ${\eta}$ is in some sense a Ornstein-Uhlenbeck process,
with a unique Gaussian invariant measure.
In addition, the parameters ${x^{\varepsilon}(t)}$ and ${c^{\varepsilon}(t)}$ may be developed up to order one in ${\varepsilon}$ and we get
$$ \left\{
\begin{array}{l}
d{x^{\varepsilon}}=c_0dt+{\varepsilon} B_1dt +{\varepsilon} dB_2+o({\varepsilon})\\
d{c^{\varepsilon}}={\varepsilon} dB_1 +o({\varepsilon}),
\end{array}
\right.$$
where $B_1$ and $B_2$ are correlated real valued Brownian motions; keeping only the order one terms in those modulation
parameters, we then obtain a diffusion result on the modulated soliton similar to the result obtained by Wadati in \cite{Wa},
but with a different time exponent (see Section 4).
In all what follows,
$(.,.)$ will denote the inner product in $L^2({\mathbb R})$,
$$
(u,v)=\int_{{\mathbb R}} u(x)v(x) dx
$$
and we denote by
${\mathcal T}_{x_0}$ the translation operator
defined for ${\varphi} \in C({\mathbb R})$ by $({\mathcal T}_{x_0} {\varphi}) (x)={\varphi}(x+x_0)$.
Note that since the process $W$ is stationary in space, for any
$x_0 \in {\mathbb R}$ the process ${\mathcal T}_{x_0} W$ is still a Wiener process with
covariance $\phi \phi^*$. Indeed by (\ref{2}),
$${\mathcal T}_{x_0} W(t,x)=\sum_{k\in {\mathbb N}} (\phi e_k) (x+x_0) {\beta}_k(t)=
\sum_{k\in {\mathbb N}} (\phi \tilde e_k)(x) {\beta}_k(t),$$
with $\tilde e_k(x)={\mathcal T}_{x_0}e_k$.
\section{Modulation and estimate on the exit time}
In this section, we prove the following theorem.
\begin{Theorem}
\label{t2}
Assume that the kernel $k$ of the noise satisfies (\ref{h1}) together with
$k\in L^1({\mathbb R})$
and let $c_0$ be fixed. For ${\varepsilon}>0$, let ${u^{\vep}}(t,x)$,
as defined above, be the solution of (\ref{1}) with $u(0,x)={\varphi}_{c_0}(x)$.
Then there exists ${\alpha}_0 >0$ such that, for each ${\alpha}$, $0<{\alpha}\leq{\alpha}_0$, there
is a stopping time ${\tau^{\varepsilon}_{\alpha}}>0$ a.s. and there are semi-martingale processes ${c^{\varepsilon}(t)}$
and ${x^{\varepsilon}(t)}$, defined a.s. for $t\leq {\tau^{\varepsilon}_{\alpha}}$, with values respectively in
${\mathbb R}^{+*}$ and ${\mathbb R}$, so that if we set ${\varepsilon} {\eta^{\varepsilon}}(t)={u^{\vep}}(t,.+{x^{\varepsilon}(t)})-{\varphi_{c^{\varepsilon}(t)}}$, then
a.s. for $t\leq {\tau^{\varepsilon}_{\alpha}}$,
$
\|{\varepsilon} {\eta^{\varepsilon}} (t)\|_1 \leq {\alpha}
$
and
$
|{c^{\varepsilon}(t)}-c_0| \leq {\alpha}.
$
In addition, for ${\alpha}_0$ sufficiently small, and any
${\alpha}\leq {\alpha}_0$, there is a constant $C>0$,
depending only on ${\alpha}$ and $c_0$, such that for any $T>0$,
there is an ${\varepsilon}_0>0$, with, for each ${\varepsilon}<{\varepsilon}_0$,
\begin{equation}
\label{2.6}
{\mathbb P}({\tau^{\varepsilon}_{\alpha}}\leq T) \leq
\exp\left(-\frac{C(\alpha,c_0)}{{\varepsilon}^2T\|k\|_{H^1}^2}\right).
\end{equation}
\end{Theorem}
It was noticed heuristically in \cite{dBD1}, and proved in \cite{dBG} that in the
additive case, the use of the modulation parameters ${x^{\varepsilon}(t)}$ and ${c^{\varepsilon}(t)}$ was
necessary in order to get the estimate (\ref{2.6}).
Indeed, it was proved in \cite{dBG} that if we denote by
${\tilde \tau}^{\varepsilon,n}_{\alpha} =\inf \{ t>0, \|u^{{\varepsilon},n}(t,.)-{\varphi_{c_0}}\|_1>{\alpha}\}$,
where $u^{{\varepsilon},n}$ is here the solution of equation (\ref{1}), but with an additive noise
that becomes stationary in space as $n$ goes to infinity (see \cite{dBG} for a precise statement)
then there
exists a constant $C(\alpha,c_0)$ which depends on $\alpha$ and
$c_0$ but not on $T$ such that
\begin{equation}
\label{ssmod}
\underline{\lim}_{n\rightarrow\infty}\underline{\lim}_{{\varepsilon}\to0}{\varepsilon}^2\log\mathbb{P}
\left(\tilde{\tau}_{\alpha}^{n,{\varepsilon}} \le
T\right)\ge-\frac{C(\alpha,c_0)}{T^3}.
\end{equation}
It is not clear that (\ref{ssmod}) is still true in the present multiplicative case,
because the proof involves a controlability problem with a potential which -- up to now --
is open.
Note also that the decomposition given in Theorem \ref{t2}
is not unique, and is determined by the choice of specific orthogonality conditions
(see the proof below). In particular, contrary to the additive case, we will
be able here to investigate the asymptotic behavior in time of the limit
process by choosing one particular decomposition of the form given in Theorem \ref{t2}.
This is the object of Section 3.3.
\noindent
{\em Proof of Theorem \ref{t2}}
The proof follows closely the proof of Theorem 2.1 in \cite{dBD1} and we refer to \cite{dBD1}
for more details. The parameters ${x^{\varepsilon}(t)}$ and ${c^{\varepsilon}(t)}$ are obtained thanks to the use of the implicit
function Theorem. These are then local semi-martingales defined as long as
$|{c^{\varepsilon}(t)}-c_0|<{\alpha}$ and $\|{u^{\vep}}(t,.+{x^{\varepsilon}(t)})-{\varphi_{c_0}}\|_1<{\alpha}$, and setting
$$ {\varepsilon} {\eta^{\varepsilon}}(t)={u^{\vep}}(t,.+{x^{\varepsilon}(t)})-{\varphi_{c^{\varepsilon}(t)}},$$
one has for each ${\varepsilon}>0$, almost surely,
\begin{equation}
\label{ortho}
({\eta^{\varepsilon}},{\varphi_{c_0}})=({\eta^{\varepsilon}}, {\partial}_x {\varphi_{c_0}})=0.
\end{equation}
In order to estimate the exit time
$$
{\tau^{\varepsilon}_{\alpha}}=\inf\{t\ge 0, \; |{c^{\varepsilon}(t)}-c_0|>{\alpha} \; \mathrm{or} \; \|{\varepsilon} {\eta^{\varepsilon}} (t) \|_1 >{\alpha}\},
$$
we make use , as in \cite{dBD1}, of the functional defined for $u \in H^1({\mathbb R})$,
\begin{equation}
\label{lyap}
Q_{c_0}(u):= H(u)+c_0 m(u)
\end{equation}
where $H$ and $m$ are defined respectively in (\ref{masse}) and (\ref{hamiltonien}).
Note that ${\varphi_{c_0}}$ is a critical point of $Q_{c_0}$. We denote by $L_{c_0}$ the linearized operator
around ${\varphi_{c_0}}$, that is
\begin{equation}
\label{linearise}
L_{c_0}=-{\partial}_x^2+c_0-2{\varphi_{c_0}}.
\end{equation}
The next lemma, which is proved with the use of the It\^o Formula, using the same regularization procedure
as in \cite{dBD0}, gives the evolution of $H$ and $m$
for the solution ${u^{\vep}}$ of (\ref{1}) with ${u^{\vep}}(0)={\varphi_{c_0}}$~:
\begin{Lemma}
\label{l2.1}
For any stopping time $\tau <+\infty$ a.s, one has
$$
m({u^{\vep}}({\tau}))= m({\varphi_{c_0}}) -{\varepsilon} \int_0^{{\tau}} (({u^{\vep}})(s), dW(s)) + {\varepsilon}^2 |k|^2_{L^2} \int_0^{{\tau}} m({u^{\vep}}(s))ds
$$
and
\begin{eqnarray}
H({u^{\vep}}({\tau}))&=& H({\varphi_{c_0}})+{\varepsilon} \int_0^{{\tau}} ({\partial}_x{u^{\vep}}, {\partial}_x({u^{\vep}} dW(s)))
-\frac{{\varepsilon}}{2} \int_0^{{\tau}} (({u^{\vep}})^3,dW(s))\\
& &+\frac{{\varepsilon}^2}{2} \int_0^{{\tau}} \left\{ |k|_{L^2}^2 |{\partial}_x {u^{\vep}}|_{L^2}^2
+ |k'|_{L^2}^2 |{u^{\vep}}|_{L^2}^2\right\}ds \\
& & - \frac{{\varepsilon}^2}{2} \displaystyle \sum_k \int_0^{\tau} \int_{{\mathbb R}} ({u^{\vep}})^3 |\phi e_k|^2 dx ds.
\end{eqnarray}
\end{Lemma}
Consider $\nu>0$ such that $(Q''_{c_0}({\varphi_{c_0}})v,v)\ge \nu \|v\|_1^2$ for any $v \in H^1$
satisfying $(v,{\varphi_{c_0}})=(v,{\partial}_x {\varphi_{c_0}})=0$. The existence of such a constant is a classical result
(see \cite{Ben} or \cite{BSS}). Then it is easy to show (see \cite{dBD1}) that there is a constant
$C({\alpha}_0)>0$ such that for any $t<{\tau^{\varepsilon}_{\alpha}}$,
\begin{equation}
\label{minq}
Q_{c_0}({u^{\vep}}(t,.+{x^{\varepsilon}(t)}))-Q_{c_0}({\varphi_{c^{\varepsilon}(t)}})\ge \frac{\nu}{4} \|{\varepsilon} {\eta^{\varepsilon}}(t)\|_1^2 -C|{c^{\varepsilon}(t)}-c_0|^2.
\end{equation}
Now, if ${\tau}={\tau^{\varepsilon}_{\alpha}}\wedge t$, then by (\ref{minq}), the
translation invariance of $Q_{c_0}$, and Lemma \ref{l2.1}
\begin{equation}
\label{majeta}
\begin{array}{rcl}
\|{\varepsilon} {\eta^{\varepsilon}}({\tau})\|_1^2 &\le & \displaystyle \frac{4}{\nu} \left[ Q_{c_0}({\varphi_{c_0}}) -Q_{c_0}({\varphi_{c^{\varepsilon}(\tau)}})\right]
+{\varepsilon} \displaystyle \int_0^{{\tau} } ({\partial}_x{u^{\vep}}(s),{\partial}_x({u^{\vep}} dW(s))) \\
& & -\displaystyle \frac{{\varepsilon}}{2} \displaystyle \int_0^{{\tau}} (({u^{\vep}})^3(s),dW(s)) +\displaystyle \frac{{\varepsilon}^2}{2} \displaystyle \int_0^{{\tau}}
(|k|_{L^2}^2 |{\partial}_x {u^{\vep}}|_{L^2}^2 +|k'|_{L^2}^2 |{u^{\vep}}|_{L^2}^2) ds \\
& & -\displaystyle \frac{{\varepsilon}^2}{2} \displaystyle \sum_k \int_0^{{\tau}} \int_{{\mathbb R}} ({u^{\vep}})^3(s) |\phi e_k|^2 dx ds -c_0 {\varepsilon} \displaystyle \int_0^{{\tau}} (({u^{\vep}})^2,dW(s)) \\
& & +c_0{\varepsilon}^2 |k|_{L^2}^2 \displaystyle \int_0^{{\tau}} m({u^{\vep}}(s))ds +C |{c^{\varepsilon}(\tau)}-c_0|^2.
\end{array}
\end{equation}
The term $|{c^{\varepsilon}(\tau)}-c_0|$ is then estimated thanks to the orthogonality condition $({\eta^{\varepsilon}},{\varphi_{c_0}})=0$
and the evolution of $m({u^{\vep}}({\tau}))$ given in Lemma \ref{l2.1}; one obtains, for some constants
$\mu>0$ and $C>0$, depending only on $c_0$ and ${\alpha}_0$ (with ${\alpha}\le {\alpha}_0$)
\begin{eqnarray*}
\mu |{c^{\varepsilon}(\tau)}-c_0| &\le & \left| |{\varphi_{c_0}}|_{L^2}^2 - |{\varphi_{c^{\varepsilon}(\tau)}}|_{L^2}^2\right| \\
& \le & |{\varepsilon} {\eta^{\varepsilon}}({\tau})|_{L^2}^2 +C{\alpha}|{c^{\varepsilon}(\tau)}-c_0|+2{\varepsilon} \left| \int_0^{{\tau}} (({u^{\vep}})^2,dW(s)) \right|\\
& & +2 {\varepsilon}^2 |k|_{L^2}^2 \int_0^{{\tau}} |{u^{\vep}}(s)|_{L^2}^2 ds.
\end{eqnarray*}
Hence, choosing ${\alpha}_0$ sufficiently small one gets
\begin{equation}
\label{majc}
\begin{array}{rcl}
|{c^{\varepsilon}(\tau)}-c_0|^2 & \le & C \Big[ |{\varepsilon} {\eta^{\varepsilon}}({\tau})|_{L^2}^4 +4{\varepsilon}^2 \Big| \displaystyle \int_0^{{\tau}}
(({u^{\vep}})^2,dW(s)) \Big|^2\\
& & +4 {\varepsilon}^4 |k|_{L^2}^4 \Big( \displaystyle \int_0^{{\tau}} |{u^{\vep}}(s)|_{L^2}^2ds\Big)^2 \Big]
\end{array}
\end{equation}
which, once inserted into (\ref{majeta}) leads to
\begin{eqnarray*}
\|{\varepsilon} {\eta^{\varepsilon}}({\tau})\|_1^2 & \le & C \Big[ |{\varepsilon} {\eta^{\varepsilon}}({\tau})|_{L^2}^4 +{\varepsilon} \Big| \int_0^{{\tau}}
({\partial}_x {u^{\vep}},{\partial}_x ({u^{\vep}} dW(s)))\Big| \\
& & +{\varepsilon} \Big| \int_0^{{\tau}} (({u^{\vep}})^3,dW(s))\Big| +c_0 {\varepsilon} \Big| \int_0^{{\tau}} (({u^{\vep}})^2,dW(s)) \Big|\\
& & +4 {\varepsilon}^2 \Big| \displaystyle \int_0^{{\tau}} (({u^{\vep}})^2, dW(s))\Big|^2 +{\varepsilon}^2 \|k\|_1^2 \int_0^{{\tau}} \|{u^{\vep}}(s)\|_1^2 ds \\
& &+ {\varepsilon}^2 |k|^2_{L^2} \int_0^{{\tau}} \|{u^{\vep}}(s)\|_1^3 ds +{\varepsilon}^4 |k|_{L^2}^4 \Big( \int_0^{{\tau}}
|{u^{\vep}}(s)|_{L^2}^2ds\Big)^2 \Big].
\end{eqnarray*}
With this estimate in hand, together with (\ref{majc}), the conclusion of Theorem \ref{t2}
follows with the same arguments as in the proof of Proposition 3.1 in \cite{dBG}.
These arguments rely on classical exponential tail estimates for stochastic integrals,
after noticing that $\|{u^{\vep}}(s)\|_1\le C$, a.s. for $s\in [0,{\tau^{\varepsilon}_{\alpha}}\wedge T]$ and ${\alpha}\le{\alpha}_0$,
so that the quadratic variation of each of the integrals involved in the above estimates are
bounded above by $CT$.
\hfill
$\square$
\section{A central limit theorem}
This section is devoted to the proof of the next theorem:
\begin{Theorem}
\label{t3}
Under the assumptions of Theorem \ref{t2}, let ${\alpha}<{\alpha}_0$ be fixed.
Then we can find ${\tilde c}^{{\varepsilon}}(t)$ and ${\tilde x}^{{\varepsilon}}(t)$
satisfying the conclusion of Theorem \ref{t2}
such that if ${\tilde {\eta}}^{{\varepsilon}}$ is defined as in Theorem \ref{t2}, for any $T>0$, the
process $({\tilde {\eta}}^{{\varepsilon}}(t))_{t \in [0,T]}$ converges in $L^2({\Omega}; L^{\infty}(0,{\tau^{\varepsilon}_{\alpha}}\wedge
T;L^2({\mathbb R})))$ to a Gaussian process $\tilde {\eta}$ satisfying
the additive linear equation
\begin{equation}
\label{eql}
d\tilde{\eta}= {\partial}_x L_{c_0}\tilde {\eta} \, dt + \tilde Q{\varphi_{c_0}} d\tilde W,
\end{equation}
with $\tilde {\eta}(0)=0$, where $ \tilde W$ is the Wiener process with covariance
$\phi\phi^*$ given by $\tilde W={\mathcal T}_{c_0 t} W$, and $\tilde Q$ is a
projection operator.
Moreover, for $a>0$ sufficiently small compared to $c_0$, the process
$w(t,x)=e^{ax}\tilde {\eta}(t,x)$ is a well defined $H^1$ valued process, of
Ornstein-Uhlenbeck type, which converges in law to an $H^1$-valued Gaussian
random variable as $t$ goes to infinity.
\end{Theorem}
The conclusion of Theorem \ref{t3} will be obtained in three steps.
The first step consists in estimating the modulation parameters obtained in
Theorem \ref{t2}, in terms of ${\eta^{\varepsilon}}$, using the equations for those parameters;
then the convergence of ${\eta^{\varepsilon}}$ as ${\varepsilon}$ tends to zero is proved, and finally
in the third step, a slight change in the modulation parameters is performed,
in order that the limit process ${\eta}$ may be written as an Ornstein-Uhlenbeck
process.
From now on, we assume that ${\alpha}$ is fixed and sufficiently small, so that the conclusion
of Theorem \ref{t2} holds, and we denote ${\tau^{\varepsilon}_{\alpha}}$ by ${\tau^{\varepsilon}}$.
\subsection{Modulation equations}
Since we know that the modulation parameters ${x^{\varepsilon}(t)}$ and ${c^{\varepsilon}(t)}$ are semi-martingale processes
adapted to the filtration generated by $(W(t))_{t\ge 0}$, we may a priori write the stochastic
evolution equations for those parameters in the form
\begin{equation}
\label{modeq}
\left\{
\begin{array}{l}
d{x^{\varepsilon}} = {c^{\varepsilon}} dt +{\varepsilon} {y^{\varepsilon}} dt +{\varepsilon} ({z^{\varepsilon}}, dW)\\
d{c^{\varepsilon}} = {\varepsilon} {a^{\varepsilon}} dt +{\varepsilon} ({b^{\varepsilon}}, dW)
\end{array}
\right.
\end{equation}
where ${y^{\varepsilon}}$ and ${a^{\varepsilon}}$ are real valued adapted processes with a.s. locally integrable paths
on $[0,{\tau^{\varepsilon}})$,
and ${b^{\varepsilon}}$, ${z^{\varepsilon}}$ are predictable processes with paths a.s. in $L^2_{loc}(0,{\tau^{\varepsilon}}; L^2({\mathbb R}))$.
We then proceed as in \cite{dBD1} : the It\^o-Wentzell Formula applied to ${u^{\vep}}(t,x+{x^{\varepsilon}(t)})$,
together with equation (\ref{1}) for ${u^{\vep}}$ and the first equation of (\ref{modeq}) for ${x^{\varepsilon}}$ give a stochastic
evolution equation for ${u^{\vep}}(t, x+{x^{\varepsilon}})$.
On the other hand, the standard It\^o Formula together with the second equation of (\ref{modeq}) for ${c^{\varepsilon}}$
give an equation for the evolution of ${\varphi_{c^{\varepsilon}(t)}}$.
Replacing then ${\varphi_{c^{\varepsilon}(t)}}+{\varepsilon} {\eta^{\varepsilon}}(t,x)$ for ${u^{\vep}}(t,x+{x^{\varepsilon}(t)})$ in the first equation leads to the
following stochastic equation for the evolution of ${\eta^{\varepsilon}}(t)$ :
\begin{equation}
\label{eqetae}
\renewcommand{\arraystretch}{1.7}
\begin{array}{rcl}
d{\eta^{\varepsilon}} & = & {\partial}_x L_{c_0} {\eta^{\varepsilon}} dt +({y^{\varepsilon}}{\partial}_x {\varphi_{c^{\varepsilon}}} -{a^{\varepsilon}} {\partial}_c {\varphi_{c^{\varepsilon}}}) dt- {\partial}_x ( ({\varphi_{c^{\varepsilon}}}-{\varphi_{c_0}}){\eta^{\varepsilon}}) dt \\
& & + ({c^{\varepsilon}}-c_0+{\varepsilon} {y^{\varepsilon}}){\partial}_x {\eta^{\varepsilon}} dt -\frac{{\varepsilon}}{2} {\partial}_x (({\eta^{\varepsilon}})^2)dt +{\varphi_{c^{\varepsilon}}} {\mathcal T}_{{x^{\varepsilon}}}dW \\
& & + {\partial}_x{\varphi_{c^{\varepsilon}}} ({z^{\varepsilon}},dW) -{\partial}_c {\varphi_{c^{\varepsilon}}} ({b^{\varepsilon}},dW) +{\varepsilon} {\eta^{\varepsilon}} {\mathcal T}_{{x^{\varepsilon}}}dW +{\varepsilon} {\partial}_x {\eta^{\varepsilon}} ({z^{\varepsilon}}, dW) \\
& & +\frac{{\varepsilon}}{2} {\partial}_x^2 {\varphi_{c^{\varepsilon}}} |\phi^*{z^{\varepsilon}}|_{L^2}^2 dt -\frac{{\varepsilon}}{2} {\partial}_c^2 {\varphi_{c^{\varepsilon}}} |\phi^* {b^{\varepsilon}}|_{L^2}^2 dt+{\varepsilon} \displaystyle \sum_{l\in {\mathbb N}} {\partial}_x({\varphi_{c^{\varepsilon}}} {\mathcal T}_{{x^{\varepsilon}}}\phi e_l) ({z^{\varepsilon}},\phi e_l) dt \\
& & +\frac12 {\varepsilon}^2 {\partial}_x^2 {\eta^{\varepsilon}} |\phi^* {z^{\varepsilon}}|_{L^2}^2 dt +{\varepsilon}^2 \displaystyle \sum_{l \in {\mathbb N}} {\partial}_x ({\eta^{\varepsilon}} {\mathcal T}_{{x^{\varepsilon}}}\phi e_l)({z^{\varepsilon}}, \phi e_l) dt
\end{array}
\end{equation}
where $L_{c_0}$ is defined in (\ref{linearise}).
Now, taking the $L^2$- inner product of equation (\ref{eqetae}) with ${\varphi_{c_0}}$, on the one hand,
and with ${\partial}_x {\varphi_{c_0}}$ on the other hand, then using the orthogonality conditions (\ref{ortho})
and the fact that $L_{c_0} {\partial}_x {\varphi_{c_0}} =0$, and finally identifying the drift parts and the martingale
parts of each of the resulting equations lead to the same kind of system that we previously obtained
in \cite{dBD1}; namely, setting
$$
Y^{{\varepsilon}}(t)= \begin{pmatrix} {y^{\varepsilon}}(t)\\ {a^{\varepsilon}}(t) \end{pmatrix}
\quad \mathrm{and} \quad Z^{{\varepsilon}}_l(t) =\begin{pmatrix} ({z^{\varepsilon}}, \phi e_l)\\({b^{\varepsilon}}, \phi e_l)\end{pmatrix}
$$
then one gets for the drift parts
\begin{equation}
\label{eqdrift}
A^{{\varepsilon}}(t) Y^{{\varepsilon}}(t) = G^{{\varepsilon}}(t)
\end{equation}
where
\begin{equation}
\label{eqA}
A^{{\varepsilon}}(t)= \begin{pmatrix} ({\partial}_x{\varphi_{c^{\varepsilon}}}+{\varepsilon} {\partial}_x {\eta^{\varepsilon}}, {\partial}_x {\varphi_{c_0}}) & -({\partial}_c {\varphi_{c^{\varepsilon}}},{\partial}_x{\varphi_{c_0}}) \\
-({\partial}_x {\varphi_{c^{\varepsilon}}}, {\varphi_{c_0}}) & ({\partial}_c {\varphi_{c^{\varepsilon}}}, {\varphi_{c_0}})
\end{pmatrix}
\end{equation}
and
$$
G^{{\varepsilon}}(t)=\begin{pmatrix}
G^{{\varepsilon}}_1(t) \\ G^{{\varepsilon}}_2(t)
\end{pmatrix},
$$
with
\begin{equation}
\label{eqG1}
\renewcommand{\arraystretch}{1.7}
\begin{array}{rcl}
G^{{\varepsilon}}_1(t) &=& ({\eta^{\varepsilon}}, L_{c_0}{\partial}_x^2 {\varphi_{c_0}})+({c^{\varepsilon}}-c_0)({\eta^{\varepsilon}}, {\partial}_x^2 {\varphi_{c_0}}) +\frac{{\varepsilon}}{2} ({\partial}_x({\eta^{\varepsilon}})^2, {\partial}_x {\varphi_{c_0}})\\
& & +({\partial}_x(({\varphi_{c^{\varepsilon}}}-{\varphi_{c_0}}){\eta^{\varepsilon}}), {\partial}_x{\varphi_{c_0}}) -\frac{{\varepsilon}}{2} ({\partial}_x^2 {\varphi_{c^{\varepsilon}}}, {\partial}_x {\varphi_{c_0}})|\phi^* {z^{\varepsilon}}|_{L^2}^2 \\
& & +\frac{{\varepsilon}}{2} ({\partial}_c^2 {\varphi_{c^{\varepsilon}}}, {\partial}_x {\varphi_{c_0}}) |\phi^* {b^{\varepsilon}}|_{L^2}^2 -{\varepsilon} \displaystyle \sum_{l\in {\mathbb N}} ({z^{\varepsilon}}, \phi e_l)({\partial}_x({\varphi_{c^{\varepsilon}}} {\mathcal T}_{{x^{\varepsilon}}}\phi e_l), {\partial}_x {\varphi_{c_0}}) \\
& & +\frac12 {\varepsilon}^2 ({\eta^{\varepsilon}}, {\partial}_x^3 {\varphi_{c_0}}) |\phi^* {z^{\varepsilon}}|_{L^2}^2 -{\varepsilon}^2 \displaystyle \sum_{l \in {\mathbb N}} ({\partial}_x({\eta^{\varepsilon}} {\mathcal T}_{{x^{\varepsilon}}}\phi e_l), {\partial}_x {\varphi_{c_0}}) ({z^{\varepsilon}}, \phi e_l)
\end{array}
\end{equation}
and
\begin{equation}
\label{eqG2}
\renewcommand{\arraystretch}{1.7}
\begin{array}{rcl}
G^{{\varepsilon}}_2(t)&=& -\frac{{\varepsilon}}{2} ({\partial}_x({\eta^{\varepsilon}})^2, {\varphi_{c_0}})-({\partial}_x(({\varphi_{c^{\varepsilon}}}-{\varphi_{c_0}}){\eta^{\varepsilon}}), {\varphi_{c_0}}) +\frac{{\varepsilon}}{2} ({\partial}_x^2 {\varphi_{c^{\varepsilon}}},{\varphi_{c_0}})|\phi^* {z^{\varepsilon}}|_{L^2}^2\\
& & -\frac{{\varepsilon}}{2} ({\partial}_c^2 {\varphi_{c^{\varepsilon}}}, {\varphi_{c_0}})|\phi^* {b^{\varepsilon}}|_{L^2}^2 +{\varepsilon} \displaystyle \sum ({z^{\varepsilon}},\phi e_l) ({\partial}_x({\varphi_{c^{\varepsilon}}}{\mathcal T}_{{x^{\varepsilon}}}\phi e_l) ,{\varphi_{c_0}}) \\
& & +\frac{{\varepsilon}^2}{2} ({\eta^{\varepsilon}},{\partial}_x^2 {\varphi_{c_0}})|\phi^* {z^{\varepsilon}}|_{L^2}^2 +{\varepsilon}^2 \displaystyle \sum_{l\in {\mathbb N}} ({\partial}_x ({\eta^{\varepsilon}} {\mathcal T}_{{x^{\varepsilon}}} \phi e_l), {\varphi_{c_0}}) ({z^{\varepsilon}}, \phi e_l);
\end{array}
\end{equation}
note that $A^{{\varepsilon}}(t)=A_0+O(|{c^{\varepsilon}}-c_0|+\|{\varepsilon} {\eta^{\varepsilon}}\|_1)$, a.s. for $t\le {\tau^{\varepsilon}}$
with
$$
A_0= \begin{pmatrix}
|{\partial}_x {\varphi_{c_0}}|^2_{L^2} & 0 \\ 0 & ({\varphi_{c_0}}, {\partial}_c {\varphi_{c_0}})
\end{pmatrix}
$$
and $O(|{c^{\varepsilon}}-c_0|+\|{\eta^{\varepsilon}}\|_1)$ is uniform in ${\varepsilon}, t$ and ${\omega}$ as long as $t\le {\tau^{\varepsilon}}$.
Concerning the martingale parts, one gets the equation
\begin{equation}
\label{eqmart}
A^{{\varepsilon}}(t) Z^{{\varepsilon}}_l(t) = F^{{\varepsilon}}_l(t), \quad \forall l \in {\mathbb N}
\end{equation}
with
\begin{equation}
\label{eqF}
F ^{{\varepsilon}}(t) =\begin{pmatrix}
-(({\varphi_{c^{\varepsilon}}}+{\varepsilon} {\eta^{\varepsilon}}){\mathcal T}_{{x^{\varepsilon}}}\phi e_l, {\partial}_x {\varphi_{c_0}}) \\ (({\varphi_{c^{\varepsilon}}}+{\varepsilon}{\eta^{\varepsilon}}) {\mathcal T}_{{x^{\varepsilon}}} \phi e_l, {\varphi_{c_0}}).
\end{pmatrix}
\end{equation}
\begin{Proposition}
\label{p1}
Under the above assumptions, there is a constant ${\alpha}_1>0$, such that if ${\alpha} \le {\alpha}_1$, then
\begin{equation}
\label{majbz}
|\phi^* {z^{\varepsilon}}(t)|_{L^2} +|\phi^* {b^{\varepsilon}} |_{L^2} \le C_1|k|_{L^2}, \quad \mathrm{a.s.}\; \mathrm{for} \; t\le {\tau^{\varepsilon}}
\end{equation}
and
\begin{equation}
\label{majay}
|{a^{\varepsilon}}(t)| +|{y^{\varepsilon}}(t)| \le C_2 |{\eta^{\varepsilon}}(t)|_{L^2} +{\varepsilon} C_3, \quad \mathrm{a.s.}\; \mathrm{for} \; t\le {\tau^{\varepsilon}}
\end{equation}
for some constants $C_1$, $C_2$, $C_3$, depending only on ${\alpha}$ and $c_0$, and for any ${\varepsilon} \le {\varepsilon}_0$.
\end{Proposition}
\noindent
{\em Proof}
The proof is exactly the same as the proof of Corollary 4.3 in \cite{dBD1}, once noticed that, a.s. for $t\le {\tau^{\varepsilon}}$,
\begin{eqnarray*}
\displaystyle \sum_{l\in {\mathbb N}} |F^{{\varepsilon}}_l(t)|^2 & \le & C \displaystyle \sum_{l \in {\mathbb N}} |({\varphi_{c^{\varepsilon}}}+{\varepsilon} {\eta^{\varepsilon}}) {\mathcal T}_{{x^{\varepsilon}}}\phi e_l|_{L^2}^2 \\
& \le& C \displaystyle \sum_{l} \int_{{\mathbb R}} ({\varphi_{c^{\varepsilon}}}+{\varepsilon} {\eta^{\varepsilon}})^2 (x) [ ({\mathcal T}_{{x^{\varepsilon}}}k)* e_l]^2(x) dx\\
& \le & \int_{{\mathbb R}} ({\varphi_{c^{\varepsilon}}}+{\varepsilon} {\eta^{\varepsilon}})^2 (x)\displaystyle \sum_l ({\mathcal T}_{{x^{\varepsilon}}} k(x-.), e_l)^2dx\\
& \le & C \int_{{\mathbb R}} ({\varphi_{c^{\varepsilon}}}+{\varepsilon} {\eta^{\varepsilon}})^2 (x) |{\mathcal T}_{{x^{\varepsilon}}} k(x-.)|_{L^2}^2 dx\\
& \le & C|k|_{L^2}^2 |{\varphi_{c^{\varepsilon}}}+{\varepsilon} {\eta^{\varepsilon}}|_{L^2}^2 \le C|k|_{L^2}^2
\end{eqnarray*}
where we have used the Parseval equality in the fourth line.
\hfill
$\square$
\subsection{Convergence of ${\eta^{\varepsilon}}$}
Let us first assume that ${\eta^{\varepsilon}}$ has a limit as ${\varepsilon}$ goes to zero, and take formally the
limit as ${\varepsilon}$ goes to zero in the preceding equations. Then, as was noticed above,
$$
\lim_{{\varepsilon} {\rightarrow} 0} A^{{\varepsilon}} =A_0= \begin{pmatrix}
|{\partial}_x {\varphi_{c_0}}|_{L^2}^2 & 0 \\ 0 & ({\varphi_{c_0}}, {\partial}_c {\varphi_{c_0}})
\end{pmatrix}
$$
hence
\begin{equation}
\label{limz}
\lim_{{\varepsilon} {\rightarrow} 0} \phi^* {z^{\varepsilon}} = -\frac{1}{|{\partial}_x{\varphi_{c_0}}|_{L^2}^2} ({\mathcal T}_{c_0 t}\phi)^* ({\varphi_{c_0}} {\partial}_x {\varphi_{c_0}}) :=z
\end{equation}
\begin{equation}
\label{limb}
\lim_{{\varepsilon} {\rightarrow} 0} \phi^* {b^{\varepsilon}}= \frac{1}{({\varphi_{c_0}},{\partial}_c {\varphi_{c_0}})} ({\mathcal T}_{c_0 t}\phi^*) ({\varphi}_{c_0}^2) :=b
\end{equation}
\begin{equation}
\label{limy}
\lim_{{\varepsilon} {\rightarrow} 0} y^{{\varepsilon}}= \frac{1}{|{\partial}_x {\varphi_{c_0}}|_{L^2}^2} ({\eta}, L_{c_0} {\partial}_x^2 {\varphi_{c_0}}):=y
\end{equation}
and
\begin{equation}
\label{lima}
\lim_{{\varepsilon} {\rightarrow} 0} {a^{\varepsilon}}=0.
\end{equation}
Moreover, formally, ${\eta}$ satisfies the equation
\begin{equation}
\label{eqeta}
\begin{array}{rcl}
d{\eta} & = & {\partial}_x L_{c_0} {\eta} dt + \frac{1}{|{\partial}_x {\varphi_{c_0}}|_{L^2}^2} ({\eta}, L_{c_0}{\partial}_x^2 {\varphi_{c_0}}) {\partial}_x {\varphi_{c_0}} dt \\
& & + {\varphi_{c_0}} {\mathcal T}_{c_0 t} dW -\frac{1}{2 |{\partial}_x {\varphi_{c_0}}|_{L^2}^2} ({\partial}_x({\varphi}_{c_0}^2), {\mathcal T}_{c_0 t} dW){\partial}_x {\varphi_{c_0}}\\
& & - \frac{1}{({\varphi_{c_0}}, {\partial}_c {\varphi_{c_0}}) } ({\varphi}_{c_0}^2, {\mathcal T}_{c_0 t} dW) {\partial}_c {\varphi_{c_0}}.
\end{array}
\end{equation}
It is easy to show that (\ref{eqeta}) has a unique adapted solution ${\eta}$ with paths a.s. in $C({\mathbb R}^+, H^1)$ satisfying
${\eta}(0)=0$. Moreover using the fact that $({\partial}_c {\varphi_{c_0}}, {\partial}_x {\varphi_{c_0}})=0$, one easily gets from the above equation that
$ ({\eta}, {\varphi_{c_0}})=({\eta},{\partial}_x {\varphi_{c_0}})=0$, $\forall t>0$.
Next, we make use of the following lemmas, whose proofs are obtained in the same way as the corresponding Lemmas in
\cite{dBD1}.
\begin{Lemma}
\label{l2}
Let ${\eta}$ be the solution of (\ref{eqeta}) with ${\eta}(0)=0$.
Then, for any $T>0$, there is a constant $C$ depending only on $c_0$, $T$ and $\|k\|_1$ such that
$$
{\mathbb E}\left( \|{\eta}(t)\|_1^4\right) \le C, \; \forall t\le T.
$$
\end{Lemma}
\begin{Lemma}
\label{l3}
Let ${\eta^{\varepsilon}}$ be the solution of (\ref{eqetae}), defined for $t \in [0,{\tau^{\varepsilon}}[$, obtained
thanks to the modulation procedure of Section 2.
Then, for any $T>0$,
$$
{\mathbb E}\Big( \sup_{t\le{\tau^{\varepsilon}}\wedge T} |{\eta^{\varepsilon}}(t) |_{L^2}^4 \Big) \le C(T,{\alpha}, c_0,\|k\|_1).
$$
\end{Lemma}
The above lemmas show that
\begin{equation}
\label{cvgencec}
\forall T>0, \; \forall q\ge 2, \; \lim_{{\varepsilon} {\rightarrow} 0} {\mathbb E} \Big( \sup_{t\le T\wedge {\tau^{\varepsilon}}} |{c^{\varepsilon}(t)} -c_0|^q\Big)=0.
\end{equation}
Indeed, the expression of ${c^{\varepsilon}(t)}-c_0$ given by (\ref{modeq}) together with (\ref{majbz}) and (\ref{majay}) imply
easily
$$
{\mathbb E} \Big( \sup_{t\le T\wedge {\tau^{\varepsilon}}} |{c^{\varepsilon}(t)} -c_0|^2\Big) \le C{\varepsilon}^2 [1+{\mathbb E} \int_0^{T\wedge {\tau^{\varepsilon}}} |{\eta^{\varepsilon}}(s)|_{L^2}^2ds]
$$
with $C=C({\alpha}, c_0,T,\|k\|_1)$. Then, (\ref{cvgencec}) is deduced form Lemma \ref{l3} for $q=2$, and follows
for all other values of $q$ from the uniform boundedness of $|{c^{\varepsilon}(t)}-c_0|$ on $[0,T\wedge{\tau^{\varepsilon}}]$.
Note that an immediate consequence of (\ref{cvgencec}) is the fact that
\begin{equation}
\forall T>0, \; \forall q\ge 2, \; \lim_{{\varepsilon} {\rightarrow} 0} {\mathbb E} \Big( \sup_{t\le T\wedge {\tau^{\varepsilon}}} \|{\varphi_{c^{\varepsilon}(t)}}-{\varphi_{c_0}}\|_2^2\Big)=0.
\end{equation}
We will finally need the next lemma.
\begin{Lemma}
\label{l6}
For any $T>0$, and any $q\ge 1$,
$$
\lim_{{\varepsilon} {\rightarrow} 0} {\mathbb E} \Big( \sup_{t\le T\wedge {\tau^{\varepsilon}}} \Big( \sum_{l\in {\mathbb N}}|Z^{{\varepsilon}}_l(t) -Z_l(t)|^2\Big)^q \Big)=0
$$
where we have set for $l \in {\mathbb N}$
$$
Z_l(t)=\begin{pmatrix}
(z,\phi e_l) \\ (b,\phi e_l)
\end{pmatrix},
$$
$z$ and $b$ being given by (\ref{limz}) and (\ref{limb}), respectively.
\end{Lemma}
\noindent
{\em Proof}
Here again, it is sufficient to consider the case $q=1$. We recall that
$Z^{{\varepsilon}}_l$ satisfies equation (\ref{eqmart}).
First, it is clear that
$$
\lim_{{\varepsilon} {\rightarrow} 0} {\mathbb E} \Big( \sup_{t\le T\wedge {\tau^{\varepsilon}}} \|(A^{{\varepsilon}}(t))^{-1} -(A_0(t))^{-1}\|^{2q}\Big) =0, \; \forall q\ge 1.
$$
On the other hand, in view of (\ref{eqF}), denoting $F^0_l(t)$ the formal limit
of $F^{{\varepsilon}}_l(t)$, one has
$$
\begin{array}{l}
{\mathbb E} \Big( \displaystyle \sup_{t\le T\wedge{\tau^{\varepsilon}}} \sum_l |F^{{\varepsilon}}_l(t)-F^0_l(t)|^2\Big) \\
\le C {\mathbb E} \Big( \displaystyle \sup_{t\le T\wedge{\tau^{\varepsilon}}} \sum_l |{\partial}_x {\varphi_{c_0}}({\mathcal T}_{{x^{\varepsilon}}}\phi-{\mathcal T}_{c_0 t} \phi)e_l|_{L^2}^2\Big)
+C {\mathbb E} \Big( \displaystyle \sup_{t\le T\wedge{\tau^{\varepsilon}}} \|{\varphi_{c^{\varepsilon}(t)}}-{\varphi_{c_0}}\|_1^2\Big)
\end{array}
$$
and
$$
\begin{array}{l}
{\mathbb E} \Big( \displaystyle \sup_{t\le T\wedge{\tau^{\varepsilon}}} \displaystyle \sum_l |{\partial}_x {\varphi_{c_0}}({\mathcal T}_{{x^{\varepsilon}}}\phi-{\mathcal T}_{c_0 t} \phi)e_l|_{L^2}^2\Big)\\
\le \|{\varphi_{c_0}}\|_1^2 {\mathbb E} \Big( \displaystyle \sup_{t\le T\wedge{\tau^{\varepsilon}}} |k(.+{x^{\varepsilon}(t)} -c_0 t)-k|_{L^2}^2\Big).
\end{array}
$$
Then, the It\^o Formula applied to the function
$$
{\mathcal K}^{{\varepsilon}}(t,x)=(k(x+{x^{\varepsilon}(t)}-c_0 t)-k(x))^2
$$
using equation (\ref{modeq}) for $d{x^{\varepsilon}(t)}$, together with (\ref{majbz}), (\ref{majay}),
and (\ref{cvgencec}) lead to the conclusion
of Lemma \ref{l6}.
\hfill $\square$
Now, in order to prove that
\begin{equation}
\label{cvgenceeta}
\lim_{{\varepsilon} {\rightarrow} 0} {\mathbb E} \Big( \sup_{t\le T\wedge{\tau^{\varepsilon}}} |{\eta^{\varepsilon}}(t)-{\eta}(t)|_{L^2}^2\Big) =0,
\end{equation}
where ${\eta}$ is the solution of (\ref{eqeta}) with ${\eta}(0)=0$, it suffices to set $v^{{\varepsilon}}={\eta^{\varepsilon}}-{\eta}$,
to deduce from (\ref{eqeta}) and (\ref{eqetae}) the equation for $dv^{{\varepsilon}}$ and to apply the
It\^o Formula to get the evolution of $|v^{{\varepsilon}}|_{L^2}^2$.
We do not give the details of those tedious, but easy computations.
Finally, the use of the following estimates :
$$
\begin{array}{l}
{\varepsilon} |({v^{\vep}}, {\partial}_x (({\eta^{\varepsilon}})^2))|={\varepsilon} |({\partial}_x {\eta},({\eta^{\varepsilon}})^2)|\le {\varepsilon} \|{\eta}\|_1 |{\eta^{\varepsilon}}|_{L^4}^2 \\
\le C {\varepsilon} \|{\eta}\|_1 |{\eta^{\varepsilon}}|_{L^2}^{3/2}|{\partial}_x{\eta^{\varepsilon}}|_{L^2}^{1/2}
\le C \sqrt{{\varepsilon}}\|{\eta}\|_1 |{\eta^{\varepsilon}}|_{L^2}^{3/2}
\end{array}
$$
on the one hand, and
$$
|y^{{\varepsilon}} -y| +|a^{{\varepsilon}}|\le C (|{v^{\vep}}|_{L^2} + |{c^{\varepsilon}} -c_0||{\eta^{\varepsilon}}|_{L^2} +{\varepsilon} |{\eta^{\varepsilon}}|_{L^2}^2
+|{\eta^{\varepsilon}}|_{L^2} \|{\varphi_{c^{\varepsilon}}} -{\varphi_{c_0}}\|_1 +{\varepsilon})
$$
which is obtained as in the proof of Lemma \ref{l6} on the other hand, together with Lemma \ref{l2}
to \ref{l6} allow to get the conclusion, that is the convergence of ${\eta^{\varepsilon}}$ to ${\eta}$ in
$L^2({\Omega},L^{\infty}(0,{\tau^{\varepsilon}} \wedge T;L^2({\mathbb R})))$.
\hfill
$\square$
\subsection{Complements on the limit equation}
First of all, we note that the modulation equations may be written at order one in ${\varepsilon}$
as
$$
\left\{
\begin{array}{l}
d{x^{\varepsilon}} = c_0 dt +{\varepsilon} y dt +{\varepsilon} W_1 dt +{\varepsilon} dW_2 +o({\varepsilon})\\
d{c^{\varepsilon}} = {\varepsilon} dW_1 +o({\varepsilon})
\end{array}
\right.
$$
where
$$
y=|{\partial}_x {\varphi_{c_0}}|_{L^2}^{-2} ({\eta},L_{c_0}{\partial}_x^2 {\varphi_{c_0}}),
$$
$$
W_1(t)=({\varphi_{c_0}},{\partial}_c {\varphi_{c_0}})^{-1} ({\varphi}_{c_0}^2,\tilde W(t))
$$
and
$$
W_2(t)=-\frac12 |{\partial}_x{\varphi_{c_0}}|_{L^2}^{-2} ({\partial}_x ({\varphi_{c_0}}^2),\tilde W(t)).
$$
Note that $W_1$ and $W_2$ are real valued Brownian motions, which are independent
since
$$
{\mathbb E}(W_1(t)W_2(s)) =-\frac12 |{\partial}_x{\varphi_{c_0}}|_{L^2}^{-2} ({\varphi_{c_0}},{\partial}_c{\varphi_{c_0}})^{-1} (\phi^*({\partial}_x({\varphi}_{c_0}^2)),\phi^*({\varphi}_{c_0}^2))(t\wedge s)
= 0
$$
because the operator $\phi^*$ commutes with spatial derivation.
Now, we want to investigate the asymptotic behavior in time of the process $\eta$.
However, in the present form, the process ${\eta}$ does not converge in law as $t$ goes to infinity;
this is due to the fact that the preceding modulation does not exactly correspond to the projection
of the solution ${u^{\vep}}$ on the (two-dimensional) center manifold, in which case the remaining term would
belong to the stable manifold around the soliton trajectory.
We now show that by slightly changing the modulation parameters, we can get a new decomposition of
the solution ${u^{\vep}}$ which is defined on the same time interval as before,
but which fits with the preceding requirements.
For that purpose, we first need to recall a few facts from \cite{PW}.
The generalized nullspace of the operator ${\partial}_x L_{c_0}$ (that is the operator arising in
the linearized evolution equation in the soliton reference frame)
is spanned by the functions ${\partial}_x {\varphi_{c_0}}$ and ${\partial}_c {\varphi_{c_0}}$, with the equality
$$
{\partial}_x L_{c_0} {\partial}_c {\varphi_{c_0}}= -{\partial}_x {\varphi_{c_0}}
$$
and there are constants $\theta_1$ and $\theta_2$ (with $\theta_1=({\varphi_{c_0}},{\partial}_c{\varphi_{c_0}})$) such that
if we set
$$
\tilde g_1(x)=-\theta_1\int_{-\infty}^x {\partial}_c{\varphi_{c_0}}(y)dy +\theta_2 {\varphi_{c_0}} \quad \mathrm{and} \quad \tilde g_2(x)=\theta_1{\varphi_{c_0}}
$$
then the generalized nullspace of $-L_{c_0}{\partial}_x$ is spanned by $\tilde g_1$ and $\tilde g_2$ and
$$
(\tilde g_1,{\partial}_x {\varphi_{c_0}})=1, \; (\tilde g_1, {\partial}_c {\varphi_{c_0}})=0, \; (\tilde g_2, {\partial}_x {\varphi_{c_0}})=0, \; (\tilde g_2, {\partial}_c {\varphi_{c_0}})=1.
$$
We also set, for $a>0$,
$$
f_1^a(x)=e^{ax}{\partial}_x {\varphi_{c_0}}, \; f_2^a(x)=e^{ax} {\partial}_c {\varphi_{c_0}}, \; g_1^a(x)=e^{-ax}\tilde g_1(x), \; g_2^a(x)=e^{-ax}\tilde g_2(x),
$$
so that $(f_i^a,g_j^a)=\delta_{ij}$. Then the operator $A_a$ defined for
$a>0$ by $A_a=e^{ax}{\partial}_x L_{c_0}e^{-ax}$ has a well defined generalized nullspace
spanned by $f_1^a, f_2^a$ and the spectral projection on this nullspace is
given by $Pw=\sum_{k=1}^2 (w,g_k^a)f_k^a$ where $w=e^{ax} v$, and $v$ is an $L^2$
function. Moreover, if $Q=I-P$, then $Q$ is the spectral projection on the stable
manifold of $A_a$, and under the condition $0<a<\sqrt{c_0/3}$, there are constants
$C>0$ and $b>0$ such that
\begin{equation}
\label{decexp}
\|e^{A_a t}Qw\|_1 \leq Ce^{-bt}\|w\|_1, \; \forall t>0, \; \forall w \in H^1,
\end{equation}
where $e^{A_a t}$ is the $C^0$-semi-group generated by $A_a$ (see Theorem 4.2 in \cite{PW}).
Now, let ${\eta}$ be the solution of (\ref{eqeta}) with ${\eta}(0)=0$, and consider $w(t,x)=e^{ax}{\eta}(t,x)$.
Note that the orthogonality condition $({\eta},{\varphi_{c_0}})=0$ implies $(w,g_2^a)=0$, so that
$Pw=\lambda (t) f_1^a$ with $\lambda(t)=(w(t),g_1^a)$ a real valued stochastic process
whose evolution is given by
\begin{equation}
\label{eqlam}
\begin{array}{rcl}
\lambda(t)&=& \displaystyle \int_0^t |{\partial}_x {\varphi_{c_0}}|_{L^2}^{-2} ({\eta}(s),L_{c_0}{\partial}_x^2{\varphi_{c_0}})ds
-\displaystyle \int_0^t |{\partial}_x {\varphi_{c_0}}|_{L^2}^{-2}({\varphi_{c_0}}{\partial}_x{\varphi_{c_0}}, d\tilde W(s)) \\ & & +\displaystyle \int_0^t (e^{ax}{\varphi_{c_0}} d\tilde W(s),g_1^a)
\end{array}
\end{equation}
where we have used (\ref{eqeta}) and the fact that $A_a Pw=0$ and $\lambda(0)=0$. Hence, $\lambda(t)$ is bounded
in $L^4({\Omega};L^{\infty}(0,T\wedge {\tau^{\varepsilon}}))$ by Lemma \ref{l2}.
Let us set ${\tilde x}^{{\varepsilon}}(t)={x^{\varepsilon}(t)}-{\varepsilon} \lambda(t)$ for $t\in [0,{\tau^{\varepsilon}}[$.
Then
\begin{equation}
\label{newmod}
{u^{\vep}}(t,x+{\tilde x}^{{\varepsilon}}(t))={\varphi_{c^{\varepsilon}(t)}}(x)+{\varepsilon} {\tilde {\eta}}^{{\varepsilon}}(t,x)
\end{equation}
with
$$
{\tilde {\eta}}^{{\varepsilon}}(t,x)=\frac{1}{{\varepsilon}} ({\varphi_{c^{\varepsilon}(t)}}(x-{\varepsilon}\lambda(t))-{\varphi_{c^{\varepsilon}(t)}}(x))+{\eta^{\varepsilon}}(t,x-{\varepsilon}\lambda(t)).
$$
Note that, a.s. for $t\le {\tau^{\varepsilon}}$ :
$$
|{\varphi_{c^{\varepsilon}(t)}}(.-{\varepsilon} \lambda(t))-{\varphi_{c^{\varepsilon}(t)}}-{\varepsilon} \lambda(t) {\partial}_x {\varphi_{c^{\varepsilon}(t)}}|_{L^2}
\le {\varepsilon}^2 \lambda^2(t) C(c_0,{\alpha}).
$$
Hence, it follows from Lemma \ref{l2}, \ref{l3} and the above bound on $\lambda$
that
\begin{equation}
\label{limetatilde}
\lim_{{\varepsilon} {\rightarrow} 0} {\mathbb E} \big( \sup_{t\le T\wedge {\tau^{\varepsilon}}} |{\tilde {\eta}}^{{\varepsilon}}(t)-\tilde {\eta}(t)|_{L^2}^2 \big)=0
\end{equation}
with $\tilde {\eta}(t)={\eta}(t)-\lambda(t){\partial}_x {\varphi_{c_0}}$.
So now, with this new decomposition, we clearly have, setting $\tilde w(t,x)=e^{ax}\tilde {\eta}(t,x)$ :
$$
P\tilde w=0, \quad Q\tilde w=Qw.
$$
Also, if $w_2=Qw$, then the equation (\ref{eqeta}) implies
\begin{equation}
\label{eqw2}
dw_2=A_aw_2 dt +Qe^{ax}{\varphi_{c_0}} d\tilde W
\end{equation}
hence
$$
w_2(t)=\int_0^t e^{A_a(t-{\sigma})}Q[e^{ax}{\varphi_{c_0}} d\tilde W({\sigma})];
$$
the trace of the covariance operator of the Gaussian process $w_2$ in $H^1$ may be easily computed and
estimated thanks to (\ref{decexp}) as
$$
\int_0^t \sum_l \|e^{A_a {\sigma}} Q e^{ax}{\varphi_{c_0}}\phi e_l\|_1^2 d{\sigma}
\le C \big(\int_0^t e^{-b{\sigma}}d{\sigma}\big) \sum_l \|e^{ax}{\varphi_{c_0}}\phi e_l\|_1^2 d{\sigma}
\le C \|k\|_1^2 \|e^{ax}{\varphi_{c_0}}\|_1^2.
$$
Moreover, this covariance operator converges as $t$ goes to infinity and it follows that $w_2$ converges
in law in $H^1$ to a Gaussian random variable.
The end of the statement of Theorem \ref{t3} follows, setting $\tilde Qv=e^{-ax}Qe^{ax}v$.
\hfill
$\square$
\section{A remark on the soliton diffusion}
Let us go back to the stochastic evolution equations for the new modulation parameters, that we may write as
\begin{equation}
\label{newmodeq}
\left\{
\begin{array}{l}
d{\tilde x}^{{\varepsilon}} = c_0 dt +{\varepsilon} B_1 dt +{\varepsilon} dB_2 +o({\varepsilon}) \\
d{c^{\varepsilon}} = {\varepsilon} dB_1 +o({\varepsilon})
\end{array}
\right.
\end{equation}
with $B_1=W_1$ and $B_2=-(e^{ax}{\varphi_{c_0}} \tilde W(t),g_1^a)=-(\tilde W(t), {\varphi_{c_0}} \tilde g_1)$.
Note that $B_1$ and $B_2$ are now correlated Brownian motions. We denote by
$${\sigma}=({\sigma}_{ij})_{i,j}=\mathrm{cov}(B_1,B_2).
$$
If we keep only the order one terms in ${\varepsilon}$ i.e. we consider the solution $(X^{{\varepsilon}}(t), C^{{\varepsilon}}(t))$ of the system of SDEs
$$
\left\{
\begin{array}{l}
dX^{{\varepsilon}}= c_0dt +{\varepsilon} B_1dt +{\varepsilon} dB_2\\
dC^{{\varepsilon}}={\varepsilon} dB_1,
\end{array}
\right.
$$
then $(X^{{\varepsilon}}(t)-c_0t, C^{{\varepsilon}}(t)-c_0)$ is a centered Gaussian vector, and it is easy to compute its covariance
matrix. Let us denote by $\mu^{{\varepsilon}}_t$ the law of $(X^{{\varepsilon}}(t)-c_0t, C^{{\varepsilon}}(t)-c_0)$; we may compute
\begin{equation}
\label{maxesp}
\begin{array}{l}
\displaystyle \max_{x\in {\mathbb R}} {\mathbb E} \Big( {\varphi}_{C^{{\varepsilon}}(t)}(x-X^{{\varepsilon}}(t)) \Big) \\
= \displaystyle \max_{x\in {\mathbb R} } \displaystyle \int \!\! \displaystyle \int {\varphi}_{c+c_0}(x-c_0 t-y)
\mu^{{\varepsilon}}_t (dy,dc)\\
= \displaystyle \max_{x\in {\mathbb R} } \displaystyle \frac{1}{(\det \Sigma)^{1/2}}\displaystyle \int\!\! \displaystyle \int {\varphi}_{c+c_0} (x-c_0 t -y) \exp \Big(-\frac12 \Sigma^{-1}\begin{pmatrix}
c\\y
\end{pmatrix}.
\begin{pmatrix}
c\\y
\end{pmatrix}\Big)dc dy
\end{array}
\end{equation}
where $\Sigma$ is the covariance matrix of $(X^{{\varepsilon}}(t)-c_0t, C^{{\varepsilon}}(t)-c_0)$, given by
$$
\Sigma = {\varepsilon}^2 \begin{pmatrix}
{\sigma}_{11} t & {\sigma}_{12} t +{\sigma}_{11} \frac{t^2}{2} \\
{\sigma}_{12} t +{\sigma}_{11} \frac{t^2}{2} & {\sigma}_{22} t +{\sigma}_{12} t^2 +{\sigma}_{11} \frac{t^3}{3}
\end{pmatrix}.
$$
It is not difficult to see that
$$
\exp \Big( -\frac12 \Sigma^{-1} \begin{pmatrix}
c\\y
\end{pmatrix}.
\begin{pmatrix}
c\\y
\end{pmatrix}\Big)
\le \exp\Big( -\frac12 \frac{{\varepsilon}^2}{\det \Sigma} \big( {\sigma}_{11} \frac{t^3}{12} +({\sigma}_{22}-\frac{{\sigma}_{12}^2}{{\sigma}_{11}}t)\big)c^2\Big).
$$
Inserting this inequality in (\ref{maxesp}), using the fact that ${\varphi}_c(x)=c{\varphi}_1(\sqrt{c}x)$ and integrating in $y$
give the bound
$$
{\mathbb E} \Big( {\varphi}_{C^{{\varepsilon}}(t)}(x-X^{{\varepsilon}}(t)) \Big) \le \frac{K}{(\det \Sigma)^{1/2}} \displaystyle \int_0^{+\infty} \sqrt{c+c_0}
e^{-\frac12 \frac{{\varepsilon}^2}{\det \Sigma} [ {\sigma}_{11} \frac{t^3}{12} +({\sigma}_{22}-\frac{{\sigma}_{12}^2}{{\sigma}_{11}}t)]c^2} dc
$$
where $K$ is a constant, and since
$$
\displaystyle \int_0^{+\infty} \sqrt{c} e^{-\frac{c^2}{2{\alpha}^2}} dc \le K {\alpha}^{3/2}
$$
for another constant $K$, it follows
\begin{equation}
\label{diffusion}
\max_{x\in {\mathbb R}} {\mathbb E} \Big( {\varphi}_{C^{{\varepsilon}}(t)}(x-X^{{\varepsilon}}(t)) \Big) \le K_0 {\varepsilon}^{-1/2} t^{-5/4}
\end{equation}
for $t$ large enough.
This inequality has to be compared to the result of \cite{Wa} where an additive equation
with a white noise in time was considered. An inequality of the form (\ref{diffusion})
was obtained, but with a power $t^{-3/2}$ instead of $t^{-5/4}$.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,207 |
Q: GET_CONTENT intent works in debug, but not release build I have an issue with an intent I am running in my Android app. It works perfectly in the debug build of my app, but not in my release version. Am I missing something?
This is the intent code and activity result callback.
protected void onActivityResult(int requestCode, int resultCode, Intent intent) {
if (requestCode != FILECHOOSER_RESULTCODE) return;
if (null == mUploadMessage) return;
mUploadMessage.onReceiveValue(intent.getData());
mUploadMessage = null;
}
private void pickFile() {
Intent chooserIntent = new Intent(Intent.ACTION_GET_CONTENT);
chooserIntent.setType("image/*");
startActivityForResult(chooserIntent, FILECHOOSER_RESULTCODE);
}
Here I call the pickFile function
echoView.setWebChromeClient(new WebChromeClient() {
@SuppressWarnings("unused")
public void openFileChooser(ValueCallback<Uri> uploadMsg, String AcceptType, String capture) {
this.openFileChooser(uploadMsg);
}
@SuppressWarnings("unused")
public void openFileChooser(ValueCallback<Uri> uploadMsg, String AcceptType) {
this.openFileChooser(uploadMsg);
}
public void openFileChooser(ValueCallback<Uri> uploadMsg) {
mUploadMessage = uploadMsg;
pickFile();
}
}
A: I found that there were two issues. One especially related to this issue regarding debug/release build, and one related to Android 4.4.
Debug/release build
Proguard was enabled for the release build, and appeared to strip a JavaScript interface, which for some reason interfered with the callback to openFileChooser.
Android 4.4
In KitKit, the Android team have removed the private API call to openFileChooser when the user tabs a input[type=file] element. I have yet to find a workaround. Until then file upload through a WebView seems impossible on 4.4.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,891 |
Accompanying the move to value-based care and reimbursement is the need to align economic and practice incentives to create accountability, says Cynthia Kilroy, senior vice president of provider strategy and business development, Optum. It is not just about volume, but about managing populations, and investing in capabilities and tools to manage populations.
We are seeing five trends in the industry, with implications for each of them.
First, there is a consolidation of the provider community that physicians are organizing, and then hospital systems or large integrated delivery networks (IDNs) are purchasing physicians. We are seeing both an affiliated and an employed model in the market right now.
Another trend is system affordability. Premiums have been increasing significantly — more than 30 percent over the last five years. The challenge that CMS and some payors are focusing on is how to make healthcare more affordable to the community at large.
A third trending area is value-based care, and aligning the economic and the practice incentives to create accountability. It is not just about volume, but about managing populations. This leads into the fourth trend, which is that provider organizations are investing in capabilities and tools to manage populations. Then the incentive models are moving more around that population care, which is more challenging to measure.
Finally, there is a significant amount of interest in finding performance metrics. There is HCAHPS®. Every other payor is asking for different performance metrics from organizations; how do we focus that into the right incentive, especially from an incentive program for physicians? Each organization will be trying to achieve something different; each market is very different. I may see one provider organization focus in particular areas and disease states around quality. In other markets there might be something completely different. It is based on what is going on in that particular market and practice.
"We transform a conversation of chronic disease into something patients can look forward to." Susan Lehrer, RN, CDE, NYCHHC House Calls.
Guided by the philosophy, "Be real to your patients, and let them be real to you," the New York City Health and Hospitals Corporation (NYCHHC) House Calls telehealth program is as committed to participants' "life bottom line" as it is to its own program ROI.
In the House Calls telehealth program for diabetics, patients' blood sugar, blood pressure and weight are transmitted via hand-size wireless modems to a team of specially trained nurses who provide feedback and education during pleasant telephone conversations at scheduled intervals.
A digital dashboard provides the telehealth nurses with a quick view of patients' vitals and individuals who may be alerting. The telehealth technology enables immediate feedback that prevents overcorrection on the part of patients, Ms. Lehrer notes, while facilitating dramatic clinical outcomes.
The telephonic exchanges augment regular patient visits and enhanced by the nurses' use of motivational interviewing. The telephonic communications are "templated" to avoid long narratives.
Ms. Lehrer presented some of House Calls' clinical outcomes for the 2,500 patients it has serviced since the program's inception during a July 2014 webinar, Diabetic Telehealth Monitoring: The Impact of Real-Time Data on High-Risk Patients sponsored by the Healthcare Intelligence Network.
Most House Calls participants are diabetics who spend an average of two years in the program, she explained. Of a random sampling of 769 participants, 76 percent improved their A1C almost every three months. Additionally, of patients in that sampling with A1Cs between 11 and 13, 91 percent improved A1Cs by an average of 2.9 percent.
House Calls, which has experienced a side benefit of fewer appointment cancellations on the part of participants, has been so successful the program already has been rolled out for patients with heart disease; its use for the chronic obstructive pulmonary disorder (COPD) population is being discussed.
However, Ms. Lehrer is quick to point out barriers to telehealth still exist. Physicians who treat a patient with diabetes for years without seeing any real change can develop "clinical inertia," she says, although this quickly dissipates once the doctor sees a patient engaged in House Calls.
There is also the occasional patient resistant to change, and the frustration of being unable to integrate patient data into an electronic medical record (EMR).
Still, despite the program's focus on technology and results, the nurses remind themselves that at its core, House Calls is about the person at the other end of the line.
Listen to an interview with Susan Lehrer here.
Between 1995 and 2010, annual emergency room visits in the United States grew by 34 percent, while the number of hospitals with ERs declined by 11 percent, according to a new infographic from the George Washington University MHA program.
The infographic also looks at the impact of overcrowding on U.S. emergency rooms, including the major causes of congested ERs and the impact on care delivery and proposed solutions to the problem of overcrowding.
Evidence is lacking to support the effectiveness of public policy interventions based on performance measurement, such as public reporting of data and pay for performance. To succeed, emergency clinicians need to understand and practice in alignment with national performance measures.
Quality And Performance Measurement: A Guide For Emergency Physicians reviews the origin and evolution of performance measurement, explains the current landscape of reporting, and discusses projections for future hospital quality measure implementation through 2014.
Many patient-centered medical home (PCMH) initiatives have added home visits to care transition management to reduce avoidable hospital readmissions and ER utilization. Jessica Simo, program manager with Durham Community Health Network for the Duke Division of Community Health, describes likely candidates for home visits, the structure of a typical home visit and recommended staff training.
HIN: Which diagnosis or patient profile benefits most from a home visit?
(Jessica Simo) As a general rule for the patient population we serve, the people who get the most home visits are middle-aged individuals with at least two chronic health conditions. These are not generally healthy individuals who had one adverse event that brought them to our attention. These are people living day in and day out with chronic health problems they struggle with managing. Those people benefit the most from the amount of time it takes to do a home visit.
HIN: What is the average length and typical format of a home visit?
(Jessica Simo) The average home visit lasts 45-60 minutes. It would be longer for the initial home visit when an assessment is being done—where the Care Partner (a partnering stakeholder from across the Duke University health system and the Durham community) collects information for the first time about medications the patient takes, their sources of support, ADL deficits, etc. Those visits tend to be a bit longer, certainly an hour at a minimum, but once that rapport has been established, the weekly visits are often less than an hour. They become briefer as a patient transitions from phase one to phase two of the Care Partners Pathway because there is less to talk about at that point. This is a good thing; it means they are improving.
The home visits are structured around assessments and protocols, but as the home visits progress and the care partner becomes more familiar with the patient, there is less reliance on assessments and more on follow-up from the previous week.
HIN: How do you prepare and train staff to conduct home visits?
(Jessica Simo) The best way to prepare somebody to do home visits is to have them shadow a more experienced staff person. There are too many independent variables at play when you go into somebody's home and you just don't have control over that environment. Nor should you. It's impossible to anticipate every possible scenario. Therefore, we do a lot of shadowing for at least a month before someone does a home visit on their own.
Without the help of a care team, physicians would not have enough hours in the day to adhere to all the protocols for chronic care patients, according to a new infographic by Phase Space.
The infographic looks at the number of individuals with chronic conditions, the capacity of providers to care for these patients appropriately and how care teams fill these gaps.
With the advent of the medical neighborhood, care coordination is no longer the sole domain of the primary care practice (PCP) but a responsibility shared among all providers that touch the patient. But how to formalize co-management of patients by PCPs and specialists ‒ in a way that both assures efficient delivery of high-quality healthcare and addresses the 'pain points' of each provider group?
Every day in America millions of young adults use illicit substances, ranging from marijuana, heroin, and cocaine, to hallucinogens and inhalants. Out of the 35.6 million young adult population (from 2012) in the United States, one fifth used an illicit drug in the past month, and the percentage of those users has increased from 2008.
The infographic below shows how often drugs are used daily in the United States and the number of first-time illicit drug users on an average day.
Bringing the most comprehensive research and information available today to the mental health field, the Dartmouth Psychiatric Research Center and Hazelden have redesigned the innovative Integrated Dual Disorders Treatment: Best Practices, Skills, and Resources for Successful Client Care curriculum.
Since healthcare is local, it's vital that health systems engage local providers, enlisting both clinical and administrative champions, advises Julie Faulhaber, vice president of enterprise Medicaid at Health Care Service Corporation. Ms. Faulhaber offers a variety of guidelines for engagement of community partners in care coordination for Medicare and Medicaid beneficiaries.
Our community care coordination partners may employ different models of care coordination. First, some may have care systems, larger accountable care organization (ACO)-type organizations; many take full financial risk, including risk on home- and community-based services. There are also waivers.
Second, some of these large care systems also have nurse practitioner (NP) models that provide mainly facility-based care. Those can be extremely successful with outcomes for the numbers, as well as from a cost perspective. Third, we also work with care management organizations and providers. Another example would be the Triple A's—Adult Areas Agencies on Aging—and other behavioral health organizations. In our experience, these organizations will take on some financial risk, but really for those care coordination services.
Fourth, there are many different financial models you can use with both groups, particularly for the care management organization providers. For example, looking at a risk on care coordination, gain sharing—potentially in a new program—helping to pay for some startup infrastructure cost, providing loans with some paybacks. There are many different opportunities to make it financially viable for those important community partners to work with health plans in order to provide community-based, social model services to the member to meet all of their needs.
Finally, when working with community partners, it is critical to have both a clinical and administrative champion for the program. Clinically, it helps to have a physician nurse who can talk with their peers in the organization to help them understand the program. Clinicians want to provide care in a very uniform way, but if there is an opportunity to provide additional benefits in lieu of services for members, it helps to have that clinician champion to be able to share that.
Administratively, it is also important to manage the enrollment and care coordination paperwork. The plans are putting significant faith in these organizations to meet their contractual obligations, so having someone to follow up for those types of things is critical. It is also important to provide reporting and feedback on the results for these groups. We have done quarterly meetings in the past, which I found to be very helpful.
It is also helpful to provide benchmarking data. We look at how one organization serving the same population in a similar environment shapes up in comparison to another. This has improved results overall; it makes those organizations leading the pack feel good, and provides those trying to catch up with some role models to look at.
Physicians are becoming more proactive in managing their incomes by being more selective about insurers and patients and providing ancillary services. In addition, a small but growing number of physicians are moving toward cash-only practices.
A new infographic from Medscape looks at these trends, along with details on how the Affordable Care Act is impacting physician practices, the income gender disparity among physicians and physician career satisfaction.
In today's value-based healthcare sphere, providers must not only shoulder more responsibility for healthcare outcomes, cost and quality but also align with emerging compensation models rewarding these efforts — models that often seem confusing or contradictory. The challenges for payors and partners in creating a common value-based vision are sizing the reimbursement model to the provider organization and engaging physicians' skills, knowledge and behaviors to foster program success. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,532 |
{"url":"https:\/\/www.transtutors.com\/questions\/estimate-cost-460374.htm","text":"# Estimate cost\n\nSpeedy Parcel Service operates a fleet of delivery trucks in a large metropolitan area. A careful study by the company\u2019s cost analyst has determined that if a truck is driven 126,000 miles during a year, the average operating cost is 13.3 cents per mile. If a truck is driven only 84,000 miles during a year, the average operating cost increases to 15.5 cents per mile.\n\nRequired\n\n1. Using the high-low method, estimate the variable and fixed cost elements of the annual cost of truck operation.\u00a0(Round the \"Variable cost per mile\" to 3 decimal places and the \"Fixed cost\" to the nearest dollar amount. Omit the \"$\" sign in your response.) Variable cost$ per mile Fixed cost $per year 2. Express the variable and fixed costs in the form Y = a + bX(Round the \"Variable cost per mile\" to 3 decimal places and the \"Fixed cost\" to the nearest dollar amount. Omit the \"$\" sign in your response.)\n Y = $+$ X\n3. If a truck were driven 105,000 miles during a year, what total cost would you expect to be incurred?\u00a0(Round the \"Variable cost\u00a0per mile\" to 3 decimal places. Round your intermediate and final answers to the nearest dollar amount. Omit the \"$\" sign in your response.) Total annual cost$\n\n## Plagiarism Checker\n\nSubmit your documents and get free Plagiarism report\n\nFree Plagiarism Checker","date":"2020-05-27 12:20:09","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.2064717561006546, \"perplexity\": 2466.589391299883}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-24\/segments\/1590347394074.44\/warc\/CC-MAIN-20200527110649-20200527140649-00216.warc.gz\"}"} | null | null |
\section{Introduction}
The atomic properties of Ba$^{+}$ ion are of particular interest owing to the prospects of studying the
parity nonconservation (PNC) with a single trapped ion \cite{fortson:93}.
Progress on the related spectroscopy with a single Ba$^+$ ion is
reported in \cite{koerber:02,koerber:03}, and precision measurements of light
shifts in a single trapped Ba$^+$ ion have been reported in
\cite{Ba:2005}.
The PNC interactions
gives rise to non-zero amplitudes
for transitions that are otherwise forbidden by the parity selection rules,
such as $6s-7s$ electric-dipole transition in Cs.
The study of parity
nonconservation in cesium \cite{wood:97,bennett:99} involving both high-precision measurements and
several high-precision calculations provided an atomic-physics test of the standard
model of the electroweak interactions and
yielded the
first measurement of the nuclear anapole moment (see \cite{Ginges:2004} for the review of
study of fundamental symmetries with heavy atoms).
The analysis of the Cs experiment, which
required a calculation of the nuclear spin-dependent PNC amplitude,
led to constraints on weak nucleon-nucleon coupling constants that
are inconsistent with constraints from deep inelastic scattering and
other nuclear experiments \cite{HAX:01}.
More PNC experiments in other
atomic systems, such as Ba$^+$, are needed to resolve this issue.
The prospects for measuring parity violation in Ba$^+$ have been recently discussed in
\cite{koerber:03}.
Ba$^+$ is also of particular interest for developing optical frequency standard \cite{Sherman:2005} and
quantum information processing \cite{Blinov,Chen:2005} owing to the extremely
long lifetimes of $5d$ states.
The accuracy of optical frequency standards is
limited by the frequency shift in the clock transitions caused by
the interaction of the ion with external fields. Therefore,
knowledge of atomic properties is needed for the analysis of the ultimate
performance of such frequency standard.
Another motivation for study of Ba$^+$ is an excellent opportunity for tests of theoretical
and experimental methods, in particular in light of recent measurements of Ba$^+$ atomic
properties \cite{koerber:02,koerber:03,snow:05,Gallagher:2006,Snow:2007,Gurell:2007}.
Ba$^+$ is a monovalent system allowing for precise
theoretical predictions, and, in some cases, for evaluation of the theoretical uncertainties
that do not directly rely on the comparison with experiment.
It is also an excellent testing case for further studies of Ra$^+$ ion, where the
correlation corrections are expected to be larger owing to a larger core. A project to measure PNC in
a single trapped radium ion recently started at the Accelerator Institute (KVI) of the University of
Groningen \cite{Ra:pnc}.
In this work, we calculate $6s-np_j$ ($n=6-9$), $6p_j-5d_{j^{\prime}}$ electric-dipole matrix elements,
$6s-nd_j$ ($n=5-9$) electric-quadrupole
matrix elements, and $5d_{5/2}-5d_{3/2}$ magnetic-dipole matrix element in Ba$^+$.
This set of matrix elements is needed for accurate
calculation of ground state dipole and quadrupole polarizabilities and lifetimes of the
$6p_{1/2}$, $6p_{3/2}$, $5d_{3/2}$, and $5d_{5/2}$ levels.
We carefully investigate the uncertainty in our values of $6s-5d_j$ matrix elements in order to
estimate the uncertainties in the quadrupole polarizability and the $5d_j$ lifetime values.
It is particularly important to independently determine these uncertainties
because of significant inconsistencies between different measurements of the
$5d_{3/2}$ and $5d_{5/2}$ lifetimes \cite{Schneider:1979,Knab-Bernardini:1992,yu:97,Gurell:2007,
Plumelle:1980,Nagourney:86,madej:90}. There are also large discrepancies between
experimental determinations of the
$5d-6s$ quadrupole matrix element from the lifetime experiments and studies of the Rydberg states of barium
\cite{snow:05,Snow:2007,Gallagher:2006}.
The experimental values of the ground state quadrupole polarizability from Refs.~\cite{Gallagher:1982,snow:05,Snow:2007}
differ by a factor of two; our value of the quadrupole polarizability is in agreement with
Ref.~\cite{Snow:2007}.
We note that there are no inconsistencies between the experimental lifetimes \cite{Andra:1976,Kuske:1978,pinnington:95}
of the $6p_j$ levels and experimental determinations of the
electric-dipole ground state polarizability \cite{Gallagher:1982,snow:05,Snow:2007}. The experimental
values of the electric-dipole polarizability of the Ba$^{+}$ ion in its ground state
\cite{Gallagher:1982,snow:05,Snow:2007} are also in agreement with each other
and our theoretical value. Our lifetimes of the $6p_{1/2}$ and $6p_{3/2}$ levels are in agreement
with experimental values \cite{Andra:1976,Kuske:1978,pinnington:95} within expected accuracy (1\%).
The paper is organized as follows.
In Section~\ref{method}, we give a short description of the
method used for the calculation of the matrix elements.
In Section~\ref{section-e1}, we discuss the calculation of the electric-dipole polarizability and conduct
comparative analysis of the correlation corrections to the $ns-np$ matrix elements in Ba$^{+}$, Cs, and Ca$^{+}$.
The $6s-5d$ quadrupole matrix elements and the ground state
quadrupole polarizability are discussed in Section~\ref{section-e2}, and the lifetimes are discussed in
Section \ref{section-lifetimes}. A consistency study of the $5d_j$ lifetime and ground state quadrupole polarizability
measurements in Ba$^{+}$ is presented in Section~\ref{section-e2}.
\section{Method}
\label{method}
We calculate the reduced multipole matrix elements
using the relativistic all-order method \cite{blundell:89,Safronova:1999,Safronova:2007}
which is a linearized coupled-cluster method where all single and double excitations of the Dirac-Fock
wave function are included to all orders of perturbation theory.
The present implementation of the method is suitable for the calculation of matrix elements of any one-body operator, i.e.,
the calculations of the E1, E2, and M1 matrix elements are carried out in the same way.
We refer the reader to the review ~\cite{Safronova:2007} and references therein
for the detailed description of the all-order method.
Briefly, our starting point is the relativistic no-pair Hamiltonian \cite{brown:51}
expressed in second quantization as
\begin{equation}
H = \sum_{i} \epsilon_{i} :a_{i}^{\dagger} a_{i}:
+ \frac{1}{2} \sum_{ijkl} g_{ijkl} :a_{i}^{\dagger} a_{j}^{\dagger} a_{l} a_{k}: ,
\end{equation}
\noindent where $a_{i}^{\dagger}, a_{j}$ are single-particle creation and annihilation
operators, respectively, $\epsilon_{i}$ is the Dirac-Fock (DF)
energy for the state $i$, $g_{ijkl}$ are the two-body Coulomb integrals, and :~: indicates normal order of the operators
with respect to the closed core.
The single-double (SD) all-order wave function is written as
\begin{eqnarray} \nonumber
&& |\Psi_v^{\rm SD} \rangle = \left( 1 + \sum\limits_{ma}^{} \rho_{ma} a_{m}^{\dagger} a_{a} + \frac{1}{2} \sum\limits_{mna
b}^{} \rho_{mnab} a_{m}^{\dagger} a_{n}^{\dagger} a_{b} a_{a} \right. \\
&& + \left. \sum\limits_{m \neq v} \rho_{mv} a_{m}^{\dagger} a_v
+ \sum\limits_{mna} \rho_{mnva} a_{m}^{\dagger} a_{n}^{\dagger} a_a a_v \right)|\Phi_v\rangle
\end{eqnarray}
\noindent where $| \Phi_v \rangle$ is the lowest-order wave function
taken to be the frozen-core DF wave function of a state $v$.
Indices at the beginning of
the alphabet, $a$, $b$, $\cdots$, refer to occupied core states, those in
the middle of the alphabet $m$, $n$, $\cdots$, refer to excited states,
and index $v$ designates the valence orbital.
The all-order equations for the excitation coefficients $\rho_{ma}$, $\rho_{mv}$, $\rho_{mnab}$, and $\rho_{mnva}$ are solved iteratively with a finite basis set, and
the correlation energy is used as a convergence parameter.
The basis set is defined in a spherical cavity on non-linear grid and
consists of single-particle basis states which are linear combinations of B-splines \cite{johnson:88}.
We use a basis set of 50 splines of order 9 in a spherical cavity of radius 80 a.u.
Such cavity size is chosen to accurately represent all orbitals of interest to the present study.
The resulting excitation coefficients $\rho_{ma}$, $\rho_{mv}$, $\rho_{mnab}$, and $\rho_{mnva}$
are used to calculate the one-body E1, M1, and E2 matrix elements.
The SD all-order method yielded results for the primary $ns-np_{j}$ E1 matrix elements of alkali-metal atoms
that are in agreement with experiment to 0.1\%-0.5\% \cite{Safronova:1999}.
We note that while the all-order expression for the matrix elements contains 20 terms that are linear or quadratic
functions of the excitation coefficients, only two terms are dominant for all matrix elements considered
in this work:
\begin{equation}
Z^{(a)} = \sum\limits_{ma} \left( z_{am} \tilde{\rho}_{wmva} + z_{ma} \tilde{\rho}^{*}_{vmwa} \right)
\end{equation}
and
\begin{equation}
Z^{(c)} = \sum\limits_{m} \left( z_{wm} \rho_{mv} + z_{mv} \rho^{*}_{mw} \right),
\end{equation}
where $\tilde{\rho}_{mnab}=\rho_{mnab}-\rho_{nmab}$ and $z_{wv}$ are lowest-order matrix elements of the corresponding operator.
In the case of the electric-quadrupole transitions studied in this work, the second term $Z^{(c)}$ is overwhelmingly
(by an order of magnitude) larger than any other term. In such cases, it was found necessary to
include at least partially triple excitations into the
wave function
\begin{equation}
|\Psi_v^{\rm SDpT} \rangle =|\Psi_v^{\rm SD} \rangle +
\frac{1}{6} \sum\limits_{mnrab}^{} \rho_{mnrvab} a_{m}^{\dagger} a_{n}^{\dagger} a_{r}^{\dagger} a_b a_a a_v | \Phi_v \rangle
\end{equation}
and to correct single excitation coefficient $\rho_{mv}$ equation for the effect of triple
excitations \cite{blundell:91,Safronova:1999,Safronova:2005,safronovacs:04}.
We have conducted such a calculation for the $6s-5d_j$, $6s-6d_j$, and $6s-7d_j$ electric-quadrupole
matrix elements and refer to the corresponding results as SDpT values (i.e. including all single, double, and partial
triple excitations).
We note that such approach works poorly when terms $Z^{(a)}$ and $Z^{(c)}$ are of similar
order of magnitude (such as all E1 transition considered here) owing most likely
to cancellation of high-order corrections to terms $Z^{(a)}$ and $Z^{(c)}$. The term $Z^{(a)}$
is not directly corrected for triple excitations in the SDpT extension of the method leading to
consistent treatment of the higher-order correlations only when the second term is overwhelmingly dominant.
We refer the reader to Ref.~\cite{Triples:2006} for the detailed discussion of triple
excitations. The results of the matrix element calculation are discussed in the following sections.
\section{Ba$^{+}$ ground state dipole polarizability}
\label{section-e1}
The ground state dipole or quadrupole polarizability can be represented as a sum of the valence
polarizability $\alpha_v$ and the polarizability of the ionic core $\alpha_{core}$ \cite{Safronova:1999}.
The calculation of the core polarizability assumes allowed excitations
to any excited state including the valence shell,
which requires the introduction of the small counter terms $\alpha_{vc}$ to subtract out $1/2$
of the contribution corresponding to the $6s$ shell excitation \cite{Safronova:1999}.
The core polarizabilities have been calculated in random-phase approximation (RPA)
in Ref.~\cite{johnson:83}. The accuracy of the RPA values is expected to be on the order of
5\% \cite{safronovacs:04}. We calculated the $\alpha_{vc}$ term the in the RPA approximation
for consistency with $\alpha_{core}$ value. The valence dipole polarizability for the $6s$ state
of Ba$^+$ is calculated as sum-over-states
\begin{eqnarray}
\label{e1}
\alpha_{v, E1} = \frac{1}{3} \sum\limits_{n}^{}
\left( \frac{| \langle 6s||d||n p_{1/2} \rangle |^{2}}{E_{np_{1/2}} - E_{6s}} +
\frac{| \langle 6s||d||n p_{3/2} \rangle |^{2}}{E_{np_{3/2}} - E_{6s}} \right).
\end{eqnarray}
The sum over the principal quantum number $n$ in Eq.~(\ref{e1}) converges
very rapidly and very few first terms have to be calculated to high precision.
In this work, we use SD all-order matrix elements and experimental energies for terms with
$n=6-9$ and evaluate the remainder $\alpha_{tail}$ in the Dirac-Fock approximation.
The contributions to the dipole polarizability are summarized in Table~\ref{tab1}.
We also list the absolute values of corresponding SD all-order reduced electric-dipole
matrix elements $d$. The contribution of the terms with $n=6$ is overwhelmingly dominant.
Therefore, the uncertainty in our calculation of the dipole polarizability
is dominated by the uncertainties in the $6s-6p_{1/2}$ and $6s-6p_{3/2}$
matrix elements.
To study the uncertainty in these values, we investigate the importance of the contributions from
various
correlation correction terms and the overall size of the correlation correction. The
contributions to the $6s-6p_{1/2}$ matrix element are summarized in Table~\ref{tab2}. The breakdown of the contributions
to the $6s-6p_{3/2}$ matrix element is essentially the same, and we do not list it here.
We also give the breakdown of the correlation correction for the same transition in Cs and
$4s-4p_{1/2}$ transition in Ca$^+$. Cs values are taken from Ref.~\cite{safronova:thesis}. Final
Ca$^{+}$ value has been published in Ref.~\cite{Arora:BBR}.
As we noted in Section~\ref{method}, only two terms
give large contributions to the correlation correction. While there are some cancellations
in the other terms, all them are at least an order of magnitude smaller. Unfortunately,
there is no straightforward way to evaluate the uncertainty in the $Z^{(a)}$ term (as we show in the
later section it can be done for $Z^{(c)}$). Therefore, we can not make an uncertainty estimate that is
independent on experimental observations. However, we note that Cs $6s-6p_{j}$ transitions are
extremely well studied by a number of different experimental approaches (see, for example, \cite{Amini:2003}
and references therein), and all-order SD data are in agreement with Cs experimental values to 0.2\%-0.4\% \cite{Safronova:1999}.
The breakdown of terms for Ba$^{+}$ is slightly different than for Cs but is very similar to Ca$^{+}$. As expected,
the size of the correlations is larger in Ba$^{+}$ than in Ca$^{+}$. Unfortunately, there is only one high-precision
measurement of the $4p_j$ Ca$^{+}$ lifetimes \cite{jin} that is in significant (2\%) disagreement with high-precision
theoretical results. Similar discrepancies existed for the alkali-metal atom measurements done
with the same technique and
later experiments confirmed the theory values. We refer the reader to Ref.~\cite{Arora:BBR} for more
detailed discussion of this issue. It would have been very interesting to see the $4p$ lifetimes in Ca$^+$
remeasured to resolve this issue. Based on the similar size of the correlation corrections for Cs and
Ba$^+$, we expect similar accuracy of our data (on the order of 0.5\%). Therefore, the
resulting accuracy of our dipole polarizability is expected to be on the order of 1\%.
We find that our value is in excellent agreement with both experimental values \cite{Snow:2007,Gallagher:1982}
when our estimated uncertainty is taken into account. Our results are in good
agreement with other theoretical calculations \cite{Lim:2004,Miadokova:1997,Patil:1997}.
We also note that the $\langle 6s|d|6p \rangle$ matrix element has been recently extracted from the $K$ splittings
of the bound $6snl$ states in Ref.~\cite{Gallagher:2006}, and the resulting value
$\langle 6s|d|6p \rangle=4.03(12)$ is in excellent agreement with our result
$\langle 6s|d|6p \rangle=4.08$ (normalized spherical harmonics $C_{1}$ is factored out here for comparison).
\begin{table}
\caption{\label{tab1} Contributions to the ground state $6s$ scalar dipole polarizability $\alpha_{E1}$ in Ba$^{+}$ in
units of $a^3_0$. Comparison with experiment and other calculations. The absolute values of corresponding SD all-order reduced electric-dipole
matrix elements $d$ (in a.u.) are also given. }
\begin{ruledtabular}
\begin{tabular}{lrr}
\multicolumn{1}{c}{Contribution} &
\multicolumn{1}{c}{$d$} &
\multicolumn{1}{c}{$\alpha_{E1}$} \\
\hline
$6s-6p_{1/2} $& 3.3357& 40.18 \\
$6s-6p_{3/2} $& 4.7065& 73.82 \\
$6s-7p_{1/2} $& 0.0621& 0.06 \\
$6s-7p_{3/2} $& 0.0868& 0.01 \\
$\alpha_{tail}$& & 0.03 \\
$\alpha_{core}$& & 10.61 \\
$\alpha_{vc}$ & & -0.51 \\
Total & & 124.15 \\
Expt.~\protect\cite{Snow:2007}&& 123.88(5)\\
Expt.~\protect\cite{Gallagher:1982}&& 125.5(10)\\
Theory~\protect\cite{Lim:2004}&& 123.07 \\
Theory~\protect\cite{Miadokova:1997} && 126.2 \\
Theory~\protect\cite{Patil:1997} && 124.7
\end{tabular}
\end{ruledtabular}
\end{table}
\begin{table}
\caption{\label{tab2} Contributions of different terms to the Ba$^+$, Ca$^+$, and Cs $ns-np_{1/2}$ reduced matrix elements in a.u.}
\begin{ruledtabular}
\begin{tabular}{lccc}
\multicolumn{1}{c}{Contribution} &
\multicolumn{1}{c}{Ba$^{+}$} &
\multicolumn{1}{c}{Cs~\cite{safronova:thesis}} &
\multicolumn{1}{c}{Ca$^{+}$} \\
\hline
\multicolumn{1}{c}{} &
\multicolumn{1}{c}{$6s-6p_{1/2}$} &
\multicolumn{1}{c}{$6s-6p_{1/2}$} &
\multicolumn{1}{c}{$4s-4p_{1/2}$} \\
DF & 3.891 & 5.278& 3.201\\
$Z^{(a)}$ & -0.387 &-0.334& -0.200\\
$Z^{(c)}$ & -0.209 &-0.485& -0.120\\
Other & 0.041 & 0.019& 0.016\\
Total & 3.336 & 4.478& 2.898\\
Correlation & 16.6\% & 17.9\%& 10.5\%
\end{tabular}
\end{ruledtabular}
\end{table}
\begin{table*}
\caption{\label{tab3} Absolute values of electric-quadrupole $6s-5d_{3/2}$ and $6s-5d_{5/2}$ reduced matrix
elements in Ba$^+$ calculated in different approximations in a.u. Columns labeled ``DF'' and ``III'' are lowest-order
Dirac-Fock and third-order MBPT values, respectively. The third-order results calculated with maximum
number of partial values $l_{max}=6$ and $l_{max}=10$ are given to illustrate the contribution of the higher partial
waves. Breit correction is given separately. The all-order \textit{ab initio} results are given in columns labeled
``SD'' and ``SDpT'', respectively;
these results include contributions from higher partial waves and Breit correction. The
corresponding scaled values are listed in columns labeled ``SD$_{sc}$'' and ``SDpT$_{sc}$''. The calculation of the
uncertainties of the final values is described in detail in text.}
\begin{ruledtabular}
\begin{tabular}{lccccccccc}
\multicolumn{1}{c}{Transition} &
\multicolumn{1}{c}{DF} &
\multicolumn{1}{c}{III ($l_{max}=6$)} &
\multicolumn{1}{c}{III ($l_{max}=10$)} &
\multicolumn{1}{c}{Breit} &
\multicolumn{1}{c}{SD} &
\multicolumn{1}{c}{SDpT} &
\multicolumn{1}{c}{SD$_{sc}$} &
\multicolumn{1}{c}{SDpT$_{sc}$} &
\multicolumn{1}{c}{Final} \\
\hline
$6s-5d_{3/2}$ &14.76 &11.82& 11.75& -0.07& 12.42& 12.66& 12.63& 12.59& 12.63(9)\\
$6s-5d_{5/2}$ &18.38 &14.86& 14.78& -0.09& 15.55& 15.84& 15.80 &15.76& 15.80(11)
\end{tabular}
\end{ruledtabular}
\end{table*}
\section{Ba$^{+}$ ground state quadrupole polarizability}
\label{section-e2}
The valence part of the quadrupole polarizability is given in the sum-over-states approach by
\begin{eqnarray}\label{e2}
\alpha_{v,E2} = \frac{1}{5} \sum\limits_{n}^{}
\left( \frac{|\langle 6s||Q||n d_{3/2} \rangle |^{2}}{E_{n d_{3/2}} - E_{6s}} +
\frac{| \langle 6s||Q||n d_{5/2} \rangle |^{2}}{E_{n d_{5/2}} - E_{6s}} \right).
\end{eqnarray}
\begin{table}
\caption{\label{tab4} Contributions to the ground state $6s$ quadrupole polarizability $\alpha_{E2}$ in Ba$^{+}$ and their uncertainties in
units of $a^5_0$. The absolute values of corresponding all-order reduced electric-quadrupole
matrix elements $Q$ (in a.u.) and their uncertainties are also given.}
\begin{ruledtabular}
\begin{tabular}{lrr}
\multicolumn{1}{c}{Contribution} &
\multicolumn{1}{c}{$Q$} &
\multicolumn{1}{c}{$\alpha_{E2}$} \\
\hline
$6s-5d_{3/2} $& 12.63(9) &1436(20) \\
$6s-6d_{3/2} $& 16.83(5) & 270(2) \\
$6s-7d_{3/2} $& 5.68(5) & 23.7(4) \\
$6s-8d_{3/2} $& 3.09(6) & 6.3(3) \\
$6s-9d_{3/2} $& 2.07(4) & 2.7(1) \\ [0.2pc]
$6s-5d_{5/2} $& 15.8(1)& 1932(27) \\
$6s-6d_{5/2} $& 20.30(6) & 392(2) \\
$6s-7d_{5/2} $& 6.98(6) & 35.7(6) \\
$6s-8d_{5/2} $& 3.83(8) & 9.6(4) \\
$6s-9d_{5/2} $& 2.57(5) & 4.192) \\[0.2pc]
$\alpha_{tail}$& & 24(6) \\
$\alpha_{core}$& & 46(2) \\
Total & & 4182(34) \\
Expt.~\protect\cite{Snow:2007}&& 4420(250)\\
Expt.~\protect\cite{snow:05}&& 2462(361)\\
Expt.~\protect\cite{Gallagher:1982}&& 2050(100) \\
\end{tabular}
\end{ruledtabular}
\end{table}
The RPA core value \cite{johnson:83} is $46a_0^5$, and the $\alpha_{vc}$
term is negligible. The terms containing the $6s-5d_{3/2}$ and $6s-5d_{5/2}$
matrix elements give overwhelmingly dominant contribution to the total values. Therefore,
we study these transitions in more detail and evaluate their uncertainties. Unlike the
case of the E1 transitions considered earlier,
$Z^{(c)}$ term contributes over 90\% of the total correlation correction.
Therefore, we carried out the calculation using both SD and SDpT approaches described in Section ~\ref{method}.
We also carried out semi-empirical scaling in both approximations by multiplying single excitation
coefficients $\rho_{mv}$ by the ratio of the ``experimental'' and corresponding (SD or SDpT)
correlation energies \cite{blundell:91}. The ``experimental'' correlation energies are determined as the
difference of the total experimental energy and the DF lowest-order values.
The calculation of the matrix elements is then repeated with the
modified excitation coefficients.
The accuracy of such scaling procedure for the similar cases was discussed in detail in
Refs.~\cite{Safronovarb:04,safronovacs:04,Safronova:2005}. The reasoning for such a scaling procedure in
third-order perturbation theory (scaling of the self-energy operator) has been discussed in Ref.~\cite{ADNDT:1996}.
We list SD, SDpT, and the corresponding scaled results (labeled ``SD$_{sc}$'' and ``SDpT$_{sc}$'')
in Table~\ref{tab3}. The lowest-order DF results are listed to illustrate the size of the correlation corrections.
We demonstrate the size of the two other corrections, contribution of the higher partial waves and
Breit correction, in the same table. The first correction results from the truncation
of the partial waves in all sums in all-order calculation at $l_{max}=6$. All-order
calculation with higher number of partial waves is unpractical. Therefore, we carry out the third-order MBPT
calculation (following Ref.~\cite{ADNDT:1996}) including all partial waves up to $l_{max}=6$ and $l_{max}=10$ and
take the difference of these two values to be the contribution of the omitted partial waves that we add to \textit{ab initio}
all-order results. We verified that the contribution of the $l=9-10$ partial waves is very small justifying the omission of
contributions from $l>10$. The Breit correction is calculated as the difference of the
third-order results with two different basis sets. The second basis set is
generated with taking into account one-body part of the Breit interaction.
We note that scaled values should not be corrected for either partial wave truncation error or
Breit interaction to avoid possible double-counting of the same effects.
We take SD$_{sc}$ values as our final results. The uncertainty of the final values is calculated as follows:
the uncertainty in the $Z^{(c)}$ term is evaluated as the spread of the most high-precision values
(SD$_{sc}$, \textit{ab initio} SDpT, and SDpT$_{sc}$), the remaining theoretical uncertainty in the Coulomb
correlation correction is taken to be the same as the uncertainty in the dominant $Z^{(c)}$ term.
We assume 100\% uncertainties in the contributions of the higher partial waves and Breit correction.
The final uncertainty of the $6s-5d_j$ matrix elements (0.7\%) is obtained by adding these four uncertainties in quadrature.
We note that this procedure for the uncertainty evaluation does not rely on the experimental values
with the exception of the experimental energies used for scaling.
The contributions to the ground state quadrupole polarizability are given in Table~\ref{tab4}.
While the $n=5$ term is dominant, the contributions of the few next terms are substantial.
Therefore, we carry out SD, SDpT, and both scaled calculations for the $6s-6d_{j}$ and $6s-7d_{j}$
matrix elements as well and repeat the uncertainty analysis described above (we omit Breit and higher-partial
wave corrections here since such precise evaluation of the uncertainties is not needed for these transitions).
The $6s-8d_{j}$ and $6s-9d_{j}$ matrix elements are calculated in third-order MBPT, and their
accuracy is taken to be 2\% based on the comparison of the third-order and all-order values of the
$6s-7d_{j}$ matrix elements. The remainder is evaluated in the DF approximation and
reduced by 23\% based on the comparison of the DF and third-order data for
$6s-8d_{j}$ and $6s-9d_{j}$ matrix elements. Its accuracy is correspondingly taken to be 23\%.
Our recommended value for the ground state quadrupole polarizability is in
agreement within the corresponding uncertainties with the most recent
experimental work~\cite{Snow:2007}. However, our value for the contribution of the
$6s-5d_{j}$ transitions to the quadrupole polarizability [3368(34)] differs by about a factor of 2
from the experimental values \cite{Snow:2007,snow:05,Gallagher:2006} obtained based
on the nonadiabatic effects on the Rydberg fine-structure
intervals. This issue and the discrepancies in the experimental values of the quadrupole
polarizabilities are addressed in detail in
Ref.~\cite{Snow:2007}.
We note that these experimental values of the $6s-5d_{j}$ contributions to the
quadrupole polarizabilities (1524(8) \cite{Snow:2007} and 1562(93) \cite{Gallagher:2006} in the two most
recent studies) are significantly inconsistent with all high-precision calculations of the $5d_j$
lifetimes \cite{Guet:1991,Guet:2007,dzuba:01,das:02,Sahoo:2006,Gurell:2007} carried out by different methods as well as with
all experimental
lifetime measurements (also carried out by different techniques)
\cite{Schneider:1979,Knab-Bernardini:1992,yu:97,Gurell:2007,Plumelle:1980,Nagourney:86,madej:90}.
For comparison, the value 1562(93) obtained from the $\langle 6s|r^2|5d \rangle=9.76(29)$
matrix element that was extracted from the $K$ splittings
of the bound $6snl$ states in Ref.~\cite{Gallagher:2006}
corresponds to the lifetime $\tau_{5d{3/2}}=170(10)$~s
that is a factor of 2 longer than all other values.
We discuss the lifetimes of the
$5d_{3/2}$ and $5d_{5/2}$ levels in the next section.
\vspace{0.3cm}
\section{Lifetimes}
\label{section-lifetimes}
The lifetime of a state $a$ is calculated as
$\tau_{a} = (\sum_{b \leq a} A_{ab})^{-1}$. The E1, E2, and M1 transition rates $A_{ab}$
are given by \cite{johnson:02}:
\begin{eqnarray}
A_{ab}^{E1} &=& \frac{2.02613 \times 10^{18}}{\lambda^{3}} \frac{S_{E1}}{2j_a+1}~s^{-1},\\
A_{ab}^{E2} &=& \frac{1.11995 \times 10^{18}}{\lambda^{5}} \frac{S_{E2}}{2j_a+1}~s^{-1},\\
A_{ab}^{M1} &=& \frac{2.69735 \times 10^{13}}{\lambda^{3}} \frac{S_{M1}}{2j_a+1}~s^{-1},
\end{eqnarray}
respectively, where $\lambda$ is the wavelength of the transition in \AA~and $S$ is the line strength.
In this work, we calculated the lifetimes of the $6p_{1/2}$, $6p_{3/2}$, $5d_{3/2}$, and $5d_{5/2}$
levels in Ba$^{+}$. The results are compared with experimental and other theoretical values in Table ~\ref{tab5}.
Since the $6p$ levels are above $5d$ levels in Ba$^+$, we also needed to calculate
the SD all-order reduced matrix elements for the $6p-5d$ E1 transitions, and our results (in atomic units) are
$d(6p_{1/2}-5d_{3/2})=3.034$, $d(6p_{3/2}-5d_{3/2})=1.325$, and $d(6p_{3/2}-5d_{5/2})=4.080$. These values include
contributions from the higher partial waves (0.6\%) and 0.1\%-0.2\% Breit correction. The correlation corrections
to these transitions are similar to the ones for the $6s-6p_j$ transitions. Therefore, similar (on the order of 0.5\%)
accuracy is expected for these matrix elements.
The $6s-6p_j$ transitions contribute about 73\% to the respective $\sum_{b \leq a} A_{ab}$ totals for the $6p_j$ lifetimes.
Based on our evaluation of the uncertainty in these matrix elements discussed in Section~\ref{section-e1}, we expect
present $6p$ lifetime values to be accurate to about 1\%.
Our results are in excellent agreement with other recent theoretical \cite{dzuba:01,das:02}
and experimental \cite{Andra:1976,Kuske:1978,pinnington:95} values.
The calculation of Refs.~\cite{Guet:1991,Guet:2007} is a third-order MBPT calculation that omits
higher-order corrections included in the present calculation, slightly different values are expected.
\begin{table}
\caption{\label{tab5} Lifetimes of the $6p_j$ and $5d_j$ states in Ba$^{+}$; comparison with experiment and other theory. The lifetimes
of the $6p_j$ states are given in ns, and the lifetimes of the $5d_j$ states are given in s. }
\begin{ruledtabular}
\begin{tabular}{lcccc}
\multicolumn{1}{c}{}&
\multicolumn{1}{c}{$\tau_{6p_{1/2}}$~(ns)}&
\multicolumn{1}{c}{$\tau_{6p_{3/2}}$~(ns)} &
\multicolumn{1}{c}{$\tau_{5d_{3/2}}$~(s)} &
\multicolumn{1}{c}{$\tau_{5d_{5/2}}$~(s)} \\
\hline
Present & 7.83 & 6.27 & 81.5(1.2)& 30.3(4)\\
Theory~\protect\cite{Guet:1991,Guet:2007}& 7.99 & 6.39 & 83.7 & 30.8 \\
Theory~\protect\cite{dzuba:01} & 7.89 & 6.30 & 81.5 & 30.3 \\
Theory~\protect\cite{das:02} & 7.92 & 6.31 & 81.4 & 36.5 \\
Theory~\protect\cite{Sahoo:2006} & & & 80.1(7) & 29.9(3) \\
Theory~\protect\cite{Gurell:2007} & & & 82.0 & 31.6 \\
Expt.~\protect\cite{Andra:1976} & & 6.312(16) & & \\
Expt.~\protect\cite{Kuske:1978} & 7.92(8) & & & \\
Expt.~\protect\cite{pinnington:95} &7.90(10)&6.32(10)& & \\
Expt.~\protect\cite{Schneider:1979} & & & 17.5(4) & \\
Expt.~\protect\cite{Knab-Bernardini:1992} & & & 48(6) & \\
Expt.~\protect\cite{yu:97} & & & 79.8(4.6)& \\
Expt.~\protect\cite{Gurell:2007} & & & 89.4(15.6)& 32.0(4.6) \\
Expt.~\protect\cite{Plumelle:1980} & & & & 47.0(16) \\
Expt.~\protect\cite{Nagourney:86} & & & & 32.0(5) \\
Expt.~\protect\cite{madej:90} & & & & 34.5(3.5) \\
\end{tabular}
\end{ruledtabular}
\end{table}
Only one transition contributes to the $5d_{3/2}$ lifetime: $6s-5p_{3/2}$ E2 transition (the
contribution of the $6s-5d_{3/2}$ M1 transition is negligible). In the case of the $5d_{5/2}$ lifetime,
M1 $5d_{5/2}-5d_{3/2}$ transition has to be included as pointed out in \cite{dzuba:01,Guet:2007,Sahoo:2006}.
Our SD all-order value for this transition (in a.u.) is 1.5493. The correlation correction
contribution is very small, and the lowest order gives essentially the same value, 1.5489.
The M1 transition contributes 17\% to the $\sum_{b \leq a} A_{ab}$ total for the $5d_{5/2}$ level.
We compare our final results for the $5d_{3/2}$ and $5d_{5/2}$ lifetimes with experimental \cite{Schneider:1979,Knab-Bernardini:1992,yu:97,Gurell:2007,Plumelle:1980,Nagourney:86,madej:90}
and other theoretical \cite{Guet:1991,Guet:2007,dzuba:01,das:02,Sahoo:2006,Gurell:2007}
values in Table~\ref{tab5}. We note that calculation \cite{das:02} omitted $5d_{5/2}-5d_{3/2}$ M1 contribution to the $5d_{5/2}$
lifetime leading to a higher value, as noted in later work \cite{Sahoo:2006}.
Our results are in agreement with other theoretical calculations, most recent values
from \cite{Gurell:2007} measured in a beam-laser experiment performed at the ion storage ring CRYRING, as
well as experimental values from \cite{yu:97,Nagourney:86,madej:90}.
\section{Conclusion}
In conclusion, we carried out the relativistic all-order calculations of
Ba$^+$ $6s-np_{j}$ ($n=6-9$), $6p_{1/2}-5d_{3/2}$, $6p_{3/2}-5d_{5/2}$,
and $6p_{3/2}-5d_{5/2}$ electric-dipole matrix elements; $6s-5d_{3/2}$, $6s-5d_{5/2}$,
$6s-6d_{3/2}$, $6s-6d_{5/2}$, $6s-7d_{3/2}$, and $6s-7d_{5/2}$
electric-quadrupole matrix elements; and $5d_{5/2}-5d_{3/2}$ magnetic-dipole matrix element.
These values are used to evaluate lifetimes of the $6p_{1/2}$, $6p_{3/2}$, $5d_{3/2}$, and $5d_{5/2}$
levels as well as dipole and quadrupole ground state polarizabilities.
Extensive comparison with other theoretical and experimental values is carried out.
The present values of the dipole polarizability and $6p_{j}$ lifetimes are in excellent agreement with experimental values.
We estimated the uncertainty of our theoretical values for these properties to be on the order of 1\%.
Our recommended value of the quadrupole ground state polarizability $\alpha_{E2} = 4182(34)a^5_0 $ is
in agreement with the most recent experimental work \cite{Snow:2007}.
Our recommended values for the $5d_j$ lifetimes $\tau_{5d_{3/2}}=81.5(1.2)$~s and
$\tau_{5d_{5/2}}=30.3(4)$~s are in agreement with other theoretical calculations, most recent values
from \cite{Gurell:2007} measured in a beam-laser experiment performed at the ion storage ring CRYRING, as
well as experimental values from \cite{yu:97,Nagourney:86,madej:90}.
\section*{Acknowledgements}
We thank Steve Lundeen and Erica Snow for many useful discussions.
This work was supported in part by the National Science Foundation Grant
No.\ PHY-04-57078.
| {
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} | 1,107 |
Davidiella cephalanthae är en svampart som först beskrevs av Sawada, och fick sitt nu gällande namn av Aptroot 2006. Davidiella cephalanthae ingår i släktet Davidiella och familjen Davidiellaceae. Inga underarter finns listade i Catalogue of Life.
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\section{\label{sec1}Introduction}
Several types of time-dependent oscillators have been studied along the past years. Examples are: (i) the harmonic oscillator\cite{1}; (ii) the pseudo-harmonic oscillator\cite{1, 2}; (iii) the parametric oscillator\cite{3}; and (iv) the inverted harmonic oscillator\cite{4}. Recently, another interesting class of time-dependent oscillators, named log-periodic oscillators, was studied\cite{5}.
In Ref. \onlinecite{5}, \"{O}zeren\cite{5} considered the time evolution of five different one-dimensional classical oscillators. The coherent states for each system were constructed by using the SU(1, 1) algebra and their time evolution was investigated.
In this work, we use the Lewis and Riesenfeld\cite{6} (LR) invariant method and a unitary transformation to obtain the exact Schr\"{o}dinger wave function for three out of the five log-periodic-type oscillators investigated by \"{O}zeren\cite{5}, namely: (i) $m(t)=m_0\frac{t}{t_0}$ and $k(t)=k_0\frac{t_0}{t}$; (ii) $m(t)=m_0$ and $k(t)=k_0\left(\frac{t_0}{t}\right)^{2}$; (iii) $m(t)=m_0\left(\frac{t}{t_0}\right)^{2}$ and $k(t)=k_0$. In all three cases $\omega(t)=\omega_0\frac{t_0}{t}$.
The wave functions $\psi_n (q,t)$ for the time dependent harmonic oscillator ($H(t)=\frac{p^2}{2m(t)}+\frac{1}{2}m(t)\omega^2(t)q^2$) obtained in Ref. \onlinecite{1} are written in terms of $\rho$, a c-number quantity satisfying the generalized Milne-Pinney equation ($\ddot{\rho}+\gamma(t)\dot{\rho}+\omega^2(t)\rho=\frac{1}{m^2(t)\rho^3}$), whose solution can be found following the procedure reported in Refs. \onlinecite{7, 8}.
Here we write the solution of the Milne-Pinney equation for each system to obtain the exact wave functions for the oscillators. This paper is outlined as follows. In Sec. \ref{sec2} we briefly review the LR invariant method for the time-dependent harmonic oscillator. In Sec. \ref{sec3} we obtain the wave functions for the oscillators considered, and calculate the correlations between position and momentum and the uncertainty product. For oscillator (i) we construct the coherent states, while for oscillators (ii) and (iii) we construct the squeezed states. The analysis of the phase diagram for the three oscillators is also presented. Finally, some concluding remarks are added in Sec. \ref{sec4}.
\section{\label{sec2}THE LEWIS AND RIESENFELD INVARIANT METHOD - WAVE FUNCTIONS FOR A TIME-DEPENDENT HARMONIC OSCILLATOR}
Consider a time-dependent harmonic oscillator described by the Hamiltonian
\begin{equation}\label{1}H(t)=\frac{p^2}{2m(t)}+\frac{1}{2}m(t)\omega^2(t)q^2,\end{equation}whose mass ($m(t)$) and angular frequency ($\omega(t)$) depend on time explicitly, and the variables $q$ and $p$ are canonical coordinates with $[q,p]=i\hbar$. From Eq. (\ref{1}), we obtain the equation of motion
\begin{equation}\label{2}\ddot{q}+\gamma(t)\dot{q}+\omega^2(t)q=0,\end{equation}where
\begin{equation}\label{3}\gamma(t)=\frac{d}{dt}\ln{m(t)}.\end{equation}
It is well known that an invariant for Eq. (\ref{1}) is given by\cite{6}
\begin{equation}\label{4}I=\frac{1}{2}\left[ \left(\frac{q}{\rho}\right)^2+(\rho p-m\dot{\rho}q)^{2} \right] \end{equation}where $q(t)$ satisfies Eq. (\ref{2}) and $\rho(t)$ satisfies the generalized Milne-Pinney\cite{7} equation
\begin{equation}\label{5}\ddot{\rho}+\gamma(t)\dot{\rho}+\omega^2(t)\rho=\frac{1}{m^2(t)\rho^3}.\end{equation}
The invariant $I(t)$ satisfies the equation
\begin{equation}\label{6}\frac{dI}{dt}=\frac{\partial I}{\partial t}+\frac{1}{i\hbar}[I, H]=0\end{equation}and can be considered hermitian if we choose only the real solutions of Eq. (\ref{5}). Its eigenfunctions, $\phi_n(q,t)$, are assumed to form a complete orthonormal set with time-independent discrete eigenvalues, $\lambda_n$. Thus
\begin{equation}\label{7}I\phi_n(q, t)=\lambda_n\phi_n(q, t),\end{equation}with $\langle\phi_n,\phi_{n^\prime}\rangle=\delta_{nn^\prime}$.
Consider the Schr\"odinger equation (SE)
\begin{equation}\label{8}i\hbar\frac{\partial\psi(q, t)}{\partial t}=H(t)\psi(q, t),\end{equation}where $H(t)$ is given by Eq. (\ref{1}) with $p=-i\hbar\frac{\partial}{\partial q}$. Lewis and Riesenfeld\cite{6} showed that the solutions $\psi_n (q,t)$ of the SE (see Eq. (\ref{8})) are related to the functions $\phi_n (q,t)$ by
\begin{equation}\label{9}\psi_n(q, t)=e^{i\theta_n(t)}\phi_n(q, t),\end{equation}where the phase functions $\theta_n(t)$ satisfy the equation
\begin{equation}\label{10}\hbar\frac{d\theta_n(t)}{dt}=\langle\phi_n(q, t)|\left[i\hbar\frac{\partial}{\partial t}-H(t)\right]|\phi_n(q, t)\rangle.\end{equation}
The general solution of the SE (Eq. (\ref{8})) may be written as
\begin{equation}\label{11}\psi_n(q, t)=\sum_nc_ne^{i\theta_n(t)}\phi_n(q, t),\end{equation}where $c_n$ are time-independent coefficients.
Next, consider the unitary transformation
\begin{equation}\label{12}\phi_n^\prime(q, t)=\mathcal{U}\phi_n(q, t)\end{equation}where
\begin{equation}\label{13}\mathcal{U}=\exp{\left\{-i\left[\frac{m(t)\dot{\rho}}{2\hbar\rho}\right]q^2\right\}}.\end{equation}Under this transformation and defining $\sigma=q/\rho$, Eq. (\ref{7}) now reads
\begin{align}\label{14}I^\prime\varphi_n(\sigma)&=\left[-\left(\frac{\hbar^2}{2}\right)\frac{\partial^2}{\partial\sigma^2}+\left(\frac{\sigma^2}{2}\right)\right]\varphi_n(\sigma)\nonumber\\
&=\lambda_n\varphi_n(\sigma),\quad\lambda_n=\left(n+\frac{1}{2}\right)\hbar,\end{align}where $I^\prime=\mathcal{U}I\mathcal{U}^\dagger$ and $\frac{\varphi_n(\sigma)}{\rho^{1/2}}=\phi_n^\prime$. The factor $\rho^{1/2}$ warrants the normalization condition
\begin{equation}\label{15}\int{\phi_n^{\prime *}(q, t)\phi_n^\prime(q, t)}dq=\int{\varphi_n^{*}(q, t)\varphi_n(q, t)}d\sigma=1.\end{equation}
The solution of Eq. (\ref{14}) corresponds to that of the time-independent harmonic oscillator with $\lambda_n=(n+\frac{1}{2})\hbar$ . Then, by using Eqs. (\ref{12}), (\ref{13}) and (\ref{15}) we obtain
\begin{equation}\label{16}\phi_n(q,t)=\left[\frac{1}{\pi^{1/2}\hbar^{1/2}n!2^n\rho}\right]^{1/2}\exp{\left[\frac{im(t)}{2\hbar}\left(\frac{\dot{\rho}}{\rho}+\frac{i}{m(t)\rho^2}\right)q^2\right]}\times H_n\left[\left(\frac{1}{\hbar}\right)^{1/2}\frac{q}{\rho}\right],\end{equation}here $H_n$ is the usual Hermite polynomial of order $n$.
Applying $\mathcal{U}$ to the right-hand side of Eq. (\ref{10}) and after some algebra, we obtain
\begin{equation}\label{17}\theta_n(t)=-\left(n+\frac{1}{2}\right)\int_{t_0}^t{\frac{1}{m(t^\prime)\rho^2(t^\prime)}}dt^\prime.\end{equation}
Finally, using Eqs. (\ref{9}) and (\ref{16}) the exact solution of the SE for the time-dependent harmonic oscillator reads
\begin{equation}\label{18}\psi_n(q,t)=e^{i\theta_n(t)}\left[\frac{1}{\pi^{1/2}\hbar^{1/2}n!2^n\rho}\right]^{1/2}\exp{\left[\frac{im(t)}{2\hbar}\left(\frac{\dot{\rho}}{\rho}+\frac{i}{m(t)\rho^2}\right)q^2\right]}\times H_n\left[\left(\frac{1}{\hbar}\right)^{1/2}\frac{q}{\rho}\right].\end{equation}
\section{\label{sec3}WAVE FUNCTIONS OF TIME-DEPENDENT LOG-PERIODIC OSCILLATORS}
In Ref. \onlinecite{5}, \"{O}zeren considered five different variations of $m(t)$ and $k(t)$, namely: (i) $m(t)=m_0$ and $k(t)=k_0\left(\frac{t_0}{t}\right)^2$; (ii) $m(t)=m_0 \left(\frac{t}{t_0}\right)^2$ and $k(t)=k_0$; (iii) $m(t)=m_0\left(\frac{t}{t_0}\right)^\alpha$ and $k(t)=k_0 \left(\frac{t_0}{t}\right)^{(\alpha+2)}$; (iv) $m(t)=m_0 \left(\frac{t}{t_0}\right)$ and $k(t)=k_0 \left(\frac{t_0}{t}\right)$; and (v) $m(t)=m_0 \left(\frac{t}{t_0}\right)^\alpha$ and $k(t)=k_0\left(\frac{t}{t_0}\right)^\alpha$. Here we consider only three ((i), (ii) and (iv)) out of the five oscillators studied by \"{O}zeren\cite{5}, for which $\omega(t)=\sqrt{\frac{k(t)}{m(t)}}=\omega_0\frac{t_0}{t}$.
\subsection{$\bm{m(t)=m_0\frac{t}{t_0}}$ and $\bm{k(t)=k_0\frac{t_0}{t}}$}
In this case Eqs. (\ref{2}) and (\ref{5}) read
\begin{equation}\label{19}\ddot{q}+\frac{1}{t}\dot{q}+\frac{\omega_0^2t_0^2}{t^2}q=0\end{equation}and
\begin{equation}\label{20}\ddot{\rho}+\frac{1}{t}\dot{\rho}+\frac{\omega_0^2t_0^2}{t^2}\rho=\frac{t_0^2}{m_0^2}\frac{1}{t^2\rho^3},\end{equation}respectively.
Following the procedure described in Ref.\onlinecite{7}, we find $\rho=c=\frac{1}{\sqrt{m_0\omega_0}}$ . From Eqs. (\ref{17}) and (\ref{18}) we have
\begin{equation}\label{21}\psi_n(q,t)=e^{-i\left(n+\frac{1}{2}\right)\omega_0t_0\ln{\frac{t}{t_0}}}\left[\frac{m_0\omega_0}{\pi\hbar(n!)^22^{2n} }\right]^{1/4}\exp{\left[-\frac{m_0^2\omega_0^2q^2}{2\hbar}\right]}\times H_n \left[\left(\frac{m_0\omega_0}{\hbar}\right)^{1/2}q \right],\end{equation}which, except for the phase factor, is similar to the well-known wave function for the time-independent harmonic oscillator.
The coherent states for the time-dependent harmonic oscillator (Eq.(\ref{1})) are constructed as follows\cite{9}. Consider the time-dependent creation ($a^\dagger(t)$) and annihilation ($a(t)$) operators defined as
\begin{equation}\label{22}a^\dagger(t)=\left(\frac{1}{2\hbar}\right)^{1/2}\left[ \left(\frac{q}{\rho}\right)-i(\rho p-m\dot{\rho}q)\right]\end{equation}
\begin{equation}\label{23}a(t)=\left(\frac{1}{2\hbar}\right)^{1/2}\left[ \left(\frac{q}{\rho}\right)+i(\rho p-m\dot{\rho}q)\right],\end{equation}where $[a^\dagger(t),a(t)]=1$. In terms of $a(t)$ and $a^\dagger(t)$ the invariant $I$ (see Eq. (\ref{4})) can be written as
\begin{equation}\label{24}I=\hbar\left(a^\dagger(t)a(t)+\frac{1}{2}\right).\end{equation}
Let $|n,t\rangle$ be the eigenstates of $I$. Therefore the following relations hold
\begin{equation}\label{25}a(t)=\sqrt{n}|n-1,t\rangle\end{equation}
\begin{equation}\label{26}a^\dagger(t)=\sqrt{n+1}|n+1,t\rangle,\end{equation}
\begin{equation}\label{27}I|n,t\rangle=\hbar\left(n+\frac{1}{2}\right)|n,t\rangle.\end{equation}
Since the coherent states for $I$ can be easily constructed, the coherent states for the time-dependent harmonic oscillator are straightforwardly obtained:
\begin{equation}\label{28}|\alpha,t\rangle=e^{-|\alpha|^2/2} \sum_{n=0}^\infty{\frac{\alpha^n}{(n!)^{1/2}}e^{i\theta_n(t)} |n,t\rangle},\end{equation}where $\theta_n$ is given by Eq. (\ref{17}), and the complex number $\alpha(t)$ satisfies the eigenvalue equation
\begin{equation}\label{29}a(t)|\alpha,t\rangle=\alpha(t)|\alpha,t\rangle,\end{equation}with
\begin{equation}\label{30}\alpha(t)=\alpha(t_0)e^{2i\theta_0(t)}\end{equation}and
\begin{equation}\label{31}\theta_0(t)=-\frac{1}{2}\int_{t_0}^t\frac{dt^\prime}{m(t^\prime)\rho^2(t^\prime)}.\end{equation}
The fluctuations in $q$ ($\Delta q$) and $p$ ($\Delta p$) and the uncertainty product ($\Delta q\Delta p$) in the coherent state $|\alpha,t\rangle$ , read
\begin{equation}\label{32}\Delta q_\alpha=\sqrt{\langle q^2\rangle_\alpha-\langle q\rangle_\alpha^2}=\sqrt{\frac{\hbar}{2}}\rho,\end{equation}
\begin{equation}\label{33}\Delta p_\alpha=\sqrt{\langle p^2\rangle_\alpha-\langle p\rangle_\alpha^2}=\sqrt{\frac{\hbar}{2}}\frac{1}{\rho}\left(1+m^2\dot{\rho}^2\rho^2\right)^{1/2}\end{equation}and
\begin{equation}\label{34}\Delta q_\alpha\Delta p_\alpha=\frac{\hbar}{2}\left(1+m^2\dot{\rho}^2\rho^2\right)^{1/2},\end{equation}respectively.
If $m(t)\dot{\rho}\rho\neq0$, $\Delta q_\alpha\Delta p_\alpha$ is not minimum, indicating that the coherent states $|\alpha,t\rangle$ are not minimum-uncertainty (coherent) states. In fact, the states $|\alpha,t\rangle$ for the time-dependent harmonic oscillator are equivalent to the well-known squeezed states, as pointed out in Refs. \onlinecite{10, 11}.
For $\rho=c$, $\dot{\rho}=0$ and $\Delta q_\alpha\Delta p_\alpha=\frac{\hbar}{2}$ , indicating that the states $|\alpha,t\rangle$ are ``true" coherent states. This is an interesting result since the minimum uncertainty product is assumed to be satisfied only for time-independent harmonic oscillator, unless the solution of Eq.(\ref{5}) is a constant\cite{1}.
Next, let us analyze the time behavior of $\langle q\rangle_\alpha$, $\langle p\rangle_\alpha$ and the phase diagram $\langle q\rangle_\alpha\times\langle p\rangle_\alpha$. By setting $\alpha(t_0 )=u+iv$ and using Eqs. (\ref{22}) and (\ref{23}), we find
\begin{equation}\label{35}\langle q\rangle_\alpha=\sqrt{2\hbar}\left[u\cos{\left(\theta_0(t)\right)}-v\sin{\left(\theta_0(t)\right)}\right]\end{equation}
\begin{equation}\label{36}\langle p\rangle_{\alpha}=\sqrt{2\hbar}\left[\left(\frac{v}{\rho}+um\dot{\rho}\right)\cos{\left(\theta_0(t)\right)}+\left(\frac{u}{\rho}-vm\dot{\rho}\right)\sin{\left(\theta_0(t)\right)}\right].\end{equation}
The constants $u$ and $v$ are determined from the initial conditions $\langle q(t_0)\rangle_\alpha=q_0$ and $\langle p(t_0)\rangle_\alpha=p_0=m(t_0)v_0$. For $q_0=1$ and $v_0=0$, we find
\begin{equation}\label{37}\langle q\rangle_\alpha=\cos{\left(t_0\omega_0\ln{\frac{t}{t_0}}\right)},\end{equation}
\begin{equation}\label{38}\langle p\rangle_\alpha=-m_0\omega_0\sin{\left(t_0\omega_0\ln{\frac{t}{t_0}}\right)}.\end{equation}
Figures \ref{fig1}(a) and (b) show the time dependent behavior of $\langle q\rangle_\alpha$ and $\langle p\rangle_\alpha$, respectively. In all plots we used $t_0=1.0$, $\omega_0=10.0$ and $m_0=1.0$. From Fig. \ref{1}(a) we observe that the system oscillates forth and back between the classical turning points with an increasing period and constant amplitude. The phase diagram is shown in Fig. \ref{1}(c). Even though that this system is dissipative (total energy $E=\frac{1}{2t}$), it behaves like the usual time-independent harmonic oscillator ($E=$ constant). This can be seen from the relation $A=\sqrt{\frac{2E}{k}}$, where $A$ is the amplitude of motion. Since $k\propto\frac{1}{t}$ and $E\propto\frac{1}{t}$, $A$ is a constant. As $t$ increases, the frequency $\omega(t)$ decreases ($\propto\frac{1}{t}$) while the period increases ($\propto\frac{t}{lnt}$) leading to the ``exact" log periodic behavior shown in Fig. \ref{1}(a).
\begin{figure}[t]
\centering
\includegraphics{104354_0_figure_202875_l9kqn9.eps}
\caption{Plots of (a) $\langle q\rangle_\alpha$, (b) $\langle p\rangle_\alpha$, and (c) the phase diagram $\langle p\rangle_\alpha$ vs $\langle q\rangle_\alpha$. In the plots we used $t_0=1.0$, $q_0=1.0$, $v_0=0.0$, $\omega_0=10.0$ and $m_0=1.0$.}
\label{fig1}
\end{figure}
Pedrosa et al\cite{12} have combined linear invariants and the LR method to obtain the exact wave function for a particle trapped by oscillating fields, which were written in terms of Mathieu functions. They calculated $\Delta q\Delta p$ and the quantum correlation between $q$ and $p$, defined by $C_{1,1}=\frac{1}{2}\langle\left(qp+pq\right)\rangle-\langle q\rangle\langle p\rangle$\cite{13}. They are related through the equation
\begin{equation}\label{39}\Delta q\Delta p=\frac{\hbar}{2}\sqrt{1+\left(\frac{2}{\hbar}C_{1,1}\right)^2},\end{equation}which shows that $\Delta q\Delta p$ is minimum whenever $C_{1,1}=0$, as it happens for $C_{1,1}$ calculated in the coherent state $|\alpha,t\rangle$, i.e., $(C_{1,1})_\alpha=0$. The fact that $C_{1,1}=0$ does not mean that $q$ and $p$ are uncorrelated. In order to verify the correlation between $q$ and $p$ one may study the function $C_{n,m}=\frac{1}{2}\langle\left(q^np^m+p^mq^n\right)\rangle-\langle q^n\rangle\langle p^m\rangle$. For the coherent state $|\alpha,t\rangle$, we find that $(C_{2,2} )_\alpha=-\frac{\hbar^2}{2}$ , indicating that $q$ and $p$ even assumed as ``classical" quantities are correlated.
The uncertainty product and correlations in the state $\psi_n$ (Eq.(\ref{21})) are more easily calculated using the relation $|\psi_n(q,t)\rangle=e^{i\theta_n(t)}|n,t\rangle$. They are given by
\begin{equation}\label{40}\Delta q_{\psi_n}\Delta p_{\psi_n}=\left(n+\frac{1}{2}\right)\hbar,\end{equation}
\begin{equation}\label{41}\left(C_{1,1}\right)_{\psi_n}=0\end{equation}and
\begin{equation}\label{42}\left(C_{2,2}\right)=-\left(n^2+n+\frac{1}{2}\right)\hbar^2.\end{equation}
We noticed that Eq. (\ref{39}) is satisfied for $n=0$. In this case $\psi_0$ given by
\begin{equation}\label{43}\psi_0(q,t)=e^{-i\frac{t_0\omega_0}{2}\ln{\left(\frac{t}{t_0}\right)}}\left(\frac{m_0\omega_0}{\pi\hbar}\right)^{1/4}\exp{\left[-\frac{m_0^2\omega_0^2 q^2}{2\hbar}\right]}\end{equation}is the coordinate representation of the coherent state\cite{14}.
\subsection{$\bm{m(t)=m_0}$ and $\bm{k(t)=k_0\left(\frac{t_0}{t}\right)^2}$}
In this case Eqs. (\ref{2}) and (\ref{5}) are, respectively, given by
\begin{equation}\label{44}\ddot{q}+\omega_0^2\frac{t_0^2}{t^2}q=0\end{equation}and
\begin{equation}\label{45}\ddot{\rho}+\omega_0^2\frac{t_0^2}{t^2}\rho=\frac{1}{m_0\rho^3}.\end{equation}
Following the procedure described in Refs. \onlinecite{7, 8}, we find $\rho=\sqrt{\frac{2}{m_0}}\frac{\sqrt{t}}{\left(4\omega_0^2 t_0^2-1\right)^{1/4}}$ and from Eqs. (\ref{17}) and (\ref{18}) we have
\begin{align}\label{46}\psi_n(q,t)=&e^{-\frac{i}{2}\left(n+\frac{1}{2}\right)\left(4\omega_0^2t_0^2-1\right)^{1/2}\ln\left(\frac{t}{t_0}\right)}\left[\frac{m_0\left(4\omega_0^2t_0^2-1\right)^{1/2}}{\pi\hbar(n!)^22^{2n+1}}\right]^{1/4}\times\frac{1}{t^{1/4}}\nonumber\\ \times&\exp{\left\{\frac{m_0}{4\hbar t}\left[i-\left(4\omega_0^2t_0^2-1\right)^{1/2}\right]q^2\right\}}
\times H_n\left[\left(\frac{m_0}{2\hbar}\right)^{1/2}\frac{\left(4\omega_0^2t_0^2-1\right)^{1/4}}{\sqrt{t}}q\right]\end{align}
The values of $\Delta q\Delta p$ and $C_{1,1}$ in the state $|\psi_n (q,t)\rangle$, are given by, respectively
\begin{equation}\label{47}\Delta q_{\psi_n}\Delta p_{\psi_n}=\frac{2\omega_0 t_0}{\left(4\omega_0^2t_0^2-1\right)^{1/2}}\left(n+\frac{1}{2}\right)\hbar,\end{equation}and
\begin{equation}\label{48}\left(C_{1,1}\right)_{\psi_n}=-\left(\frac{1}{4\omega_0^2t_0^2-1}\right)^{1/2}\left(n+\frac{1}{2}\right)\hbar.\end{equation}
For $n=0$, $\Delta q_{\psi_0}\Delta p_{\psi_0}=\frac{\omega_0 t_0}{\left(4\omega_0^2t_0^2-1\right)^{1/2}}\hbar$, and the state
\begin{align}\label{49}\psi_0(q,t)=e^{-\frac{i}{4}\left(4\omega_0^2t_0^2-1\right)^{1/2}\ln{\left(\frac{t}{t_0}\right)}}&\left[\frac{m_0\left(4\omega_0^2t_0^2-1\right)^{1/2}}{2\pi\hbar}\right]^{1/4}\times\frac{1}{t^{1/4}}\nonumber\\
\times&\exp{\left\{\frac{m_0}{4\hbar t}\left[i-\left(4\omega_0^2t_0^2-1\right)^{1/2}q^2\right]\right\}}\end{align}corresponds to the coordinate representation of the squeezed state\cite{14}.
For the sake of comparison with case \textbf{A}, let us discuss the behavior of the classical variables $q$ and $p$ on time, as well as the phase diagram. By solving Eq.(\ref{44}), the solutions for $q$ and $p$ satisfying the initial conditions $q_0=1$ and $v_0=0$ are, respectively, given by
\begin{equation}\label{50}q(t)=\sqrt{\frac{t}{t_0}}\left[\cos{\left(\frac{\left(4\omega_0^2t_0^2-1\right)^{1/2}}{2}\ln{\frac{t}{t_0}}\right)}-\frac{1}{\left(4\omega_0^2t_0^2-1\right)^{1/2}}\sin{\left(\frac{\left(4\omega_0^2t_0^2-1\right)^{1/2}}{2}\ln{\frac{t}{t_0}}\right)}\right]\end{equation}
And
\begin{equation}\label{51}p(t)=-\sqrt{\frac{t_0}{t}}\frac{2m_0\omega_0^2t_0}{\left(4\omega_0^2t_0^2-1\right)^{1/2}}\sin{\left(\frac{\left(4\omega_0^2t_0^2-1\right)^{1/2}}{2}\ln{\frac{t}{t_0}}\right)}.\end{equation}
Figures \ref{fig2}(a) and \ref{fig2}(b) show the variation of $q$ and $p$ on time, respectively. Unlike from case \textbf{A} where the system oscillates back and forth between the turning points with constant amplitude, here $q$ increases while that $p$ decreases as time increases. Figure \ref{fig2}(c) shows the phase diagram for this oscillator. Initially at rest, the particle is speeded up, and then slowed down, indicating that the system is also dissipative. Since $E\propto 1/t$ and $k\propto 1/t^2$, the amplitude $A$ increases as $A\propto \sqrt{t}$. Due to the presence of the factor $\sqrt{t}$ in Eq. (\ref{50}) this oscillator exhibits a pseudo-log-periodic behavior.
\begin{figure}[t]
\centering
\includegraphics{104354_0_figure_202876_ldkqnd.eps}
\caption{Plots of (a) q, (b) p, and (c) the phase diagram $p$ vs $q$. In the plots we used $t_0=1.0$, $q_0=1.0$, $v_0=0.0$, $\omega_0=10.0$ and $m_0=1.0$.}
\label{fig2}
\end{figure}
\subsection{$\bm{m(t)=m_0\left(\frac{t}{t_0}\right)^2}$ and $\bm{k(t)=k_0$}}
Now Eqs. (\ref{2}) and (\ref{5}) are, respectively, given by
\begin{equation}\label{52}\ddot{q}+\frac{2}{t}\dot{q}+\omega_0^2\frac{t_0^2}{t^2}q=0\end{equation}and
\begin{equation}\label{53}\ddot{\rho}+\frac{2}{t}\dot{\rho}+\omega_0^2\frac{t_0^2}{t^2}\rho=\frac{t_0^4}{m_0^2}\frac{1}{t^4\rho^3}.\end{equation}
Again, by following the procedure described in Refs. \onlinecite{7, 8}, we find $\rho=\sqrt{\frac{2}{m_0}}\frac{t_0}{\left(4\omega_0^2 t_0^2-1\right)^{1/4}}\frac{1}{\sqrt{t}}$ and from Eqs. (\ref{17}) and (\ref{18}) we obtain
\begin{align}\label{54}\psi_n(q,t)=&e^{-\frac{i}{2}\left(n+\frac{1}{2}\right)\frac{\left(4\omega_0^2t_0^2-1\right)^{1/2}}{t_0}\ln\left(\frac{t}{t_0}\right)}\left[\frac{m_0\left(4\omega_0^2t_0^2-1\right)^{1/2}}{\pi\hbar t_0(n!)^22^{2n+1}}\right]^{1/4}\times t^{1/4}\nonumber\\ \times&\exp{\left\{\frac{m_0t}{4\hbar }\left[i-\frac{\left(4\omega_0^2t_0^2-1\right)^{1/2}}{t_0}\right]q^2\right\}}
\times H_n\left[\left(\frac{m_0}{2\hbar t_0}\right)^{1/2}\left(4\omega_0^2t_0^2-1\right)^{1/4}\sqrt{t}q\right]\end{align}
The values of $\Delta q\Delta p$ and $C_{1,1}$ in the state $|\psi_n (q,t)\rangle$, are respectively given by
\begin{equation}\label{55}\Delta q_{\psi_n}\Delta p_{\psi_n}=\frac{2\omega_0 t_0}{\left(4\omega_0^2t_0^2-1\right)^{1/2}}\left(n+\frac{1}{2}\right)\hbar\end{equation}and
\begin{equation}\label{56}\left(C_{1,1}\right)_{\psi_n}=-\left(\frac{1}{4\omega_0^2t_0^2-1}\right)^{1/2}\left(n+\frac{1}{2}\right)\hbar.\end{equation}
The expression of the coordinate representation of the squeezed state for $n=0$, reads
\begin{align}\label{57}\psi_0(q,t)=e^{-\frac{i}{4}\frac{\left(4\omega_0^2t_0^2-1\right)^{1/2}}{t_0}\ln{\left(\frac{t}{t_0}\right)}}&\left[\frac{m_0\left(4\omega_0^2t_0^2-1\right)^{1/2}}{2\pi\hbar t_0^2}\right]^{1/4}\times\frac{1}{t^{1/4}}\nonumber\\
\times&\exp{\left\{\frac{m_0t}{4\hbar }\left[i-\frac{\left(4\omega_0^2t_0^2-1\right)^{1/2}}{t_0}\right]q^2\right\}}.\end{align}
By solving Eq. (\ref{52}) and using the initial conditions $q_0=1$ and $v_0=0$, we find
\begin{equation}\label{58}q(t)=\sqrt{\frac{t_0}{t}}\left[\cos{\left(\frac{\left(4\omega_0^2t_0^2-1\right)^{1/2}}{2}\ln{\frac{t}{t_0}}\right)}-\frac{1}{\left(4\omega_0^2t_0^2-1\right)^{1/2}}\sin{\left(\frac{\left(4\omega_0^2t_0^2-1\right)^{1/2}}{2}\ln{\frac{t}{t_0}}\right)}\right]\end{equation}and
\begin{equation}\label{59}p(t)=\sqrt{\frac{t}{t_0}}\frac{2m_0\omega_0^2t_0}{\left(4\omega_0^2t_0^2-1\right)^{1/2}}\sin{\left(\frac{\left(4\omega_0^2t_0^2-1\right)^{1/2}}{2}\ln{\frac{t}{t_0}}\right)}.\end{equation}
\begin{figure}[t]
\centering
\includegraphics{104354_0_figure_202877_lckqnc.eps}
\caption{Plots of (a) $q$, (b) $p$, and (c) the phase diagram $p$ vs $q$. In the plots we used $t_0=1.0$, $q_0=1.0$, $v_0=0.0$, $\omega_0=10.0$ and $m_0=1.0$.}
\label{fig3}
\end{figure}
The behavior of the classical $q$ and $p$ variables on time is displayed in Figs. \ref{fig3} (a) and (b), respectively. Despite the oscillating ($\cos{\left(\frac{\sqrt{3}}{2}\ln{t}\right)}$ and ($\sin{\left(\frac{\sqrt{3}}{2}\ln{t}\right)}$) terms, $q$ and $p$ exhibit an opposite behavior compared to those calculated in case \textbf{B}. Here $q$ decreases while $p$ increases as $t$ increases. Figure \ref{fig3}(c) shows the phase diagram. We observe that the amplitude $A$ decreases as $\propto 1/\sqrt{t}$. This oscillator also exhibits a pseudo-log-periodic character.
\section{\label{sec4}CONCLUDING REMARKS}
In this paper we have used a unitary transformation and the LR invariant method in the Schr\"{o}dinger picture to obtain the exact wave functions for oscillators exhibiting either log-periodic or pseudo-log-periodic behavior. It is well-known that a challenge in obtaining the exact solution (see Eq. (\ref{18})) for the SE (Eq.(\ref{8})) for $H(t)$ given in Eq. (\ref{1}), is the solution of the auxiliary equation for the c-number quantity $\rho$ (see Eq. (\ref{5})). Here we find $\rho$ for each case using the methods described in Refs. \onlinecite{7, 8}.
For case $A$, we find $\rho=c$ and, as a consequence, the solution for $\psi_n (q,t)$ (see Eq. (\ref{21})) except for the phase factor ($e^{-i\left(n+\frac{1}{2}\right)t_0\omega_0\ln{\frac{t}{t_0}}}$), is similar to the well-known wave function for the time-independent harmonic oscillator of mass $m_0$ and frequency $\omega_0$. In Ref. \onlinecite{1}, we observed that when $m(t)=m_0$, $\omega(t)=\omega_0$, and $\rho(t)=\left(\frac{1}{m_0\omega_0}\right)^{1/2}$, which is a particular solution of Eq. (\ref{5}), the wave function obtained also corresponds to that of the time-independent harmonic oscillator. In case \textbf{A} even with $m\propto t$ and $\omega\propto \frac{1}{t}$, we obtain the same solution for $\rho$ ($\rho= c$), indicating that this oscillator behaves as the harmonic oscillator with $m$ and $\omega$ constant.
We have constructed the ``true" coherent states, $|\alpha,t\rangle$, whose coordinate representation is given by Eq. (\ref{43}). We verified that Eq. (\ref{39}) holds for $|\alpha,t\rangle$. We calculated the quantum fluctuations in the coordinate and momentum as well as the quantum correlations between the coordinate and momentum in the state $\psi_n (q,t)$.
We analyzed the time behavior of $\langle q\rangle_\alpha$ and $\langle p\rangle_\alpha$ , as well as the phase diagram $\langle q\rangle_\alpha\times\langle p\rangle_\alpha$ (see Fig.\ref{fig1}(a)-(c)). We observed that $\langle q\rangle_\alpha$ and $\langle p\rangle_\alpha$ exhibits the exact log-periodic behavior, and that the phase diagram indicates, as already anticipated, that the log-periodic oscillator behaves as the classical harmonic oscillator with $m(t)=m_0$ and $\omega(t)=\omega_0$.
For cases \textbf{B} and \textbf{C}, we obtained the wave functions given by Eqs. (\ref{46}) and (\ref{54}), respectively.
| {
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{"url":"https:\/\/apboardsolutions.in\/inter-1st-year-maths-1a-products-of-vectors-solutions-ex-5a\/","text":"Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(a)\n\nPracticing the Intermediate 1st Year Maths 1A Textbook Solutions Inter 1st Year Maths 1A Products of Vectors Solutions Exercise 5(a) will help students to clear their doubts quickly.\n\nIntermediate 1st Year Maths 1A Products of Vectors Solutions Exercise 5(a)\n\nI.\n\nQuestion 1.\nFind the angle between the vectors $$\\bar{i}+2 \\bar{j}+3 \\bar{k}$$ and $$3 \\bar{i}-\\bar{j}+2 \\bar{k}$$.\nSolution:\nLet $$\\overline{\\mathrm{a}}=\\overline{\\mathrm{i}}+2 \\overline{\\mathrm{j}}+3 \\overline{\\mathrm{k}}$$ and $$\\overline{\\mathrm{b}}=3 \\overline{\\mathrm{i}}-\\overline{\\mathrm{j}}+2 \\overline{\\mathrm{k}}$$ and \u2018\u03b8\u2019 be the angle between them (i.e.,) $$(\\bar{a}, \\bar{b})$$ = \u03b8\n\nQuestion 2.\nIf the vectors $$\\mathbf{2} \\overline{\\mathbf{i}}+\\lambda \\overline{\\mathbf{j}}-\\overline{\\mathbf{k}}$$ and $$4 \\bar{i}-2 \\bar{j}+2 \\bar{k}$$ are perpendicular to each other, then find \u03bb.\nSolution:\n\nQuestion 3.\nFor what values of \u03bb, the vectors $$\\overline{\\mathbf{i}}-\\lambda \\overline{\\mathbf{j}}+2 \\overline{\\mathbf{k}}$$ and $$8 \\overline{\\mathbf{i}}+6 \\overline{\\mathbf{j}}-\\overline{\\mathbf{k}}$$ are at right angles?\nSolution:\n\nQuestion 4.\n$$\\overline{\\mathbf{a}}=2 \\overline{\\mathbf{i}}-\\overline{\\mathbf{j}}+\\overline{\\mathbf{k}}, \\overline{\\mathbf{b}}=\\overline{\\mathbf{i}}-3 \\overline{\\mathbf{j}}-5 \\overline{\\mathbf{k}}$$. Find the vector C such that $$\\overline{\\mathbf{a}}$$, $$\\overline{\\mathbf{b}}$$ and $$\\overline{\\mathbf{c}}$$ form the sides of a triangle.\nSolution:\n\nQuestion 5.\nFind the angle between the planes $$\\bar{r} \\cdot(2 \\bar{i}-\\bar{j}+2 \\bar{k})=3$$ and $$\\overline{\\mathrm{r}} \\cdot(3 \\overline{\\mathrm{i}}+6 \\bar{j}+\\bar{k})=4$$.\nSolution:\n\nQuestion 6.\nLet $$\\overline{\\mathbf{e}}_{1}$$ and $$\\overline{\\mathbf{e}}_{2}$$ be unit vectors makingangle \u03b8. If $$\\frac{1}{2}\\left|\\bar{e}_{1}-\\bar{e}_{2}\\right|=\\sin \\lambda \\theta$$, then find \u03bb.\nSolution:\n\nQuestion 7.\nLet $$\\overline{\\mathbf{a}}=\\overline{\\mathbf{i}}+\\overline{\\mathbf{j}}+\\overline{\\mathbf{k}}$$ and $$\\overline{\\mathbf{b}}=\\mathbf{2} \\overline{\\mathbf{i}}+3 \\overline{\\mathbf{j}}+\\overline{\\mathbf{k}}$$. Find\n(i) The projection vector of $$\\overline{\\mathbf{b}}$$ on $$\\overline{\\mathbf{a}}$$ and its magnitude.\n(ii) The vector components of $$\\overline{\\mathbf{b}}$$ in the direction of a and perpendicular to $$\\overline{\\mathbf{a}}$$.\nSolution:\n\nQuestion 8.\nFind the equation of the plane through the point (3, -2, 1) and perpendicular to the vector (4, 7, -4).\nSolution:\n\nQuestion 9.\nIf $$\\overline{\\mathbf{a}}=2 \\bar{i}+2 \\bar{j}-3 \\bar{k}$$; $$\\overline{\\mathbf{b}}=3 \\overline{\\mathbf{i}}-\\overline{\\mathbf{j}}+2 \\overline{\\mathbf{k}}$$, then find the angle between $$2 \\overline{\\mathbf{a}}+\\overline{\\mathbf{b}}$$ and $$\\bar{a}+2 \\bar{b}$$.\nSolution:\n\nII.\n\nQuestion 1.\nFind unit vector parallel to the XOY- plane and perpendicular to the vector $$4 \\bar{i}-3 \\bar{j}+\\bar{k}$$.\nSolution:\nAny vector parallel to XOY-plane will be of the form $$p \\bar{i}+q \\bar{j}$$\n\u2234 The vector parallel to the XOY-plane and perpendicular to the vector $$4 \\bar{i}-3 \\bar{j}+\\bar{k}$$ is $$3 \\bar{i}+4 \\bar{j}$$\nIts magnitude = $$|3 \\bar{i}+\\overline{4 j}|=\\sqrt{9+16}=5$$\n\u2234 Unit vector parallel to the XOY-plane and perpendicular to the vector $$4 \\bar{i}-3 \\bar{j}+\\bar{k}$$ is $$\\pm \\frac{(3 \\overline{\\mathrm{i}}+4 \\overline{\\mathrm{j}})}{5}$$\n\nQuestion 2.\nIf $$\\overline{\\mathbf{a}}+\\overline{\\mathbf{b}}+\\overline{\\mathrm{c}}=0,|\\overline{\\mathbf{a}}|=3,|\\overline{\\mathbf{b}}|=5$$ and $$|\\bar{c}|=7$$, then find the angle between $$\\overline{\\mathbf{a}}$$ and $$\\overline{\\mathbf{b}}$$.\nSolution:\n\nQuestion 3.\nIf $$|\\overline{\\mathbf{a}}|$$ = 2, $$|\\overline{\\mathbf{b}}|$$ = 3 and $$|\\overline{\\mathbf{c}}|$$ = 4 and each of $$\\overline{\\mathbf{a}}, \\overline{\\mathbf{b}}, \\overline{\\mathbf{c}}$$ is perpendicular to the sum of the other two vectors, then find the magnitude of $$\\overline{\\mathbf{a}}+\\overline{\\mathbf{b}}+\\overline{\\mathbf{c}}$$.\nSolution:\n\nQuestion 4.\nFind the equation of the plane passing through the point $$\\overline{\\mathbf{a}}=\\mathbf{2} \\overline{\\mathbf{i}}+3 \\bar{j}-\\overline{\\mathbf{k}}$$ and perpendicular to the vector $$3 \\bar{i}-2 \\bar{j}-2 \\bar{k}$$ and the distance of this plane from the origin.\nSolution:\nEquation of the plane passing through the point $$\\overline{\\mathbf{a}}=\\mathbf{2} \\overline{\\mathbf{i}}+3 \\bar{j}-\\overline{\\mathbf{k}}$$ and perpendicular to the vector $$\\bar{n}=3 \\bar{i}-2 \\bar{j}-2 \\bar{k}$$ is\n\nQuestion 5.\n$$\\overline{\\mathbf{a}}, \\overline{\\mathbf{b}}, \\overline{\\mathbf{c}}$$ and $$\\overline{\\mathbf{d}}$$ are the position vectors of four coplanar points such that $$(\\mathbf{a}-\\overline{\\mathbf{d}}) \\cdot(\\bar{b}-\\bar{c})=(\\bar{b}-\\bar{d}) \\cdot(\\bar{c}-\\bar{a})=0$$. Show that the point $$\\bar{d}$$ represents the orthocentre of the triangle with $$\\bar{a}$$, $$\\bar{b}$$ and $$\\bar{c}$$ as its vertices.\nSolution:\nPosition vectors of A, B, C, D are $$\\bar{a}$$, $$\\bar{b}$$, $$\\bar{c}$$ and $$\\bar{d}$$ respectively.\n\n\u21d2 BD is perpendicular to AC\n\u2234 BD is another altitude of \u2206ABC.\nAltitudes AD and BD intersect at D.\n\u2234 D is the orthocentre of \u2206ABC.\n\nIII.\n\nQuestion 1.\nShow that the points (5, -1, 1), (7, -4, 7), (1, -6, 10) and (-1, -3, 4) are the vertices of a rhombus.\nSolution:\nLet A(5, -1, 1), B(7, -4, 7), C(1, -6, 10) and D(-1, -3, 4) are the given points.\n\n\u2235 AB = BC = CD = DA = 7 units\nAC \u2260 BD\n\u2234 A, B, C, D points are the vertices of a rhombus.\n\nQuestion 2.\nLet $$\\bar{a}=4 \\bar{i}+5 \\bar{j}-\\bar{k}, \\quad \\bar{b}=\\bar{i}-4 \\bar{j}+5 \\bar{k}$$ and $$\\overline{\\mathbf{c}}=\\mathbf{3} \\overline{\\mathbf{i}}+\\overline{\\mathbf{j}}-\\overline{\\mathbf{k}}$$. Find the vector which is perpendicular to both $$\\overline{\\mathbf{a}}$$ and $$\\overline{\\mathbf{b}}$$ and whose magnitude is twenty one times the magnitude of $$\\overline{\\mathbf{c}}$$.\nSolution:\n\nQuestion 3.\nG is the centroid of \u0394ABC and a, b, c are the lengths of the sides BC, CA and AB respectively prove that a2 + b2 + c2 = 3 (OA2 + OB2 + OC2) \u2013 9(OG)2 where O is any point.\nSolution:\nGiven that $$\\overline{\\mathrm{BC}}=\\overline{\\mathrm{a}}, \\overline{\\mathrm{CA}}=\\overline{\\mathrm{b}}, \\overline{\\mathrm{AB}}=\\overline{\\mathrm{c}}$$\nLet \u2018O\u2019 be the origin and let $$\\overline{\\mathrm{OA}}=\\overline{\\mathrm{p}}, \\overline{\\mathrm{OB}}=\\overline{\\mathrm{q}} \\text { and } \\overline{\\mathrm{OC}}=\\overline{\\mathrm{r}}$$\nThen P.V. of the centroid of \u0394ABC is\n\nFrom (1) and (2)\n\u2234 a2 + b2 + c2 = 3(OA2 + OB2 + OC2) \u2013 9(OG)2.\n\nQuestion 4.\nA line makes angles \u03b81, \u03b82, \u03b83, and \u03b84 with the diagonals of a cube. 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\section{Introduction}
\label{sec:intro}
Whether in CPU code, in GPU kernels, or in the inter-node communication --
performance bottlenecks in High-Performance Computing (HPC) applications may be
hidden in any part of the program.
There have been many attempts to ease the development of HPC applications and
to indicate and to avoid as many bottlenecks as possible.
One example is the asynchronous many-task (AMT) system HPX~\cite{Kaiser2020},
which aims to solve some of the more common problems. It makes it easier to
overlap communication and computation, to employ both CPUs and GPUs, and to
avoid overhead for fine-grained parallelism using lightweight threading.
Even with HPX, of course, it is still perfectly possible to introduce
performance bottlenecks into one's application. Being able to collect
performance measurements to profile an HPX application remains important. For
AMT systems such as HPX, it is beneficial to have a profiling tool that
understands the task-based nature of the runtime system -- for example, call stacks
themselves are less useful: A system thread may jump back and
forth between various HPX tasks as they are yielded and resumed, and the call
stack itself may be dozens of levels of runtime functions that have no
particular interest to the application developer.
Collecting these measurements in a profiling run is challenging, as one not
only needs a profiling framework that supports both CPU and GPU code as well as
the distributed collection of profiling data across many compute nodes. One
also needs to keep any overheads introduced by the profiling itself to an
absolute minimum. Otherwise, it would not only distort the collected
measurements, but also make large, distributed runs infeasible, rendering us
unable to detect potential performance bottlenecks that only appear at scale.
HPX is integrated with a performance measurement library, APEX (Automatic
Performance for Exascale), which was designed specifically for the HPX
runtime and the above requirements.
In previous work, APEX was used together with the HPX performance counters to
collect performance measurements for Octo-Tiger \cite{marcello2021octo}, an
astrophysics application which is built upon HPX and contains optimized kernels
for both CPUs and GPUs~\cite{pfander2018accel,daiss2021beyond}. Octo-Tiger is
capable of running the same kernels on the CPUs and GPUs simultaneously (on
different data) depending on the load. In that work, profiling data of
Octo-Tiger was gathered with APEX in distributed CPU-only runs where the energy
usage, the idle rate, and overhead of the HPX AGAS (Active Global Address
Space) was analyzed~\cite{diehl2021performance}.
Furthermore, combined CPU-GPU profiling runs have been performed on Summit,
analyzing the performance behavior of Octo-Tiger's new CUDA\textsuperscript{\texttrademark} hydro module in
different configurations for simple benchmark
scenarios~\cite{diehl2021octotigers}.
All these previous efforts inspire this new work, collecting performance
measurements on both CPU and GPU during a full-scale production-scenario run on
Piz Daint (Cray\textsuperscript{\texttrademark} XC50 with one 12-core Intel\textsuperscript{\textregistered} Xeon\textsuperscript{\textregistered} E5-2690 + one NVIDIA\textsuperscript{\textregistered} Tesla\textsuperscript{\textregistered} P100 per node)\cite{pizdaintweb}
and Summit (IBM\textsuperscript{\textregistered} AC922 with two 22-core Power9\textsuperscript{\texttrademark} + six NVIDIA\textsuperscript{\textregistered} Tesla\textsuperscript{\textregistered} V100 per node)\cite{summitweb}.
The purpose of this work is thus twofold: First, we collect data that we can actually use to improve Octo-Tiger by identifying potential bottlenecks.
To do so, we collect measurements running the production scenario for 40 time-steps both on Summit and Piz Daint, using 48 compute nodes on Piz Daint and 8 compute nodes on Summit (resulting in 48 GPUs in either case).
With those measurements, we can investigate the specific parts of Octo-Tiger on two distinct architectures in a distributed CPU/GPU run, providing insights into the different runtime behavior of Octo-Tiger regarding GPU-performance, CPU performance, and communication. For example, the communication seems to have a larger overhead on Piz Daint.
Second, we are showcasing the feasibility of APEX for large-scale runs,
collecting combined CPU and GPU performance measurements, showing that the
overhead introduced by the profiling itself is small enough to handle large
production-scale scenarios.
To this end, we are running the scenario both with and without APEX profiling
enabled for a scaling run on each machine, to see both the overhead on a few
compute nodes, and the runtime behavior when scaling to more nodes (with up to 2000 compute nodes on Piz Daint).
Furthermore, we repeat these overhead measurements on Piz Daint for a CPU-only run to determine the performance impact of the NVIDIA\textsuperscript{\textregistered} CUDA\textsuperscript{\texttrademark} Profiling Tools Interface (CUPTI) which is used to collect the GPU performance data.
To highlight the need for low profiling overhead, we can look at the short
runtimes for each time-step of Octo-Tiger: During the test runs on Summit, we
gather \num{5} GB of data during all eight runs. For each run \num{40} time-steps were executed. Each time-step takes
about \num{0.720325}\si{\second} on \num{128} Summit nodes and
consists of 6 iterations of the gravity solver, 3 iterations of the hydro
solver, and all required communication. On Piz Daint we collected \num{55} GB of data in total. Here, each time step on \num{2000} nodes took \num{0.7872625}\si{\second}. As time steps are serial in nature, these iterations are our smallest parallel
unit. As the time-steps only run for a few hundred milliseconds overall,
overheads introduced by the profiling can be very noticeable even if they only
take a few milliseconds in total as well.
The remainder of this work is structured as follows: In Section \ref{sec:related}, we take a
brief look at profiling solutions in other AMT frameworks.
We then introduce the scientific scenario which we are simulating with
Octo-Tiger in Section \ref{sec:application}. This is the scenario we also used
to collect the profiling data by running it for 40 timesteps with APEX enabled.
Section \ref{sec:framework} in turn introduces Octo-Tiger itself, as well as the utilized software stack.
In Section \ref{sec:performance}, we show and discuss the collection of the
performance measurements for Octo-Tiger. In Section~\ref{sec:performance:improvements}, we test communication optimization and analyze the performance improvement. Finally, we conclude the paper in Section \ref{sec:conclusion}.
\section{Related work}
\label{sec:related}
For the related work, we focus on AMTs with distributed capabilities which are:
Legion~\cite{bauer2012legion}, Charm++~\cite{kale1993charm++},
Chapel~\cite{chamberlain2007parallel}, and UPC++~\cite{bachan2017upc++}. For a
more detailed review, we refer to~\cite{thoman2018taxonomy}.
Legion~\cite{bauer2012legion} provides \texttt{Legion Prof} for combined CPU and GPU
profiling which is compiled into all builds. Enabling the profiler produces log
files which can be viewed using the profiler.
Charm++~\cite{kale1993charm++} provides \texttt{Charm
debug}~\cite{debugger04} and the \texttt{Projections}
framework~\cite{TauICPP09} for performance analysis and visualization.
Chapel~\cite{chamberlain2007parallel} provides
\texttt{ChplBlamer}~\cite{zhang2018chplblamer} for profiling. UPC++ seems not
to have some dedicated tool for profiling, and any profiler supporting C++ is
recommended in their documentation.
Like HPX, nearly all of these runtimes provide a specialized tool that has been
designed to deal with the particular challenges of AMTs in general, and the
needs of the runtime system in particular. APEX is a specialized tool in the case
of HPX, and provides similar measurement and analysis abilities of the
above tools, including flat profiling, tracing, sampling, taskgraphs/trees,
and concurrency graphs. In addition, APEX provides support for several
programming models/abstractions with or without HPX, including CUDA\textsuperscript{\texttrademark}, HIP,
OpenMP, OpenACC, Kokkos, POSIX threads, and C++ threads.
APEX does not provide analysis tools directly, but rather uses commonly
accepted formats and targets both HPC performance analysis tools
(ParaProf\cite{bell2003paraprof}, Vampir\cite{knupfer2008vampir},
Perfetto\cite{perfettoweb}) and standard data analysis tools (Python,
Graphviz\cite{graphvizweb}).
\section{Scientific application}
\label{sec:application}
Stellar mergers are mysterious phenomena that pack a broad range of physical processes into a small volume and a fleeting time duration. With the proliferation of deep wide-field, time-domain surveys, we have been catching on camera a vastly increased number of outbursts, many of which have been interpreted as stellar mergers. The best case, so far, of an observed merger is V1309 Sco, a contact binary identified using a recent survey database, the Optical Gravitational Lensing Experiment~\cite{Udalski1992}. Fortunately, not only the merger itself was observed, but archival data from other observing programs enabled the reconstruction of the light curve years before the merger. During the merger itself, the system brightness increased by 4 magnitudes, with a peak luminosity in the red visible light~\cite{Tylenda2011}. This complete record of observations has led to term V1309 Sco the "Rosetta Stone" of mergers. Previous attempts to model this merger included
semi-analytical calculations \cite{pejcha2014} and hydrodynamic simulations (e.g.,~\cite{nandez2014}).
However, the hydrodynamic simulations fail to adequately resolve the atmosphere, the rapid transition between the optically thick merger fluid and the optically thin, nearly empty space, surroundings of the simulated stellar material. To overcome this barrier, computational scientists intend to use the adaptive mesh-refinement hydrodynamics code Octo-Tiger. Using Octo-Tiger's dynamic mesh refinement, the simulations are able to resolve the atmosphere at a higher resolution than ever before. Simulation of the V1309 merger in high resolution provide greater insight into the nature of the mass flow and the consequential angular momentum losses.
In this paper, we analyze the performance of Octo-Tiger to identify potential bottlenecks in the combined CPU and GPU long-term production runs, where the atmosphere is maximally resolved. Analyzing the performance is essential at this stage since this model will serve as the necessary baseline for extending Octo-Tiger to include radiation transport to the V1309 model, as well as to other binary merger models.
\section{Software framework}
\label{sec:framework}
\subsection{C++ standard library for parallelism and concurrency}
HPX is the C++ standard library for parallelism and concurrency~\cite{Kaiser2020} and one of the distributed asynchronous many-task runtime systems) AMT. Other notable AMTs with distributed capabilities are: Uintah~\cite{germain2000uintah},
Chapel~\cite{chamberlain2007parallel}, Charm++~\cite{kale1993charm++}, Legion~\cite{bauer2012legion}, and PaRSEC~\cite{bosilca2013parsec}. However, according to~\cite{thoman2018taxonomy} HPX is one with a higher technology
readiness level. HPX's API is fully conforming with the recent C++ standard~\cite{cxx14_standard,cxx17_standard,cxx20_standard} which is the major difference to the other mentioned AMTs. For more details about HPX, we refer to~\cite{hpx_pgas_2014,Kaiser:2015:HPL:2832241.2832244,Heller2016,Kaiser2020}. In this work, HPX has two purposes: \textit{1)} the coordination of the synchronous execution of a multitude of heterogeneous tasks (both on CPUs and GPUs), thus managing local and distributed parallelism while observing all necessary data dependencies and \textit{2)} as the parallelization infrastructure for executing CUDA\textsuperscript{\texttrademark}-kernels on the GPUs via the asynchronous HPX backend.
\subsection{APEX}
APEX~\cite{huck2015autonomic} is a performance measurement library for
distributed, asynchronous multitasking systems. It provides lightweight
measurements without perturbing high concurrency through synchronous and
asynchronous interfaces.
To support performance measurement in systems that employ user-level threading,
APEX uses a dependency chain in addition to the call stack to produce traces
and task dependency graphs.
The synchronous APEX instrumentation application programming interface (API)
can be used to add instrumentation to a runtime, library or application, and
includes support for timers and counters.
For measuring kernels on NVIDIA\textsuperscript{\textregistered} accelerated platforms, APEX is integrated with
the NVIDIA\textsuperscript{\textregistered} CUDA\textsuperscript{\texttrademark} Profiling Tools Interface~\cite{cuptiweb} that provides CUDA\textsuperscript{\texttrademark} host
callback and device activity measurements.
Similarly, on AMD accelerated platforms, APEX is integrated with
the Roctracer library~\cite{roctracerweb} providing HIP host
callback and device activity measurements.
In addition to timer measurements, the hardware and operating system are
monitored through an asynchronous measurement that involves the periodic or
on-demand interrogation of the operating system, hardware states, or runtime
states (e.g., CPU use, resident set size, memory ``high water mark'').
The NVIDIA\textsuperscript{\textregistered} Management Library interface~\cite{nvmlweb} provides periodic CUDA\textsuperscript{\texttrademark}
device monitoring to APEX, and the ROCm SMI API provides periodic HIP device
monitoring.
APEX has been extended to capture additional timers and counters related to
CUDA\textsuperscript{\texttrademark} device-to-device memory transfers, as well as tracking memory consumption
on both device and host when requested with the \lstinline{CUDAMalloc*}/\lstinline{cudaFree*} or
\lstinline{hipMalloc*}/\lstinline{hipFree*} API calls~\cite{wei2021memory}.
APEX supports tracing in both the OTF2 and Google Trace Events formats, but
for comparing results between platforms, profile data can be easier to work with.
To complement the profile data which collapses the time axis, APEX also captures task
and counter scatter plot data, indicating on the $x$ axis when the task started
or the counter was captured, and the $y$ axis contains the duration of the task
or the value of the counter. The tasks are sampled using a user-specified
fraction (default 1\%) whereas the counters are sampled at every value capture.
This data collection allows the application developer to capture a time
sequence of data without the file system overhead of a full event trace.
\subsection{Octo-Tiger}
\label{sec:framework-octotiger}
Octo-Tiger is a 3D adaptive mesh refinement (AMR) hydrodynamics finite volume code with Newtonian gravity designed specifically for the study of interacting stellar binaries~\cite{octotiger_apcs_2016}, \cite{marcello2021octo}. The AMR grid rotates with the initial orbital frequency of the binary, which reduces inaccuracies due to numerical viscosity. The gravitational potential and force are computed using a modified version of the Fast Multipole Method that eliminates the gravitational field as a source of angular momentum conservation violation \cite{octotiger_fmm}. This enables Octo-Tiger to conserve energy and linear momentum to machine precision in the rotating frame. The astrophysical fluid is modeled using the inviscid Euler equations, which are solved with a finite volume central scheme~\cite{kurganov2000}.
The computational domain is based on a properly nested three-dimensional octree structure. Each node in the structure is an $N^3$ Cartesian sub-grid (usually, $N=8$), and may be further refined into eight child nodes, each containing its own $N^3$ sub-grid with twice the resolution of the parent.
In~\cite{marcello2021octo}, an improved hydro solver for Octo-Tiger that includes a full three-dimensional reconstruction technique has been introduced.
The performance of this improved hydro solver (after porting it to GPUs) on Summit's ORNL has been tested as well as its accuracy~\cite{diehl2021octotigers}, showing a good GPU speedup (the exact amount depending on the chosen sub-grid size), and a greater accuracy in maintaining a rotating star in equilibrium than the old hydro solver.
It has been fully benchmarked demonstrating superior angular momentum conservation and extreme scalability properties allowing it to compute larger problems in a shorter wall-clock time~\cite{marcello2021octo}.
A convergence study in a real production run of a white dwarf merger has been also performed~\cite{diehl2021performance}.
Octo-Tiger's CUDA\textsuperscript{\texttrademark} implementation is worth elaborating on in a bit more detail here to understand the later performance results in Section~\ref{sec:performance}, in particular the short runtimes of the GPU kernels.
Usually, each of the compute kernels in the solver (both hydro solver and gravity solver) operates on a single sub-grid at a time with one CPU core. Multi-core usage is achieved by launching many of those methods on different sub-grids concurrently as HPX tasks.
In a GPU run, an issue arises where a single sub-grid does not provide enough work to utilize the entire GPU. However, the scheduler can simply launch multiple kernels on different sub-grids concurrently.
They are launched in different CUDA\textsuperscript{\texttrademark} streams, which are drawn from a pre-allocated pool to avoid the overhead of stream creation. Usually, we use $128$ streams per GPU.
As managing multiple CUDA\textsuperscript{\texttrademark} streams in each of the CPU threads might become unwieldy, the code uses an HPX-CUDA\textsuperscript{\texttrademark} integration. With it, HPX futures can be obtained from CUDA\textsuperscript{\texttrademark} kernel launches, allowing HPX to treat them as any other HPX task. Using this strategy, HPX can simply launch a CUDA\textsuperscript{\texttrademark} kernel, define subsequent post-processing tasks to be run once it is done, and then suspend the current HPX task that starts the kernel. The current CPU thread can then work on another task, potentially launching another GPU kernel in a different stream.
This implementation scheme has the implication that HPX runs many GPU kernels and CPU tasks concurrently, allowing the application to interleave GPU kernels, CPU tasks, CPU-GPU memory transfers, and inter-node communication seamlessly. Each of the tasks, both on CPU and GPU, has a rather short runtime as a result of only dealing with a single sub-grid at a time. For example, the individual GPU kernel execution time ranges from $100$ microseconds (less for smaller auxiliary kernels) to about $2$ milliseconds.
\section{Performance measurements}
\label{sec:performance}
Octo-Tiger's dependencies are listed in Table~\ref{tab:octo:dependencies} in the Artifact Description of the appendix. On both systems, we used Octo-Tiger's git hash (\textit{4c38f3bf}) as the baseline. However, we had to do some minor changes with regard to compilation, which do not affect the astrophysics kernels. Therefore, we used a slightly diverged git hash (\textit{b091fd26}) on Piz Daint and git hash (\textit{fd7faf5e}) on Summit. To quantify the overhead $o(n)$ introduced by APEX in percent, we define the metric
\begin{align}
o(n) = \frac{comp\_apex(n)}{comp\_time\_no\_apex(n)} \cdot 100 - 100\,,
\label{eq:overhead}
\end{align}
where $n$ is the number of nodes, $comp\_time\_no\_apex(n)$ without APEX, and $comp\_apex(n)$ the computation time with APEX. Note that we only measure the time for the actual computation and not the IO.
\subsection{Distributed scaling}
The largest V1309 scenario (\num{18} million cells) fitting on four CSCS's Piz Daint nodes was chosen for this scaling test. Note that the largest scenario for a single node was too small to scale out up to \num{2000} nodes with \num{2000} NVIDIA\textsuperscript{\textregistered} P\num{100} GPUs and \num{24000} CPU cores. However, due to the large amount of memory per node, the same scenario fits on one of ORNL's Summit node. On both machines, we executed the same runs with the APEX + CUDA\textsuperscript{\texttrademark} profiling on and off to investigate the overhead introduced by the profiling. Figure~\ref{fig:distributed:scaling} shows the cells processed per second for increasing amount of nodes. On Piz Daint the scaling was done using Octo-Tiger's new hydro solver~\cite{diehl2021octotigers} using a three-dimensional reconstruction scheme and scaling results using the old solver for a different scenario are shown here~\cite{daiss2019piz}. The \textcolor{plot1}{blue} lines show the scaling on Piz Daint. For both configurations, the scenario scales up to \num{2000} nodes. However, for \num{1400} Piz Daint nodes the problem size gets too small, and the scaling starts to flatten out. The overhead $o(n)$ in Equation~\ref{eq:overhead} for Piz Daint is shown in Figure~\ref{fig:overhead:daint}. The overhead is the largest on \num{4} nodes and seems to decline with increasing node counts.
\begin{figure}
\centering
\subfloat[\label{fig:distributed:scaling}]{
\begin{tikzpicture}
\begin{axis}[xlabel={\# nodes},ylabel={Cells processed per second},title={Distributed scaling (CPU + GPU)},grid,legend pos=north west,xmode=log,log basis x={2},xtick={1,2,4,8,16,32,64,128,256,512,1024,2048},ymode=log,log basis y={2},legend columns=2,legend style={at={(0.5,-0.2)},anchor=north}]
\addplot[thick,mark=*,plot1] table [x expr=\thisrowno{0},y expr={512*(35855+35904)/2*40/\thisrowno{1}}, col sep=comma] {distributed-scaling-no-hyper.csv};
\addplot[thick,mark=square*,plot1] table [x expr=\thisrowno{0},y expr={512*(35855+35904)/2*40/\thisrowno{1}}, col sep=comma] {distributed-scaling-no-apex-no-hyper.csv};
%
\addplot[thick,mark=*,plot2] table [x expr=\thisrowno{0},y expr={512*(35855+35904)/2*40/\thisrowno{1}}, col sep=comma] {distributed-scaling-summit.csv};
\addplot[thick,mark=square*,plot2] table [x expr=\thisrowno{0},y expr={512*(35855+35904)/2*40/\thisrowno{1}}, col sep=comma] {distributed-scaling-summit-no-apex.csv};
\legend{Piz Daint with APEX,Piz Daint,Summit with APEX,Summit};
\end{axis}
\end{tikzpicture}
}
\\
\subfloat[\label{fig:distributed:speedup}]{
\begin{tikzpicture}
\begin{axis}[xlabel={\# nodes},ylabel={Speedup},title={Speedup (CPU + GPU)},grid,legend pos=north west,xmode=log,log basis x={2},xtick={1,2,4,8,16,32,64,128,256,512,1024,2048},ymode=log,log basis y={2},legend columns=2,ymin=1]
\addplot[thick,mark=*,plot1] table [x expr=\thisrowno{0},y expr={1915.98/\thisrowno{1}}, col sep=comma] {distributed-scaling-no-hyper.csv};
\addplot[thick,mark=square*,plot1] table [x expr=\thisrowno{0},y expr={886.772/\thisrowno{1}}, col sep=comma] {distributed-scaling-no-apex-no-hyper.csv};
\addplot[thick,mark=*,plot2] table [x expr=\thisrowno{0},y expr={1111.07/\thisrowno{1}}, col sep=comma] {distributed-scaling-summit.csv};
\addplot[thick,mark=square*,plot2] table [x expr=\thisrowno{0},y expr={1034.04/\thisrowno{1}}, col sep=comma] {distributed-scaling-summit-no-apex.csv};
\addplot[domain=1:2000]{x};
\addplot [domain=1:2000,dashed,shift={(axis cs:4,1)},legend image post style={shift={(0,0)}}]{x};
\legend{Piz Daint with APEX,Piz Daint, Summit with APEX,Summit,Optimal (Summit),Optimal};
\end{axis}
\end{tikzpicture}
}
\caption{Cells processed per second \protect\subref{fig:distributed:scaling} and speedup \protect\subref{fig:distributed:speedup}. On Piz Daint (\textcolor{plot1}{blue line}) we were able to use \num{4}, \num{8}, \num{16}, \num{32}, \num{64}, \num{128}, \num{256}, \num{512}, \num{1024}, \num{1400}, \num{1600}, \num{1800}, and \num{2000} nodes. On Summit (\textcolor{plot2}{violet line}) we used \num{1}, \num{2}, \num{4}, \num{8}, \num{16}, \num{32}, \num{64}, and \num{128} nodes. The speedup was obtained with respect to the smallest amount of nodes the scenario (\num{18} Million cells) fitted on. Note that for the runs with and without APEX a different time on the smallest nodes were used.}
\end{figure}
The \textcolor{plot2}{violet} lines show the scaling on Summit. Here, again, the code scales well up to \num{128} nodes (using \num{768} NVIDIA\textsuperscript{\textregistered} V\num{100} GPUs overall) and the work is not sufficient for more nodes. The overhead $o(n)$ in Equation~\eqref{eq:overhead} on Summit is shown in Figure~\ref{fig:overhead:summit}. The overhead is less prominent on Summit.
We have seen an introduced overhead by the combined profiler on both systems. In a previous study using a different problem and the old version of the hydro module, the overhead introduced by pure APEX CPU profiling within HPX was around \num{1}\si{\percent}~\cite{diehl2021performance}. To verify if this still holds with the new hydro module and the V1309 scenario, we ran the V1309 scenario on Piz Daint with pure APEX CPU profiling. First, we observed scaling for the CPU kernels up to 2000 Piz Daint nodes. Second, we observed see that the difference is again around \num{1}\si{\percent} as in the previous study. It seems that the overhead is mostly introduced by enabling CUPTI measurement in APEX for both systems. However, it seems that the overhead for smaller node counts on Piz Daint is larger, which needs to be investigated. Regardless, CUPTI provides varying levels of detail/support, and APEX should be refactored to enable the minimum amount of useful support by default, and allow the user to request additional details as needed.
\begin{figure}[tb]
\centering
\subfloat[Piz Daint\label{fig:overhead:daint}]{
\begin{tikzpicture}
\begin{axis} [ybar,height=6cm,width=8cm,xlabel=\# nodes,ylabel=$o(n)$ in \%,symbolic x coords={4,8,16,32,64,128,256,512,1024,1400,1800,2000}
,xtick=data,x tick label style={rotate=45,anchor=east},x label style={at={(axis description cs:0.5,-0.1)},anchor=north}]
\addplot [draw=black] coordinates {
(4,116.08)
(8,107.89)
(16,87.88)
(32,86.52)
(64,66.85)
(128,57.62)
(256,47.22)
(512,83.85)
(1024,57.03)
(1400,65.79)
(1800,54.08)
(2000,53.41)
};
\end{axis}
\end{tikzpicture}
}
\\
\subfloat[Summit\label{fig:overhead:summit}]{
\begin{tikzpicture}
\begin{axis} [ybar,height=6cm,width=8cm,xlabel=\# nodes,ylabel=$o(n)$ in \%,symbolic x coords={1,2,4,8,16,32,64,128},xtick=data]
\addplot [draw=black] coordinates {
(1,20.08)
(2,19.81)
(4,22.26)
(8,20.68)
(16,21.19)
(32,19.19)
(64,33.08)
(128,31.85)
};
\end{axis}
\end{tikzpicture}
}
\caption{Overhead $o(n)$ in Equation~\eqref{eq:overhead} for the runs on Piz Daint and Summit, respectively.}
\label{fig:overhead}
\end{figure}
%
\subsection{Profiling of Octo-Tiger}
\subsubsection{Setup}
Because APEX is directly integrated into the HPX runtime, annotating HPX actions (tasks) is simply a matter of providing an annotation for
the task when it is instantiated in the code. Not all actions are annotated (anonymous lambdas launched through \lstinline{hpx::async()} calls, for example), but major operations in the Octo-Tiger code and in the HPX runtime are annotated. When HPX is configured and built with APEX support, all annotations are provided to APEX, and APEX times the life cycle of each task. HPX is designed to yield and resume tasks when resources are unavailable (futures, system calls, other dependencies), so APEX also provides the ability to yield (and resume) timing a task when it is not executing. For GPU kernels, the kernel name is obtained using the instruction pointer address and debug information in the executable provided by the compiler.
For the following Octo-Tiger profiling results, we use $48$ HPX localities (processes).
As we use one HPX locality per GPU, this results in using $48$ compute nodes on Piz Daint and 8 compute nodes on Summit (as each node contains $6$ GPUs here).
This ensures that we use the same number of GPUs on both systems. We are still using the same V1309 scenario as before.
Below is a short description of the tasks and kernels of interest in Octo-Tiger.
The main CPU tasks include \lstinline{node_server::non-refined_step::compute_fluxes}: directs the computation of a single Runge-Kutta substep in the hydro and gravity solvers for a given sub-grid. It directs the GPU to compute hydrodynamic fluxes, corrects these fluxes on coarse-fine boundaries, works with other invocations of the same action to compute the global time-step size, computes hydrodynamic sources, then lastly updates the hydrodynamic variables and directs the GPU to update the gravitational variables;
\lstinline{local_step::execute_step}: directs the execution of an entire time-step for a given sub-grid. This involves multiple calls to \lstinline{node_server::nonrefined_step::compute_fluxes};
\lstinline{diagnostics_actions_type}: performs some measurements on the grid, for example, computing total mass on the grid, total angular momentum, and center of masses of each star (for binaries);
\lstinline{solve_gravity_action_type}: solves for gravity. This is called when an additional gravity solve is needed (other than what \lstinline{node_server::nonrefined_step::compute_fluxes} calls), such as after grid refinement;
\lstinline{check_for_refinement_action_type} flags each sub-grid in need of refinement;
\lstinline{regrid_gather_action_type} gathers information about the sub-grid structure and determines boundaries for the global decomposition;
\lstinline{regrid_scatter_action_type} uses the information computed by \lstinline{regrid_gather_action_type} to redistribute the sub-grids.
The GPU kernels include \lstinline{cuda multipole interactions kernel}: Computes the cell to cell interactions for the gravity solver in refined sub-grids (non-rho is without the angular momentum correction);
\lstinline{cuda p2p interactions kernels}: Computes the cell to cell interactions for the gravity solver in non-refined sub-grids;
Special case: \lstinline{cuda p2m interactions kernel}: Computes the cell to cell interactions for the gravity solver in non-refined sub-grids with refined neighbor sub-grids (non-rho is without the angular momentum correction);
Special case: \lstinline{multipole root}: computes the remaining cell to cell interactions in the root sub-grid;
\lstinline{reconstruct cuda kernel} reconstructs the evolution variables using the PPM method, see \cite{diehl2021octotigers};
\lstinline{flux cuda kernel} the flux method computes the fluxes and the Newtonian quadrature to get the final flux.
\subsubsection{Results / Analysis}
In this section, we analyze the collected profiling results and draw conclusions regarding Octo-Tiger's performance on two different architectures. These insights are important for further optimization on the code. Some of them are straightforward and others are quite puzzling. Let us start with the insights on Summit:
\begin{figure}[tb]
\centering
\includegraphics[width=0.48\textwidth]{figures/paraprof-cpu.png}
\caption{Comparison of top CPU tasks with TAU ParaProf.}
\label{fig:comparison:cpu}
\end{figure}
\begin{figure}[tb]
\centering
\includegraphics[width=0.48\textwidth]{figures/paraprof-gpu.png}
\caption{Comparison of top GPU kernels with TAU ParaProf.}
\label{fig:comparison:gpu}
\end{figure}
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{figures/top_task_scatterplot_comparison_cpu_ross.png}
\caption{Comparison of sampled profiles of CPU tasks on 48 localities between Summit (\textcolor{red}{red}) and Piz Daint (\textcolor{blue}{blue}). The overall runtime on Summit is shorter, but interestingly, some CPU tasks took longer (per call) to execute on Summit than on Piz Daint.}
\label{fig:cpu:scatterplot}
\end{figure*}
\begin{figure*}[tb]
\centering
\includegraphics[width=\textwidth]{figures/top_task_scatterplot_comparison_gpu_ross.png}
\caption{Comparison of sampled GPU kernels and memory transfers on 48 localities between Summit (\textcolor{red}{red}) and Piz Daint (\textcolor{blue}{blue}). Not surprisingly, in all cases, the performance of the V100s on Summit outperformed that of the P100s on Piz Daint. However, not all kernels saw a significant improvement in performance.}
\label{fig:gpu:scatterplot}
\end{figure*}
\begin{enumerate}
\item Figure~\ref{fig:comparison:cpu} shows that on Summit, the CPU tasks are not always faster with respect to Piz Daint. However, the overall computational time is lower due to the benefit of the newer GPUs. Looking at the scatterplot data for sampled timers in Figure~\ref{fig:cpu:scatterplot}, we see that while the average time \textit{per task} on Summit is slower for some tasks (\lstinline{compute_fluxes}, \lstinline{execute_step}), the aggregated mean profile is approximately the same.
\item Figure~\ref{fig:comparison:gpu} shows that on Summit, the GPU tasks are nearly always faster on NVIDIA\textsuperscript{\textregistered} V100 GPUs. This is not surprising, since the GPU kernels are working better on the newer NVIDIA\textsuperscript{\textregistered} V100 than on the older NVIDIA\textsuperscript{\textregistered} P100. We found two exception, both the \lstinline{cuda_multipole_interactions_kernel_no_rho()} and \lstinline{cuda_interactions_kernel__rho()} were faster on the NVIDIA\textsuperscript{\textregistered} P100. These tasks computes the monopole-multipole gravity interactions in the case that the leave nodes have different refinements. More investigation is needed to determine if it is possible to improve this compute kernel for the NVIDIA\textsuperscript{\textregistered} V100.
\end{enumerate}
On Piz Daint we gather the following insights:\\
Unrelated to the application itself, we found some differences within the HPX run time system. HPX provides the function \lstinline{hpx::async} to asynchronously launch functions and lambda functions. Figures~\ref{fig:comparison:cpu} and~\ref{fig:cpu:scatterplot} show that this operation's mean was $4.6\times$ more expensive on Piz Daint as on Summit. Here, the HPX main developers need to investigate this behavior.
Figure~\ref{fig:comparison:cpu} also shows that an another more expensive HPX operation on Piz Daint was \lstinline{schedule_parcel} which mean was nearly $5.6\times$ higher as on Summit. A delay in the \lstinline{schedule_parcel} potentially happens when some \lstinline{hpx::future} is not ready, which \lstinline{schedule_parcel} depends on.
Our primary focus here is distributed combined CPU and GPU profiling. We anticipate future research concentrating on the interpretation of these findings and major optimization. However, we will show one minor optimization.
\section{Performance improvements}
\label{sec:performance:improvements}
\begin{figure*}[tb]
\centering
\includegraphics[width=\textwidth]{figures/top_task_scatterplot_comparison_optimized_ross.png}
\caption{Comparison of sampled CPU and GPU kernels and 48 localities between the original code (\textcolor{blue}{blue}) and the optimized code (\textcolor{orange}{orange}) on Piz Daint.}
\label{fig:scatterplot:optimized}
\end{figure*}
On Piz Daint the CPU tasks \lstinline{async} and \lstinline{schedule_parcle}, see Figure~\ref{fig:cpu:scatterplot}, took longer as on Summit.
We further noticed that this seems to be worse when running scenarios that included the Octo-Tiger hydro solver on Piz Daint when doing more tests.
Consequently, we took a look at the boundary communication within the Hydro module, and a way to reduce the overall number of messages. For sub-grids located on the same HPX locality, we now access the memory of these directly (foregoing HPX actions and temporary communication buffers) to fill the ghost-cells. This requires some more overhead within the \lstinline{node_server::collect_hydro_boundaries::set_hydro_boundary} method (and other associated methods) itself as we need to make sure that the results of those direct neighbors is up-to-date, which is done with local HPX promises/future pairs. This increases the mean time of said communication methods as they need to handle the promises, but reduces the overall HPX action calls and thus reduces the calls to schedule parcel. We gained a noticeable speedup on Daint (reduced total runtime from $400$ to $320$ seconds) and will be become helpful for other machines in the future as well.
This becomes evident in, Figure~\ref{fig:scatterplot:optimized} compares the run on 48 Piz Daint nodes with the optimization enabled (\textcolor{orange}{orange}) and the previous run (\textcolor{blue}{blue}).
This does not address the root issue, which seem to be the longer runtimes for \lstinline{async} and \lstinline{schedule_parcle} on Daint, but it reduces the symptoms considerably and is furthermore a useful optimization for other machines as well.
Unfortunately, the test-bed allocation on Summit ended before we could finish implementing the optimization. Therefore, we could not show results here.
\section{Conclusion}
\label{sec:conclusion}
In this work, we have analyzed the overhead of performing combined CPU and GPU performance measurements with APEX in a large-scale HPX application distributed across up to \num{2000} compute nodes.
We have demonstrated that Octo-Tiger easily scales to that many nodes on Piz Daint with the APEX profiling enabled. Profiling is thus feasible for real-world production-size runs on high-performance systems equipped with GPUs.
However, we encountered a noticeable profiling overhead at scale (\num{52.408539388924}\% on 2000 Piz Daint nodes). This seems to be due to the GPU measurements with CUPTI, as a subsequent CPU-only run exhibited a smaller profiling overhead. Note that the overhead on Piz Daint with very few nodes is about two times higher and requires further investigation.
On Summit, there is a noticeable overhead at scale from profiling, too, but significantly less (\num{31.848570683336}\% on \num{128} Summit nodes). Overall, regarding the APEX profiling overhead, the distributed profiling with both CPU and GPU measurements works, scales up, and is ready to use; yet more investigation is needed to work on minimizing the overhead of the GPU measurements, if possible.
While we focused on evaluating how suitable APEX is for these large-scale, distributed analyses, there are some interesting results regarding Octo-Tiger itself as well.
It is notable that the speedup of the average GPU kernel runtime from a P100 to a newer V100 varies a lot between the kernels. There are many factors that may influence the average kernel runtime difference between the devices: For instance, going to the V100, there is an increase in the number of Streaming Multiprocessors (SM), an increase in L1 cache available per SM, an increase in global memory bandwidth and a slight increase in clock speed. This does not even take into account the fact that we compile for different architectures.
Considering that we use concurrent kernel execution via $128$ CUDA\textsuperscript{\texttrademark} streams per GPU to achieve device utilization (as outlined in Section~\ref{sec:framework-octotiger}), it is unlikely that the increased numbers of SMs in the V100 has a major impact on the average kernel execution time (as those would rather facilitate more concurrent kernel execution). Thus, it seems more likely that the speedup is due to a combination of the other factors, the exact speedup depending on what is currently limiting the kernel. The larger speedups indicate that the kernels benefit from the larger L1 cache available, however, determining the exact cause is subject of future work, especially as the kernels are currently still undergoing changes. However, the results here give us an idea which kernels need more attention during this process, particularly when targeting older architectures, thus helping us to steer our development focus.
Beyond the GPU results, the profiling uncovered that there are some crucial methods that run significantly slower on Piz Daint than on Summit, e.g. \lstinline{schedule_parcel}. We consequently optimized the communication of the hydro solver to alleviate this issue. Using the APEX measurements we could identify the parts of the code which benefited from the optimizations. This shows the need for distributed performance measurements on a production system.
Of course, while we focused on the usability of APEX for these kinds of analyses in this work and fixed some of the issues, the uncovered issues still need to be further investigated and addressed. Consequently, we will examine these remaining issue in future work, hopefully further improving the runtime of future simulations with Octo-Tiger.
A radiation module for Octo-Tiger is currently being implemented by its developers, and it is in the testing phase. Our performance analysis of the current modules will be
crucial to estimate the performance impact of the new module prior to its inclusion.
Including radiation in the simulations of V1309 together with resolving the star atmosphere at a higher resolution than ever before will enable one to self-consistently compute the light curve and directly compare it with the observed one of V1309.
If one is able to accurately reproduce the light curve of the "Rosetta Stone of mergers", it will be possible to reliably simulate the outburst light curves of other mergers.
\section*{Acknowledgment}
\label{sec:acknowledgement}
{\footnotesize This work was supported by a grant from the Swiss National
Supercomputing Centre (CSCS) under project ID s1078. This research used resources of the Oak Ridge Leadership Computing
Facility, which is a DOE Office of Science User Facility supported under Contract DE-AC05-00OR22725.
The APEX work was supported by the Scientific Discovery through Advanced Computing (SciDAC) program funded by U.S. Department of Energy, Office of Science, Advanced Scientific Computing Research (ASCR) under contract DE-SC0021299.}
\section{Supplementary materials}
The build scripts are available on GitHub\footnote{\url{https://github.com/STEllAR-GROUP/OctoTigerBuildChain}} and the input files are available on Zenodo~\cite{marcello_2021_5213015}.
\cleardoublepage
\newpage
\bibliographystyle{./bibliography/IEEEtran}
\section{Introduction}
\label{sec:intro}
Whether in CPU code, in GPU kernels, or in the inter-node communication --
performance bottlenecks in High-Performance Computing (HPC) applications may be
hidden in any part of the program.
There have been many attempts to ease the development of HPC applications and
to indicate and to avoid as many bottlenecks as possible.
One example is the asynchronous many-task (AMT) system HPX~\cite{Kaiser2020},
which aims to solve some of the more common problems. It makes it easier to
overlap communication and computation, to employ both CPUs and GPUs, and to
avoid overhead for fine-grained parallelism using lightweight threading.
Even with HPX, of course, it is still perfectly possible to introduce
performance bottlenecks into one's application. Being able to collect
performance measurements to profile an HPX application remains important. For
AMT systems such as HPX, it is beneficial to have a profiling tool that
understands the task-based nature of the runtime system -- for example, call stacks
themselves are less useful: A system thread may jump back and
forth between various HPX tasks as they are yielded and resumed, and the call
stack itself may be dozens of levels of runtime functions that have no
particular interest to the application developer.
Collecting these measurements in a profiling run is challenging, as one not
only needs a profiling framework that supports both CPU and GPU code as well as
the distributed collection of profiling data across many compute nodes. One
also needs to keep any overheads introduced by the profiling itself to an
absolute minimum. Otherwise, it would not only distort the collected
measurements, but also make large, distributed runs infeasible, rendering us
unable to detect potential performance bottlenecks that only appear at scale.
HPX is integrated with a performance measurement library, APEX (Automatic
Performance for Exascale), which was designed specifically for the HPX
runtime and the above requirements.
In previous work, APEX was used together with the HPX performance counters to
collect performance measurements for Octo-Tiger \cite{marcello2021octo}, an
astrophysics application which is built upon HPX and contains optimized kernels
for both CPUs and GPUs \cite{pfander2018accel,daiss2021beyond}. Octo-Tiger is
capable of running the same kernels on the CPUs and GPUs simultaneously (on
different data) depending on the load. In that work, profiling data of
Octo-Tiger was gathered with APEX in distributed CPU-only runs where the energy
usage, the idle rate, and overhead of the HPX AGAS (Active Global Address
Space) was analyzed~\cite{diehl2021performance}.
Furthermore, combined CPU-GPU profiling runs have been performed on Summit,
analyzing the performance behavior of Octo-Tiger's new CUDA\textsuperscript{} hydro module in
different configurations for simple benchmark
scenarios~\cite{diehl2021octotigers}.
All these previous efforts inspire this new work, collecting performance
measurements on both CPU and GPU during a full-scale production-scenario run on
Piz Daint (Cray\textsuperscript{\texttrademark} XC50 with one 12-core Intel\textsuperscript{\textregistered} Xeon\textsuperscript{\textregistered} E5-2690 + one NVIDIA\textsuperscript{\textregistered} Tesla\textsuperscript{\textregistered} P100 per node)\cite{pizdaintweb}
and Summit (IBM\textsuperscript{\textregistered} AC922 with two 22-core Power9\textsuperscript{\texttrademark} + six NVIDIA\textsuperscript{\textregistered} Tesla\textsuperscript{\textregistered} V100 per node)\cite{summitweb}.
The purpose of this work is thus twofold: First, we collect data that we can actually use to improve Octo-Tiger by identifying potential bottlenecks.
To do so, we collect measurements running the production scenario for 40 time-steps both on Summit and Piz Daint, using 48 compute nodes on Piz Daint and 8 compute nodes on Summit (resulting in 48 GPUs in either case).
With those measurements, we can investigate the specific parts of Octo-Tiger on two distinct architectures in a distributed CPU/GPU run, providing insights into the different runtime behavior of Octo-Tiger regarding GPU-performance, CPU performance, and communication. For example, the communication seems to have a larger overhead on Piz Daint.
Second, we are showcasing the feasibility of APEX for large-scale runs,
collecting combined CPU and GPU performance measurements, showing that the
overhead introduced by the profiling itself is small enough to handle large
production-scale scenarios.
To this end, we are running the scenario both with and without APEX profiling
enabled for a scaling run on each machine, to see both the overhead on a few
compute nodes, and the runtime behavior when scaling to more nodes (with up to 2000 compute nodes on Piz Daint).
Furthermore, we repeat these overhead measurements on Piz Daint for a CPU-only run to determine the performance impact of the NVIDIA\textsuperscript{\textregistered} CUDA\textsuperscript{\texttrademark} Profiling Tools Interface (CUPTI) which is used to collect the GPU performance data.
To highlight the need for low profiling overhead, we can look at the short
runtimes for each time-step of Octo-Tiger: During the test runs on Summit, we
gather \num{5} GB of data during all eight runs. For each run \num{40} time-steps were executed. Each time-step takes
about \num{0.720325}\si{\second} on \num{128} Summit nodes and
consists of 6 iterations of the gravity solver, 3 iterations of the hydro
solver, and all required communication. On Piz Daint we collected \num{55} GB of data in total. Here, each time step on \num{2000} nodes took \num{0.7872625}\si{\second}. As time steps are serial in nature, these iterations are our smallest parallel
unit. As the time-steps only run for a few hundred milliseconds overall,
overheads introduced by the profiling can be very noticeable even if they only
take a few milliseconds in total as well.
The remainder of this work is structured as follows: In Section \ref{sec:related}, we take a
brief look at profiling solutions in other AMT frameworks.
We then introduce the scientific scenario which we are simulating with
Octo-Tiger in Section \ref{sec:application}. This is the scenario we also used
to collect the profiling data by running it for 40 timesteps with APEX enabled.
Section \ref{sec:framework} in turn introduces Octo-Tiger itself, as well as the utilized software stack.
In Section \ref{sec:performance}, we show and discuss the collection of the
performance measurements for Octo-Tiger, followed by the conclusions and a
short outlook in Section \ref{sec:conclusion}.
\section{Related work}
\label{sec:related}
For the related work, we focus on AMTs with distributed capabilities which are:
Legion~\cite{bauer2012legion}, Charm++~\cite{kale1993charm++},
Chapel~\cite{chamberlain2007parallel}, and UPC++~\cite{bachan2017upc++}. For a
more detailed review, we refer to~\cite{thoman2018taxonomy}.
Legion~\cite{bauer2012legion} provides \texttt{Legion Prof} for combined CPU and GPU
profiling which is compiled into all builds. Enabling the profiler produces log
files which can be viewed using the profiler.
Charm++~\cite{kale1993charm++} provides \texttt{Charm
debug}~\cite{debugger04} and the \texttt{Projections}
framework~\cite{TauICPP09} for performance analysis and visualization.
Chapel~\cite{chamberlain2007parallel} provides
\texttt{ChplBlamer}~\cite{zhang2018chplblamer} for profiling. UPC++ seems not
to have some dedicated tool for profiling, and any profiler supporting C++ is
recommended in their documentation.
Like HPX, nearly all of these runtimes provide a specialized tool that has been
designed to deal with the particular challenges of AMTs in general, and the
needs of the runtime system in particular. APEX is a specialized tool in the case
of HPX, and provides similar measurement and analysis abilities of the
above tools, including flat profiling, tracing, sampling, taskgraphs/trees,
and concurrency graphs. In addition, APEX provides support for several
programming models/abstractions with or without HPX, including CUDA\textsuperscript{\texttrademark}, HIP,
OpenMP, OpenACC, Kokkos, POSIX threads, and C++ threads.
APEX does not provide analysis tools directly, but rather uses commonly
accepted formats and targets both HPC performance analysis tools
(ParaProf\cite{bell2003paraprof}, Vampir\cite{knupfer2008vampir},
Perfetto\cite{perfettoweb}) and standard data analysis tools (Python,
Graphviz\cite{graphvizweb}).
\section{Scientific application}
\label{sec:application}
Stellar mergers are mysterious phenomena that pack a broad range of physical processes into a small volume and a fleeting time duration. With the proliferation of deep wide-field, time-domain surveys, we have been catching on camera a vastly increased number of outbursts, many of which have been interpreted as stellar mergers. The best case, so far, of an observed merger is V1309 Sco, a contact binary identified using a recent survey database, the Optical Gravitational Lensing Experiment \cite{Udalski1992}. Fortunately, not only the merger itself was observed, but archival data from other observing programs enabled the reconstruction of the light curve years before the merger. During the merger itself, the system brightness increased by 4 magnitudes, with a peak luminosity in the red visible light \cite{Tylenda2011}. This complete record of observations has led to term V1309 Sco the "Rosetta Stone" of mergers. Previous attempts to model this merger included
semi-analytical calculations \cite{pejcha2014} and hydrodynamic simulations (e.g., \cite{nandez2014}).
However, the hydrodynamic simulations fail to adequately resolve the atmosphere, the rapid transition between the optically thick merger fluid and the optically thin, nearly empty space surrounding the simulated stellar material. To overcome this barrier, computational scientists intend to use the adaptive mesh-refinement hydrodynamics code Octo-Tiger. Using Octo-Tiger's dynamic mesh refinement, the simulations will be able to resolve the atmosphere at a higher resolution than ever before. Simulation of the V1309 merger in high resolution will provide greater insight into the nature of the mass flow and the consequential angular momentum losses.
In this paper, we analyze the performance of Octo-Tiger to identify potential bottlenecks in the combined CPU and GPU simulations since the scientists aim for long-term production runs on Piz Daint, where the atmosphere is maximally resolved. Analyzing the performance is essential at this stage since this model will serve as the necessary baseline for extending Octo-Tiger to include radiation transport for the V1309 model, as well as other binary merger models.
\section{Software framework}
\label{sec:framework}
\subsection{C++ standard library for parallelism and concurrency}
HPX is the C++ standard library for parallelism and concurrency~\cite{Kaiser2020} and one of the distributed asynchronous many-task runtime systems) AMT. Other notable AMTs with distributed capabilities are: Uintah~\cite{germain2000uintah},
Chapel~\cite{chamberlain2007parallel}, Charm++~\cite{kale1993charm++}, Legion~\cite{bauer2012legion}, and PaRSEC~\cite{bosilca2013parsec}. However, according to~\cite{thoman2018taxonomy} HPX has the highest technology
readiness level. HPX's API is fully conforming with the recent C++ standard~\cite{cxx14_standard,cxx17_standard,cxx20_standard} which is the major difference to the other mentioned AMTs. For more details about HPX, we refer to~\cite{hpx_pgas_2014,Kaiser:2015:HPL:2832241.2832244,Heller2016,Kaiser2020}. In this work, HPX has two purposes: \textit{1)} the coordination of the synchronous execution of a multitude of heterogeneous tasks (both on CPUs and GPUs), thus managing local and distributed parallelism while observing all necessary data dependencies and \textit{2)} as the parallelization infrastructure for executing CUDA\textsuperscript{\texttrademark}-kernels on the GPUs via the asynchronous HPX backend.
\subsection{APEX}
APEX~\cite{huck2015autonomic} is a performance measurement library for
distributed, asynchronous multitasking systems. It provides lightweight
measurements without perturbing high concurrency through synchronous and
asynchronous interfaces.
To support performance measurement in systems that employ user-level threading,
APEX uses a dependency chain in addition to the call stack to produce traces
and task dependency graphs.
The synchronous APEX instrumentation application programming interface (API)
can be used to add instrumentation to a runtime, library or application, and
includes support for timers and counters.
For measuring kernels on NVIDIA\textsuperscript{\textregistered} accelerated platforms, APEX is integrated with
the NVIDIA\textsuperscript{\textregistered} CUDA\textsuperscript{\texttrademark} Profiling Tools Interface~\cite{cuptiweb} that provides CUDA\textsuperscript{\texttrademark} host
callback and device activity measurements.
Similarly, on AMD accelerated platforms, APEX is integrated with
the Roctracer library~\cite{roctracerweb} providing HIP host
callback and device activity measurements.
In addition to timer measurements, the hardware and operating system are
monitored through an asynchronous measurement that involves the periodic or
on-demand interrogation of the operating system, hardware states, or runtime
states (e.g., CPU use, resident set size, memory ``high water mark'').
The NVIDIA\textsuperscript{\textregistered} Management Library interface~\cite{nvmlweb} provides periodic CUDA\textsuperscript{\texttrademark}
device monitoring to APEX, and the ROCm SMI API provides periodic HIP device
monitoring.
APEX has been extended to capture additional timers and counters related to
CUDA\textsuperscript{\texttrademark} device-to-device memory transfers, as well as tracking memory consumption
on both device and host when requested with the \lstinline{CUDAMalloc*}/\lstinline{cudaFree*} or
\lstinline{hipMalloc*}/\lstinline{hipFree*} API calls~\cite{wei2021memory}.
APEX supports tracing in both the OTF2 and Google Trace Events formats, but
for comparing results between platforms, profile data can be easier to work with.
To complement the profile data which collapses the time axis, APEX also captures task
and counter scatter plot data, indicating on the $x$ axis when the task started
or the counter was captured, and the $y$ axis contains the duration of the task
or the value of the counter. The tasks are sampled using a user-specified
fraction (default 1\%) whereas the counters are sampled at every value capture.
This data collection allows the application developer to capture a time
sequence of data without the file system overhead of a full event trace.
\subsection{Octo-Tiger}
\label{sec:framework-octotiger}
Octo-Tiger is a 3D adaptive mesh refinement (AMR) hydrodynamics finite volume code with Newtonian gravity designed specifically for the study of interacting stellar binaries \cite{octotiger_apcs_2016}, \cite{marcello2021octo}. The AMR grid rotates with the initial orbital frequency of the binary, which reduces inaccuracies due to numerical viscosity. The gravitational potential and force are computed using a modified version of the Fast Multipole Method that eliminates the gravitational field as a source of angular momentum conservation violation \cite{octotiger_fmm}. This enables Octo-Tiger to conserve energy and linear momentum to machine precision in the rotating frame. The astrophysical fluid is modeled using the inviscid Euler equations, which are solved in a finite volume central scheme \cite{kurganov2000}.
The computational domain is based on a properly nested three-dimensional octree structure. Each node in the structure is an $N^3$ Cartesian sub-grid (usually, $N=8$), and may be further refined into eight child nodes, each containing its own $N^3$ sub-grid with twice the resolution of the parent.
In \cite{marcello2021octo}, an improved hydro solver for Octo-Tiger that includes a full three-dimensional reconstruction technique has been introduced.
The performance of this improved hydro solver (after porting it to GPUs) on Summit's ORNL has been tested as well as its accuracy \cite{diehl2021octotigers}, showing a good GPU speedup (the exact amount depending on the chosen sub-grid size), and a greater accuracy in maintaining a rotating star in equilibrium than the old hydro solver.
It has been fully benchmarked demonstrating superior angular momentum conservation and extreme scalability properties allowing it to compute larger problems in a shorter wall-clock time~\cite{marcello2021octo}.
A convergence study in a real production run of a white dwarf merger has been also performed \cite{diehl2021performance}.
Octo-Tiger's CUDA\textsuperscript{} implementation is worth elaborating on in a bit more detail here to understand the later performance results in Section \ref{sec:performance}, in particular the short runtimes of the GPU kernels.
Usually, each of the compute kernels in the solver (both hydro solver and gravity solver) operates on a single sub-grid at a time with one CPU core. Multi-core usage is achieved by launching many of those methods on different sub-grids concurrently as HPX tasks.
In a GPU run, an issue arises where a single sub-grid does not provide enough work to utilize the entire GPU. However, the scheduler can simply launch multiple kernels on different sub-grids concurrently.
They are launched in different CUDA\textsuperscript{\texttrademark} streams, which are drawn from a pre-allocated pool to avoid the overhead of stream creation. Usually, we use $128$ streams per GPU.
As managing multiple CUDA\textsuperscript{\texttrademark} streams in each of the CPU threads might become unwieldy, the code uses an HPX-CUDA\textsuperscript{\texttrademark} integration. With it, HPX futures can be obtained from CUDA\textsuperscript{\texttrademark} kernel launches, allowing HPX to treat them as any other HPX task. Using this techique, HPX can simply launch a CUDA\textsuperscript{\texttrademark} kernel, define subsequent post-processing tasks to be run once it is done, and then suspend the current HPX task that starts the kernel. The current CPU thread can then work on another task, potentially launching another GPU kernel in a different stream.
This implementation scheme has the implication that HPX runs many GPU kernels and CPU tasks concurrently, allowing the application to interleave GPU kernels, CPU tasks, CPU-GPU memory transfers, and inter-node communication seamlessly. Each of the tasks, both on CPU and GPU, has a rather short runtime as a result of only dealing with a single sub-grid at a time. For example, the individual GPU kernel execution time ranges from 100 microseconds (less for smaller auxiliary kernels) to about 2 milliseconds.
\section{Performance measurements}
\label{sec:performance}
Octo-Tiger's dependencies are listed in Table~\ref{tab:octo:dependencies}. On both systems, we used Octo-Tiger's git hash (\textit{4c38f3bf}) as the baseline. However, we had to do some minor changes with regard to compilation, which do not affect the astrophysics kernels. Therefore, we used a slightly diverged git hash (\textit{b091fd26}) on Piz Daint and git hash (\textit{fd7faf5e}) on Summit. To quantify the overhead $o(n)$ introduced by APEX in percent, we define the metric
\begin{align}
o(n) = \frac{comp\_apex(n)}{comp\_time\_no\_apex(n)} \cdot 100 - 100\,,
\label{eq:overhead}
\end{align}
where $n$ is the number of nodes, $comp\_time\_no\_apex(n)$ without APEX, and $comp\_apex(n)$ the computation time with APEX. Note that we only measure the time for the actual computation and not the IO.
\begin{table}[tb]
\centering
\begin{tabular}{ll|ll}
\toprule
clang/gcc & 11/9.1.0 & hdf5 & 1.8.12 \\
cray-mpich & 7.7.16 & jemalloc & 5.1.0 \\
hpx & 1.6.0 & CUDA\textsuperscript{\texttrademark} & 11.0.2 \\
silo & 4.10.2 & boost & 1.76.0 \\
hwloc & 1.11.12 & VC & 1.4.1 \\
IBM\textsuperscript{\textregistered} spectrum & 10.4.0.3 & CMake & 3.19.5/3.18.0\\
\bottomrule
\end{tabular}
\caption{Octo-Tiger's depencencies used in this study. First numbers show the versions on Piz Daint and second ones on Summit if different versions were used.}
\label{tab:octo:dependencies}
\end{table}
\subsection{Distributed scaling}
The largest V1309 scenario (\num{18} million cells) fitting on four CSCS's Piz Daint nodes was chosen for this scaling test. Note that the largest scenario for a single node was too small to scale out up to \num{2000} nodes with \num{2000} NVIDIA\textsuperscript{\textregistered} P\num{100} GPUs and \num{24000} CPU cores. However, due to the large amount of memory per node, the same scenario fits on one of ORNL's Summit node. On both machines, we executed the same runs with the APEX + CUDA\textsuperscript{\texttrademark} profiling on and off to investigate the overhead introduced by the profiling. Figure~\ref{fig:distributed:scaling} shows the cells processed per second for increasing amount of nodes. On Piz Daint the scaling was done using Octo-Tiger's new hydro solver~\cite{diehl2021octotigers} using a three-dimensional reconstruction scheme and scaling results using the old solver for a different scenario are shown here~\cite{daiss2019piz}. The \textcolor{plot1}{blue} lines show the scaling on Piz Daint. For both configurations, the scenario scales up to \num{2000} nodes. However, for \num{1400} Piz Daint nodes the problem size gets too small, and the scaling starts to flatten out. The overhead $o(n)$ in Equation~\ref{eq:overhead} for Piz Daint is shown in Table~\ref{tab:overhead:daint}. The overhead is the largest on \num{4} nodes and seems to decline with increasing node counts.
\begin{figure}[tb]
\centering
\subfloat[Summit]{
\begin{tikzpicture}
\begin{axis} [ybar,height=6cm,width=8cm,xlabel=Nodes,ylabel=$o(n)$ in \%]
\addplot [draw=black] coordinates {
(1,20.08)
(2,19.81)
(3,22.26)
(4,20.68)
(5,21.19)
(6,19.19)
(7,33.08)
(8,31.85)
};
\end{axis}
\end{tikzpicture}
}
\\
\subfloat[Piz Daint]{
\begin{tikzpicture}
\begin{axis} [ybar,height=6cm,width=8cm,xlabel=Nodes,ylabel=$o(n)$ in \%]
\addplot [draw=black] coordinates {
(1,116.08)
(2,107.89)
(3,87.88)
(4,86.52)
(5,66.85)
(6,57.62)
(7,47.22)
(8,83.85)
(9,57.03)
(10,65.79)
(11,54.08)
(12,53.41)
};
\end{axis}
\end{tikzpicture}
}
\caption{Overhead $o(n)$ in Equation~\eqref{eq:overhead} for the runs on Piz Daint and Summit, respectively.}
\label{fig:my_label}
\end{figure}
The \textcolor{plot2}{violet} lines show the scaling on Summit. Here, again, the code scales well up to \num{128} nodes (using \num{768} NVIDIA\textsuperscript{\textregistered} V\num{100} GPUs overall) and the work is not sufficient for more nodes. The overhead $o(n)$ in Equation~\eqref{eq:overhead} on Summit is shown in Table~\ref{tab:overhead:daint}. The overhead is less prominent on Summit.
We have seen an introduced overhead by the combined profiler on both systems. In a previous study using a different problem and the old version of the hydro module, the overhead introduced by pure APEX CPU profiling within HPX was around \num{1}\si{\percent}~\cite{diehl2021performance}. To verify if this still holds with the new hydro module and the V1309 scenario, we ran the V1309 scenario on Piz Daint with pure APEX CPU profiling. First, we observed scaling for the CPU kernels up to 2000 Piz Daint nodes. Second, we observed see that the difference is again around \num{1}\si{\percent} as in the previous study. It seems that the overhead is mostly introduced by enabling CUPTI measurement in APEX for both systems. However, it seems that the overhead for smaller node counts on Piz Daint is larger, which needs to be investigated. Regardless, CUPTI provides varying levels of detail/support, and APEX should be refactored to enable the minimum amount of useful support by default, and allow the user to request additional details as needed.
\begin{figure}
\centering
\subfloat[\label{fig:distributed:scaling}]{
\begin{tikzpicture}
\begin{axis}[xlabel={\# nodes},ylabel={Cells processed per second},title={Distributed scaling (CPU + GPU)},grid,legend pos=north west,xmode=log,log basis x={2},xtick={1,2,4,8,16,32,64,128,256,512,1024,2048},ymode=log,log basis y={2},legend columns=2,legend style={at={(0.5,-0.2)},anchor=north}]
\addplot[thick,mark=*,plot1] table [x expr=\thisrowno{0},y expr={512*(35855+35904)/2*40/\thisrowno{1}}, col sep=comma] {distributed-scaling-no-hyper.csv};
\addplot[thick,mark=square*,plot1] table [x expr=\thisrowno{0},y expr={512*(35855+35904)/2*40/\thisrowno{1}}, col sep=comma] {distributed-scaling-no-apex-no-hyper.csv};
%
\addplot[thick,mark=*,plot2] table [x expr=\thisrowno{0},y expr={512*(35855+35904)/2*40/\thisrowno{1}}, col sep=comma] {distributed-scaling-summit.csv};
\addplot[thick,mark=square*,plot2] table [x expr=\thisrowno{0},y expr={512*(35855+35904)/2*40/\thisrowno{1}}, col sep=comma] {distributed-scaling-summit-no-apex.csv};
\legend{Piz Daint with APEX,Piz Daint,Summit with APEX,Summit};
\end{axis}
\end{tikzpicture}
}
\\
\subfloat[\label{fig:distributed:speedup}]{
\begin{tikzpicture}
\begin{axis}[xlabel={\# nodes},ylabel={Speedup},title={Speedup (CPU + GPU)},grid,legend pos=north west,xmode=log,log basis x={2},xtick={1,2,4,8,16,32,64,128,256,512,1024,2048},ymode=log,log basis y={2},legend columns=2,ymin=1]
\addplot[thick,mark=*,plot1] table [x expr=\thisrowno{0},y expr={1915.98/\thisrowno{1}}, col sep=comma] {distributed-scaling-no-hyper.csv};
\addplot[thick,mark=square*,plot1] table [x expr=\thisrowno{0},y expr={886.772/\thisrowno{1}}, col sep=comma] {distributed-scaling-no-apex-no-hyper.csv};
\addplot[thick,mark=*,plot2] table [x expr=\thisrowno{0},y expr={1111.07/\thisrowno{1}}, col sep=comma] {distributed-scaling-summit.csv};
\addplot[thick,mark=square*,plot2] table [x expr=\thisrowno{0},y expr={1034.04/\thisrowno{1}}, col sep=comma] {distributed-scaling-summit-no-apex.csv};
\addplot[domain=1:2000]{x};
\addplot [domain=1:2000,dashed,shift={(axis cs:4,1)},legend image post style={shift={(0,0)}}]{x};
\legend{Piz Daint with APEX,Piz Daint, Summit with APEX,Summit,Optimal (Summit),Optimal};
\end{axis}
\end{tikzpicture}
}
\caption{Cells processed per second \protect\subref{fig:distributed:scaling} and speedup \protect\subref{fig:distributed:speedup}. On Piz Daint (\textcolor{plot1}{blue line}) we were able to use \num{4}, \num{8}, \num{16}, \num{32}, \num{64}, \num{128}, \num{256}, \num{512}, \num{1024}, \num{1400}, \num{1600}, \num{1800}, and \num{2000} nodes. On Summit (\textcolor{plot2}{violet line}) we used \num{1}, \num{2}, \num{4}, \num{8}, \num{16}, \num{32}, \num{64}, and \num{128} nodes. The speedup was obtained with respect to the smallest amount of nodes the scenario (\num{18} Million cells) fitted on. Note that for the runs with and without APEX a different time on the smallest nodes were used.}
\end{figure}
\subsection{Hyper-threading vs. non-hyper-threading on Piz Daint}
On Piz Daint we investigated the effect of enabling hyper-threading on the scaling and on the performance profiling. In Figure~\ref{fig:distributed:scaling:hyperthread} we ran same scenario as in Figure~\ref{fig:distributed:scaling} on Piz Daint, however, we enabled hyper-threading. Thus, instead of \num{12} physical cores on each node, HPX sees \num{24} virtual cores. The \textcolor{plot1}{blue lines} are the same values as in the previous figure. The \textcolor{plot2}{violet lines} are the new values with hyper-threading enabled. First, we can see no performance benefit of the hyper-threading for both configurations, and instead we obtain some performance loss. Second, the scaling shows the same trend. Due to the performance loss, we have not investigated the APEX measurements of the runs with hyper-threading enabled.
\begin{figure}
\centering
\begin{tikzpicture}
\begin{axis}[xlabel={\# nodes},ylabel={Cells processed per second},title={Distributed scaling (CPU + GPU)},grid,legend pos=north west,xmode=log,log basis x={2},xtick={1,2,4,8,16,32,64,128,256,512,1024,2048},ymode=log,log basis y={2},legend columns=2,legend style={at={(0.5,-0.2)},anchor=north}]
\addplot[thick,mark=*,plot2] table [x expr=\thisrowno{0},y expr={512*(35855+35904)/2*40/\thisrowno{1}}, col sep=comma] {distributed-scaling.csv};
\addplot[thick,mark=square*,plot2] table [x expr=\thisrowno{0},y expr={512*(35855+35904)/2*40/\thisrowno{1}}, col sep=comma] {distributed-scaling-no-apex.csv};
%
\addplot[thick,mark=*,plot1] table [x expr=\thisrowno{0},y expr={512*(35855+35904)/2*40/\thisrowno{1}}, col sep=comma] {distributed-scaling-no-hyper.csv};
\addplot[thick,mark=square*,plot1] table [x expr=\thisrowno{0},y expr={512*(35855+35904)/2*40/\thisrowno{1}}, col sep=comma] {distributed-scaling-no-apex-no-hyper.csv};
\legend{APEX + hyper-threading,hyper-threading,APEX,No APEX};
\end{axis}
\end{tikzpicture}
\caption{Cells processed per second on Piz Daint for \num{4}, \num{8}, \num{16}, \num{32}, \num{64}, \num{128}, \num{256}, \num{512}, \num{1024}, \num{1400}, \num{1600}, \num{1800}, and \num{2000} nodes. The \textcolor{plot1}{blue lines} show the scaling without hyper-threading, and the \textcolor{plot2}{violet lines} show the scaling with hyper-threading.}
\label{fig:distributed:scaling:hyperthread}
\end{figure}
\subsection{Profiling of Octo-Tiger}
\subsubsection{Setup}
Because APEX is directly integrated into the HPX runtime, annotating HPX actions (tasks) is simply a matter of providing an annotation for
the task when it is instantiated in the code. Not all actions are annotated (anonymous lambdas launched through \lstinline{hpx::async()} calls, for example), but major operations in the Octo-Tiger code and in the HPX runtime are annotated. When HPX is configured and built with APEX support, all annotations are provided to APEX, and APEX times the life cycle of each task. HPX is designed to yield and resume tasks when resources are unavailable (futures, system calls, other dependencies), so APEX also provides the ability to yield (and resume) timing a task when it is not executing. For GPU kernels, the kernel name is obtained using the instruction pointer address and debug information in the executable provided by the compiler.
For the following Octo-Tiger profiling results, we use $48$ HPX localities (processes).
As we use one HPX locality per GPU, this results in using $48$ compute nodes on Piz Daint and 8 compute nodes on Summit (as each node contains $6$ GPUs here).
This ensures that we use the same number of GPUs on both systems. We are still using the same V1309 scenario as before.
Below is a short description of the tasks and kernels of interest in Octo-Tiger.
The main CPU tasks include \lstinline{node_server::non-refined_step::compute_fluxes}: directs the computation of a single Runge-Kutta substep in the hydro and gravity solvers for a given sub-grid. It directs the GPU to compute hydrodynamic fluxes, corrects these fluxes on coarse-fine boundaries, works with other invocations of the same action to compute the global time-step size, computes hydrodynamic sources, then lastly updates the hydrodynamic variables and directs the GPU to update the gravitational variables;
\lstinline{local_step::execute_step}: directs the execution of an entire time-step for a given sub-grid. This involves multiple calls to \lstinline{node_server::nonrefined_step::compute_fluxes};
\lstinline{diagnostics_actions_type}: performs some measurements on the grid, for example, computing total mass on the grid, total angular momentum, and center of masses of each star (for binaries);
\lstinline{solve_gravity_action_type}: solves for gravity. This is called when an additional gravity solve is needed (other than what \lstinline{node_server::nonrefined_step::compute_fluxes} calls), such as after grid refinement;
\lstinline{check_for_refinement_action_type} flags each sub-grid in need of refinement;
\lstinline{regrid_gather_action_type} gathers information about the sub-grid structure and determines boundaries for the global decomposition;
\lstinline{regrid_scatter_action_type} uses the information computed by \lstinline{regrid_gather_action_type} to redistribute the sub-grids.
The GPU kernels include \lstinline{cuda multipole interactions kernel}: Computes the cell to cell interactions for the gravity solver in refined sub-grids (non-rho is without the angular momentum correction);
\lstinline{cuda p2p interactions kernels}: Computes the cell to cell interactions for the gravity solver in non-refined sub-grids;
Special case: \lstinline{cuda p2m interactions kernel}: Computes the cell to cell interactions for the gravity solver in non-refined sub-grids with refined neighbor sub-grids (non-rho is without the angular momentum correction);
Special case: \lstinline{multipole root}: computes the remaining cell to cell interactions in the root sub-grid;
\lstinline{reconstruct cuda kernel} reconstructs the evolution variables using the PPM method, see \cite{diehl2021octotigers};
\lstinline{flux cuda kernel} the flux method computes the fluxes and the Newtonian quadrature to get the final flux.
\subsubsection{Results / Analysis}
In this section, we analyze the collected profiling results and draw conclusions regarding Octo-Tiger's performance on two different architectures. These insights are important for further optimization on the code. Some of them are straightforward and others are quite puzzling. Let us start with the insights on Summit:
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{figures/paraprof-cpu.png}
\caption{Comparison of top CPU tasks with TAU ParaProf.}
\label{fig:comparison:cpu}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{figures/paraprof-gpu.png}
\caption{Comparison of top GPU kernels with TAU ParaProf.}
\label{fig:comparison:gpu}
\end{figure}
\begin{figure*}
\centering
\includegraphics[width=0.8\textwidth]{figures/top_task_scatterplot_comparison_cpu.png}
\caption{Comparison of sampled profiles of CPU tasks on 48 localities between Summit (red) and Piz Daint (blue). The overall runtime on Summit is shorter, but interestingly, some CPU tasks took longer (per call) to execute on Summit than on Piz Daint.}
\label{fig:cpu:scatterplot}
\end{figure*}
\begin{figure*}[tb]
\centering
\includegraphics[width=\textwidth]{figures/top_task_scatterplot_comparison_gpu.png}
\caption{Comparison of sampled GPU kernels and memory transfers on 48 localities between Summit (red) and Piz Daint (blue). Not surprisingly, in all cases, the performance of the V100s on Summit outperformed that of the P100s on Piz Daint. However, not all kernels saw a significant improvement in performance.}
\label{fig:gpu:scatterplot}
\end{figure*}
\begin{enumerate}
\item Figure~\ref{fig:comparison:cpu} shows that on Summit, the CPU tasks are not always faster with respect to Piz Daint. However, the overall computational time is lower due to the benefit of the newer GPUs. Looking at the scatterplot data for sampled timers in Figure ~\ref{fig:cpu:scatterplot}, we see that while the average time \textit{per task} on Summit is slower for some tasks (\lstinline{compute_fluxes}, \lstinline{execute_step}), the aggregated mean profile is approximately the same.
\item Figure~\ref{fig:comparison:gpu} shows that on Summit, the NVIDIA\textsuperscript{\textregistered} V100 the GPU tasks are nearly always faster. This is not surprising, since the GPU kernels are working better on the newer NVIDIA\textsuperscript{\textregistered} V100 than on the older NVIDIA\textsuperscript{\textregistered} P100. We found one exception, and the \lstinline{cuda_p2m_interaction_rho()} GPU task was faster on the NVIDIA\textsuperscript{\textregistered} P100. This task computes the monopole-multipole gravity interactions in the case that the leave nodes have different refinements. More investigation is needed to determine if it is possible to improve this compute kernel for the NVIDIA\textsuperscript{\textregistered} V100.
\item Figures~\ref{fig:daint:devicememory} and~\ref{fig:summit:devicememory} show that the device memory consumption on Summit is greater than on Piz Daint. However, in average it is not much, but we could identify some big outliers. More investigation is needed to look into the asynchronous GPU kernel launch scheduler for multiple GPUs to investigate these outliers.
\end{enumerate}
On Piz Daint we gather the following insights:
\begin{enumerate}
\item Figures~\ref{fig:daint:hostmemory} and~\ref{fig:summit:hostmemory} show that the host memory consumption on Piz Daint is greater which needs to be further investigated.
\item Figures~\ref{fig:daint:user}, \ref{fig:summit:user}, \ref{fig:daint:system}, and~\ref{fig:summit:system} show that on Piz Daint we find that much of the CPU utilization is spent in system calls which we could not observe on Summit. This could indicate some contention since the many small launched tasks compete for resources. However, some follow-up profiling on other x86 processors is needed for further understanding.
\item Unrelated to the application itself, we found some differences within the HPX run time system. HPX provides the function \lstinline{hpx::async} to asynchronously launch functions and lambda functions. Figures~\ref{fig:comparison:cpu} and~\ref{fig:cpu:scatterplot} show that this operation was $13\times$ more expensive on Piz Daint as on Summit. Here, the HPX main developers need to investigate this behavior.
Figure~\ref{fig:comparison:cpu} also shows another more expensive HPX operation on Piz Daint was \lstinline{schedule_parcel} which took nearly $10\times$ longer as on Summit. A delay in the \lstinline{schedule_parcel} potentially happens when some \lstinline{hpx::future} is not ready, which \lstinline{schedule_parcel} depends on. Another possible cause is that APEX needs to do some look-up for some global ID when creating and scheduling a task. This needs further investigation to identify if APEX or the depending \lstinline{hpx::future} is causing the delay. For the second case we would identify the delaying task which might seem to have some larger computational time on Piz Daint.
\end{enumerate}
Our primary focus here is distributed combined CPU and GPU profiling. We anticipate future research concentrating on the interpretation of these findings.
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{figures/daint/GPU:_Total_Bytes_Occupied_on_Device.pdf}
\caption{Piz Daint total memory allocated on each device. The largest occupied amounts are around 250MB of the available 16GB memory per device.}
\label{fig:daint:devicememory}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{figures/summit/GPU:_Total_Bytes_Occupied_on_Device.pdf}
\caption{Summit total memory allocated on each device. Some outliers occupied around 600MB of the available 16GB memory per device.}
\label{fig:summit:devicememory}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{figures/daint/status:VmHWM.pdf}
\caption{Piz Daint memory high water mark for each process. Some processes allocated nearly 10GB each.}
\label{fig:daint:hostmemory}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{figures/summit/status:VmHWM.pdf}
\caption{Summit memory high water mark for each process. Allocations ranged between 4-5GB per process.}
\label{fig:summit:hostmemory}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{figures/daint/CPU_User_.pdf}
\caption{Piz Daint user \% of CPU (normal processes executing in user mode). Because hyperthreading is enabled in the operating system, but only one thread per core was used, the maximum expected value is 50\%.}
\label{fig:daint:user}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{figures/summit/CPU_User_.pdf}
\caption{Summit user \% of CPU. Because four hardware threads are supported per core, but only one thread per core was used, the maximum expected value is 25\%.}
\label{fig:summit:user}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{figures/daint/CPU_System_.pdf}
\caption{Piz Daint system \% of CPU (processes executing in kernel mode). Peaks of 40\% represent an unreasonable amount of system level utilization, reflecting high contention in the hardware or operating system for limited resources.}
\label{fig:daint:system}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{figures/summit/CPU_System_.pdf}
\caption{Summit system \% of CPU. 2\% represents a reasonable amount of system level utilization, reflecting little contention in the hardware or operating system for limited resources.}
\label{fig:summit:system}
\end{figure}
\section{Conclusion}
\label{sec:conclusion}
In this work, we have analyzed the overhead of performing combined CPU and GPU performance measurements with APEX in a large-scale HPX application distributed across up to \num{2000} compute nodes.
We have demonstrated that Octo-Tiger easily scales to that many nodes on Piz Daint with the APEX profiling enabled. Profiling is thus feasible for real-world production-size runs on high-performance systems equipped with GPUs.
However, we encountered a noticeable profiling overhead at scale (\num{52.408539388924}\% on 2000 Piz Daint nodes). This seems to be due to the GPU measurements with CUPTI, as a subsequent CPU-only run exhibited a smaller profiling overhead. Note that the overhead on Piz Daint with very few nodes is about two times higher and requires further investigation.
On Summit, there is a noticeable overhead at scale from profiling, too, but significantly less (\num{31.848570683336}\% on \num{128} Summit nodes). Overall, regarding the APEX profiling overhead, the distributed profiling with both CPU and GPU measurements works, scales up, and is ready to use; yet more investigation is needed to work on minimizing the overhead of the GPU measurements, if possible.
While we focused on evaluating how suitable APEX is for these large-scale, distributed analyses, there are some interesting results regarding Octo-Tiger itself as well.
It is notable that the speedup of the average GPU kernel runtime from a P100 to a newer V100 varies a lot between the kernels. There are many factors that may influence the average kernel runtime difference between the devices: For instance, going to the V100, there is an increase in the number of Streaming Multiprocessors (SM), an increase in L1 cache available per SM, an increase in global memory bandwidth and a slight increase in clock speed. This does not even take into account the fact that we compile for different architectures.
Considering that we use concurrent kernel execution via $128$ CUDA\textsuperscript{\texttrademark} streams per GPU to achieve device utilization (as outlined in Section~\ref{sec:framework-octotiger}), it is unlikely that the increased numbers of SMs in the V100 has a major impact on the average kernel execution time (as those would rather facilitate more concurrent kernel execution). Thus, it seems more likely that the speedup is due to a combination of the other factors, the exact speedup depending on what is currently limiting the kernel. The larger speedups indicate that the kernels benefit from the larger L1 cache available, however, determining the exact cause is subject of future work, especially as the kernels are currently still undergoing changes. However, the results here give us an idea which kernels need more attention during this process, particularly when targeting older architectures, thus helping us to steer our development focus.
Beyond the GPU results, the profiling uncovered that there are some crucial methods that run significantly slower on Piz Daint than on Summit, such as \lstinline{schedule_parcel}.
As the method is part of the communication of Octo-Tiger in the HPX parcelport backend, this indicates that there is some deeper issue regarding the communication on this machine.
Furthermore, we have measured that much of the CPU utilization on Piz Daint is spent in system calls and, surprisingly, we have observed a higher memory consumption on Summit.
Uncovering issues and oddities like these highlights the need for distributed performance measurements on a production system.
Of course, while we focused on the usability of APEX for these kinds of analyses in this work, the uncovered issues still need to be further investigated and addressed.
Consequently, we will examine these issue in future work, hopefully further improving the runtime of future simulations with Octo-Tiger.
A radiation module for Octo-Tiger is currently being implemented by its developers, and it is in the testing phase. Our performance analysis of the current modules will be
crucial to estimate the performance impact of the new module prior to its inclusion.
Including radiation in the simulations of V1309 together with resolving the star atmosphere at a higher resolution than ever before will enable one to self-consistently compute the light curve and directly compare it with the observed one of V1309.
If one is able to accurately reproduce the light curve of the "Rosetta Stone of mergers", it will be possible to reliably simulate the outburst light curves of other mergers.
\section{Supplementary materials}
\label{sec:suppliementary}
Will be added after review to keep the paper double-blind.
The build scripts are available on GitHub\footnote{\url{https://github.com/STEllAR-GROUP/OctoTigerBuildChain}}, the input files are available on Zenodo~\cite{marcello_2021_5213015}, and the scripts to generate the slurm files are available on Zenodo as well.
\section*{Acknowledgment}
\label{sec:acknowledgement}
Will be added after review to keep the paper double-blind.
\cleardoublepage
\newpage
\bibliographystyle{siamplain}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 410 |
Yuki Okada (født 4. oktober 1983) er en tidligere japansk fodboldspiller.
Han har spillet for flere forskellige klubber i sin karriere, herunder Consadole Sapporo og Tochigi SC.
Eksterne henvisninger
Fodboldspillere fra Japan | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,672 |
Elizabeth Gilbert est une romancière, essayiste et biographe américaine, née le .
Biographie
En 2006, elle a publié Eat, Pray, Love : One Woman's Search for Everything Across Italy, India and Indonesia, traduit en français en 2008, sous le titre Mange, Prie, Aime. Elle y relate son voyage d'un an en Italie (Mange), en Inde (Prie) et à Bali (Aime) à la recherche de son moi profond.
Le livre devient un véritable best-seller (10 millions d'exemplaires vendus).
Une adaptation cinématographique avec Julia Roberts dans le rôle principal est sortie en 2010 sous le titre Mange, prie, aime.
Elle raconte la vie du survivaliste américain Eustace Conway (protagoniste entre autres de la série télévisée Seuls face à l'Alaska) dans The Last American Man (2002) .
Vie privée
En , Elizabeth Gilbert annonce sur Facebook avoir divorcé de son mari José Nunes (héros de Mange, Prie, Aime) après être tombée amoureuse de sa meilleure amie, l'écrivaine d'origine syrienne Rayya Elias. Après s'être mariée avec elle en 2017, Rayya Elias décède en 2018 d'un cancer du foie et du pancréas. En 2019, Gilbert annonce s'être mis en couple avec Simon MacArthur, un photographe qu'elle comptait depuis longtemps parmi ses amis proches et qui se trouvait lui-même être un proche de Rayya Elias pour avoir été son colocataire trente ans plus tôt.
Ouvrages
Comme par magie, traduction de l'anglais Big Magic: Creative Living Beyond Fear, 2015, Éditions Calmann-Lévy, 320p, 2016,
Au bonheur des filles, Éditions Calmann-Lévy, 544p, 2020,
Notes et références
Liens externes
Site personnel
Romancière américaine du XXe siècle
Romancière américaine du XXIe siècle
Naissance en juillet 1969
Étudiant de l'université de New York
Personnalité ayant fait son coming out
Essayiste américaine du XXIe siècle
Biographe américaine | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,949 |
Kingman (- ) est un cheval de course qui participe aux courses hippiques de plat. Né en Angleterre, propriété de son éleveur Khalid Abdullah, il est entraîné par John Gosden et monté par James Doyle.
Carrière de courses
Kingman effectue de spectaculaires débuts le 29 juin 2013 sur l'hippodrome Newmarket, s'imposant par six longueurs. La forte impression laissée par le poulain lui vaut d'être aussitôt consacré favori des 2000 guinées, qui se dérouleront un an plus tard, même si son jockey d'un jour, l'Irlandais Ryan Moore, tenta de tempérer les ardeurs. Mais sa sortie suivante, soldée par une victoire aisée dans les Solario Stakes, un groupe 3 disputé à Sandown, ne désenfla pas la rumeur. Annoncé au départ des Dewhurst Stakes, Kingman devra y renoncer en raison d'un problème de santé qui lui vaudra de subir une intervention chirurgicale pour lui retirer un éclat d'os. Il s'en tint à ces deux performances pour sa saison de 2 ans.
Très attendu pour son retour, qui a pour théâtre les Greenham Stakes (Gr.3) à Newbury, Kingman confirme qu'il est bien le favori naturel des classiques sur le mile en l'emportant par quatre longueurs. Dans les 2000 Guinées, on lui oppose l'Irlandais Australia, un élève de Coolmore, qui fait sa rentrée directement dans le classique et sera sacré, un mois plus tard, dans le Derby d'Epsom. Mais c'est un troisième larron qui arbitre le duel, Night of Thunder, dauphin pourtant lointain de Kingman dans les Greenham Stakes, qui s'impose d'une demi-longueur devant Kingman, lui-même passant une tête devant Australia. Ce sera la seule défaite de la carrière de Kingman.
En appel dans les 2000 Guinées irlandaises, le protégé de John Gosden ne fait pas dans le détail : cinq longueurs le séparent du deuxième, Shifting Power. Rendez-vous est pris avec Night of Thunder pour la belle dans les St. James's Palace Stakes, et c'est Kingman qui a le dernier mot, terrassant son rival de deux grandes longueurs. Il confirme cette victoire face aux chevaux d'âge dans les Sussex Stakes, qu'il remporte sans coup férir, puis à Deauville, dans le Prix Jacques Le Marois. Son programme pour la fin de la saison, qui passe par les Queen Elizabeth II Stakes et le Breeders' Cup Mile, ne sera en revanche jamais accompli : une infection à la gorge a raison de la carrière de Kingman, qui se voit tout de même sacré meilleur 3 ans de l'année en Europe, et même cheval de l'année tout court.
Résumé de carrière
Au haras
Rentré prématurément au haras, Kingman s'y annonce d'emblée comme une recrue d'envergure pour l'élevage. Son prix de saillie reflète la confiance que lui accorde les éleveurs, puisqu'il démarre d'emblée à £ 55 000, un tarif qui ne tarde pas à s'envoler, jusqu'à £ 150 000 en 2020. Il faut dire que le jeune étalon, stationné à Banstead Manor Stud, l'antenne européenne de Juddmonte Farms, où séjourne aussi un certain Frankel, voit très vite sa progéniture briller en courses. Il est, avec ce dernier, l'un des jeunes reproducteurs les plus côtés de la planète.
Parmi ses meilleurs produits, citons (avec entre parenthèse le nom du père de mère) :
Palace Pier (Nayef) : St. James's Palace Stakes, Prix Jacques Le Marois (deux fois), Lockinge Stakes, Queen Ann Stakes. Meilleur 3 ans d'Europe (2020). Cheval d'âge de l'année en Europe (2021).
Persian King (Dylan Thomas) : Poule d'Essai des Poulains, Prix d'Ispahan Prix du Moulin de Longchamp, 2e Prix du Jockey-Club, 3e Prix de l'Arc de Triomphe.
Kinross (Selkirk) : Prix de la Forêt, British Champions Sprint Stakes.
Domestic Spending (Street Cry) : Hollywood Derby, Turf Classic, Manhattan Handicap.
Schnell Meister (Soldier Hollow) : NHK Mile Cup.
Commissioning (Galileo) : Fillies' Mile.
Origines
Kingman est par le top étalon Invincible Spirit, champion sprinter issu d'une lauréate du Prix de Diane, qui fut l'un des meilleurs continuateurs de Green Desert au haras, père de nombreux champions dont Moonlight Cloud. Côté maternel, il ressort de l'une des plus belles lignées de l'élevage mondiale, celle de sa quatrième mère, la très influente Sorbus, véritable poulidor des classiques en 1975. Sa descendance contient une multitude de champions, dont d'importants étalons tels Oasis Dream ou désormais Kingman, qui en assurent la continuité.
Sorbus (Busted) : 2e Irish Oaks, Irish 1000 Guineas, Irish St. Leger, Yorkshire Oaks.
Beldarian (Last Tycoon) : 2e Ballyroan Stakes (Gr.3). 4e Oaks.
Dariana (Redoute's Choice) : Queensland Derby (Gr.1, Australie). 2e Underwood Stakes (Gr.1)
Klarifi (Habitat)
Fracas (In the Wings) : Derrinstown Stud Derby Trial Stakes (Gr.2), Classic Trial (Gr.3), Meld St (Gr.3). 2e Rheinland-Pokal, Mooresbridge Stakes (Gr.3), Ballyroan Stakes (Gr.3). 4e Derby.
Bahamian (Mill Reef)
Coraline (Sadler's Wells)
Reefscape (Linamix) : Prix du Cadran, Hubert de Chaudenay (Gr.2), Gladiateur (Gr.3). 2e Gold Cup, Prix Ganay, Royal-Oak, Kergorlay (Gr.2), de Lutèce (Gr.3). 3e Coronation Cup, Prix de la Vicomtesse Vigier (Gr.2), de Barbeville (Gr.3).
Martaline (Linamix) : Prix Maurice de Nieuil (Gr.2), d'Hédouville (Gr.3). 2e Grand Prix de Chantilly, Prix Foy. 3e Prix Jean de Chaudenay.
Costal Path (Halling) : Prix Hubert de Chaudenay (Gr.2), Prix Vicomtesse Vigier (Gr.2), de Lutèce (Gr.3), de Barbeville (Gr.3). 2e Prix Kergorlay (Gr.2), Prix d'Hédouville (Gr.3). 3e Gold Cup.
Clear Thinking (Rainbow Quest) : 2e Prix Berteux (Gr.3). 3e Prix Maurice de Nieuil (Gr.2), Prix Vicomtesse Vigier (Gr.2), Hubert de Chaudenay (Gr.2).
Trellis Bay (Sadler's Wells)
Bellamy Cay (Kris) : Prix Maurice de Nieuil (Gr.2), Prix d'Hédouville (Gr.3). 2e Prix Royal-Oak. 3e Grand Prix de Chantilly.
Cinnamon Bay (Zamindar)
New Bay (Dubawi) : Prix du Jockey Club, Guillaume d'Ornano, Prix Niel, Prix Gontaut Biron. 2e Poule d'Essai des Poulains, 3e Prix de l'Arc de Triomphe
Wemyss Bight (Dancing Brave) : Irish Oaks, Prix de Malleret (Gr.2), Prix Cléopâtre (Gr.3), Prix Pénélope (Gr.3). 2e Prix Vermeille.
Beat Hollow (Sadler's Wells) : Grand Prix de Paris, Arlington Million, Woodford Reserve Turf Classic (Gr.1, USA), Manhattan Handicap (Gr.1, USA). 2e Eddie Read Handicap (Gr.1, USA). 3e Derby, Keeneland Turf Mile Stakes (Gr.1, USA).
Hope (Dancing Brave)
Oasis Dream (Green Desert) : Middle Park Stakes, July Cup, Nunthorpe Stakes.
Zenda (Zamindar) : Poule d'Essai des Pouliches. 2e Coronation Stakes, Queen Elizabeth II Challenge Cup (Gr.1, USA).
Remote (Dansili) : Tercentenary Stakes (Gr.3)
First Eleven (Frankel) : 3e Cumberland Lodge Stakes (Gr.3)
Kingman
Pedigree
Références
Cheval de course
Cheval de course né en 2011
Animal né en 2011
Animal né au Royaume-Uni
Cheval de l'année en Europe
Lauréat d'un Cartier Racing Award
Étalon pur-sang anglais | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,237 |
Review of Wikia website with the positives and negatives shown. I can honestly say that I strongly dislike Wikia, which is basically a website that profits off other people's content, but I will try to review it in as balanced a way as I possibly can in the circumstances.
One of the major issues is that Wikia expects people to contribute for free but makes a profit off their services by displaying advertisements. It is the perfect business as Wikia gets a legion of free workers, however that is also a problem. As Wikia makes money, unlike Wikipedia, those volunteers are not just contributing to a cause they are contributing to a business. Many of those volunteers are underage, so essentially they are underaged workers. On this point I must congratulate Wikia: it has managed to hire underaged workers for no cost in first world nations, underming professional journalism and worker's rights in one swoop. A job well done.
The site does work very well and if you want to set up a community quickly and do not mind the fact that you are basically working for someone else for free, then the software is fairly intuitive and easy to use. Wikia takes care of hosting fees and the like, which are small anyway and they make the advertising profits which cover them fairly easily.
I am trying to be positive but I really, really dislike Wikia as it takes the popularity of Wikipedia and turns it towards making a profit. The site is good: it functions well. Its just a little unfair it does not share some of its profits with the people responsible for them.
If all you wish to do is create a website for a community of some kind and you do not mind advertisements provided the website is easy to create and intuitive, then Wikia is a good option. If you are turned off by the idea that other people profit from your work, consider hosting a free blog on WordPress. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,147 |
Through Colby's relationships with premier research facilities including MDI Biological Laboratory, the Jackson Laboratory in Bar Harbor, and Bigelow Laboratory for Ocean Sciences in East Boothbay, Colby students like Farrington and Krasniak are working closely with accomplished professors and researchers on world-changing biomedical projects that will prepare them for science at Colby and beyond.
Down the road at the Jackson Laboratory, a sprawling genetics research facility, research assistant Dan Sunderland '14 was explaining his work on the possible genetic causes of glaucoma.
That path is expected to widen as Colby reaps the success of grants and donor funding, and strengthens and builds its connections to three of Maine's world-class research facilities.
In June MDI Biological Laboratory was awarded an $18 million grant to fund its work with Colby and a dozen other Maine universities, colleges, and research facilities. Included in that grant was $500,000 for work done by Assistant Professor of Biology Tariq Ahmad, who is researching genetic causes of degenerative diseases like Parkinson's. | {
"redpajama_set_name": "RedPajamaC4"
} | 5,856 |
{"url":"https:\/\/giasutamtaiduc.com\/eulers-formula.html","text":"# \u2705 Euler\u2019s Formula \u2b50\ufe0f\u2b50\ufe0f\u2b50\ufe0f\u2b50\ufe0f\u2b50\n\n5\/5 - (1 b\u00ecnh ch\u1ecdn)\n\nEuler\u2019s formula, either of two important mathematical theorems of\u00a0Leonhard Euler. The first formula, used in\u00a0trigonometry\u00a0and also called the Euler identity, says\u00a0eix\u00a0= cos\u00a0x\u00a0+\u00a0isin\u00a0x, where\u00a0e\u00a0is the base of the natural\u00a0logarithm\u00a0and\u00a0i\u00a0is the\u00a0square root\u00a0of \u22121 (see\u00a0irrational number). When\u00a0x\u00a0is equal to \u03c0 or 2\u03c0, the formula yields two elegant expressions relating \u03c0,\u00a0e, and\u00a0iei\u03c0\u00a0= \u22121 and\u00a0e2i\u03c0\u00a0= 1, respectively. The second, also called the Euler polyhedra formula, is a topological invariance (see\u00a0topology) relating the number of faces, vertices, and edges of any\u00a0polyhedron. It is written\u00a0F\u00a0+\u00a0V\u00a0=\u00a0E\u00a0+ 2, where\u00a0F\u00a0is the number of faces,\u00a0V\u00a0the number of vertices, and\u00a0E\u00a0the number of edges. A cube, for example, has 6 faces, 8 vertices, and 12 edges and satisfies this formula.\n\nM\u1ee5c L\u1ee5c\n\n## Euler\u2019s Formula for Complex Numbers\n\n(There is another \u201cEuler\u2019s Formula\u201d about Geometry,\n\nFirst, you may have seen the famous \u201cEuler\u2019s Identity\u201d:\n\nei\u03c0\u00a0+ 1 = 0\n\nIt seems absolutely magical that such a neat equation combines:\n\n\u2022 e\u00a0(Euler\u2019s Number)\n\u2022 i\u00a0(the unit\u00a0imaginary number)\n\u2022 \u03c0\u00a0(the famous number\u00a0pi\u00a0that turns up in many interesting areas)\n\u2022 1 (the first counting number)\n\u2022 0 (zero)\n\nAnd also has the basic operations of add, multiply, and an exponent too!\n\nBut if you want to take an interesting trip through mathematics, you will discover how it comes about.\n\n## Discovery\n\nIt was around 1740, and mathematicians were interested in\u00a0imaginary\u00a0numbers.\n\nHe must have been so happy when he discovered this!\n\nAnd it is now called\u00a0Euler\u2019s Formula.\n\nLet\u2019s give it a try:\n\nThe answer is a combination of a Real and an Imaginary Number, which together is called a\u00a0Complex Number.\n\nWe can plot such a number on the\u00a0complex plane\u00a0(the real numbers go left-right, and the imaginary numbers go up-down):\n\nHere we show the number\u00a00.45 + 0.89\u00a0i\n\nWhich is the same as\u00a0e1.1i\n\nLet\u2019s plot some more!\n\n## A Circle!\n\nYes, putting Euler\u2019s Formula on that graph produces a circle:\n\nAnd when we include a radius of\u00a0r\u00a0we can turn any point (such as\u00a03 + 4i) into\u00a0reix\u00a0form by finding the correct value of\u00a0x\u00a0and\u00a0r:\n\n## It is Another Form\n\nIt is basically another way of having a complex number.\n\nThis turns out to very useful, as there are many cases (such as multiplication) where it is easier to use the\u00a0reix\u00a0form rather than the\u00a0a+bi\u00a0form.\n\n## Euler\u2019s Formula\n\n(There is another \u201cEuler\u2019s Formula\u201d about complex numbers,\n\n## Euler\u2019s Formula\n\nFor any polyhedron\u00a0that doesn\u2019t intersect itself,\u00a0the\n\n\u2022 Number of Faces\n\u2022 plus the\u00a0Number of Vertices\u00a0(corner points)\n\u2022 minus the\u00a0Number of Edges\n\nalways equals 2\n\nThis can be written:\u00a0F + V \u2212 E = 2\n\n## Example With Platonic Solids\n\nLet\u2019s try with the 5 Platonic Solids:\n\nIt is still an icosahedron (but no longer convex).\n\nIn fact it looks a bit like a drum where someone has stitched the top and bottom together.\n\nNow, there are the same number of edges and faces \u2026\u00a0but one less vertex!\n\nSo:\n\nF + V \u2212 E = 1\n\nOh No! It doesn\u2019t always add to 2.\n\nThe reason it didn\u2019t work was that this new shape is basically different \u2026 that joined bit in the middle means that two vertices get reduced to 1.\n\n## Euler Characteristic\n\nSo, F+V\u2212E can equal 2, or 1, and maybe other values, so the more general formula is\n\nF + V \u2212 E =\u00a0\u03c7\n\nWhere\u00a0\u03c7\u00a0is called the \u201cEuler Characteristic\u201c.\n\nHere are a few examples:\n\n## Donut and Coffee Cup\n\nLastly, this discussion would be incomplete without showing that a Donut and a Coffee Cup are really the same!\n\nWell, they can be deformed into one another.\n\nWe say the two objects are \u201chomeomorphic\u201d (from Greek\u00a0homoios\u00a0= identical and\u00a0morphe\u00a0= shape)\n\nJust like the platonic solids are homeomorphic to the sphere.\n\nAnd your body is homeomorphic to a torus if you pinch your nose closed.\n\n## Euler\u2019s Formula\n\nEuler\u2019s formula was given by\u00a0Leonhard Euler, a Swiss mathematician. There are two types of Euler\u2019s formulas: a)\u00a0For complex analysis, b) For polyhedra. a)\u00a0Euler\u2019s formula used in complex analysis: Euler\u2019s formula\u00a0is a key formula used to solve complex exponential functions.\u00a0Euler\u2019s formula is also\u00a0sometimes known as\u00a0Euler\u2019s identity.\u00a0It\u00a0is used to establish the relationship between trigonometric functions and complex exponential functions. b) Euler\u2019s formula for polyhedra: For any polyhedron\u00a0that does not self-intersect, the number of faces, vertices, and edges is related in a particular way, and that is given by\u00a0Euler\u2019s formula or also known as Euler\u2019s characteristic. Let us learn this formula along with a few solved examples.\n\n## What is Euler\u2019s Formula?\n\na) For complex analysis:\u00a0The Euler\u2019s form of a complex number is important enough to deserve a separate section. It is an extremely convenient representation that leads to simplifications in a lot of calculations.\u00a0Euler\u2019s formula in complex analysis is used for establishing the relationship between trigonometric functions and complex\u00a0exponential functions. Euler\u2019s formula is defined for any real number x and can be written as:\n\neiix\u00a0= cos x +\u00a0iisin x\n\nHere, cos and sin are trigonometric functions,\u00a0ii\u00a0is the imaginary unit, and\u00a0e\u00a0is the base of the natural logarithm. The interpretation of this formula can be taken\u00a0in a complex plane, as a unit complex function\u00a0eii\u03b8\u00a0tracing\u00a0a unit circle, where\u00a0\u03b8 is a real number and is measured in radians.\n\nThis representation might seem confusing at first. What sense does it make to raise a real number to an imaginary number? However, you may rest assured that a valid justification for this relation exists.\u00a0 Although we will not discuss rigorous proof for this, you may observe the following approximate proof to see why it should be true.\n\nWe use the following expansion series for\u00a0 ex\u00a0:\n\nb) For polyhedra\n\nPolyhedra are 3D\u00a0solid shapes whose surfaces are flat and edges are straight.\u00a0 For example cube,\u00a0cuboid, prism, and pyramid. For any polyhedron\u00a0that does not self-intersect, the number of faces, vertices, and edges are related in a particular way.\n\nEuler\u2019s formula for polyhedra tells us that the number of vertices and faces together is exactly two more than the number of edges. Euler\u2019s formula for a polyhedron can be written as:\n\nFaces + Vertices \u2013 Edges = 2\n\nHere,\n\n\u2022 F is the number of faces,\n\u2022 V the number of vertices, and\n\u2022 E the number of edges.\n\n## Euler\u2019s Formula Proof\n\nWhen we draw dots and lines alone, it\u00a0becomes a graph. We obtain a planar\u00a0graph when no lines or edges cross each other. We can represent a cube as a planar graph by projecting the vertices and edges onto a plane.\n\nAccording to Euler\u2019s formula graph theory, the number of dots \u2212 the number of lines + the number of regions the plane is cut into = 2.\n\n### Solution for the Utilities Problem\n\nEuler\u2019s formula is proved using the utility problem: The three houses are to be connected to the 3 utility gas, water, and electricity. They are to be connected in such a way that no pipe passes over the other pipe.\u00a0To get a complete cycle with no intersection in any planar graph,\u00a0we remove an edge to create a tree. This brings down both edges and faces by one, leaving vertices \u2212 edges + faces = a constant. We repeat this process until the remaining graph is a tree. Finally, we obtain\u00a0vertices \u2212 edges + faces = 2, i.e. the Euler characteristic. Consider our utility graph and apply Euler\u2019s formula graph theory.\n\nIn order to prove that we can\u2019t represent this graph in the form without any intersecting edges, we need to use Euler\u2019s Formula graph theory on this. We find that there are 6\u00a0vertices and 9\u00a0edges. We need\u00a0to verify\u00a0Euler\u2019s formula and check for the number of faces.\n\nF + V \u2013 E = 2\n\nF + 6\u00a0\u2013 9 = 2\n\nF = 5\n\nIf each of the 5\u00a0faces had 4\u00a0edges bounding them, we get the graph as below.\n\nWe notice that we need 10\u00a0edges. However, the problem has only 9\u00a0edges. By this contradiction, we obtain Euler\u2019s formula proof. This two-dimensional planar graph\u00a0when inflated into a solid becomes an octahedron. An octahedron has 8\u00a0faces, 6\u00a0vertices, and 12\u00a0edges. Thus with the help of\u00a0Euler\u2019s formula proof,\u00a0it is impossible to make the utility connections.\n\nFaces + Vertices \u2212 Edges = 28 + 6 \u2212 12 = 2\n\n## What Is Euler\u2019s Formula Used For?\n\nF + V \u2212 E can equal 2\u00a0or 1 and have other values, so the more generic formula is\u00a0F + V \u2212 E = X, where X\u00a0is the Euler characteristic.\u00a0We verify Euler\u2019s formula to\u00a0study\u00a0any three-dimensional space\u00a0and not just polyhedra. Euler\u2019s graph theory proves that there are exactly 5\u00a0regular polyhedra. We can use Euler\u2019s formula calculator and verify if there is a\u00a0simple polyhedron with 10\u00a0faces and 17\u00a0vertices. The prism,\u00a0which has an octagon as its base, has 10\u00a0faces, but the number of vertices\u00a0is 16.\n\n## Verification of Euler\u2019s Formula for Solids\n\nEuler\u2019s formula examples include solid shapes\u00a0and\u00a0complex polyhedra. Let\u2019s verify the formula for\u00a0a few simple polyhedra such as a square pyramid and a triangular prism.\n\nA square pyramid has 5\u00a0faces, 5\u00a0vertices, and 8\u00a0edges.\n\nF + V \u2212 E = 5 + 5 \u2212 8 = 2\n\nA triangular prism\u00a05\u00a0faces, 6\u00a0vertices, and 9\u00a0edges.\n\nF + V \u2212 E = 5 + 6 \u2212 9 = 2\n\n## Euler\u2019s Formula Explanation\n\nThere are 5 platonic solids for which Euler\u2019s formula can be proved. They are cube, tetrahedron, octahedron, dodecahedron, and icosahedron. Let\u2019s verify Euler\u2019s formula on these complex polyhedra which serve as\u00a0Euler\u2019s formula examples.\n\n## Solved Examples Using Euler\u2019s Formula\n\nExample 1:\u00a0Express\u00a0ei(\u03c0\/2)\u00a0in\u00a0the\u00a0(a +\u00a0ib) form by using \u00a0Euler\u2019s formula.\n\nSolution:\n\nGiven: \u03b8 = \u03c0\/2\nUsing Euler\u2019s\u00a0formula,\n\nei\u03b8\u00a0= cos\u03b8 +\u00a0isin\u03b8\n\n\u27f9\u00a0ei(\u03c0\/2)\u00a0= cos(\u03c0\/2) +\u00a0isin(\u03c0\/2) = 0 +\u00a0i\u00a0\u00d7 1 =\u00a0i\n\nAnswer:\u00a0Hence\u00a0ei(\u03c0\/2)\u00a0in\u00a0the a +\u00a0ib\u00a0form is\u00a0i.\n\nExample 2:\u00a0Express\u00a03e5i\u00a0in\u00a0the\u00a0(a +\u00a0ib)\u00a0form by using\u00a0Euler\u2019s formula.\nSolution:\n\nGiven: \u03b8 =\u00a05\nUsing Euler\u2019s\u00a0formula,\n\nei\u03b8\u00a0= cos\u03b8 +\u00a0isin\u03b8\n\n\u27f9\u00a0e5i\u00a0= cos5 +\u00a0i\u00a0sin5 = 0.284 +\u00a0i(\u22120.959) = 0.284 \u2212 0.959i\n\nNow,\n3e5i\u00a0= 0.852 \u2013 2.877i\n\nAnswer:\u00a0Hence, 3e5i\u00a0in\u00a0the a + ib\u00a0form is 3e5i\u00a0= 0.852 \u2013 2.877i.\n\nExample 3:\u00a0Jack knows that a\u00a0polyhedron has 12 vertices and 30 edges. How can he find the number of faces?\nSolution:\n\nUsing Euler\u2019s formula:\n\nF + V \u2212 E = 2\n\nF + 12 \u2212 30 = 2\n\nF \u2212 18 = 2\n\nF = 20\n\nAnswer:\u00a0Number of faces = 20.\n\nExample 4:\u00a0Sophia finds a pentagonal prism in the laboratory. What do you think the value of F + V \u2212 E\u00a0is for it?\nSolution:\n\nA pentagonal prism has 7 faces, 15\u00a0edges, and 10\u00a0vertices.\n\nLet\u2019s apply Euler\u2019s formula here,\n\nF + V \u2212 E = 7 + 10 \u2212 15 = 2\n\nAnswer:\u00a0F + V \u2212 E for a pentagonal prism = 2.\n\n## FAQs on Euler\u2019s Formula\n\n### What is Euler\u2019s Formula for Solids?\n\nFor solid shapes, especially polyhedra,\u00a0the sum of the faces and vertices will be 2 more than their edges. Faces + vertices = edges + 2\n\n### What is Euler\u2019s Formula for the Cube?\n\nA cube, also known as a hexahedron has 6 faces, 8 vertices, and 12 edges, and satisfies Euler\u2019s formula. According to Euler\u2019s formula, F + V \u2212 E\u00a0= 6 + 8 \u2212 12 = 2\n\n### What is Euler\u2019s formula for Complex Numbers?\n\nEuler\u2019s formula for complex numbers is ei\u03b8\u00a0= icos\u03b8 + isin\u03b8\u00a0where\u00a0i is an imaginary number. Many trigonometric identities are derived from this formula.\n\n### What is Euler\u2019s Number?\n\nEuler\u2019s number or \u2018e\u2019, is an important constant, used across different branches of mathematics has a value of 2.71828.\n\n### What Is Euler\u2019s Formula Used For?\n\nEuler\u2019s formula in geometry is used for determining the relation between the faces and vertices of polyhedra. And in trigonometry, Euler\u2019s formula is used for tracing the unit circle.\n\n### What Are the Limitations of Euler\u2019s Formula?\n\nIn the field of civil engineering, the crippling stress increases with the decrease\u00a0in the slenderness ratio. In case it reaches zero, the crippling stress will touch infinity which isn\u2019t practically possible.\n\n### What Is the Purpose of Euler\u2019s Formula?\n\nThe purpose of Euler\u2019s formula in a polyhedron is to find the relationship between the number of vertices and edges. This further helps in solving problems related to this property.\n\n### What Does Euler\u2019s Formula Mean?\n\nWhen talking in terms of complex numbers, Euler\u2019s formula states that an imaginary or exponential growth will trace out a circle.\n\n## Euler\u2019s Formula and Trigonometry\n\nMath Formulas \u2b50\ufe0f\u2b50\ufe0f\u2b50\ufe0f\u2b50\ufe0f\u2b50","date":"2022-05-20 19:33:14","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8482915163040161, \"perplexity\": 848.0395963085092}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-21\/segments\/1652662534669.47\/warc\/CC-MAIN-20220520191810-20220520221810-00031.warc.gz\"}"} | null | null |
El municipio de Liberty (en inglés: Liberty Township) es un municipio ubicado en el condado de Susquehanna en el estado estadounidense de Pensilvania. En el año 2000 tenía una población de 1.266 habitantes y una densidad poblacional de 16 personas por km².
Geografía
El municipio de Liberty se encuentra ubicado en las coordenadas .
Demografía
Según la Oficina del Censo en 2000 los ingresos medios por hogar en la localidad eran de $33,750 y los ingresos medios por familia eran $38,333. Los hombres tenían unos ingresos medios de $26,771 frente a los $21,898 para las mujeres. La renta per cápita para la localidad era de $14,664. Alrededor del 13,6% de la población estaban por debajo del umbral de pobreza.
Referencias
Enlaces externos
Municipios de Pensilvania
Localidades del condado de Susquehanna | {
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Lukwago Rebecca Nalwanga is a Ugandan politician. She was the former female Member of Parliament in the eighth Parliament of Uganda representing Luweero District under the National Resistance Movement (NRM) political party.
Political life
The former Luweero district Member of the Parliament asked for a cash bailout from the President Museveni to assist her in settling legal fees piled up from a host of court petitions she filed. She was as a result of her 2011 election getting challenged to the Parliament by her political rivals.
Rebecca requested that every one who aids in crime to be charged and brought to book. She added and emphasized the need to restore the public's trust in courts before they resort to mob justice. In 2014, she was reported to have been one of the five contestants from the NRM including Elizabeth Lugudde, Ramlah Kadala, Lilian Nakate, Jimiya Ssenkanja and Rita Mugalu who declared their interest in the MP seat.
See also
List of members of the eighth Parliament of Uganda
References
Living people
Year of birth missing (living people)
People from Luweero District
National Resistance Movement politicians
Members of the Parliament of Uganda
Women members of the Parliament of Uganda | {
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Q: Allow negative decimal input on Android with the nice 3X3 numeric keypad I want to allow only negative numeric and decimal input with the nice numeric keypad that shows all the numbers in a 3x3 keypad. This keypad has a negative button, but it doesn't do anything, and I can't figure out how to activate it. The following is what I have, but it only allows numeric input:
<EditText
android:id="@+id/expense_amount"
android:layout_width="match_parent"
android:layout_height="wrap_content"
style="@style/FieldNumeric"
android:text="0.00"/>
I've seen several answers already that say to add
android:inputType="numberSigned"
or to use
TYPE_NUMBER_FLAG_SIGNED
among other suggestions. The problem with these answers is they lose the nice 3x3 numeric keypad and bring in the alphabetic keyboard.
Any suggestions would be awesome.
Thanks,
Devin
A: This worked for android 4.2.2 (Jellybean):
android:inputType="numberSigned|numberDecimal"
| {
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4327 Ries è un asteroide della fascia principale del diametro medio di circa 14,8 km. Scoperto nel 1982, presenta un'orbita caratterizzata da un semiasse maggiore pari a 2,7729358 UA e da un'eccentricità di 0,2243790, inclinata di 16,66722° rispetto all'eclittica.
Collegamenti esterni
Corpi celesti scoperti nel 1982
Ries | {
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Hazel Osborne Byford, Baroness Byford DBE (* 14. Januar 1941 in Rothley) ist eine britische Politikerin (Conservative Party).
Leben und Karriere
Byford wurde als Tochter von Sir Cyril Osborne (1898–1969) geboren. Ihr Vater war Politiker der Conservative Party und von 1945 bis zu seinem Tode im Jahre 1969 Mitglied des House of Commons für den Wahlkreis Louth in Lincolnshire. Byford besuchte zunächst die Portland House School in der London Road in Leicester, später dann die private St. Leonard's School in St Andrews. Nach ihrem Schulabschluss studierte sie Landwirtschaft am Northamptonshire Agricultural College in Moulton. Später war sie als Geflügelzüchterin tätig; sie züchtete Elterntiere für die Geflügelfirma Thornbers.
Von 1961 bis 1976 war Byford für den Women's Royal Voluntary Service tätig, die letzten fünf Jahre dieser Zeit als "County Organiser" in Leicestershire. Von 1989 bis 1994 war sie Mitglied des Transport Users' Consultative Committee und von 1994 bis 1995 beim Rail Users' Consultative Committee.
Sie hatte mehrere Parteiämter inne: Von 1986 bis 1993 war sie Vorsitzende des East Midlands Women's Conservative Committee, von 1989 bis 1990 stellvertretende Vorsitzende und ab 1990 bis 1993 Vorsitzende (Chairman) des National Committee der Conservative Women. 1996 war sie Vorsitzende beim Parteitag der Conservative Party (Conservative Party Conference) in Bournemouth. Von 1996 bis 1997 war sie Präsidentin der National Union of Conservative and Unionist Associations.
Byford übte zahlreiche weitere Ämter und Ehrenämter aus. 2007 wurde sie zur Präsidentin der Royal Association of British Dairy Farmers gewählt; dieses Amt hatte sie bis Juni 2010 inne. 2010 war sie Präsidentin des Royal Smithfield Club. Am 2. März 2010 wurde sie bei der Livery Company The Worshipful Company of Farmers als "Under Warden" in die Ratsversammlung (Court) gewählt.
Sie ist Mitglied des Domkapitels (Canon) der Kathedrale von Leicester. Sie ist Schirmherrin (Patron) von verschiedenen landwirtschaftlichen Organisationen: Women Food and Farming Union, National Farm Attractions Network und des Institute of the Agricultural Secretaries and Administrators. Sie ist Präsidentin der Leicestershire Clubs for Young People und der Organisation Linking Environment and Farming (LEAF). Sie ist außerdem Präsidentin der St Leonard's School in St Andrews.
Seit 2003 ist sie assoziiertes Mitglied (Fellow) der Royal Agricultural Society.
Mitgliedschaft im House of Lords
Byford wurde am 15. Oktober 1996 zum Life Peer als Baroness Byford, of Rothley in the County of Leicestershire, erhoben und am 19. November 1996 offiziell ins House of Lords eingeführt. Ihre Antrittsrede dort hielt sie am 27. November 1996.
Als Themen von politischem Interesse nennt sie auf der Webseite des Oberhauses das Landwirtschaft, ländliche Gebiete und ländliche Themen.
Von 1997 bis 1998 war sie Whip der Opposition. Von 1998 bis 2007 war sie Sprecherin der Opposition für Ernährung, Landwirtschaft und Ländliche Fragen. Byford war von Dezember 1998 bis 2002 Sprecherin der Opposition für Landwirtschaft und von 1998 bis 2003 für Umwelt.
Ihre Anwesenheit bei Sitzungstagen liegt im mittleren Bereich.
Wirken in der Öffentlichkeit
Im Januar 2009 hinterfragte sie in einer parlamentarischen Anfrage im House of Lords die Position der Regierung in Hinblick auf zu hohe Zahlungen an landwirtschaftliche Unternehmen im Rahmen der sog. 'Hill Farm Allowance' (Hochlandsprämie).
Im Rahmen der Veranstaltungsreihe Peers in Schools besuchte Byford im März 2012 das Guthlaxton College und berichtete dort unter anderem von ihrer Arbeit im House of Lords.
Ehrungen
Sie wurde 1994 als Dame Commander des Order of the British Empire geadelt. 2008 erhielt sie die Ehrendoktorwürde als Doctor of Science der Nottingham Trent University in Anerkennung ihrer außergewöhnlichen Verdienste um die Landwirtschaft. 2010 erhielt sie von der University of Leicester den Ehrendoktor als Doctor of Laws.
Familie
Sie heiratete 1962 C. Barrie Byford. Sie ist Mutter von zwei Kindern, ein Sohn (†) und eine Tochter.
Weblinks
Hazel Byford DBE, DL – Offizielle Webseite der Conservative Party
Hazel Byford, Baroness Byford – Offizielle Webseite des House of Lords
The Rt Hon the Baroness Byford, DBE – Biografie bei Debretts
Hazel Byford, Baroness Byford bei theyworkforyou
Hazel Byford, Baroness Byford bei The Public Whip
Einzelnachweise
Agrarpolitiker
Mitglied des House of Lords
Conservative-Party-Mitglied
Politiker (20. Jahrhundert)
Politiker (21. Jahrhundert)
Life Peer
Dame Commander des Order of the British Empire
Ehrendoktor der University of Leicester
Ehrendoktor der Nottingham Trent University
Brite
Geboren 1941
Frau | {
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} | 251 |
What is "mikey"
Crossword clues for mikey
Boy in Life cereal ads
Cereal eater of old ads
Mikey may refer to:
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Mikey Way, a bassist, formerly of the band My Chemical Romance
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Little Mikey, fictional character in a commercial for Life cereal
Mikey Simon, main character of the television series Kappa Mikey
Mikeyy, a computer worm
A nickname for Michael
An alien in the movie Men in Black
Mikey (film)
Mikey is a 1992 horror- thriller film directed by Dennis Dimster and starring Brian Bonsall. The film centers on the character of Mikey, a young boy who is adopted by a family after his previous adoptive family dies. Rather than the darling child they expected, however, Mikey turns out to be a violent psychopath and a budding serial killer.
Mikey (singer)
Mikey (Cho Myung-ik) ( Hangul: 마이키 ; Hanja: 趙明翼; born 30 Jan 1980) is a member of a popular South Korean band Turbo.
words rhyming with mikey, words from word "mikey", words starting with "m", words starting with "mi", words starting with "mik", words starting with "mike", words ending with "y", words ending with "ey", words ending with "key", words ending with "ikey", words containing "i", words containing "ik", words containing "ike", | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,423 |
Category Archives: Veronica Mars
Quadruple Z Episode #017 – Review of Veronica Mars S1x1 – Pilot
QZPodcast Veronica Mars No Comments
Join Tabz, Kim and Emma as they discuss one of their favorite shows Veronica Mars.
Veronica's junior year at Neptune High gets off to a rocky start as she is targeted by the P.C.H. Biker Gang for helping a snitch who later becomes one of her best friends.
Meanwhile, her dad, Keith, forbids Veronica to investigate Jake Kane, a local billionaire and father of Duncan Kane, her ex-boyfriend. Ignoring his orders, she makes a shocking discovery about her family's past (from TV.com)
This episode was edited by John N.
https://media.blubrry.com/quadz/p/quadruplez.com/mp3/QZ017_VeronicaMarsS1Ep1.mp3
TV Review: Veronica Mars 2×10 One Angry Veronica
John Pavlich
Reviews TV Veronica Mars No Comments
Buy Veronica Mars – The Complete Second Season on Amazon.com
It's the end of the school day, just before Christmas vacation, and Veronica Mars is severely lacking in holiday cheer. Not even the cafeteria's Christmas Cake has the power to lift her spirits. Who could blame her? She's dating Duncan Kane, again. I kid, but seriously, little Miss Mars has good reason to be all "Bah, Humbug!"
Her friend Meg is in the hospital. What's more, the bed-ridden girl is pregnant with Duncan's baby. It's a long story. Meanwhile, Veronica's gumshoe Dad, Keith has been assigned by the Mayor (Steve Guttenberg. That's right.) to investigate Neptune's finest. It seems someone broke into the evidence room and stole some incriminating video footage. The tapes feature a roll in the sheets between Aaron Echolls, the movie star Father of Veronica's previous boyfriend, and Lilly Kane, Veronica's dead best friend and Duncan's Sister. That's an even longer story.
All of that pertains to the continued, over arching mystery of the season proper. However, the real treat of this episode comes in the form of a subplot involving Veronica getting stuck with Jury Duty. Once she is bestowed the title of Foreman, it's off to the races. In a smart, delightful tribute to Sidney Lumet's classic film adaptation of Reginald Rose's TV play, 12 Angry Men, One Angry Veronica deals with a curious case about two stoners, a hooker and her pimp. This side story takes place in just one room, and is completely dialogue and character driven, making it my favorite episode of the entire second season. In fact, one could almost edit everything else out and safely watch just these scenes in the jury room.
Though Christmas first seems like a simple, inconsequential backdrop, the holiday setting does actually manage to have relevance to the episode, one of two scenes in particular actually directly influences the jury room B-plot. First, in what is my personal favorite scene of the episode (as Father/Daughter scenes in the Mars household usually are), Keith comes home from his investigation at the precinct, having not much luck. He opens the door to a "Winter Wonderland" of colored lights, candy canes and tinsel. At the center of it all, Veronica has prepared a feast fit for a family of two with a middle-class income. The both of them decide to spend this important time together, before they go back to dealing with violence, sex and theft during Christmas. I love it when Keith refers to Veronica as simply, "Elf."
This scene is crucial in that it not only provides much needed warmth and positivity for the characters and the audience, but it also helps Veronica save the day in the jury room. Her Dad upgrades her computer's hard drive and processing power as an early Christmas present, but he also conveniently leaves substantial information on the screen that leads Veronica to later solve the case. Mars Investigations strikes again.
The second instance of Christmas magic comes at the end of the episode, when Veronica needs it the most. Though she solves the case, she learns some devastating news about one of her dear friends. Fed up with the recent events of this depressing holiday, Veronica accepts defeat and settles for New Year's pizza with her Dad. That is, until a Christmas miracle happens and Veronica gets a surprise visit from a long-lost best friend. This finally renews her sense of Christmas spirit, and Veronica looks ahead to the new year with hope.
I bet that Christmas Cake would be much more appreciated now.
(Photo: Warner Brothers)
TV Review: Veronica Mars 1×10 "An Echoll's Family Christmas"
Buy Veronica Mars – The Complete First Season on Amazon.com
This Christmas episode from season one of Veronica Mars starts out with Veronica and Keith decorating the tree. This caring family environment is quickly brought into stark contrast when we head to the Echoll's house. Logan is hosting a poker game. Duncan appears to be drunk. Weirdly, Weevil is playing with them. We've come in at the end of the game. It's down to Logan and Weevil. Weevil wins and Logan goes to pay him, however the money box is empty. This does not go over well with Weevil. Through Weevil's efforts to find the money, we find out Duncan and Logan both own the same reindeer boxer shorts.
Weevil starts stealing the 09ers possessions, saying they'll get them back when he gets his money. Duncan's laptop is missing, so Veronica goes to Weevil and says she'll solve the case. Weevil tells Veronica how he bought into the poker game.
Meanwhile, Keith is hired by Lynn Echolls to find out who is stalking her husband, Aaron. They're throwing a big Christmas party and she doesn't want anything to go wrong. This doesn't end up working out too well, and Aaron's affairs are brought into the open.
Duncan thinks Logan stole the money, so he and Logan aren't getting along too well. Veronica talks to Sean and Connor, who were also there. The clues keep coming. Everyone had an opportunity to steal the money, and everyone seems to have been hiding something.
I won't give away the answer to the mystery, but it's a good one. Veronica makes them all gather for another poker game and buy her in, she wants to send her dad to baseball camp for Christmas. Before the game, she unravels the mystery, then takes the place of the guilty party.
This is one of my favorite episodes of Veronica Mars. I often have no idea who commits the crimes on this show, but this case was extremely surprising to me. I also really enjoy the scene with Veronica and Logan. It is a hint at what's to come and has the great line, "Annoy tiny blonde one, annoy like the wind."
This is also a big episode in the overall continuity of the show. This isn't just a throwaway holiday episode. There are things in this episode that affect the rest of the series. I think this is an excellent example of a great holiday episode of television.
(photo: Warner Brothers/vm-caps.com) | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 6,883 |
\section{Strong Stability Preserving Runge--Kutta methods}
\vspace{-0.05in}
The exact solution of a hyperbolic conservation law of the form
\begin{eqnarray}
\label{pde}
U_t + f(U)_x = 0,
\end{eqnarray}
frequently develops sharp gradients or discontinuities, which may cause significant difficulties
in numerical simulations. For this reason, the development of high order spatial discretizations
that can handle such discontinuities or sharp gradients has been an area of active research for the past
few decades. Such methods have special nonlinear non-inner-product stability properties that mimic
some significant physical properties of the exact solution, such as total variation stability, a maximum principle,
or positivity. These properties ensure that when \eqref{pde} is discretized in space, the spatial discretization
$F(u)$ satisfies
\begin{equation} \label{eqn:FEcond}
\| u^n + \Delta t F(u^{n}) \| \leq \| u^n \|, \quad 0 \leq \Delta t \leq \Delta t_{FE},
\end{equation}
where $u^n$ is a discrete approximation to $U$ at time $t^n$ and
$\| \cdot \|$ is the desired norm, semi-norm, or convex functional. In other words, when the semi-discretized system
\begin{eqnarray}
\label{ode}
u_t = F(u),
\end{eqnarray}
is evolved forward in time using a first order explicit Euler method, the numerical
solution satisfies the desired strong stability property, as long as the time-step is suitably limited.
In practice we want to use a higher order time integrator instead of the first order forward Euler method
\eqref{eqn:FEcond}, but we still need the strong stability property
$ \| u^{n+1} \| \le \|u^n\| $
to be satisfied, perhaps under a modified time-step restriction.
Some higher order Runge--Kutta methods can be decomposed into convex combinations
of forward Euler steps \cite{shu1988b}, so that any convex functional property satisfied by \eqref{eqn:FEcond}
is automatically {\em preserved}, usually under a different time-step restriction.
For example, if the $s$-stage implicit Runge--Kutta method is written in the form \cite{shu1988, SSPbook2011},
\begin{eqnarray}
\label{rkSO}
y^{(i)} & = & v_i u^n + \sum_{j=1}^{s} \left( \alpha_{i,j} y^{(j)} + \Delta t \beta_{i,j} F(y^{(j)}) \right),
\; \; \; \; i=1, . . ., s \\% \quad \aik \geq 0, \qquad i=1 ,..., m \\
u^{n+1} & = & v_{s+1} u^n + \sum_{j=1}^{s} \left( \alpha_{s+1,j} y^{(j)} + \Delta t \beta_{s+1,j} F(y^{(j)}) \right), \nonumber
\end{eqnarray}
it is obvious that each stage can be written as a linear combination of the solution at the
previous step and forward Euler steps of the stages
\begin{subequations}
\begin{equation}
y^{(i)} = v_i u^n + \sum_{j=1}^{s} \alpha_{i,j} \left( y^{(j)} + \Delta t \frac{\beta_{i,j}}{\alpha_{i,j}} F(y^{(j)}) \right),
\end{equation}
\mbox{and}
\begin{equation}
u^{n+1} = v_{s+1} u^n + \sum_{j=1}^{s} \alpha_{s+1,j} \left( y^{(j)} + \Delta t \frac{\beta_{s+1,j}}{\alpha_{s+1,j}} F(y^{(j)}) \right).
\end{equation}
\end{subequations}
If the coefficients $v_i$, $\alpha_{i,j}$ and $\beta_{i,j}$ are all non-negative and
$\beta_{i,j}$ is zero whenever the corresponding $\alpha_{i,j}$ is zero, the stages decompose into a
convex combination of forward Euler steps of the form \eqref{eqn:FEcond},
with each time step replaced by $ \frac{\beta_{i,j}}{\alpha_{i,j}} \Delta t$. Each stage is then bounded by
\begin{subequations}
\begin{equation} \label{ccbound}
\| y^{(i)}\|
\leq v_i \left\| u^n \right\| + \sum_{j=1}^{s} \alpha_{i,j} \, \left\| y^{(j)} + \Delta t \frac{\beta_{i,j}}{\alpha_{i,j}} F(y^{(j}) \right\|
\end{equation}
\mbox{with the new solution value}
\begin{equation} \label{ccbound2}
\| u^{n+1} \|
\leq v_{s+1} \left\| u^n \right\| + \sum_{j=1}^{s} \alpha_{s+1,j} \, \left\| y^{(j)} + \Delta t \frac{ \beta_{s+1,j} }{ \alpha_{s+1,j} } F(y^{(j}) \right\| .
\end{equation}
\end{subequations}
Recall that $v_i+ \sum_{j=1}^{s} \alpha_{i,j} =1$ for consistency, and that
each $\| y^{(j)} + \Delta t \frac{\beta_{i,j}}{\alpha_{i,j}} F(y^{(j)}) \| \leq \| y^{(j)} \|$ for
$ \frac{\beta_{i,j}}{\alpha_{i,j}} \Delta t \leq \dt_{\textup{FE}}$ from \eqref{eqn:FEcond}, so we have
\begin{eqnarray}
\label{rkSSP}
\| y^{(i)}\| \leq \| u^{n}\| \; \; \; \mbox{and} \; \; \; \; \|u^{n+1} \| \leq \| u^n \| \; \; \; \; \mbox{for any} \; \; \;
\Delta t \leq \mathcal{C} \dt_{\textup{FE}}
\end{eqnarray}
where $\mathcal{C} = \min_{i,j} \frac{\alpha_{i,j}}{\beta_{i,j}}$.
(If any of the $\beta_{i,j}$'s are equal to zero, we consider the corresponding
ratios to be infinite.)
Any method that can be written in this form with $\mathcal{C}>0$ is called a
{\em strong stability preserving (SSP)} method.
Strong stability preserving second and third order explicit Runge--Kutta methods \cite{shu1988}
and fourth order methods \cite{SpiteriRuuth2002,ketcheson2008}
were developed using this convex combination approach. These methods ensure that any
strong stability property satisfied by the spatial discretization when using the forward Euler condition
\eqref{eqn:FEcond} is preserved by the higher order strong stability preserving Runge--Kutta method.
Furthermore, the convex combination decomposition above ensures that the strong stability property is also
satisfied by the intermediate stages in a Runge--Kutta method. This may be desirable in many applications,
notably in simulations that require positivity. Clearly, the condition above is sufficient for preservation of strong stability;
in \cite{ferracina2004, ferracina2005,higueras2004a, higueras2005a} it was shown that this condition is necessary, as well.
While the time-step depends on both the spatial and temporal discretizations,
we isolate the contribution of the temporal discretization to the time-step
restriction by considering the ratio of the allowable time-step of the high order method to the
forward Euler time-step. This ratio is called the {\em strong stability preserving coefficient}.
Using this approach, we view the time-step restriction \eqref{rkSSP} as a combination of two
factors: the forward Euler time-step $\dt_{\textup{FE}}$ that comes from the spatial discretization, and
the SSP coefficient $\mathcal{C}$ that is a property only of the time-discretization. For this reason,
the research on SSP methods focuses on optimizing the allowable time-step $\Delta t \le \mathcal{C} \dt_{\textup{FE}}$ by maximizing
the {\em SSP coefficient} $\mathcal{C}$ of the method. Among methods of similar types, a more
relevant quantity is the {\em effective SSP coefficient} $\sspcoef_{\textup{eff}} = \frac{\mathcal{C}}{s}$, which takes into account
the computational cost of the method at each iteration, defined by the number of stages $s$ (typically also the number
of function evaluations per time-step). However, this measure is not applicable when comparing explicit and implicit methods.
Table \ref{table:RK_ceff} gives SSP coefficients for
explicit SSP Runge--Kutta methods of orders $p \leq 4$ and implicit methods of orders $p \leq 6$ and different stages, and the
ratio of the SSP methods of corresponding implicit and explicit methods.
Explicit Runge--Kutta methods (and in fact all explicit general linear methods)
have a bound on the SSP coefficient $\mathcal{C} \leq s$ \cite{SSPbook2011}, while
all optimal implicit Runge--Kutta methods have been observed to have $ \mathcal{C} \ \leq 2s$ \cite{ketcheson2009},
which is only twice that of the explicit method.
However, an implicit methods of a {\em given number} of stages and order typically
has an SSP coefficient that is greater than twice that of the corresponding explicit Runge--Kutta method, as shown in
Table \ref{table:RK_ceff}.
\begin{table}
\begin{center}
\begin{tabular}{|c|ccc|ccccc|ccc|} \hline
& \multicolumn{3}{c|}{Explicit Methods} &\multicolumn{5}{c|}{Implicit Methods} & \multicolumn{3}{c|}{Ratio of Im/Ex} \\ \hline
$s$ $\backslash$ $p$ & 2 & 3 & 4 & 2 & 3 & 4 & 5 & 6 & 2 & 3 & 4 \\ \hline
1 & - & - & - & 2 & - & - & - & - & & & \\
2 & 1 & - & - & 4 & 2.73 & - & - & - & 4 & & \\
3 & 2 & 1 & - & 6 & 4.83 & 2.05 & - & - & 3 & 4.83 & \\
4 & 3 & 2 & - & 8 & 6.87 & 4.42 &1.14 & & 2.67 & 3.44 & \\
5 & 4 & 2.65 & 1.51 & 10 & 8.90 & 6.04 & 3.19 & & 2.50 & 3.36 & 4.00 \\
6 & 5 & 3.54 & 2.28 & 12& 10.92 & 7.80 & 4.97 & 0.18 & 2.40 & 3.08 & 3.42 \\
7 & 6 & 4.27 & 3.29 & 14 & 12.93& 9.19 & 6.21 & 0.26 & 2.33 & 3.03 & 2.79\\
8 & 7 & 5.12 & 4.15 & 16 & 14.94 & 10.67 & 7.56 & 2.25 &2.29 & 2.92 & 2.57 \\
9 & 8 & 6.03 & 4.86 & 18 & 16.94& 12.04 & 8.90 & 5.80 & 2.25 & 2.81 & 2.48 \\
10 & 9 & 6.80 & 6.00 & 20 & 18.95 & 13.64& 10.13 & 8.10 & 2.22 & 2.79 & 2.27 \\
11 & 10 &7.59 & 6.50 & 22 & 20.95 & 15.18 & 11.33 & 8.85 & 2.20 & 2.76 & 2.34 \\
\hline
\end{tabular}}
\caption{SSP coefficients of optimal explicit and implicit Runge--Kutta methods and the ratio of the implicit coefficient to the explicit
coefficient.
A dash indicates that SSP methods of this type cannot exist, a blank space indicates none were found.}
\label{table:RK_ceff}
\end{center}
\end{table}
In addition to bounds on the effective SSP coefficients, SSP Runge--Kutta methods also suffer from barriers on their
order: explicit Runge--Kutta methods with positive SSP coefficient cannot be more than
fourth-order accurate \cite{kraaijevanger1991,ruuth2001} and implicit Runge--Kutta methods with positive SSP coefficient cannot be more than
sixth-order accurate \cite{ketcheson2009,SSPbook2011}.
These restrictive order barriers of SSP Runge--Kutta methods is a result of
the nonlinearity of the ODEs. However, if we are only interested in the order of accuracy on linear autonomous ODE systems,
Runge--Kutta methods need only satisfy a smaller set of order conditions.
If we only want the method to have high {\em linear} order ($p_{lin}$), then the order barrier is
broken and Runge--Kutta methods with positive SSP coefficients exist for arbitrarily high linear orders.
These methods are of interest because their SSP coefficients
serve as upper bounds for nonlinear methods. However, they are also useful in their own right for linear problems where the
strong stability preserving property is required, such as Maxwell's equations and the equations of linear elasticity.
In \cite{LNL} the observation that the order barrier is only applicable to the nonlinear order
led to the study of explicit SSP Runge--Kutta methods with high
{\em linear order} and optimal nonlinear order $p=4$, which demonstrated that
where high linear order is desired, going to higher nonlinear order costs little or nothing in terms of the SSP coefficient.
We refer to methods that may have a higher linear order than nonlinear order ($p_{lin} \geq p$)
as {\em linear/nonlinear (LNL) methods}.
Table \ref{table:LNL} shows the optimized SSP coefficients of these explicit SSP LNL Runge--Kutta methods.
These methods may be of particular value for problems that require high linear order throughout but also have
some nonlinear features in some regions which benefit from order greater than two.
The explicit LNL methods found in \cite{LNL} suggest that implicit methods in this class could be beneficial as well.
Implicit SSP Runge--Kutta methods with very high linear order have not been widely studied, and these methods could
be very useful for linear problems. In addition, requiring that these methods have a nonlinear order $p>2$ would allow
these methods to be more useful if the problem is generally linear but has regions that feature nonlinearities.
In this work, we expand upon the explicit methods in \cite{LNL} and seek optimal implicit SSP methods with very high linear order
and nonlinear orders $p\leq 6$.
We first consider implicit SSP Runge--Kutta methods that have high order for linear problems and nonlinear order $p=2$.
We then consider optimal SSP methods with nonlinear orders $p=3,4,5,6$ and compare the SSP coefficients of these
optimal methods. These methods are presented in Section \ref{Implicit_LNL}. These implicit SSP LNL Runge--Kutta methods
have SSP coefficients that are up to six times the size of the corresponding explicit SSP LNL Runge--Kutta methods.
In Section \ref{Numerical1} we verify the convergence of these methods as well as the sharpness of the SSP coefficient on sample problems.
Next, in Section \ref{IMEX_LNL} we consider implicit-explicit Runge--Kutta methods, for use with problems of the form
\[u_t = F(u) + G(u) \] where we wish to treat $F$ explicitly and $G$ implicitly.
We formulate an optimization problem that depends on the ratio of the forward Euler
condition of the two components $F$ and $G$. Using the optimization routine we developed, we found SSP IMEX Runge--Kutta
methods where the explicit component has nonlinear order $p^e=3$ and $p^e=4$
and higher linear order $p_{lin}>4$, and the implicit component has nonlinear orders $p^i=2,3,4$ with higher linear order $p_{lin}$
that matches the linear order of the explicit part.
In Section \ref{IMEX_LNL_methods} we present the optimal methods of this type and
in Section \ref{Numerical2} we verify the convergence of these methods and show the behavior of these methods
on sample problems that require the SSP property.
\section{Implicit Runge--Kutta with high linear order} \label{Implicit_LNL}
\subsection{The SSP optimization problem}
Our goal is to find implicit SSP Runge--Kutta methods of a given order and number of stages with the largest possible SSP coefficient $\mathcal{C}$.
While the most convenient formulation for directly observing the SSP coefficient of a Runge--Kutta method
is the Shu-Osher form \eqref{rkSO}, for formulating the optimization problem
it is more convenient to use the Butcher form
\begin{eqnarray} \label{butcher}
u^{(i)} & = & u^{n} + \Delta t \sum_{j=1}^{s} a_{ij} F(u^{(j)}) \; \; \; \; (1 \leq i \leq s)
\\ \nonumber
u^{n+1} & = & u^n + \Delta t \sum_{j=1}^s b_j F(u^{(j)}).
\end{eqnarray}
(where the coefficients $a_{ij}$ are place into the matrix $\m{A}$ and $b_{j}$ into the row vector $\vb$) \cite{ketcheson2008}.
The reason this approach is preferable is that the Shu-Osher form of a Runge--Kutta method is not unique,
while the Butcher form is, so that rather than perform a search for an optimal convex combination of the Shu-Osher form,
we seek the unique optimal Butcher coefficients \eqref{rkSO}.
Following the approach developed by Ketcheson \cite{ketcheson2008},and used in
\cite{ketcheson2008, ketcheson2009, Ketcheson2010, SSPbook2011, tsrk, LNL} to find
optimal SSP methods, we aim to maximize the value of $r$ under the following constraints:
\begin{subequations} \label{optimization}
\begin{align}
\left( \begin{array}{ll} \m{A} & 0 \\ \vb & 0 \\ \end{array} \right)
\left(\m{I} + r
\left( \begin{array}{ll} \m{A} & 0 \\ \vb & 0 \\ \end{array} \right)
\right)^{-1} \geq 0
& \; \; \; \mbox{component wise.} \\
\left\| r \left( \begin{array}{ll} \m{A} & 0 \\ \vb & 0 \\ \end{array} \right)
\left(\m{I} + r \left( \begin{array}{ll} \m{A} & 0 \\ \vb & 0 \\ \end{array} \right)
\right)^{-1} \right\|_\infty \leq 1 \\
\tau_k(\m{A}, b) = 0 \; \; \; \mbox{for} \; \; \; k=1, . . ., P, \label{tau_conditions}
\end{align}
\end{subequations}
where $\tau_k$ are the order conditions, described below.
This optimization gives the Butcher coefficients $\m{A}$ and $\vb$ and an optimal value
of the SSP coefficient $\mathcal{C} = r$.
The conversion from the optimal Butcher form to the
canonical Shu-Osher form is given in \cite{SSPbook2011}, and given the matrix $\m{A}$, the vector $\vb$ and the SSP coefficient
$r$, the Shu-Osher coefficients $\alpha$ and $\beta$ can be easily accomplished in MATLAB by
\begin{verbatim}
s=size(A,1);
K=[A;b'];
G=eye(s)+r*A;
beta=K/G;
alpha=r*beta;
\end{verbatim}
where the coefficients $v_i$ are then computed by the consistency condition $v_i + \sum_{j=1}^s \alpha_{i,j} =1$.
Some of the major constraints in the optimization problem come from the
order conditions $\tau_k(\m{A}, \vb)$ above.
For to demonstrate the correct order of accuracy for nonlinear problems,
it must satisfy the order conditions:
\smallskip
\noindent $\tau_1(\m{A}, \vb)$: \; $\vb^T \ve = 1$ \\
$\tau_2(\m{A}, \vb)$: \; $ \vb^T \vc = \frac{1}{2}$ \\
$\tau_3(\m{A}, \vb)$: \; $ \vb^T\m{A} \vc = \frac{1}{6}$ \; and \; $ \vb^T \vc^2 = \frac{1}{3}$ \\
$\tau_4(\m{A}, \vb)$: \; $\vb^T\m{A}^2 \vc = \frac{1}{24}$ \; and \; $ \vb^T \m{A} \vc^2 = \frac{1}{12}$ \; and \;
$\vb^T \vc\m{A} \vc = \frac{1}{8}$ \; and \;
$ \vb^T \vc^3 = \frac{1}{4} $ , \\
where $\vc= \m{A} \ve$ and $\ve$ is a vector of ones. Thus, for first order there is only one condition ($P=1$ in Equation
\eqref{tau_conditions} above),
for second order two ($P=2$), for third order four ($P=4$), and for fourth order eight conditions are needed ($P=8$).
For fifth order, we require a total of $P=17$ conditions, and for sixth order $P=37$ conditions must be satisfied.
It is well-known that there are no implicit SSP Runge--Kutta methods greater than sixth order. In this work, we will consider
only implicit methods up to order $p=6$, but we will allow the methods to satisfy higher order $p_{lin}\geq p$
when applied to a linear problem.
In this case, the order conditions simplify, and can be expressed as
\begin{eqnarray}
\tau^{lin}_q(\m{A}, \vb) = \vb^T \m{A}^{q-2} \vc = \vb^T \m{A}^{q-1} \ve = \frac{1}{q!} \; \; \; \; \; \forall q=1, . . . , p_{lin}.
\end{eqnarray}
In the following sections we give the SSP conditions of the optimized methods resulting from the optimization problem given in
\eqref{optimization} with order conditions
\[ \tau_k(\m{A}, \vb) \; \; \; \mbox{for} \; \; 1 \leq k \leq p \]
and
\[ \tau^{lin}_q(\m{A}, \vb) \; \; \; \mbox{for} \; \; p \leq q \leq p_{lin} \]
which we implemented in a MATLAB code \cite{SSPimplicitLNL_github}. The coefficients of these optimized methods
can be downloaded from \cite{SSPimplicitLNL_github}.
\subsection{Optimal SSP LNL implicit Runge--Kutta methods} \label{implicit_LNL}
We started our search by considering fully implicit Runge--Kutta methods.
However, as in \cite{LNL} these methods had SSP coefficients that were identical to those of the
optimized diagonally implicit Runge--Kutta (DIRK) methods, so we proceeded to consider only DIRK methods.
We first found SSP DIRK methods of up to $s=9$ stages with
with linear order $2 \leq p_{lin} \leq s+1 $ and nonlinear order $p=2$.
These methods tend to have nice low-storage forms when written in their Shu-Osher arrays.
Our first observation is that the methods with nonlinear order $p=2$ and linear order $p_{lin}=3$
the same SSP coefficient as the third order ($p=3$) optimal implicit SSP Runge--Kutta methods listed
\cite{ketcheson2009,SSPbook2011}.
Clearly, the additional nonlinear order condition imposed did not constrain the methods any further.
A similar behavior is observed for SSP DIRK methods with $p_{lin}\geq 4$ and nonlinear orders $p=2$ and $p=3$,
which has SSP coefficients that are the same up to two decimal places,
except for the case of $s=3$ and $p_{lin}=4$ for which there was a difference of $\approx 1.43 \times 10^{-2}$,
and the $s=4$ and $p_{lin}=5$ where there was a difference of $\approx 2 \times 10^{-2}$. The SSP coefficients are
given in Table \ref{tab:ceff} (on left). Finding optimized DIRK SSP Runge--Kutta methods with $p=4$ and
$\leq 4 p_{lin}\geq 9$, we see that the SSP coefficients in this case are smaller than the methods with nonlinear
orders $p=2,3$, but as the number of stages increases this difference diminishes, as can be seen on
Table \ref{tab:ceff} (on right). This is also clear in Figure \ref{fig:SSPiRKcoef1}, which
plots the SSP coefficients of methods of nonlinear order $p=2$ (and therefore of $p=3$ as well)
and $p=4$ with different linear orders $3 \leq p_{in} \leq 9$.
In this figure we observe that if we compare methods with a given $p_{lin}$,
the SSP coefficient of the methods with nonlinear order $p=2$ in does not have significantly
higher SSP coefficient than the methods with nonlinear order $p=4$. This would seem to suggest that
the linear order is the main constraint on the SSP coefficient, as was the case for
the explicit methods in \cite{LNL}. However, Figure \ref{fig:SSPiRKcoef2}, which looks at methods with
higher nonlinear orders $p=5$ and $p=6$ tells a different story.
In this figure we compare three methods compare three methods with linear order $p_{lin}=5$ and nonlinear orders
$p=2,4,5$ (in blue solid, dot-dashed with circle marker, and dashed with + markers) and similarly,
three methods with linear order $p_{lin}=6$ and nonlinear orders
$p=2,4,6$ (in red solid, dot-dashed with circle marker, and dashed with + markers)
We say that in this case, as we go to higher nonlinear orders the SSP coefficient is significantly reduced.
For example, a six stage ($s=6$) sixth nonlinear order method has an SSP coefficient
$\mathcal{C}=0.18$ whereas the six stage LNL method with nonlinear order $p=4$ and linear order $p_{lin}=6$
has an SSP coefficient $\mathcal{C}=5.14$.
The SSP coefficients of the implicit SSP LNL Runge--Kutta methods in \ref{tab:ceff} should be compared to those of the
explicit SSP LNL Runge--Kutta methods \ref{table:LNL}. we see that the implicit methods have SSP coefficients that are
significantly larger than the corresponding explicit methods: up to six times larger. This suggests that the cost of the implicit
solver may be offset by the larger allowable time-step in some cases.
While the methods in this section are not necessarily suitable for every application,
they are valuable in that they provide an upper bound on the possible SSP coefficient for a given number of
stages and linear and nonlinear orders, and demonstrate the effect of increasing the nonlinear order.
\begin{table}{\normalsize
\begin{center}
\begin{tabular}{|l|ccccc||} \hline
&\multicolumn{5}{|c||}{Explicit $p=2,p=3$ methods} \\ \hline
$s \backslash p_{lin}$ &5&6&7&8&9\\\hline
5 &1.00&--&--&--&--\\ \hline
6 &2.00&1.00&--&--&-- \\ \hline
7 &2.65&2&1.00&--&---\\ \hline
8 &3.37&2.65&2.00&1.00&--\\ \hline
9 &4.1&3.37&2.65&2.00&1.00\\ \hline
\end{tabular} \hspace{-.1in}
\begin{tabular}{|ccccc|} \hline
\multicolumn{5}{|c|}{Explicit $p=4$ methods} \\ \hline
5 &6&7&8&9 \\ \hline
0.76 &--&--&--&--\\ \hline
1.81 &0.87&--&--&--\\ \hline
2.57 &1.83 &{\bf 1.00} &--&--\\ \hline
3.36 &2.56&1.93&{\bf 1.00}&--\\ \hline
4.03 &3.35&2.62&1.95&{\bf 1.00}\\ \hline
\end{tabular}
\caption{SSP coefficients of LNL explicit Runge--Kutta methods from \cite{LNL}.
A dash indicates that SSP methods of this type cannot exist, and the boldface indicates that the SSP coefficients are equal to the
corresponding ones of lower order.}
\label{table:LNL}
\end{center}}
\end{table}
\begin{table}
\begin{center}
\begin{tabular}{|l|cccccc||cccccc|}
\hline
&\multicolumn{6}{|c||}{Implicit $p=2,p=3$ methods} &\multicolumn{6}{|c|}{Implicit $p=4$ methods} \\ \hline
$s \backslash p_{lin}$ &4&5&6&7&8&9 &4&5&6&7&8&9\\ \hline
3 &3.24&-- & --&--&--&-- &2.05&-&-&-&-&- \\ \hline
4 & 4.56&3.64&--&--&--&-- & 4.42&3.54&--&--&--&-- \\ \hline
5 & 6.15&5.01&3.97&--&--&-- & 6.04&4.63 & 3.81&--&--&-- \\ \hline
6 & 7.85&6.66&5.17&4.27&--&-- & 7.80&6.49&5.14&4.14&--&-- \\ \hline
7 & 9.60&8.42&6.54&5.51&4.54&-- & 9.19&7.86&6.42&5.42&4.46&-- \\ \hline
8 & 11.23&10.21&7.92&6.91&5.82 & 4.79 &10.67&9.25&7.86&6.82&5.80&4.74 \\ \hline
9 & 12.81&11.82&9.48&8.33&7.03&6.10
&12.04&11.15&9.46&8.32&6.98&6.08 \\ \hline
\end{tabular}
\caption{\label{tab:ceff} The value of $\mathcal{C}$ for LNL iRK with linear order $p_{lin}$, and nonlinear order $p=2$ on left and $p=4$ on right.
Note along the diagonal these the SSP coefficients are close to six times larger than the corresponding explicit LNL methods.}
\end{center}
\end{table}
\begin{figure}[htb]
\centering
\begin{minipage}{0.45\textwidth}
\centering
\includegraphics[width=1.025\linewidth]{SSPCoeff_figure}
\caption{SSP coefficients of methods with nonlinear order $p=2$ (solid lines) and nonlinear order $p=4$ (dashed lines)
for linear orders $4 \leq p_{lin} \leq 9$, for increasing stages. We note that the SSP coefficients of $p=2$ and $p=3$
are close too identical for all methods with $s$ stages, and are not
significantly different for the two nonlinear orders $p=2$ and $p=4$, and get closer as $s$ increases.}
\label{fig:SSPiRKcoef1}
\end{minipage}%
\hspace{.2in}
\begin{minipage}{0.45\textwidth}
\centering
\includegraphics[width=0.95\linewidth]{SSPIRKVLNL}
\caption{Comparison of SSP coefficients of methods with linear order $p_{lin}=5$ (blue) and $p_{lin}=6$ (red) for several nonlinear orders.
The $p=2$ lines are solid, $p=4$ lines are dot-dashed with circle marker, and $p=p_{lin}$ lines are dashed with + markers.
Clearly, the SSP coefficient of methods with the same number of stages and linear order is significantly smaller for larger nonlinear orders.}
\label{fig:SSPiRKcoef2}
\end{minipage}
\end{figure}
\subsubsection{Coefficients of selected optimal methods}
The coefficients of all the methods listed in the section above can be downloaded as {\tt .mat} files from \cite{SSPimplicitLNL_github}.
In this section we list, for the user's convenience, the non-zero coefficients of three methods of particular interest.
We use the canonical Shu--Osher form here, where all the forward-Euler steps are of the same size
$\frac{\Delta t}{r}$ and $r=\mathcal{C}$:
\begin{eqnarray}
\label{cS-Oform}
y^{(i)} & = & v_i u^n + \sum_{j=1}^{s} \alpha_{i,j} \left( y^{(j)} + \frac{\Delta t}{r} F(y^{(j)}) \right),
\; \; \; \; i=1, . . ., s \\% \quad \aik \geq 0, \qquad i=1 ,..., m \\
u^{n+1} & = & v_{s+1} u^n + \sum_{j=1}^{s} \alpha_{s+1,j} \left( y^{(j)} + \frac{dt}{r} \beta_{s+1,j} F(y^{(j)}) \right), \nonumber
\end{eqnarray}
\noindent {\bf SSP iRK LNL $p=4$, $s=p_{lin}=6$:} One of the most efficient method produced was
the $s=p_{lin}=6$ LNL method of nonlinear order $p=4$ that has SSP coefficient $\mathcal{C} =5.138$.
The corresponding implicit Runge--Kutta method in \cite{SSPbook2011} which has $s=p=p_{lin}=6$ has a much smaller
SSP coefficient of $\mathcal{C}=0.18$. The non-zero coefficients are:
\smallskip
{\small
\begin{tabular}{lll}
$\alpha_{1,1} = 0.227696764527492,$ & $ \alpha_{5,2}= 0.273146312340082, $ & $ \alpha_{7,3}= 0.140604847510042,$ \\
$\alpha_{2,1} = 0.773299008278988, $& $\alpha_{5,4}= 0.468182990851259, $& $\alpha_{7,4}= 0.134029552181827,$ \\
$\alpha_{2,2}= 0.226700991721012, $& $\alpha_{5,5}= 0.226105041192215, $ & $ \alpha_{7,6}= 0.703213057832428,$ \\
$\alpha_{3,2}= 0.566850708114719, $& $\alpha_{6,5}= 0.772671881656312, $ & $v_1 = 0.772303235472508,$ \\
$\alpha_{3,3}= 0.245119620891410, $& $\alpha_{6,6}= 0.227328118343688, $ & $v_3 = 0.188029670993872,$ \\
$\alpha_{4,3}= 0.589123375926120, $& $\alpha_{7,1}= 0.005835455470528, $ & $v_4 =0.165787716189488,$ \\
$\alpha_{4,4}= 0.245088907884392, $& $\alpha_{7,2}= 0.016317087005175, $ & $v_5 = 0.032565655616444,$ \\
\end{tabular}
}
\bigskip
\noindent {\bf SSP iRK LNL $p=4$, $s=8$, $p_{lin}=9$:} A very high order LNL method with
$s=8$, $p_{lin}=9$, and $p=4$ has SSP coefficient $\mathcal{C} =4.735$. This method has the following non-zero coefficients
given in the canonical Shu-Osher form \eqref{cS-Oform}:
\smallskip
\noindent {\small \begin{tabular}{llll}
$\alpha_{1,1} = 0.146943975728437,$ & $\alpha_{5,4}= 0.796548121452431 ,$ & $\alpha_{7,7} =0.205840405060996 ,$ &
$\alpha_{9,8} =0.662855611847356,$ \\
$\alpha_{2,1} = 0.854796464970015 ,$ &$\alpha_{5,5} = 0.136561808924711 ,$ & $\alpha_{8,5} = 0.510718712707677 ,$ &
$v_1= 0.853056024271563,$ \\
$\alpha_{2,2} = 0.145203535029985 ,$ & $\alpha_{6,1} = 0.260577803576825 ,$ & $\alpha_{8,7} =0.353463620808626 ,$ &
$v_3= 0.251640022410203,$ \\
$\alpha_{3,2} = 0.612204675611763 ,$ & $\alpha_{6,5} = 0.269626835933091 ,$ & $\alpha_{8,8} =0.135817666483696 ,$ &
$v_4= 0.122149807578787,$ \\
$\alpha_{3,3} = 0.136155301978034 ,$ & $\alpha_{6,6} =0.206284522717965 ,$ & $\alpha_{9,1} = 0.003486997034287 ,$ &
$v_5= 0.066890069622858,$ \\
$\alpha_{4,3}= 0.742598809241823 ,$ & $\alpha_{7,2} =0.198036604411651 ,$ & $\alpha_{9,2} =0.067521279383993 ,$ &
$v_6= 0.263510837772119,$ \\
$\alpha_{4,4}= 0.135251383179389 ,$ &$\alpha_{7,6} =0.596122990527354 ,$ & $\alpha_{9,4} =0.256478057637965 ,$ &
$v_9= 0.009658054096400,$ \\
\end{tabular}}
\bigskip
\noindent {\bf SSP iRK LNL $p=2$, $s=10$, $p_{lin}=11$: }
A very high order LNL method with
$s=10$, $p_{lin}=11$, and $p=2$ has SSP coefficient $\mathcal{C} =5.2306$. This method has a very low storage form, where the non-zero coefficients
are given in the canonical Shu-Osher form \eqref{cS-Oform}:
\noindent{\small
\begin{tabular}{llll}
$\alpha_{1,1} = 0.193277114534410, $ & $\alpha_{5,5} =0.117235708890556, $ & $\alpha_{9,9} = 0.117235191356250, $ & $v_3 = 0.790541538474273, $ \\
$\alpha_{2,1} = 0.806723199562524, $ & $\alpha_{6,5} = 0.718962893859175, $ & $\alpha_{10,9} = 0.880317745035338, $ & $v_5 = 0.640785491489029, $ \\
$\alpha_{2,2} = 0.193276800437476, $ & $\alpha_{6,6} =0.117234259419046, $ & $\alpha_{10,10} =0.117236158521012, $ & $v_6 = 0.163802846721778, $ \\
$\alpha_{3,2} = 0.080009844643863, $ & $\alpha_{7,6} = 0.546025511754727, $ & $\alpha_{11,1} = 0.028409070825259, $ & $v_7 = 0.336736924143727, $ \\
$\alpha_{3,3} =0.129448616881864, $ & $\alpha_{7,7} =0.117237564101546, $ & $\alpha_{11,2} = 0.043364313791996, $ & $v_8 = 0.122161789752829, $ \\
$\alpha_{4,3} = 0.870552299752962, $ & $\alpha_{8,7} = 0.760604303914880, $ & $\alpha_{11,3} = 0.001158601801210, $ & $v_9 = 0.060130956027420, $ \\
$\alpha_{4,4} = 0.129447700247038, $ & $\alpha_{8,8} = 0.117233906332291, $ & $\alpha_{11,10} = 0.921532831100178, $ & $v_{10} = 0.002446096443650, $ \\
$\alpha_{5,4} = 0.241978799620415, $ & $\alpha_{9,8} = 0.822633852616330, $ & $v_1 = 0.806722885465590, $ & $v_{11} = 0.005535182481357, $ \\
\end{tabular}
}
\normalsize
\subsection{Optimizing for Additional Properties}
In the section above we optimized the implicit Runge--Kutta methods for the largest possible SSP coefficients.
However, in many cases the motivation for using an implicit method is not only the SSP coefficient; this is especially true
in cases where the additional time-step allowed for the implicit SSP methods is not enough to offset the cost of
the implicit solver needed. If we wish to optimize for other properties, such as linear stability regions,
alongside the SSP property, we can add a condition to the inequality constraints in the optimization routine.
The methods we found above did not have large linear stability regions, and so we wish to explore whether one can find
methods that have a large SSP coefficient as well as large linear stability region. In this section we present our optimized methods
using this approach. The resulting methods are suboptimal in terms of SSP coefficients
but benefit, as desired, from larger regions of absolute stability.
In Figures \eqref{fig:OptNegReal} and \eqref{fig:OptImag} we show the linear stability regions of methods
found from this co-optimization approach compared to those found in the subsection above.
Figure \eqref{fig:OptNegReal} shows several five stage ($s=5$) methods with $p=3$ and $p_{lin}=4$ resulting from optimization runs
that required increasing linear stability regions that included more of the real axis. We show four methods: in blue is the linear stability
of the optimal SSP method in the subsection above. This method has SSP coefficient $\mathcal{C}=6.1472$, and crosses the negative real axis
at $x=-26.3$. In red is the linear stability region of a method that has SSP coefficient $\mathcal{C}= 5.9537$ but allows slightly more of the negative
real axis, crossing it at $x=-29.766$. Next, in green is the linear stability of a method with a slightly smaller SSP coefficient of $\mathcal{C}=5.6249$
but which allows much more of the negative real axis, crossing it at $x=-56.247$. Finally, in black is the linear stability region of a method with
$\mathcal{C}=5.4459$ but which allows significantly more of the negative real axis, crossing it at $x=-81.68$. Thus we see that we can co-optimize
for additional linear stability properties while balancing the need for a large SSP coefficient.
Another frequently desirable feature of time-stepping methods is that they include the imaginary axis or points near the imaginary axis. This is particularly
desirable when solving hyperbolic PDEs. In our case, the optimal SSP five stage ($s=5$) method with $p=3$ and $p_{lin}=4$ does not include the imaginary axis
at all ($y=0$). Even when we consider the close neighborhood of the imaginary axis (points which are closer than $10^{-5}$ to the imaginary axis) it only allows these values
up to a value of $y=.9099$. Figure \eqref{fig:OptImag} shows the linear stability of this region in blue.
If we wish to increase these values, we can obtain a method whose linear stability region is shown in red. This method clearly captures much less
of the real axis, but it includes the imaginary axis up to the value of $y=5.28$. The zoomed image of the region of linear stability in Figure \eqref{fig:OptImag}
shows this difference clearly. The SSP coefficient of the second method is significantly less, at $\mathcal{C} =4.1322$, but this may be a worthwhile trade-off where needed.
The methods in this subsection are meant to represent a small slice of the range of possible properties for which one may co-optimize.
As in \cite{KubatkoKetcheson}, it is possible to optimize SSP methods for particular linear stability regions,
and it is also possible to consider other desirable properties with respect to which we can to co-optimize. Furthermore,
other optimization routes are possible, such as starting with a known linear stability polynomial or a particular form of a method that is known to have
desirable properties.
\begin{figure}[htb]
\centering
\begin{minipage}{0.475\textwidth}
\centering
\includegraphics[width=0.975\linewidth]{OptimizeNegativeRealStabilityRegion}
\caption{\small Linear stability regions in the complex plane of the optimal SSP method (blue line) and co-optimized methods
(in red, green, and black) with increasing real axis linear stability. Shown in the legend are the values of the SSP coefficients for
these methods. \vspace{.25in}
}
\label{fig:OptNegReal}
\end{minipage}%
\hspace{.2in}
\begin{minipage}{0.475\textwidth}
\centering
\includegraphics[width=0.675\linewidth]{ComparingImagAxis}
\includegraphics[width=0.2\linewidth, height=.705\linewidth]{ComparingImagAxisZoomed}
\caption{\small Linear stability regions in the complex plane of the optimal SSP method (blue line) and co-optimized methods
in red with larger imaginary axis linear stability, but at the cost of a smaller real axis region and SSP coefficient. On the right is
the same image zoomed in on the Imaginary axis.
}
\label{fig:OptImag}
\end{minipage}
\end{figure}
\normalsize
\subsection{Numerical Results} \label{Numerical1}
The methods above were found by the optimization code \cite{SSPimplicitLNL_github}, where the order conditions are imposed as equality constraints. It is a
good idea to check that the methods indeed perform as designed on a series of linear and nonlinear test cases. These convergence studies are
reported in Section \ref{conv1}. Next, we wish to test how the methods perform in terms of the predicted SSP time-step. Clearly, the SSP coefficient
gives a guarantee that the desired property is preserved by the time-stepping method at the corresponding time-step. In Section \ref{tvd1} we study
the difference between the guaranteed time-step and the actual time-step at which the desired nonlinear stability property (in this case the total variation
diminishing property) still holds. A close agreement between these two time-steps demonstrates the relevance of the SSP property for typical cases.
\subsubsection{Verification of the linear and nonlinear orders of convergence} \label{conv1}
\noindent{\bf Example 1.1: Study of the convergence rate for a nonlinear ODE}
To verify the nonlinear order of the implicit Runge--Kutta methods with nonlinear order $p$ and linear order $p_{lin}$
we us a nonlinear system of ODEs
\begin{eqnarray}
&u_1' = u_2 \nonumber \\
&u_2' = \frac{1}{\epsilon} (-u_1 + (1-u_1^2) u_2)
\end{eqnarray}
known as the van der Pol problem.
We use $\epsilon = 10$ and initial conditions $u_0 = (0.5; 0)$.
This problem was tested with methods with number of stages $s$, linear order $p_{lin}$ and nonlinear order $p$.
Each methods is designated by $(s,p_{lin}, p)$. Of each nonlinear order we test two methods: one with low
stages and $p_{lin} = p$ and one with high number of stages and linear order.
The methods we test are $(s,p_{lin}, p)= (2,2,2) , (7,7,2) , (3,3,3) , (7,7,3) , (4,4,4) , (7,7,4)$.
We use $\Delta t = \frac{1}{250}, \frac{1}{350}, \frac{1}{450}, \frac{1}{550}, \frac{1}{650}$
and step forward to time $T_{final} = 1.0$.
The errors are calculated by using a very accurate solution calculated by MATLAB's
ODE45 routine with tolerances set to {\tt AbsTol=}{\tt RelTol=} $10^{-14}$.
In Figure \ref{fig:VDPconv} we show that the $log_{10}$ of the errors
in the first component vs. the $log_{10}$ of the time-step $\Delta t$.
The orders (slope of the line) are taken by taking a linear fit using MATLAB's {\tt polyfit}.
We observe the rate of convergence expected for the nonlinear order of the method.
\begin{figure}[htb]
\centering
\begin{minipage}{0.45\textwidth}
\centering
\includegraphics[width=0.99\linewidth]{vdp_convergence}
\caption{\small Convergence study in Example 1.1, the nonlinear van der Pol problem.
Methods tested are $(s,p_{lin}, p)= (2,2,2) , (7,7,2) , (3,3,3) ,$ $ (7,7,3) , (4,4,4) , (7,7,4)$, using time-step
$\Delta t = \frac{1}{250}, \frac{1}{350}, \frac{1}{450}, \frac{1}{550}, \frac{1}{650}$
to step forward to time $T_{final} = 1.0$. The rate of convergence is, as predicted, the designed
nonlinear order of the method.}
\label{fig:VDPconv}
\end{minipage}
\hspace{.02\textwidth}
\begin{minipage}{0.45\textwidth}
\centering
\includegraphics[width=0.99\linewidth]{buckley_convergence}
\caption{\small Convergence study in Example 1.3, the Buckley-Leverett problem.
Methods tested are $(s,p_{lin}, p)= (2,2,2) , (7,7,2) , (3,3,3) ,$ $ (7,7,3) , (4,4,4) , (7,7,4)$, using time-step
$\Delta t = \lambda \Delta x$ with $\lambda = \frac{1}{2}, \frac{1}{4}, \frac{1}{8} , \frac{1}{16}, \frac{1}{32}$, and spatial step
$\Delta x=\frac{\pi}{4} $ to step forward to time $T_{final} = 2.0$.
The slopes of the lines show the desired order of accuracy. }
\label{fig:conv_BLspectral}
\end{minipage}
\end{figure}
\noindent{\bf Example 1.2: Study of the convergence rate for a linear PDE}
To verify the linear order of convergence of these methods we solve a linear advection problem $U_t + U_x=0$
with periodic boundary conditions and initial conditions $U_0(x) = \sin(x)$ on the spatial domain $x \in [0,2 \pi]$.
We discretize the spatial grid with $N=11$ equidistant points and use the Fourier pseudospectral
differentiation matrix $\m{D}$ \cite{HGG2007} to compute $F \approx - U_x$.
In this case, the solution is a sine wave, so that the pseudospectral method is exact, and the
spatial discretization contributes no errors. For this reason, we use grid refinement in time only,
and use a range of time steps, $\Delta t = \lambda \Delta x$
where
$\lambda =\frac{1}{10}, \frac{2}{10}, \frac{3}{10}, \frac{4}{10}, \frac{5}{10}, \frac{6}{10},
\frac{7}{10}, \frac{8}{10}, \frac{9}{10}$
to compute the solution to final time $T_f = 5.0$.
The errors are measured in the $\ell_2$ norm.
Two methods from each order were tested for convergence on this problem.
In Figure \ref{fig:conv_spectral} we show the convergence plots for the $s=p_{lin} $ methods
the $s=p_{lin} -1$ methods, both with nonlinear order $p=4$.
The slopes were measured in the region before round-off error dominates.
We observe that the design-order $p_{lin}$ of each method for linear problems is apparent.
\begin{figure}[htb]
\includegraphics[width=0.475\linewidth]{ConvergenceStudy4plnlSpectral.jpg}
\includegraphics[width=.475\textwidth]{4pLNLConvergenceStudy.jpg}
\caption{\small
Convergence study in Example 1.2, the linear advection problem with pseudospectral differentiation of the spatial
derivatives. Here we use $N=11$ equidistant points between $(0, 2 \pi)$, and $\Delta t = \lambda \Delta x$. The solution is
evolved forward to time $T_f=5.0$.
Convergence plots for linear methods with $s=p_{lin}$ (left) and $s=p_{lin}-1$ (right). The slopes of the lines
are calculated before round-off error ruins the convergence.}
\label{fig:conv_spectral}
\end{figure}
\noindent{\bf Example 1.3: Study of the convergence rate for a nonlinear PDE}
To verify the nonlinear order on PDE, we solve the Buckley-Leverett equation which is
commonly used to model two-phase flow through porous media:
\begin{align*}
u_t+f(u)_x & = 0, & \text{ where } f(u) = \frac{u^2}{u^2 +a(1-u)^2},
\end{align*}
on $x\in[0,2\pi]$, with periodic boundary conditions.
We take $a=\frac{1}{3}$ and initial condition $ u(x,0) = \sin(x) $.
For the spatial discretization, we use a Fourier pseudospectral method as above.
$\Delta t = \lambda \Delta x$ with
$\lambda = \frac{1}{32}, \frac{1}{16}, \frac{1}{8} , \frac{1}{4}, \frac{1}{2}$, and
$\Delta x=\frac{\pi}{4} $, evolved to final time $T=2.0$.
The errors are calculated by using a very accurate solution calculated by MATLAB's
ODE45 routine with tolerances set to {\tt AbsTol=}{\tt RelTol=} $10^{-14}$.
In Figure \ref{fig:conv_BLspectral} we show that the $log_{10}$ of the errors
in the first component vs. the $log_{10}$ of the time-step $\Delta t$.
The orders (slope of the line) are taken by taking a linear fit using MATLAB's {\tt polyfit}.
We observe the rate of convergence expected for the nonlinear order of the method.
\subsubsection{Verification of the SSP property} \label{tvd1}
\noindent{\bf Example 2:} The methods above are selected to optimize the SSP coefficient
$\mathcal{C}$. This value guarantees that the desired strong stability property that is satisfied by the forward Euler method
will be preserved under the condition $\Delta t \leq \mathcal{C} \dt_{\textup{FE}}$. It is interesting to see
whether for typical problems this value is predictive of the actual time-step at which the desired strong stability
properties are violated.
As an example, we consider the linear advection equation
\[ U_t + U_x =0,\]
on the domain $x \in [-1,1]$ with periodic boundary conditions and the step function initial condition
\[ U_0(x) = \left\{ \begin{array}{rc}
1 & -0.1 \leq x \leq 0.1 \\
0 & otherwise \\
\end{array} \right. \]
We approximate the spatial derivative with a first order finite difference method, which is total variation diminishing (TVD)
for all values $\Delta t \leq \Delta x$.
This simple example is chosen as our experience has shown \cite{SSPbook2011} that this problem often
demonstrates the sharpness of the SSP time-step.
For all of our simulations, we use a fixed grid of size $\Delta x = \frac{1}{300}$, and a time-step $\Delta t = \lambda \Delta x$,
where we choose values of $\lambda$ starting from the minimum of $\mathcal{C}/10$ or $0.1$. We then increase the value
of $\lambda$ until the TVD property is violated.
To measure the effectiveness of these methods, we consider the maximum observed rise in total variation defined by
\begin{equation}
\label{eqn:increase-in-tv}
\max_{0 \leq n \leq N-1} \left( \| u^{n+1} \|_{TV} - \| u^n \|_{TV} \right).
\end{equation}
over the first $N=20$ steps.
We are interested in the time-step in which this rise becomes evident, so we consider a violation if the TV
rises $10^{-10}$ or more at any time-step. The value of $\lambda$ for which this violation in TV occurs is refined to the level
of $10^{-12}$.
In Figure \ref{fig:TVtest} we plot the maximal rise in total variation over different values of $\lambda = \frac{\Delta t}{\Delta x}$
for a selection of methods. It is interesting to observe that the change in behavior of the methods is quite sharp -- once a
certain threshold is reached, the SSP coefficient goes from being very small ($<10^{-14}$)
to being much larger ($> 10^{-4}$) with
small changes in the time-step.
\begin{figure}[t]
\includegraphics[width=0.475\linewidth]{SSPCoeff_iLNL65}
\includegraphics[width=0.475\linewidth]{SSPCoeff_iLNL66}
\caption{\small The observed rise in total variation over the first $20$ steps for the $s=6$, $p_{lin}=5$ methods (left) and
the $s=6$, $p_{lin}=6$ methods. }
\label{fig:TVtest}
\end{figure}
We are also interested in difference between the predicted and observed value of $\mathcal{C}$ (or equivalently, $\Delta t$)
at which this violation happens. Table \ref{fig:TVD_violationp2} provides these values for the case $p=2$. We observe that
for all the methods with $p=2$ and $p_{lin} < s$, the observed time-step for which the TVD property is
preserved matches the predicted time-step up to $ \approx 10^{-10}$. For these cases, the SSP coefficient
is an excellent predictor of the actual value for which the TVD property breaks down.
\bigskip
\begin{table}[h]
{\small
\begin{tabular}{|l|l|l|l|l|l|l|l|l|}
\hline
$s \backslash p_{lin} $&2&3&4&5&6&7&8&9 \\\hline
2& 2.5$\times 10^{-12}$&1.5$\times 10^{-12}$ &&&&&&\\\hline
3&4.0$ \times 10^{-12}$&2.5$ \times 10^{-12}$&3.8$\times 10^{-2}$&&&&&\\\hline
4&8.1$ \times 10^{-12}$&4.5$ \times 10^{-12}$&2.2$\times 10^{-11}$&6.4$\times10^{-2}$&&&&\\\hline
5&1.6$ \times 10^{-11}$&9.1$ \times 10^{-12}$&1.2$\times 10^{-11}$&7.6$ \times 10^{-12}$
&1.7 $\times10^{-1}$ &&&\\\hline
6&3.2$ \times 10^{-11}$&1.8$ \times 10^{-11}$&1.1$\times 10^{-11}$&6.6$ \times 10^{-12}$&2.6$ \times 10^{-11}$
&3.0$\times 10^{-1}$&&\\\hline
7&6.4$ \times 10^{-11}$&3.7$ \times 10^{-11}$&1.4$\times 10^{-11}$&7.6$ \times 10^{-12}$
&2.6$ \times 10^{-10}$&7.1$ \times 10^{-12}$&3.6$\times10^{-1}$&\\\hline
8&1.2$ \times 10^{-10}$&7.4$ \times 10^{-11}$&3.7$\times 10^{-10}$&1.0$ \times 10^{-11}$
&5.5$ \times 10^{-11}$&5.5$ \times 10^{-11}$&1.4$ \times 10^{-2}$&4.7$\times10^{-1}$\\\hline
9&2.5$ \times 10^{-10}$&1.5$ \times 10^{-10}$&1.4$\times 10^{-10}$&1.7$ \times 10^{-10}$&3.4$ \times 10^{-11}$&
2.1$ \times 10^{-11}$&2.2$ \times 10^{-10}$& 3.4$ \times 10^{-2}$ \\ \hline
\end{tabular}
}
\caption{The difference between the theoretical and observed SSP coefficient at which the maximal rise in total
variation is about the threshold of $10^{-10}$ for the $p=2$ methods.}
\label{fig:TVD_violationp2}
\end{table}
For the cases where $p_{lin} = s \leq 7$, and $p=2$, the observed time step is also within $\approx 10^{-10}$ of the predicted value.
This is true also for the methods with $p=3$.
However, when $p_{lin} = s =8,9$, and $p=2,3,4$, the observed time step is much larger than the predicted time-step
($\approx 10^{-1}$ or $10^{-2}$). Also, for $p=4$, the $s=p_{lin}=5$, we also have a larger difference between the observed and
predicted time-step.
Finally, in the cases where $s = p_{lin}-1$, the observed time step is usually
much larger than the predicted time-step ($\approx 10^{-1}$ or $10^{-2}$),
and the SSP coefficient is not sharp at all.
\section{Optimal SSP IMEX Runge--Kutta methods with $p_{lin} \geq p$} \label{IMEX_LNL}
For cases in which the problem has a linear component which restricts the time-step significantly, and a nonlinear component which does not,
it may be advantageous to treat the two components differently by using an implicit-explicit method.
Consider an initial value problem of the form
\begin{equation}
U'(t) = f(U(t)) + g(U(t)), \qquad U(t_0) = U_0 \qquad t \geq t_0
\label{eq:SC_additiveODE}
\end{equation}
which is semi-discretized to give an ODE of the form
\begin{equation}
u'(t) = F(u) + G(u).
\label{eq:SC_semi_discrteAdditiveODE}
\end{equation}
To step the two components forward in time, one explicitly and one implicitly,
we use an Implicit-Explicit (IMEX) Runge--Kutta method, which is a particular kind of additive Runge--Kutta method.
An $s$-stage Additive Runge--Kutta (ARK) method is defined by two $s \times s$
real matrices $\m{A}, \tilde{\m{A}}$ and two real vectors $\vb, \vbt$ such that:
\vspace{0.1em}
\begin{equation}
\begin{aligned}
u^{(i)} &= u^n + \Delta t \sum_{j=1}^s a_{ij} F(u^{(j)} + \Delta t \sum_{j=1}^s \at_{ij} G(u^{(j)} ) \qquad i= 1, \dots,s \\
u^{n+1} &= u^n + \Delta t \sum_{i=1}^s b_i F(u^{(i)} + h \sum_{i = 1}^s \bt_i G(u^{(i)}), \qquad i = 1, \dots, s .
\label{eq:SC_ark_butcher_form}
\end{aligned}
\end{equation}
We select $\m{A}$ to be a strictly lower triangular matrix, so that this additive Runge--Kutta
method contains an explicit method (represented by ($\m{A}, \vb$)) used for the non-stiff part $F$.
As above, we limit ourselves to diagonally implicit methods and require
$\tilde{\m{A}}$ to be a lower triangular matrix, where ($\tilde{\m{A}}, \vbt$) represents an implicit Runge--Kutta (DIRK)
method used for the stiff part $G$, as in \cite{Higueras2009,Ascher1997IMEX_ARK, CarpenterKennedyIMEX2003}.
IMEX methods were first introduced by Crouziex in 1980 \cite{CrouziexIMEX} for evolving parabolic equations. In 1997, Ascher, Ruuth, and Wetton \cite{Ascher1997IMEX_ms} introduced IMEX multi-step methods for time dependent PDEs, notably convection-diffusion equations,
and in the same year Ascher, Ruuth and Spiteri \cite{Ascher1997IMEX_ARK} presented the IMEX Runge--Kutta schemes for such problems.
Although the authors were not focused on designing SSP pairs, some of methods are in fact known SSP methods.
For example, the method in \cite[Section 2.4]{Ascher1997IMEX_ARK} is the midpoint explicit and implicit SSP methods.
In this work, all the implicit methods are SDIRK methods, and most have nice properties such as L-stability, with diagonal values
$\gamma$ that are those mentioned in the books of Hairer, Norsett, and Wanner \cite{Hairer1, Hairer2}.
Implicit methods are often particularly desirable when applied to a linear component. In this case, the order conditions simplify.
In \cite{calvo2001}, Calvo, Frutos, and Novo developed IMEX pairs for the case where the implicit component $G$ is linear.
This was the first work where the linear and nonlinear orders were separated. The methods they produced had nonlinear implicit order
$p_{im} = 2$, nonlinear explicit order and linear order $ p_{ex} = p_{lin}=3,4$.
Kennedy and Carpenter \cite{CarpenterKennedyIMEX2003}
derived IMEX Runge--Kutta methods based on singly diagonally implicit
Runge--Kutta (SDIRK) methods. This work introduced sophisticated IMEX methods with good accuracy and stability properties,
as well as high quality embedded methods for error control and other features that make these methods usable in complicated
applications.
The first IMEX methods that had SSP properties were considered by Pareschi and Russo in \cite{PareschiRusso2005}.
In this work, explicit SSP Runge--Kutta methods L-stable implicit components. These schemes were designed to be asymptotically preserving.
The authors listed a $p_{ex} = p_{im} = p = 3$ order conditions, and also presented a table depicting the number of coupling conditions under
each underlying assumptions. In this work, the authors designed their methods while enforcing the condition that the abscissas of the explicit
and implicit methods were equal $\vc = \tilde{\vc}$. They observed that under the assumption $\vc=\vct$ and $\vb=\vbt$ the
coupling conditions were redundant for the $p = 3$. It is worth noting the SDIRK implicit pairs from \cite{PareschiRusso2005} have the same
$\gamma$ value listed in the books of Hairer, Norsett, and Wanner \cite{Hairer1, Hairer2}. Further work by Boscarino, Pareschi, and Russo
\cite{BoscarinoPareschiRusso2013} observed that the order reduction phenomenon disappears when $\vb = \vbt$. In this work
they presented a globally stiffly accurate, third order method with non-negative explicit coefficients (named BPR(3,5,3).
Higueras \cite{Higueras2006} was first to consider SSP IMEX methods where both explicit and implicit pairs of the methods were SSP.
In this work, she presented conditions for an IMEX Runge--Kutta methods to be SSP, and listed some previously known methods,
such as methods from \cite{Ascher1997IMEX_ARK}, that satisfy the SSP conditions. The methods that appear in this work are
of order $p \leq 3$, and there is no distinction made between the explicit, implicit and linear order of the methods.
In \cite{Higueras2009} Higueras presented order barriers and other characteristics of strong stability
preserving additive Runge--Kutta methods. This work contains several very important properties of SSP IMEX pairs
and is necessary to the understanding of the structure of these methods.
In later work, \cite{Higueras_TVD_IMEX2012} several known methods were analyzed in terms
of their linear stability region and SSP performance on astrophysics-type problems. The implicit second order methods considered
are L-stable (and thus A-stable as well), but the third order method was not A-stable.
In their work in \cite{higuerasOptSSPIMEXRK2014}, the authors mainly focused on second order, three-stage explicit methods that
implicit SSP pairs. In this work, the main focus was on methods with {\em suboptimal} SSP coefficients that offered significant
additional benefits (including large linear stability regions).
Our approach to deriving optimal SSP methods is slightly different from \cite{Higueras2006,higuerasOptSSPIMEXRK2014}.
First, we use a slightly different formulation of the SSP condition, which facilitates the construction of
an optimization problem. Next, we introduce a coefficient which relates the strong stability condition of $F$ to
that of $G$, and make an observation about time step of optimal methods using this coefficient,
which also simplifies the optimization routine. But the main distinction between our work and prior work is the
fact that our investigations focus on higher order methods, and especially methods with higher linear order.
To analyze an IMEX method in the SSP context, we assume as in \cite{Higueras2006,higuerasOptSSPIMEXRK2014}
that when each component ($F$ or $G$)
of the ODE is stepped forward individually with a forward Euler method, the new solution
will satisfy a strong stability property in some desired convex functional $\| \cdot \|$,
but under very different time-step restrictions:
\begin{eqnarray}
\| u^n + \Delta t F(u^{n}) \| \leq \| u^n \|, \quad 0 \leq \Delta t \leq \Delta t_{FE},
\label{eq:F_fe_condition}
\end{eqnarray}
and
\begin{eqnarray}
\| u^n + \Delta t G(u^{n}) \| \leq \| u^n \|, \quad 0 \leq \Delta t \leq K \Delta t_{FE}.
\label{eq:G_fe_condition}
\end{eqnarray}
If $K$ is small, the $G$ component will make the overall allowable time-step of the method very small,
so it may be worthwhile to treat this component implicitly. We know \cite{ketcheson2009, SSPbook2011} that the allowable time-step
for an $s$-stage SSP implicit Runge--Kutta method of order $p>1$ can only be expected to be, at most, $2s$ times the allowable
explicit forward Euler time-step $\dt_{\textup{FE}}$, which does not typically offset the additional cost of solving the implicit system.
However, in the previous discussions we show that the allowable time-step of an implicit method is in fact more than twice,
and in the new methods presented in Section \ref{Implicit_LNL} up to six times,
as large as that of an explicit method with the same order and
number of stages and order.
Furthermore, in the case where $G$ is a linear operator, the cost of the implicit solver is much smaller and under such circumstances
it may be worthwhile to use an implicit SSP Runge--Kutta method for this part of the problem. However, if
$F$ is nonlinear, or if for any other reason the cost of implicitly solving the system for $F$ is large,
we wish to use an explicit time-stepping method for this part.
The IMEX approach may also be desirable if the value of $K$ is not small (perhaps even infinitely large)
but other factors, such as linear stability requirements, greatly limit the
step size if $G$ is treated explicitly.
The main contribution of our work is to present SSP IMEX methods with $p_{lin} \geq 4$. These methods have not appeared
in the literature, and while it is not clear yet how useful they will be in actual applications, we believe that
understanding the SSP bounds on such methods and its dependence on the linear and nonlinear orders is of value.
\subsection{Formulating the Optimization Problem}
To formulate the optimization method for this problem, we stack the matrix $\m{A}$ and the vector $\vb$, padded with zeroes,
into a square matrix $\m{S}$ (as we did in Equation \eqref{optimization}). Similarly, we convert $\tilde{\m{A}}$ and $\vbt$ into a matrix $\tilde{\m{S}}$.
We now rewrite the method \eqref{eq:SC_ark_butcher_form} as
\begin{eqnarray} \label{MSMDvector}
Y = \ve u^n + \Delta t \m{S} F(Y) + \Delta t \tilde{\m{S}} G(Y).
\end{eqnarray}
Now, we add the terms
\begin{eqnarray*}
\left( I +r \m{S} + \tilde{r} \tilde{\m{S}} \right) Y &=&
\ve u^n + r \m{S} \left( Y + \frac{\Delta t}{r} F(Y) \right) + \tilde{r} \tilde{\m{S}} \left( Y + \frac{\Delta t}{ \tilde{r}} G(Y) \right), \\
Y & = & R (\ve u^n) + P \left( Y + \frac{\Delta t}{r} F(Y) \right) + Q \left( Y + \frac{\Delta t}{ \tilde{r}} G(Y) \right),
\end{eqnarray*}
where \[
R = \left( I +r \m{S} + \tilde{r} \tilde{\m{S}} \right)^{-1}, \; \; \; \; \; \;
P = r R \m{S}, \; \; \; \; \; \;
Q = \tilde{r} R \tilde{\m{S}}.
\]
From this formulation we see that if $R \ve$, $P$, and $Q$ are all positive component-wise, then the resulting method
is simply a convex combinations of forward Euler steps $Y + \frac{\Delta t}{r} F(Y)$ and $Y + \frac{\Delta t}{ \tilde{r}} G(Y)$,
and therefore will preserve the strong stability property in the desired convex functional $\| \cdot \|$,
under the time-step restriction $\Delta t \leq \min \left(r \dt_{\textup{FE}}, \tilde{r} K \dt_{\textup{FE}} \right) $. We observe that the optimal methods will have
$r = \tilde{r} K$, so the optimization problem becomes: \\
{\em Maximize $r$ such that}
\begin{subequations}
\begin{align}
\left( I +r \m{S} + \frac{r}{K} \tilde{\m{S}} \right)^{-1} \ve \geq 0 \label{Opt_ARK1} \\
r \left( I +r \m{S} + \frac{r}{K} \tilde{\m{S}} \right)^{-1} \m{S} \geq 0 \label{Opt_ARK2} \\
\frac{r}{K} \left( I +r \m{S} + \frac{r}{K} \tilde{\m{S}} \right)^{-1} \tilde{\m{S}} \geq 0 \label{Opt_ARK3}\\
\tau_k(\m{A}, \vb, \tilde{\m{A}}, \vbt) = 0 \; \; \; \mbox{for} \; \; \; k=1, . . ., P, \label{ARKtau_conditions}
\end{align}
\end{subequations}
where the first three inequalities are understood component-wise, and
$\tau_k(\m{A}, \vb, \tilde{\m{A}}, \vbt)$ are the order conditions, described in the sections below.
This optimization gives the Butcher coefficients $\m{A}$, $\vb$, $\tilde{\m{A}}$, and $\vbt$, and an optimal value
of the SSP coefficient $\mathcal{C} = r$. Of course, for each value of $K$, a different method will be optimal.
Note that although this process defines a sufficient condition for the resulting method to be SSP with SSP coefficient $\mathcal{C} = r$,
there is no reason to expect this condition to be necessary. In particular, this formulation ignores the possible interactions between $F$ and $G$
that would result in more relaxed SSP conditions.
Once again, the order conditions \eqref{ARKtau_conditions} are a key piece of the optimization, and serve as equality constraints.
These conditions will be described in the next two subsections.
\subsubsection{Linear Order Conditions}
\noindent{\bf First order $p_{lin}=1$:} The order conditions for an additive method \eqref{eq:SC_ark_butcher_form} to be first order are
\begin{equation}
\vb^T \ve = 1, \; \; \; \; \; \vbt^T \ve = 1 .
\end{equation}
This has two conditions: one for the explicit part and one for the implicit part. To generalize these order conditions, we define
$\Phi_1 = \{ \phi_{1,1}, \; \phi_{1,2}\}= \{ \vb^T , \; \vbt^T \} ,$ and the order conditions become
$\vb^T \ve = 1, \; \; \; \vbt^T \ve = 1 .$
\noindent{\bf Second order $p_{lin}=2$:} Define
$\Phi_2 = \Phi_1 \otimes \{ \m{A} , \; \tilde{\m{A}} \} = \{ \phi_{1,1} \m{A}, \; \phi_{1,1} \tilde{\m{A}}, \; \phi_{1,2} \m{A}, \; \phi_{1,2} \tilde{\m{A}} \}, $
and the order conditions are each of these right-multiplied by $\ve$ and set equal to $\frac{1}{2}$. Recall that $\vc = \m{A} \ve$ and
$\vct = \tilde{\m{A}} \ve$
The order conditions then become
\[ \vb^T \vc = \frac{1}{2}, \; \; \; \vb^T \vct = \frac{1}{2}, \; \; \;
\vbt^T \vc = \frac{1}{2} , \; \; \; \vbt^T \vct = \frac{1}{2} .\]
\noindent{\bf Third order $p_{lin}=3$:}
use $\Phi_3 = \Phi_2 \otimes \{ \m{A} , \; \tilde{\m{A}} \} $ and right-multiply by $\ve$ to obtain:
\[ \vb^T \m{A} \vc = \frac{1}{6}, \; \; \; \vbt^T \m{A} \vct = \frac{1}{6}, \; \; \; \vbt^T \m{A} \vc = \frac{1}{6} , \; \; \; \vb^T \m{A} \vct = \frac{1}{6} ,\]
\[ \vb^T \tilde{\m{A}} \vc = \frac{1}{6}, \; \; \; \vbt^T \tilde{\m{A}} \vct = \frac{1}{6}, \; \; \; \vbt^T \tilde{\m{A}} \vc = \frac{1}{6} , \; \; \; \vb^T \tilde{\m{A}} \vct = \frac{1}{6} .\]
To obtain higher linear order, we define
\[\Phi_{q} = \Phi_{q-1} \otimes \{\m{A} , \; \tilde{\m{A}}\} = \{ \phi_{q,1} \m{A}, \phi_{q,1} \tilde{\m{A}}, \phi_{q,2} \m{A}, \phi_{q,2} \tilde{\m{A}}, ... \phi_{q,2^q} \m{A}, \phi_{q,2^q} \tilde{\m{A}} \} \]
and the resulting new order conditions for this order take the form
\[\phi_{q,j} \; \ve = \frac{1}{q!} \ve \; \; \; \; \forall q, j .\]
To go from linear order $p_{lin} = q-1$ to linear order $p_{lin}=q$ we require an additional $2^q$ order conditions. We observe that unlike in
the implicit case, in the IMEX case the number of order conditions grows very rapidly even when both $F$ and $G$ are linear.
This is due to the many {\em coupling} order conditions that are required for the two components to interact properly.
\subsubsection{Nonlinear Order Conditions}
The order conditions for first and second order are the same for both linear and nonlinear problems.
In this section, we look at the order conditions for nonlinear orders $p^e =3,4$ for the explicit part and $p^i=3,4$
for the implicit part.
\noindent{\bf Nonlinear order $p=3$:} For third order, we have the linear order conditions for the explicit and implicit
parts, and the nonlinear coupling conditions given in the section above.
In addition, we have the following nonlinear condition on the explicit part:
\begin{subequations}
\begin{equation}
\vb^T \vc^2 = \frac{1}{3}, \label{eq:p3_ex_nl}
\end{equation}
\mbox{the nonlinear conditions on the implicit part,}
\begin{equation}
\vbt^T \vct^2 = \frac{1}{3} \label{eq:p3_im_nl}
\end{equation}
\mbox{and the nonlinear coupling conditions}
\begin{equation} \label{eq:p3_coupled_nla}
\vb^T \m{C} \vct = \frac{1}{3},
\quad \vb^T \m{\tilde{C}} \vct = \frac{1}{3}
\end{equation}
\begin{equation} \label{eq:p3_coupled_nlb}
\vbt^T \m{C} \vc = \frac{1}{3},
\quad \vbt^T \m{\tilde{C}} \vc = \frac{1}{3}.
\end{equation}
\end{subequations}
When we wish to have $p^e=3$ but use a linear $G$ (i.e $p^i = 2$) we need to include all the linear order conditions for $p_{lin} =3$, as well as
the nonlinear equations for the explicit part \eqref{eq:p3_ex_nl}, and we must include the coupled order conditions
in \eqref{eq:p3_coupled_nla}. We can neglect the nonlinear implicit conditions \eqref{eq:p3_im_nl} and \eqref{eq:p3_coupled_nlb}.
\noindent{\bf Nonlinear order $p=4$:} We include all the linear and nonlinear order conditions above, for the explicit and implicit parts as
well as the coupled conditions. In addition, we have the nonlinear order condition Equation~\eqref{eq:p4_ex_nl} for the explicit method:
\begin{subequations}
\begin{equation}
\vb^T \m{A} \vc^2 = \frac{1}{12} , \quad \vb^T \vc^3 = \frac{1}{4}, \quad \vb^T \m{C} \m{A} \vc = \frac{1}{8}. \label{eq:p4_ex_nl}
\end{equation}
\mbox{The nonlinear order condition for the implicit part is}
\begin{equation}
\vbt^T \tilde{\m{A}} \vct^2 = \frac{1}{12} , \quad \vbt^T \vct^3 = \frac{1}{4}, \quad \vbt^T \m{\tilde{C}} \tilde{\m{A}} \vct = \frac{1}{8}. \label{eq:p4_im_nl}
\end{equation}
\end{subequations}
And the nonlinear coupled order conditions
{\small
\begin{eqnarray} \label{eq:p4_coupled_nla}
\begin{aligned}
& \vb^T \m{A}\m{C}\vct = \frac{1}{12} ,
\quad \vb^T \m{A}\m{\tilde{C}}\vct = \frac{1}{12} ,
\quad \vb^T\m{C}\m{\tilde{C}}\vc = \frac{1}{4},
\quad \vb^T\m{C}\m{\tilde{C}}\vct = \frac{1}{4},
\quad \vb^T \m{\tilde{C}}\mCt\vct = \frac{1}{4},
\quad \vb^T \m{C} \tilde{\m{A}} \vc= \frac{1}{8}, \\
&
\vb^T \m{C} \tilde{\m{A}} \vct= \frac{1}{8},
\quad \vb^T \m{C} \m{A} \vct= \frac{1}{8},
\quad \vb^T \m{\tilde{C}} \m{A} \vc = \frac{1}{8},
\quad \vb^T \m{\tilde{C}} \m{A} \vct = \frac{1}{8} ,
\quad \vb^T \m{\tilde{C}} \tilde{\m{A}} \vc = \frac{1}{8},
\quad \vb^T \m{\tilde{C}} \tilde{\m{A}} \vct = \frac{1}{8}, \\
\end{aligned}
\end{eqnarray}
\begin{eqnarray} \label{eq:p4_coupled_nlb}
\begin{aligned}
& \vb^T \tilde{\m{A}}\m{C}\vc = \frac{1}{12},
\quad \vb^T \tilde{\m{A}}\m{C}\vct = \frac{1}{12},
\quad \vb^T \tilde{\m{A}}\m{\tilde{C}}\vct = \frac{1}{12},
\quad \vbt^T \m{A}\m{\tilde{C}}\vc = \frac{1}{12},
\quad \vbt^T \m{A}\m{\tilde{C}}\vct = \frac{1}{12},
\quad \vbt^T \tilde{\m{A}}\m{C}\vc = \frac{1}{12} , \\
& \vbt^T \tilde{\m{A}}\m{C}\vct = \frac{1}{12},
\quad \vbt^T \m{A}\m{C}\vc = \frac{1}{12},
\quad \vbt^T\m{C}\m{\tilde{C}}\vc = \frac{1}{4},
\quad \vbt^T\m{C}\m{\tilde{C}}\vct = \frac{1}{4},
\quad \vbt^T \m{C} \m{C} \vc = \frac{1}{4},
\quad \vbt^T \m{C} \tilde{\m{A}} \vc= \frac{1}{8}, \\
& \vbt^T \m{C} \tilde{\m{A}} \vct= \frac{1}{8},
\quad \vbt^T \m{C} \m{A} \vct= \frac{1}{8},
\quad \vbt^T \m{\tilde{C}} \m{A} \vc = \frac{1}{8},
\quad \vbt^T \m{\tilde{C}} \m{A} \vct = \frac{1}{8} ,
\quad \vbt^T \m{\tilde{C}} \tilde{\m{A}} \vc = \frac{1}{8},
\quad \vbt^T \m{C} \m{A} \vc = \frac{1}{8}, \\
\end{aligned}
\end{eqnarray}
}
In the case where $G$ is linear and we wish to have $p^i=2$, we must satisfy all the linear order conditions,
the nonlinear explicit conditions \eqref{eq:p4_ex_nl}, and the coupling conditions \eqref{eq:p4_coupled_nla},
but we can neglect the nonlinear implicit conditions \eqref{eq:p4_im_nl} and the nonlinear coupling conditions
\eqref{eq:p4_coupled_nlb}
\subsection{Optimal SSP IMEX Methods} \label{IMEX_LNL_methods}
We formulate the optimization problem above in a MATLAB routine following \cite{ketchcodes}
and use it to generate optimized methods for different choices of $K$.
We focus primarily on cases in which the number of stages $s$ is the same as the linear order $p_{lin}$ or
a little larger. The nonlinear orders of interest are $p=2, 3, 4$. We note that in finding optimal methods,
we frequently observed convergence to higher order than we required or expected. For example, in the case where
we only require $p^i=2$ and $p^e>2$ the optimal SSP methods generally have $\vb=\vbt$ and $\vc=\vct$, as we would
expect from \cite{Higueras2009}
and due to this the nonlinear implicit order is higher than required, $p^i=3$. However, we also find that in some
cases we required only $p^i=2$ and $p^e=4$, but the methods converged to $p^i=p^e=4$. For this reason,
in the following sections we describe mostly optimal methods that have $p=p^i=p^e$ (though some exceptions
exist as is noted below). In some cases, especially where the number of stages $s$ is large and where there are more order constraints,
the optimization routine had difficulties converging and we are not comfortable stating that the methods found are optimal.
For this reason, we refer to the methods as {\em optimized} rather than optimal.
\noindent{\bf IMEX pairs for $K=\infty$}
First, we look at the case where there is no constraint resulting from the $G$ component, which is equivalent to
setting $K=\infty$. We reformulate the optimization problem to account for this by shutting off the $\tilde{\m{S}}$ terms
in \eqref{Opt_ARK1} -- \eqref{Opt_ARK3}. The case of $K=\infty$ is equivalent to the case where the implicit method
is non-SSP while the explicit method is SSP. In many such cases, researchers have focused on finding methods
where the explicit component is SSP and the implicit component is A-stable, L-stable, or stiffly stable. These methods
were mostly of order $p\leq 3$ with few $p=4$ methods. It is known from the order barrier on explicit SSP Runge--Kutta methods
that methods of order $p>4$ cannot exist. In our case, we focus primarily on methods that have $p_{lin} \geq 4$.
We first study methods with $p_{lin} \geq p$, and we do not optimize for linear stability: our rationale for studying
these methods is to demonstrate that where there is no SSP constraint on $G$
we are able to find implicit pairs for optimal explicit SSP methods for $F$.
Using our optimization routine we found methods with $s$ stages, linear order $p_{lin}$ and nonlinear orders $p=p^i = p^e = 2,3$ with
$K=\infty$ with $s=p_{lin}, p_{lin} +1 , p_{lin}+2$. Initially, we imposed a non-negativity conditions on all the coeffiicents, and were
able to find many good methods. Later, we relaxed this condition and allowed negative coefficients in $\tilde{\m{S}}$, which allowed us to find
additional methods. Table \ref{IMEXKinf} contains the SSP coefficients of these methods,
which clearly match those of the explicit LNL SSP Runge--Kutta methods found in \cite{LNL} and repeated in Table \ref{table:LNL}.
The SSP coefficients of methods that have negative values in $\tilde{\m{S}}$ are listed in bold.
These methods do not have large linear stability regions, as they were not optimized for this purpose. However, these
results show that the addition of an implicit component that does not have its own SSP constraint does not have an adverse effect
on the size of the SSP IMEX method despite the addition of many new order conditions (coupling conditions) which must be satisfied.
\begin{table}
\begin{center}
\begin{tabular}{|ll|ccccc|}
\hline
nonlinear order & stages & $p_{lin}= 3$ & $p_{lin}= 4$ & $p_{lin}= 5$ & $p_{lin}= 6$ & $p_{lin}= 7$ \\ \hline
$p=2,3$ & $s=p_{lin}$ &1.0000 &1.0000 &1.0000 & 1.0000 & 1.0000 \\
$p=2,3$ &$s=p_{lin}+1$ &2.0000 &2.0000 &2.0000 & 2.0000 & 2.000 \\
$p=2,3$ & $s=p_{lin}+2$ &2.6505 &2.6505&2.6505 & 2.6505 & {\bf 2.6505} \\\hline
$p=4 $ & $s=p_{lin}$ & --- & --- & {\bf 0.7603} & 0.8677 & 1.0000 \\
$p=4$ &$s=p_{lin}+1$ & --- & 1.5082 & 1.8091 & 1.8269 & 1.9293 \\
$p=4$ & $s=p_{lin}+2$ & --- & 2.2945 & 2.5753 & 2.5629 & {\bf 2.6192} \\ \hline
\end{tabular}
\caption{ the SSP coefficient of optimal SSP IMEX methods for nonlinear order $p= p^e=p^i=2,3,4$ with $K=\infty$.
These match with the SSP coefficients of the explicit methods previously reported in \cite{SSPbook2011,LNL},
which verifies that the coupling conditions do not adversely affect the SSP coefficient in the case $K=\infty$.
Note that the SSP coefficients in bold correspond to methods that have negative values in $\tilde{\m{S}}$, while the other methods
have only nonnegative coefficient matrices.
}
\label{IMEXKinf}
\end{center}
\end{table}
Next, we look for methods that pair with popular SSP explicit Runge--Kutta methods and have large linear stability regions.
We found a family of methods that pair with the three-stage third order Shu-Osher method:
\[ \m{A}=\left( \begin{array}{lll}
0 & 0 & 0 \\
1 & 0 & 0 \\
\frac{1}{4} & \frac{1}{4} & 0 \\
\end{array} \right) \qquad \vb = \left(\frac{1}{6}, \frac{1}{6}, \frac{2}{3}\right)^T.\]
These implicit pair methods are defined by $\vb =\vbt$, and
\[ \tilde{\m{A}}=\left( \begin{array}{rrr}
0 & 0 & 0 \\
4 \gamma+2 \beta & 1- 4 \gamma - 2 \beta & 0 \\
\frac{1}{2} - \beta - \gamma & \gamma & \beta \\
\end{array} \right) \]
with
\[\gamma = \frac{2 \beta^2 - \frac{3}{2} \beta + \frac{1}{3}}{2- 4\beta} .\]
The methods are third order and for $\beta > \frac{1}{2}$ are all A-stable.
In particular, one member of this family is the nice looking method
\[ \tilde{\m{A}}=\left( \begin{array}{rrr}
0 & 0 & 0 \\
0 & 1 & 0 \\
\frac{1}{6} & - \frac{1}{3} & \frac{2}{3} \\
\end{array} \right) \qquad \vbt = \left(\frac{1}{6}, \frac{1}{6}, \frac{2}{3}\right)^T.\]
Another appealing method in this family occurs for the value $\beta= \frac{\sqrt{3}}{6} + \frac{1}{2} $.
In this case the two non-zero values on the diagonal are equal, and so we have a type of SDIRK method
(though with the first diagonal equal to zero).
Note that this value of the diagonal may look familiar: in fact, it is the same as
the diagonal value of the $s=2$, $p=3$ SDIRK method value in \cite[Table 7.2]{Hairer1},
but that method cannot be paired with the Shu-Osher method.
This family of methods can be compared with two of Pareschi and Russo's methods in \cite{PareschiRusso2005}
that also pair with the explicit SSPRK(3,3), Shu Osher method.
The first is a three stage method that is singly implicit and L-stable and has both nonzero elements
on the diagonal of $\tilde{A}$ but is only of overall order $p=2$. The second method is a four stage
method that is singly implicit, has four nonzero elements on the diagonal of $\tilde{A}$ but is of overall order
$p=3$. Depending on the cost of inverting the operator and the need for L-stability, the family of methods
we present, which require only two inverse solves (and possibly with the same diagonal value)
and are A-stable but not L-stable, may be preferable.
A useful and efficient method is Ketcheson's explicit SSP Runge--Kutta method with ten stages and nonlinear order $p^e=4$.
We found an SDIRK method which pairs well with Ketcheson's SSPRK(10,4) method and has a very large linear stability region.
The pair has linear order $p_{lin}=4$, and although it was produced by the optimizer under the assumption that $G$ is linear, we
obtained better-than-expected nonlinear order $p^i=3$ due to the equality $\vc=\vct$.
The coefficients of this method are given in the appendix and the linear stability region is plotted in Figure \ref{Ketch104stability}.
Note that the stability region this method crosses the real axis at $r_1 \approx -102,775$ and the imaginary axis at $r_2\approx \pm138,891$.
The optimization code was quickly able to match any value of $r= \min\{|r_1|, |r_2|\}$ that we requested. This leads us to suspect
that there may well be an A-stable method that pairs with Ketcheson's explicit SSPRK(10,4) method; however, the number of coefficients
here is prohibitive and unlike the case of the Shu-Osher SSPRK(3,3) method solving for this analytically seems difficult.
\begin{figure}[htb]
\centering
\begin{minipage}{0.45\textwidth}
\centering
\includegraphics[width=0.95\linewidth]{Ketcheson104CoupleStabRegion} \vspace{.125in}
\caption{Linear stability region in the complex plane of the method that pairs with Ketcheson's SSPRK(10,4) method
which has $p^e=p_{lin}= 4$, $p^i=3$.}
\label{Ketch104stability}
\end{minipage}%
\hspace{.2in}
\begin{minipage}{0.45\textwidth}
\centering
\includegraphics[width=0.995\linewidth]{LinStab106}
\caption{Linear stability region in the complex plane of the method that pairs with the LNL SSPRK with $s=10$ stages, and $p^e=4$, $p^i=3$ and $p_{lin} =6$.}
\label{10s6pIMEX_linstab}
\end{minipage}
\end{figure}
It is important to note that the explicit methods in the pair above all have an optimal SSP coefficient among methods
of its number of stages and order, and the coupling with an implicit method under the assumption $K = \infty$
does not adversely affect the SSP coefficient, so these methods are the best possible in their class
in terms of SSP coefficient.
Finally, we found an IMEX pair with $s=10$ stages, linear order $p_{lin}=6$, and nonlinear order
$p=3$. The explicit part has SSP coefficient $\mathcal{C}=3.3733$.
This method has an SDIRK implicit part, with a good linear stability region as shown in in Figure \ref{10s6pIMEX_linstab}.
The stability region crosses the real axis at $r_1 \approx -1,350 $ and the imaginary axis at $r_2\approx \pm 600$.
\noindent{\bf IMEX pairs for small $K$}
An interesting case that has not been considered extensively in the literature (except indirectly in \cite{Higueras2006}),
is the situation where the implicit component $G$ introduces a very tight time-step restriction for the strong stability
property to be satisfied. This is the case where the value of the parameter $K$ is very small.
We expect that in these cases the SSP coefficient of the IMEX pair will be correspondingly limited, and indeed the
coefficients are small. Table \ref{SSPIMEX_Ktable1}
gives the SSP coefficients of some methods we found using our optimization
routine. We see in these tables that for IMEX SSP Runge--Kutta methods the SSP coefficient is reduced as
both the linear and nonlinear orders rise; however, in general the SSP coefficients for $p=2$ and for $p=3$ are very similar,
and for these cases the increasing linear order causes the decrease in the SSP coefficient.
In addition, we see that there may be some modest benefit to tailoring the method to
the actual value of $K$ in the problem; whether this is true in practice is investigated in Section \ref{IMEX_SSP_verify}.
The coefficients $\m{A}, \tilde{\m{A}}, \vb$ and $\vbt$ of these methods can be downloaded at \cite{SSPIMEX_github}.
\begin{table}
\begin{center}
\begin{tabular}{|c|c|ccccc|} \hline
\multicolumn{7}{|c|}{$s=p_{lin}$} \\ \hline
$ \frac{1}{K}$ & p & \multicolumn{5}{|l|}{$p_{lin}$} \\
& & 3 & 4 & 5 & 6 & 7 \\ \hline
10 & 2 & 2.030$\times 10^{-1}$ & 1.729$\times 10^{-1}$& 1.520$\times 10^{-1}$& 1.323$\times 10^{-1}$ & 1.109$\times 10^{-1}$ \\
& 3 & 1.492$\times 10^{-1}$ & 1.727$\times 10^{-1}$ & 1.520 $\times 10^{-1}$ &1.323$\times 10^{-1}$ & 0.932$\times 10^{-1}$ \\
& 4 & -- & -- & -- & 1.132$\times 10^{-1}$ & 0.849$\times 10^{-1}$ \\ \hline
100 & 2 & 2.23$\times 10^{-2}$ & 2.06$\times 10^{-2}$ & 1.58$\times 10^{-2}$ & 1.51$\times 10^{-2}$ & 0.99$\times 10^{-2}$ \\
& 3 & 1.63$\times 10^{-2}$ & 2.05$\times 10^{-2}$ & 1.58$\times 10^{-2}$ & 1.51$\times 10^{-2}$ & 0.99$\times 10^{-2}$ \\
& 4 & -- & -- & --& 1.18$\times 10^{-2}$ & 0.89$\times 10^{-2}$ \\ \hline
\end{tabular}
\begin{tabular}{|c|c|ccccc|} \hline
\multicolumn{7}{|c|}{$s=p_{lin}+1$} \\ \hline
$ \frac{1}{K}$ & p & \multicolumn{5}{|l|}{$p_{lin}$} \\
& & 3 & 4 & 5 & 6 & 7 \\ \hline
10 & 2 & 3.570$\times 10^{-1}$ & 3.099$\times 10^{-1}$ & 2.891$\times 10^{-1}$ & 2.307$\times 10^{-1}$ & 1.756$\times 10^{-1}$ \\
& 3 & 2.837$\times 10^{-1}$ & 3.084$\times 10^{-1}$ & 2.767$\times 10^{-1}$ & 2.288$\times 10^{-1}$ & 1.756$\times 10^{-1}$ \\
& 4 &-- & 2.001$\times 10^{-1}$ & 2.164$\times 10^{-1}$ & 1.986$\times 10^{-1}$ & 1.755$\times 10^{-1}$ \\ \hline
100 & 2 & 3.97$\times 10^{-2}$ & 3.65$\times 10^{-2}$ & 3.26$\times 10^{-2}$ & 2.98$\times 10^{-2}$ & 1.85$\times 10^{-2}$ \\
& 3 & 3.23$\times 10^{-2}$ & 3.56$\times 10^{-2}$ & 3.20$\times 10^{-2}$ & 2.98$\times 10^{-2}$ & 1.79$\times 10^{-2}$ \\
& 4 & -- & 2.26$\times 10^{-2}$ & 2.28$\times 10^{-2}$ & 1.94$\times 10^{-2}$ & 1.66$\times 10^{-2}$ \\ \hline
\end{tabular}
\end{center}
\caption{SSP coefficients of some optimized IMEX methods with $s=p_{lin}$ and $s=p_{lin}+1$
stages optimized for different values of $K$.}
\label{SSPIMEX_Ktable1}
\end{table}
}
These methods are not at all optimized in terms of the linear stability region, as the main constraint of interest for these
methods comes from the SSP condition, and indeed their linear stability region is similar to explicit methods.
We believe these methods may be useful where the cost of the implicit solution is negligible
while the SSP condition is needed for small values of $K$. However, even in cases where these methods are not directly useful,
they give us an upper bound on the possible SSP coefficients of SSP IMEX methods for typical values of $K$, and so serve as a guide
to what we can expect when we co-optimize these methods with other desirable properties.
The numerical tests, particularly those in Section \ref{IMEX_SSP_verify}, show the performance of some of these
methods for the types of problem that require the SSP property for both the explicit $F$ and implicit $G$.
\subsection{Numerical Experiments} \label{Numerical2}
\subsubsection{Convergence studies} \label{convergence}
To test the accuracy of the SSP IMEX Runge--Kutta methods, we consider a linear or nonlinear explicit part. As our main motivation for these
methods are where the implicit component is linear, we test the methods on problems where $G$ is linear.
In these tests we confirm that our methods give us the expected linear and nonlinear orders
for the explicit and implicit parts.
\noindent{\bf Example 3.1: convergence study with explicit linear advection and implicit linear diffusion.}
We approximate the solution to the equation \[ U_t + U_x = \epsilon U_{xx} \; \; \; \; \; \; x \in [0, 2 \pi] \]
where $\epsilon = 0.01$ with periodic boundary conditions and sine wave initial conditions.
We discretize the spatial grid with $N=8$ equidistant points and use the Fourier pseudospectral
differentiation matrix for both the first and second derivative \cite{HGG2007}.
The advection part is dealt with explicitly and the diffusion implicitly (and exactly using MATLAB's backslash operator).
Temporal grid refinement with pseudospectral approximation of the spatial derivative.
We use a range of time steps, $\Delta t = \lambda \Delta x$, where $\Delta x = \frac{\pi}{4}$ and
we pick $\lambda = \frac{1}{16},\frac{1}{8}, \frac{1}{4}, \frac{1}{2}$ to compute the solution to final time $T_f = 5.0$.
The methods we test are our IMEX SSP methods with $s$ stages, nonlinear order $p=p^i=p^e$ and linear order $p_{lin}$
$(s, p,p_{lin} ) =(3,3,3), (4,3,4), (5,4,4), (6,2,5),(6,3,6), (7,2,7)$ designed for $K=\infty$ and for $K=10^{-2}$.
The $\ell_2$ errors are measured compared to the exact solution and shown in
Figure \ref{fig:IMEXconv_lin_lin}. We note that the methods designed for $K=\infty$ and those designed for $K=10^{-2}$
perform essentially the same on convergence studies. The orders (slope $m$ of the line) are taken by taking a linear fit
using MATLAB's {\tt polyfit} and given in the caption.
We see that, as expected, the linear design order is apparent in this linear problem.
It is interesting to note that the $(s,p,p_{lin} ) =(3,4,4)$ and the $(s, p,p_{lin}) =(4,4,5)$ have the same slope, because their linear order
is the same, but the latter method has a smaller error constant so the errors are smaller.
\begin{figure}[htb]
\includegraphics[width=0.475\linewidth]{AdvDiffConvergenceK0}
\includegraphics[width=0.475\linewidth]{AdvDiffConvergenceK100}
\caption{Convergence plots for an advection diffusion problem in Example 3.1.
Methods shown are $(s, p^i,p^e,p_{lin}) =(3,3,3), (4,3,4), (5,4,4), (6,2,5),(6,3,6), (7,2,7)$.
On the left are the methods created for $K=\infty$ and on the right for $K=10^{-2}$.}
\label{fig:IMEXconv_lin_lin}
\end{figure}
\noindent{\bf Example 3.2: convergence study with explicit Burgers' and implicit linear advection.}
We approximate the solution to the equation \[ U_t + \left( \frac{1}{2} U^2 \right)_x + U_x = 0 \; \; \; \; \; \; x \in [0, 2 \pi] \]
with periodic boundary conditions and sine wave initial conditions.
We discretize the spatial grid with $N=24$ equidistant points and use the Fourier pseudospectral
differentiation matrix \cite{HGG2007}.
The Burgers' flux $\left( \frac{1}{2} U^2 \right)_x$
is dealt with explicitly and the linear advection implicitly (and exactly using MATLAB's backslash operator).
We perform grid refinement with pseudospectral approximation of the spatial derivative,
we use a range of time steps, $\Delta t = \lambda \Delta x$
where we pick $\lambda = \frac{1}{128}, \frac{1}{64}, \frac{1}{32} , \frac{1}{16}, \frac{1}{8} $ to
compute the solution to final time $T_f = 0.8$ before the shock forms.
The methods we test are our IMEX SSP methods with
$(s, p,p_{lin} ) =(3,3,3), (4,3,4), (5,4,4), (6,2,5),(6,3,6), (7,2,7)$
designed for $K=\infty$ and for $K=10^{-2}$.
The $\ell_2$ errors are measured compared to the the results from MATLAB's {\tt ode45} and graphed in
Figure \ref{fig:IMEXconv_nl_lin}. The orders (slope $m$ of the line) are taken by taking a linear fit using MATLAB's {\tt polyfit}
and given in the caption. We see that the nonlinear order $p$ typically dominates: this occurs since the
linear order $p_{lin}$ is no smaller than the nonlinear orders. In the case $(7,2,7)$ for $K=10^{-2}$
we get a better order than expected by one, due to a small error constant ($\approx 10^{-8}$) for this method.
\begin{figure}[htb]
\includegraphics[width=0.475\linewidth]{convergence_advectionBurgersK0}
\includegraphics[width=0.475\linewidth]{convergence_advectionBurgersK100}
\caption{Convergence plots for an the explicit Burgers' with implicit linear advection
problem in Example 3.2.
Methods shown are $(s, p,p_{lin} ) =(3,3,3), (4,3,4), (5,4,4), (6,2,5),(6,3,6), (7,2,7)$.
On the left are the methods created for $K=\infty$ and on the right for $K=10^{-2}$.
}
\label{fig:IMEXconv_nl_lin}
\end{figure}
\subsubsection{Numerical verification of the SSP properties of these methods} \label{IMEX_SSP_verify}
In this section we focus on the behavior of our time-stepping methods on problems which require the
SSP property. In particular, we use out IMEX SSP methods for problems where the desired property
to be preserved is the total variation diminishing (TVD) property and where the spatial discretization is simple
enough that the theoretical bound on the TVD condition is easy to see.
\noindent{\bf Example 4.1: Explicit linear advection with implicit linear advection. }
Consider the linear equation
\[ U_t + U_x + 100 U_x = 0 \; \; \; \; \; \; x \in [-1, 1] \]
with periodic boundary conditions and step function initial conditions
\begin{equation}
\label{eqn:sq_wave_ic}
u_0(x) = \left\{ \begin{array}{ll}
1 & \text{if}\ \frac{1}{4} \leq x \leq \frac{1}{2}, \\
0 & \text{otherwise},
\end{array} \right.
\end{equation}
We use a first order upwind differencing the linear advection terms,
with $N=301$ points in space. The time-step is set to $\Delta t = \lambda \Delta x$, where different values of
$\lambda$ are tested to observe the value at which the discretization is no longer TVD
(i.e. the total variation rises by more than $10^{-10}$ at any given time-step). We first compared the
Predicted $\Delta t$ and Observed $\Delta t$ for which the TVD property is
violated. The results are very similar to those in Table \ref{NonlinearDT} and so we do not report them here.
We consider several IMEX methods that were optimized for $K=1, \frac{1}{10}, \frac{1}{100}$
and see how their observed time-step for the TVD to be satisfied compares to the time-step predicted
by the theory, even when the methods were designed for different values of $K$.
We also compare the performance of these methods to the situation where both components are
advected by the optimal explicit LNL SSP Runge--Kutta method of the corresponding number of
stages, nonlinear order, and linear order. For completeness, we also include a comparison to
treating both components implicitly using an optimal implicit LNL SSP Runge--Kutta method.
Table \ref{LinearDT} contains the observed and predicted time-step for which the TVD property is
violated. Clearly, the observed $\Delta t$ is never smaller than the predicted $\Delta t$, as the
SSP property provides a guarantee of this property. We note that while the predicted and observed TVD time-step
are always {\em exact} when both components are treated implicitly, and frequently so when they are both
treated explicitly, this is not as often the case where we use an IMEX method (though in many cases, particularly those
noted in Table \ref{NonlinearDT}, we do have sharp agreement). It is notable that using an IMEX method
that was optimized for a value of $K$ close to the actual value of $\frac{1}{\omega}$ we generally see larger
allowable time-step, though this difference may not be large enough to generate optimal methods for many different values.
The main question we wish to answer is whether these methods are useful as replacements to the fully explicit
method. In these cases we see that using an IMEX scheme allows us to take a time-step that is between $1.5$ and $2$
times the fully explicit time-step (see the ratio column). There are very limited circumstances in which this larger time-step
is truly enough to offset the cost of the implicit solver; however, this is what one would have expected from the fact that
the SSP coefficient of the explicit methods are bounded by the number of stages while the
SSP coefficient of the implicit methods are bounded by twice the number of stages. The more interesting
observation in this table may be the ratio of the fully implicit allowable time-step to the fully explicit allowable time-step:
these can be quite large, possibly large enough to offset the implicit solver cost in some cases.
\begin{table}[t]
\hspace{0.8in} \begin{tabular}{|cccccccc|} \hline
Type & p & $p_{lin}$ & s &$ \frac{1}{K}$ & Predicted $\Delta t$ & Observed $\Delta t$ & ratio \\ \hline
Explicit & 2 & 5 & 6 & -- & 1.98$\times 10^{-2} $ & 1.98$\times 10^{-2} $ & --\\
IMEX & & & & 1 & 1.25$\times 10^{-2}$ & 3.07$\times 10^{-2} $ & 1.55\\
IMEX & & & & 10 & 2.89$\times 10^{-2}$ & 3.33$\times 10^{-2} $ & 1.68\\
IMEX & & & & 100 & 3.26$\times 10^{-2}$ & 3.37$\times 10^{-2} $ & 1.70 \\
Implicit & & & & -- & 6.59$\times 10^{-2}$ & 6.59$\times 10^{-2}$ & 3.32 \\ \hline
Explicit & 2 & 6 & 6 & -- & 9.90$\times 10^{-3} $ & 9.90$\times 10^{-3} $ & -- \\
IMEX & & & & 1 & 6.00$\times 10^{-3} $ & 1.47$\times 10^{-2} $ & 1.48 \\
IMEX & & & & 10 & 1.32$\times 10^{-2} $ & 1.94$\times 10^{-2} $ & 1.96 \\
IMEX & & & & 100 & 1.51$\times 10^{-2} $ & 1.97$\times 10^{-2} $ & 1.99 \\
Implicit & & & & -- & 5.12$\times 10^{-2} $ & 5.12$\times 10^{-2} $ & 5.17 \\ \hline
Explicit & 3 & 5 & 5 & -- & 9.90$\times 10^{-3} $ & 9.90$\times 10^{-3} $ & -- \\
IMEX & & & & 1 & 6.34$\times 10^{-3} $ & 2.00$\times 10^{-2} $ & 2.02 \\
IMEX & & & & 10 & 1.52$\times 10^{-2} $ & 2.24$\times 10^{-2} $ & 2.26 \\
IMEX & & & & 100 & 1.58$\times 10^{-2} $ & 2.38$\times 10^{-2} $ & 2.40 \\
Implicit & & & & -- & 4.96$\times 10^{-2} $ & 4.96$\times 10^{-2} $ & 5.01 \\ \hline
Explicit & 3 & 5 & 6 & -- & 1.98$\times 10^{-2} $ & 1.98$\times 10^{-2} $ & -- \\
IMEX & & & & 1 & 1.23$\times 10^{-2} $ & 3.10$\times 10^{-2} $ & 1.56 \\
IMEX & & & & 10 & 2.77$\times 10^{-2} $ & 3.30$\times 10^{-2} $ & 1.66 \\
IMEX & & & & 100 & 3.20$\times 10^{-2} $ &3.50$\times 10^{-2} $ & 1.76 \\
Implicit & & & & -- & 6.59$\times 10^{-2} $ & 6.59$\times 10^{-2} $ & 3.32 \\ \hline
Explicit & 4 & 6 & 6 & -- & 8.59$\times 10^{-3} $ & 9.90$\times 10^{-3} $ & -- \\
IMEX & & & & 1 & 5.12$\times 10^{-3} $ & 1.91$\times 10^{-2} $ & 1.93 \\
IMEX & & & & 10 & 1.13$\times 10^{-2} $ & 2.14$\times 10^{-2} $ & 2.16 \\
IMEX & & & & 100 & 1.18$\times 10^{-2} $ & 2.09$\times 10^{-2} $ & 2.11 \\
Implicit & & & & -- & 5.09$\times 10^{-2} $ & 5.09$\times 10^{-2} $ & 5.92 \\ \hline
\end{tabular}
\caption{Comparison of the theoretical and observed allowable time-step before an SSP violation
of $10^{-12}$ occurs for the linear problem in Example 4.1
with wavespeed $\omega=100$ and for method optimized for the value of $K$ above.}
\label{LinearDT}
\end{table}
\noindent{\bf Example 4.2: Explicit Burgers' with implicit linear advection. }
We consider the equation
\[ U_t + \left( \frac{1}{2} U^2 \right)_x + \frac{1}{K} U_x = 0 \; \; \; \; \; \; x \in [-1, 1] \]
with periodic boundary conditions and step function initial conditions
\begin{equation}
\label{eqn:sq_wave_ic}
u_0(x) = \left\{ \begin{array}{ll}
1 & \text{if}\ \frac{1}{4} \leq x \leq \frac{1}{2}, \\
0 & \text{otherwise},
\end{array} \right.
\end{equation}
We use a first order upwind differencing for both the Burgers' term and the linear advection term,
with $N=301$ points in space. The methods used are those optimized for the value of $K$ that corresponds to the
wavespeed $\omega=\frac{1}{K}$, and these IMEX methods have
$s$ stages, nonlinear order $p^e=p^i=p$ and nonlinear order $p_{lin}$.
The time-step is set to $\Delta t = \lambda \Delta x$, where different values of
$\lambda$ are tested to observe the value at which the discretization is no longer TVD. This value of $\Delta t$ is then called
the ``Observed $\Delta t$'' while the value we would expect from the SSP coefficient is called the
``Predicted $\Delta t$''.
Table \ref{NonlinearDT} contains the observed and predicted time-step for which the TVD property is
violated. Clearly, the observed $\Delta t$ is never smaller than the predicted $\Delta t$, as the
SSP property provides a guarantee of this property. As seen in the table, in some cases, the
methods feature close agreement between the predicted and observed SSP coefficient, while in others
the actual time-step allowed for this problem is greater than predicted. We note that these
methods performed very similarly on the linear problem in Example 4.2 above, so the results were not repeated
in a separate table. These results, and in particular the sharpness of the predicted allowable time-step in
several cases, demonstrate that the theoretical SSP property is a good predictor of the actual behavior of the method.
\begin{table}[t]
\hspace{0.8in} \begin{tabular}{|cccccc|} \hline
p & $p_{lin}$ & s &$ \omega=\frac{1}{K}$ & Predicted $\Delta t$ & Observed $\Delta t$ \\ \hline
2 & 4 & 5 & 10 & 3.099$\times 10^{-1} $ & 3.099$\times 10^{-1} $ \\
& & & 100 & 3.65$\times 10^{-2} $ & 3.65$\times 10^{-2} $ \\ \hline
3 & 4 & 5 & 10 & 3.084$\times 10^{-1} $ & 3.084$\times 10^{-1} $ \\
& & & 100 & 3.56$\times 10^{-2} $ & 3.56$\times 10^{-2} $ \\ \hline
3 & 5 & 5 & 10 & 1.520$\times 10^{-1} $ & 2.135$\times 10^{-1} $ \\
& & & 100 & 1.58$\times 10^{-2} $ & 2.39$\times 10^{-2} $ \\ \hline
2 & 4 & 6 & 10 & 4.088$\times 10^{-1} $ & 4.088$\times 10^{-1} $ \\
& & & 100 & 4.67$\times 10^{-2} $ & 4.67$\times 10^{-2} $ \\ \hline
3 & 4 & 6 & 10 & 4.171$\times 10^{-1} $ & 4.171$\times 10^{-1} $ \\
& & & 100 & 4.59$\times 10^{-2} $ & 4.59$\times 10^{-2} $ \\\hline
3 & 6 & 7 & 10 & 2.29$\times 10^{-1} $ & 3.00$\times 10^{-1} $ \\
& & & 100 & 2.98$\times 10^{-2} $ & 3.50$\times 10^{-2} $ \\ \hline
4 & 6 & 6 & 10 & 1.13$\times 10^{-1} $ & 2.01$\times 10^{-1} $ \\
& & & 100 & 1.18$\times 10^{-2} $ & 2.10$\times 10^{-2} $ \\ \hline
\end{tabular}
\caption{Comparison of the theoretical and observed SSP coefficients that preserve the nonlinear stability properties in Example 4.2
with the methods optimized for the value $\frac{1}{K}$ to match the wavespeed $\omega$.}
\label{NonlinearDT}
\end{table}
\vspace{-.15in}
\section{Conclusions}
\vspace{-.1in}
In this work, we investigated implicit and IMEX SSP methods with very high linear order.
We first considered implicit SSP Runge--Kutta methods that have high order for linear problems and order $p=2$
for nonlinear problems. We show that as is the case with explicit methods we are able to find implicit methods with $p_{lin} > 6$
with no linear order barrier. We were further able to show that (as in \cite{LNL}) the optimal methods had no better SSP coefficient
that optimal methods that are diagonally implicit with a low storage formulation. Thus we can say that optimal implicit SSP Runge--Kutta methods
for any linear order $p_{lin}$ and nonlinear order $p=2$ are diagonally implicit and low storage.
We continued our study of implicit linear/nonlinear (LNL) methods for larger nonlinear orders $p>2$
following the approach used in \cite{LNL}. We designed SSP implicit Runge--Kutta methods where the
linear order exceeded the nonlinear order (called LNL methods), and observed that the SSP coefficient
for methods with linear order $p_{lin} \leq 9$ was similar whether we chose nonlinear order $p=2$ and $p=3$,
and that if we went up to $p=4$, the SSP coefficient was not typically significantly reduced as long as we had
sufficiently many stages. This implies that just as the case for the explicit LNL methods, as we increase the
number of stages and maintain a moderate nonlinear order $p=3,4$, the linear order is the main constraint
on the SSP coefficient. However, if we increase the nonlinear order to $p=5,6$, this did have a strong adverse
effect on the size of the SSP coefficient. We also observed that the SSP coefficients of the implicit LNL SSP Runge--Kutta methods
were up to six times as large as those of the corresponding explicit LNL SSP Runge--Kutta of the same number of stages and linear
and nonlinear orders.
We report the coefficients of methods with high linear order
and moderate number of stages that have reasonable SSP coefficients, for example the LNL implicit RK
$(s,p,p_{lin})=(6,4,6)$ method with SSP coefficient $\mathcal{C}=0.856$. These methods may be useful when a
high linear order method is desired, while still reasonable for use with nonlinear problems.
We then verified the convergence of these methods as well as the sharpness of the SSP coefficient on sample problems.
The second part of this paper focused on implicit-explicit methods, where as above the methods have higher linear order than nonlinear order.
These methods are of value when we desire both the explicit and implicit parts to have the SSP property, and are particularly beneficial
when the linear part of a problem is treated implicitly, and the nonlinear part explicitly.
We found diagonally implicit non-SSP Runge--Kutta methods with large regions of linear stability that pair with well-known
explicit SSP methods. Next, we found pairs of IMEX SSP methods, with high linear order, that were optimized for the relationship
between the forward Euler conditions of each component.
We verified the order of these methods on a variety of problems, and the sharpness of the SSP coefficient on a typical test cases,
and conclude that these IMEX LNL methods demonstrate high order convergence and perform as desired in terms of SSP.
This work shows that it is possible to produce implicit and IMEX SSP methods of very high order if we consider only the linear order.
For implicit methods, we found methods of up to order $p_{lin}=9$ and for the IMEX methods up to order $p_{lin}=7$, which show that
the order barriers of the implicit and IMEX SSP Runge--Kutta methods apply only to the nonlinear order.
The approach described in this work can be used to produce optimal SSP implicit and IMEX methods with additional
desired properties, as well. The SSP coefficients presented in this work give a baseline with which to compare any
methods that are SSP and also have other properties, and give a value to strive toward.
As expected at the outset, the allowable time-step for these methods is very restricted, especially for the IMEX schemes and therefore
the cost of the implicit solvers is still the major issue that arises in the solution of these methods.
However, in particular cases (such as where the implicit term is a constant coefficient linear term) where the cost of the implicit solver
is controllable, these methods may be of more than theoretical value.
{\bf Acknowledgements} This work was supported by AFOSR grant FA9550-15-1-0235, and
was partially supported by DOE NNSA ASC Algorithms effort, the DOE Office of Science AMR program
at Sandia National Laboratory under contract DE-AC04-94AL85000
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,375 |
bettyboop11
Heterosporous
Producing two kinds of sports: mega sporangia produce megasporangia that give rise to female gametophyte and microsporangia produce microspores that give rise to male gametophyte.
Homosporous
They produced one kind of spor which usually gives rise to a bisexual gametophyte
Integument
A layer of sporophyte tissue the envelopes and protects the megasporangium. Gymnosperm megasporangia are surrounded by one in integument, whereas those in angiosperm usually have two integuments
The whole structure which includes the megasporangium, megaspore, and their integuments. Inside and I love you is a female gametophyte develops from a megaspore and produces one or more eggs
Pollen grain
A micro sport develops into this, that consists of a male gametophyte enclosed within the pollen wall.
The transfer of pollen to the part of a seed plant that contains the ovule. If a pollen grain germinates, it gives rise to a pollen tube that discharges sperm into the female gametophyte within the ovule,
What advantages do seeds provide over spores?
Sports are usually single celled, whereas seats are multicellular, consisting of an employee protected by a layer of tissue, the seed coat. A seed can retain dormant for days, months, or even years after being released from the parent plan, whereas most boards have shorter life time. Also, unlike sports, seats have a supply of stored food.
Cone bearing plants such as Pines fir and redwoods most gymnosperm are conifers. they are heterosporous.
What three key reproductive adaptation to seed plant evolution include?
1. The miniaturization of their gametophyte
2. The appearance of paulin as an airborne agent that brings gametes together
3. The advent of the seed as a resistant, dispersible stage in the life cycle
name the four plant phyla that are gymnosperm
1. Cycadophyta
2. Ginkgophyta
3. Gnetophyta
4. Coniferophyta
300 species of living have large cones and palm like leaves. Unlike most seed plants, they have flagellated sperm, indicating their descent from seedless vascular plants that had multiple sperm. They thrive during the Mesozoic era, known as the age of cycads as well as the age of the dinosaurs. Today however they are the most endangered of all plant groups.
Ginko biloba is the only surviving species of this phylum: like cycads Ginko have flagellated sperm also known as the maidenhair tree, ginkgo biloba has deciduous fun like leaves that turns cold in autumn. It is a popular ornamental tree and cities because it tolerates air pollution well. Landscapers often plan only paulin producing trees because the fleshy seed smell rancid as they decay
Phylum Gnetophyta includes plants in three genera: Gnetum, ephedra, and welwitschia. Some species are tropical, whereas others live in deserts. Although very different in appearance the genera are grouped together based on molecular data
Coniferophyta
The largest gymnosperm phyla consists of about 600 species of conifers including many large trees. Most species have woody cones but if you have fleshy cones. Some such as Pines have needle - like leaves. Others, such as redwoods, have skill - like leaves. Some species dominate vast northern forest, whereas others are native to the southern hemisphere. Most are evergreens: they retain their leaves throughout the year. Even during winter, a limited amount of photosynthesis occurs on sunny days. When spring comes conifers already have fully developed leaves that can take advantage of the sunny here, warmer days. Some conifers such as the dawn redwood, tamarack and larch are deciduous trees that lose leaves each autumn.
What are two key adaptations of the Angiosperm in the phylum Anthophyta
Its a unique angiosperm structure specialized for sexual reproduction. In many species, insects or other animals transfer pollen from one flower to the sex organs on another flower it can have up to 4 types of modified leave called floral organs: sepals, pedals, stamens, carpels
Sepals
Are usually green and enclose the flower before it opens are sterile floral organs that do not produce sperm or eggs.
Are brightly colored in most flowers and aid in attracting pollinators. are sterile floral organs that do not produce sperm or eggs.
Produce microspores that develop into pollen grain containing male gametophyte. Consists of a stock called the filament and an internal sac which is the anther , where pollen is produced.
Carpels
Make megaspores and their products, female gametophyte. It is the container mentioned earlier in which the seeds are enclosed.
Located at the tip of the carpel. It is sticky and this is where pollin is received
Leads from the stigma to a structure at the base of the carpel, the ovary: the ovary contains one or more ovules. If fertile eggs, and develops into a seed
Complete flowers
Flowers that have all four organs
Incomplete flowers
Those that like one or more of these organs
Us seats develop from ovules after fertilization, the ovary walk the kids and the ovary matures. Fruits protect seeds and aid in their dispersal.
Embryo sac
Each of you, which develops into the ovary contains a female gametophyte also known as an embryo sac
Cross pollination
Which in angiosperm is the transfer of pollen from an answer of a flower on one plant to the stigma of a flower on another plant of the same species cross pollination enhances genetic variability.
Micropyle
A port in the integument of the ovule and discharges two sperm cells into the female gametophyte.
Double fertilization
And which one for the PlayStation event produces a cycle and the other produces a triploid cell, it is unique to angiosperm.
Endosperm
Tissue rich in starch and other food reserves that nourish the developing embryo
Cotyledons
the zygote develops into a sporophyte embryo with a rudimentary root and one or two seed leaves
What is the function of double fertilization in angiosperm?
What type of this is is a double fertilization synchronizes the development of food storage in the seat with the development of the embryo. perhaps it is an adaptation that prevents flowering plants from squandering nutrients on unfertile ovules.
Species with one cotyledon. Cotyledons are sead leaves in the embryo.
species with two cotyledon. They are paraphyletic
Eudicots
The vast majority of species once categorized as dicots form a larger clade.
Basal angiosperms
Surviving basil angeles perm are currently stuck to consist of religious compromising only about 100 species. The oldest lineage seems to be represented by a single species amborella trichopoda.
Magnoliid
Consists of about 8,000 species, most notably magnolias, laurels, and black pepper plants. They include both woody and herbaceous species. Although they share some traits with basal angiosperm, such as a typical spiral rather than whorled arrangement of floral organs. They are more closely related to eudicots and monocots
About one-quarter of angiosperm species are monocots - about 70,000 species. Some of the largest groups are the orchids, grasses, and palms. Grasses include some of the most important crops, such as maize rice and wheat
More than two-thirds of angiosperm species - roughly 170,000 species. The largest group in the legume family which includes such crops as peas and beans. Also important economically is the rose family which includes many plants with ornamental flowers as well as some species with edible fruits, such as strawberry plants and apple and pear trees. Most of the familiar flowering trees such as oak, walnut, maple, willow, and birch.
Plants 2: Seed Plants
mpleasur
Bio 2: Chapter 30
Morgan_Conn3
The evolution of seed plants
krsomersPLUS
Chapter 30 Notes (Exam III)
Brooke_McMinn
mixology lesson #2 wine&coffee
Chapter 29 Bio 2
sandelice
Ch 31 BioOrg
mohib_ibrahim
Plant Growth and Development
Samuel4378
Chapter 54 Mastering Biology: Community Ecology
Aclark700 | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,480 |
{"url":"https:\/\/en.pythonmana.com\/2022\/01\/202201312038445924.html","text":"# Deeply understand pandas to read excel, TXT, CSV files and other commands\n\n2022-01-31 20:38:47\n\n## pandas Read the document provided by the file official\n\nIn the use of pandas Before reading the file , Essential content , It must be an official document , Refer to the address of the official document\n\npandas.pydata.org\/pandas-docs\u2026\n\nDocument operation belongs to pandas Inside Input\/Output That is to say IO operation , Basic API It's all on the website , The core of this article takes you through some of the common commands\n\nRead txt The file needs to be determined txt Whether the file conforms to the basic format , That is, whether it exists \\t,,, And a special separator\n\ncommonly txt The file looks like this\n\ntxt File, for example,\n\nThe following files are Spaces apart\n\n1 2019-03-22 00:06:24.4463094 Chinese test\n2 2019-03-22 00:06:32.4565680 You need to edit encoding\n3 2019-03-22 00:06:32.6835965 ashshsh\n4 2017-03-22 00:06:32.8041945 eggg\nCopy code \n\nimport pandas as pd\nprint(df)\n\nimport pandas as pd\nprint(df)\nCopy code \n\nhowever , Be careful , The content of the data read from this place is 3 That's ok 1 Column DataFrame type , We didn't get it as we wanted 3 That's ok 4 Column\n\nimport pandas as pd\nprint(type(df))\nprint(df.shape)\n\n<class 'pandas.core.frame.DataFrame'>\n(3, 1)\nCopy code \n\nDefault : From file \u3001URL\u3001 The delimited data is loaded in the new object of the file , The default separator is a comma .\n\nAbove txt Documents are not separated by commas , So you need to increase it as you read sep Separator parameter\n\ndf = pd.read_csv(\".\/test.txt\",sep=' ')\nCopy code \n\nParameter description , official Source :\u00a0github.com\/pandas-dev\/\u2026 Description in Chinese and key function cases\n\nParameters Chinese meaning\nfilepath_or_buffer It can be URL, You can use URL Types include \uff1ahttp, ftp, s3 And documents , The local file reads the instance \uff1afile:\/\/localhost\/path\/to\/table.csv\nsep str type , Default ',' Specify the separator . If no parameters are specified , Instead, you will try to use the default value, comma . The separator is longer than one character and is not \u2018\\s+\u2019, Will use python Parsers of . And ignore commas in the data . Regular expression example \uff1a'\\r\\t'\ndelimiter delimiter , Alternative separator \uff08 If this parameter is specified , be sep Parameters of the failure \uff09 Generally do not use\ndelimiter_whitespace True or False Default False, Using a space as a separator is equivalent to spe=\u2019\\s+\u2019 If the parameter is called , be delimite It won't work\nheader Specify the row as the column name ( Ignore comment lines ), If no column name is specified , Default header=0; If the column name is specified header=None\nnames Specifies the column name , If not in the file header The line of , Should be explicit header=None ,header It could be a list of integers , Such as 0,1,3. An unspecified intermediate line is deleted ( for example , Skip this example 2 That's ok )\nindex_col( Case study 1) The default is None Use the column name DataFrame The line of label , If I give you a sequence , Then use MultiIndex. If a file is read , The file has a delimiter at the end of each line , Consider using index_col=False send panadas Do not use the first column as the row name .\nusecols Default None You can use column sequences or column names , Such as 0, 1, 2 or \u2018foo\u2019, \u2018bar\u2019, \u2018baz\u2019 , Use this parameter to speed up loading and reduce memory consumption .\nsqueeze The default is False, True The type returned is Series, If the data is parsed to contain only one line , Then return to Series\nprefix Automatically generates a prefix for the column name number , Such as \uff1a \u2018X\u2019 for X0, X1, ... When header =None Or no Settings header In force\nmangle_dupe_cols The default is True, Duplicate columns will be specified as \u2019X.0\u2019\u2026\u2019X.N\u2019, instead of \u2019X\u2019\u2026\u2019X\u2019. If you pass in False, When a duplicate name exists in a column , Will cause the data to be overwritten .\ndtype Example \uff1a {\u2018a\u2019: np.float64, \u2018b\u2019: np.int32} Specify the data type for each column ,a,b Said column names\nengine The analysis engine used . You can choose C Or is it python,C The engine is fast but Python The engine has more features\nconverters( Case study 2) Sets the handler for the specified column , It can be used \" Serial number \" You can also use \u201c Name \u201d Specifies the column\ntrue_values \/ false_values No actual application scenario was found , Note the , The late perfect\nskipinitialspace Ignore the space after the separator , Default false\nskiprows The default value is None Number of rows to ignore \uff08 Start at the beginning of the file \uff09, Or a list of line Numbers to skip \uff08 from 0 Start \uff09\nskipfooter Start at the end of the file and ignore it . (c Engine not supported )\nna_values A null value definition , By default , \u2018#N\/A\u2019, \u2018#N\/A N\/A\u2019, \u2018#NA\u2019, \u2018-1.#IND\u2019, \u2018-1.#QNAN\u2019, \u2018-NaN\u2019, \u2018-nan\u2019, \u20181.#IND\u2019, \u20181.#QNAN\u2019, \u2018N\/A\u2019, \u2018NA\u2019, \u2018NULL\u2019, \u2018NaN\u2019, \u2018n\/a\u2019, \u2018nan\u2019, \u2018null\u2019. Are characterized by NAN\nkeep_default_na If specified na_values Parameters , also keep_default_na=False, So the default NaN Will be overwritten , Otherwise, add\nna_filter Whether to check for missing values \uff08 An empty string or a null value \uff09. Not in the dataset for large files N\/A Null value , Use na_filter=False Can improve the read speed .\nverbose Whether to print the output of various parsers , for example \uff1a\u201c The number of missing values in a nonnumeric column \u201d etc. .\nskip_blank_lines If True, Skip the blank line ; Otherwise the record for NaN.\nparse_dates There are the following operations 1. boolean. True -> Analytical index 2. list of ints or names. e.g. If 1, 2, 3 -> analysis 1,2,3 The value of the column as a separate date column ;3. list of lists. e.g. If [1, 3] -> Merge 1,3 The column is used as a date column 4. dict, e.g. {\u2018foo\u2019 : 1, 3} -> take 1,3 columns , And name the merged column \"foo\"\ninfer_datetime_format If set to True also parse_dates You can use , that pandas Converts an attempt to a date type , If you can switch , Transform the method and parse it . In some cases it is 5~10 times\nkeep_date_col Resolve dates if multiple columns are concatenated , Keeps the columns that participate in the join . The default is False\ndate_parser The function that parses the date , By default dateutil.parser.parser To do the conversion .Pandas Try parsing in three different ways , Use the next method if you encounter problems .1. Use one or more arrays\uff08 from parse_dates Appoint \uff09 As a parameter ;2. Concatenation specifies a multi-column string as a single column as an argument ;3. Each line is called once date_parser Function to parse one or more strings \uff08 from parse_dates Appoint \uff09 As a parameter .\ndayfirst DD\/MM Format of the date type\nchunksize The size of the file block\ncompression Use the compressed file directly on disk . If you use infer Parameters , Then use gzip, bz2, zip Or extract the file name \u2018.gz\u2019, \u2018.bz2\u2019, \u2018.zip\u2019, or \u2018xz\u2019 These files are suffixed , Otherwise, no decompression . If you use zip, that ZIP Package China must contain only one file . Set to None Do not unzip .\nThe new version 0.18.1 Versioning support zip and xz decompression\nthousands Thousandths sign , Default \u2018,\u2019\ndecimal Decimal symbol , Default \u2018.\u2019\nlineterminator Line separator , Only in C Used under the parser\nquotechar quotes , A character used to identify the beginning and interpretation , The delimiter inside the quote is ignored\nquoting control csv Quote constant in . Optional QUOTE_MINIMAL (0), QUOTE_ALL (1), QUOTE_NONNUMERIC (2) or QUOTE_NONE (3)\ndoublequote Double quotes , When a single quote has been defined , also quoting The parameter is not QUOTE_NONE When , Using double quotation marks indicates that the element within the quotation marks is used as an element .\nescapechar When quoting by QUOTE_NONE when , Specifies an undelimited value for a character to cause .\ncomment Indicates that extra lines are not parsed . If the character appears at the beginning of the line , This line will be completely ignored . This parameter can only be one character , Blank line \uff08 It's like skip_blank_lines=True\uff09 Comment lines are header and skiprows Ignore the same . For example, if you specify comment='#' analysis \u2018#empty\\na,b,c\\n1,2,3\u2019 With header=0 Then the return result will be \u2019a,b,c' As header\nencoding Encoding mode , Specified character set type , Usually specified as 'utf-8'\ndialect If no specific language is specified , If sep Larger than one character is ignored . Specific to see csv.Dialect file\nerror_bad_lines If a row contains too many columns , Then the default will not return DataFrame , If I set it to false, So you're going to get rid of the change \uff08 Only in C Used under the parser \uff09\nwarn_bad_lines If error_bad_lines =False, also warn_bad_lines =True So all \u201cbad lines\u201d Will be output \uff08 Only in C Used under the parser \uff09\nlow_memory Block load into memory , Resolve at low memory consumption . However, type confusion can occur . To ensure that the type is not obfuscated need to be set to False. Or use dtype Parameter specified type . Pay attention to chunksize perhaps iterator Parameter block reading reads the entire file into one Dataframe, And ignore the type \uff08 Only in C Valid in the parser \uff09\ndelim_whitespace New in version 0.18.1: Python Valid in the parser\nmemory_map If filepath_or_buffer A file path is provided , The file object is mapped directly to memory , And access the data directly from there . Use this option to improve performance , Because there's no more I \/ O expenses , Using this method prevents the file from doing it again IO operation\nfloat_precision Appoint C The engine applies to converters of floating point values\n\nPart of the table is for reference Blog \u00a0www.cnblogs.com\/datablog\/p\/\u2026\u00a0 Thanks to the blogger for the translation ,O(\u2229_\u2229)O ha-ha ~\n\nCase study 1\n\nindex_col Use\n\nFirst prepare a txt file , The biggest problem with this file is that it has one more at the end of each line ',' , Follow the instructions to interpret as , If there is a separator at the end of each line , There will be problems , But in the actual test, I found that I needed to cooperate names Parameters , To have an effect\n\ngoof,1,2,3,ddd,\nu,1,3,4,asd,\nas,df,12,33,\nCopy code \n\nPart of the table is for reference Blog \u00a0www.cnblogs.com\/datablog\/p\/\u2026\u00a0 Thanks to the blogger for the translation ,O(\u2229_\u2229)O ha-ha ~\n\nCase study 1\n\nindex_col Use\n\nFirst prepare a txt file , The biggest problem with this file is that it has one more at the end of each line ',' , Follow the instructions to interpret as , If there is a separator at the end of each line , There will be problems , But in the actual test, I found that I needed to cooperate names Parameters , To have an effect\n\ngoof,1,2,3,ddd,\nu,1,3,4,asd,\nas,df,12,33,\nCopy code \n\nWrite the following code\n\ndf = pd.read_csv(\".\/demo.txt\",header=None,names=['a','b','c','d','e'])\nprint(df)\n\nprint(df)\nCopy code \n\nIn fact, the significance of the discovery is not very much , Maybe the document doesn't make it clear what it does . Let's move on index_col Common USES\n\nWhile reading the file , If not set index_col Column index , The default is to use from 0 Integer index to begin with . When a row or column in a table is manipulated , When saving to file you will find that there is always an extra column from 0 The start of the column , If you set index_col Parameter to set the column index , There would be no such problem .\n\nCase study 2\n\nconverters Sets the handler for the specified column , It can be used \" Serial number \" You can also use \u201c Name \u201d Specifies the column\n\nimport pandas as pd\n\ndef fun(x):\nreturn str(x)+\"-haha\"\n\nprint(type(df))\nprint(df.shape)\nprint(df)\nCopy code \n\n### read_csv A common problem with functions\n\n1. yes , we have IDE In the use of Pandas Of read_csv When the function imports a data file , If the file path or file name contains Chinese , Will report a mistake .\n\nterms of settlement\n\nimport pandas as pd\n#df=pd.read_csv('F:\/ Test folder \/ Test data .txt')\nf=open('F:\/ Test folder \/ Test data .txt')\nCopy code \n1. Exclude certain rows Use Parameters skiprows. Its function is to exclude a row . It should be noted that \uff1a Rule out before 3 Line is skiprows=3 Out of the first 3 Line is skiprows=3\n2. For irregular separators , Use Regular expressions Read the file The separator in the file is a space , So we just set it up sep=\" \" Just read the file . When the separator is not a single space , Maybe one space or more Spaces , If we still use it at this point sep=\" \" To read the file , Maybe you get a really weird number , Because it will use the space as data as well . data = pd.read_csv(\"data.txt\",sep=\"\\s+\")\n3. Read the file if there is a Chinese coding error Need to set encoding Parameters\n4. Add indexes for rows and columns With the parameters names Add column index , use index_col Add row index\n\nread_csv The command has a number of arguments . Most of them are unnecessary , Because most of the files you download are in standard format .\n\nThe basic usage is consistent , The difference lies in separator Separator .\n\ncsv Is a comma separated value , Can only read correctly \u201c,\u201d Segmented data ,read_table The default is '\\t'( That is to say tab) Cutting data sets\n\nReads a file with a fixed - width column , Such as files\n\nid8141 360.242940 149.910199 11950.7\nid1594 444.953632 166.985655 11788.4\nid1849 364.136849 183.628767 11806.2\nid1230 413.836124 184.375703 11916.8\nid1948 502.953953 173.237159 12468.3\nCopy code \n\ncolspecs \uff1a\n\nYou need to give a list of tuples , The list of tuples is half-open ,[from,to) , By default it will go before 100 Row data is inferred .\n\nExample \uff1a\n\nimport pandas as pd\ncolspecs = [(0, 6), (8, 20), (21, 33), (34, 43)]\nCopy code \n\nwidths\uff1a\n\nJust use a width list , Can replace colspecs Parameters\n\nwidths = [6, 14, 13, 10]\nCopy code \n\nread_fwf It's not used very often , You can refer to \u00a0pandas.pydata.org\/pandas-docs\u2026\u00a0 Study\n\npandas A new serializable data format supported , This is a lightweight, portable binary format , Similar to binary JSON, This data space utilization is high , In the writing \uff08 serialize \uff09 And read \uff08 Deserialization \uff09 Aspects provide good performance .\n\nReads the data in the clipboard , Can be seen as read_table Clipboard version . Useful in converting web pages into tables\n\nThis place appears as follows BUG\n\nmodule 'pandas' has no attribute 'compat'\n\nI updated it pandas It can be used normally\n\nThere is one more pitfall , That's when you read the clipboard , If you copy Chinese , It's easy not to read the data\n\nterms of settlement\n\n1. open site-packages\\pandas\\io\\clipboard.py This file needs to be retrieved by itself\n2. stay text = clipboard_get() Behind a line To join this \uff1a text = text.decode('UTF-8')\n3. preservation , Then you can use it\n\nThe official document is still the first \uff1apandas.pydata.org\/pandas-docs\u2026\n\nParameters Chinese meaning\nio File class object ,pandas Excel File or xlrd workbook . The string might be one URL.URL Include http,ftp,s3 And documents . for example , The local file can be written file:\/\/localhost\/path\/to\/workbook.xlsx\nsheet_name The default is sheetname by 0, Returns multiple table usage sheetname=0,1, if sheetname=None Is to return the full table . Be careful \uff1aint\/string The return is dataframe, and none and list The return is dict of dataframe, Table names are represented as strings , Index table positions are expressed as integers ;\nheader Specifies the row as the column name , Default 0, So let's take the first row , The data is below the column name row ; If the data does not contain column names , Is set header = None;\nnames Specifies the name of the column , Pass in a list data\nindex_col Specifies the column to be listed as the index column , You can also use u\u201dstrings\u201d , If I pass a list , These columns will be combined into one MultiIndex.\nsqueeze If the parsed data contains only one column , Returns a Series\ndtype The data type of the data or column , Reference resources read_csv that will do\nengine If io Not a buffer or a path , You must set it as an identity io. The acceptable value is None or xlrd\nconverters reference read_csv that will do\nThe rest of the parameters Basic and read_csv Agreement\n\npandas Read excel File error , The general treatment is\n\nError for \uff1aImportError: No module named 'xlrd'\n\npandas Read excel file , Need separate xlrd Module support pip install xlrd that will do\n\nParameters Chinese meaning\npath_or_buf A valid JSON file , The default value is None, The string can be URL, for example file:\/\/localhost\/path\/to\/table.json\norient \uff08 Case study 1\uff09 The expected json String format ,orient Has the following values \uff1a1. 'split' : dict like {index -> index, columns -> columns, data -> values}2. 'records' : list like {column -> value}, ... , {column -> value}3. 'index' : dict like {index -> {column -> value}}4. 'columns' : dict like {column -> {index -> value}}5. 'values' : just the values array\ntyp Return format (series or frame), The default is \u2018frame\u2019\ndtype The data type of the data or column , Reference resources read_csv that will do\nconvert_axes boolean, Try to convert the axis to the correct one dtypes, The default value is True\nconvert_dates Resolve the column list of dates ; If True, Attempts to resolve a date-like column , The default value is True Reference column label it ends with '_at',it ends with '_time',it begins with 'timestamp',it is 'modified',it is 'date'\nkeep_default_dates boolean,default True. If you parse the date , Resolves the default date sample column\nnumpy Direct decoding as numpy Array . The default is False; Support only digital data , But labels can be nonnumeric . Also pay attention to , If numpy=True,JSON Sort MUST\nprecise_float boolean, Default False. Set to enable higher precision when decoding a string to a double value \uff08strtod\uff09 Use of functions . The default value is \uff08False\uff09 Is to use quick but imprecise built-in features\ndate_unit string, A timestamp unit used to detect the conversion date . The default value is no . By default , The timestamp accuracy will be detected , If you don't need to , Through 's','ms','us' or 'ns' One of them forces the timestamp precision to be seconds respectively , millisecond , Microseconds or nanoseconds .\nencoding json code\nlines Each line reads one file json object .\n\nIf JSON Cannot be resolved , The parser will be generated ValueError\/TypeError\/AssertionError One of .\n\nCase study 1\n\n1. orient='split'\n import pandas as pd\ns = '{\"index\":[1,2,3],\"columns\":[\"a\",\"b\"],\"data\":[[1,3],[2,5],[6,9]]}'\nCopy code \n1. orient='records' Member for dictionary list\nimport pandas as pd\ns = '[{\"a\":1,\"b\":2},{\"a\":3,\"b\":4}]'\nCopy code \n1. orient='index' Based on the index key, Takes the dictionary of column fields as the key value . Such as \uff1a s = '{\"0\":{\"a\":1,\"b\":2},\"1\":{\"a\":2,\"b\":4}}'\n2. orient='columns' perhaps values You can extrapolate\n\nPartial Chinese translation , You can refer to github>\u00a0github.com\/apachecn\/pa\u2026\n\nRead json The file appears \u00a0ValueError: Trailing data\u00a0,JSON Format problem\n\nThe original format for\n\n{\"a\":1,\"b\":1},{\"a\":2,\"b\":2}\nCopy code \n\n[{\"a\":1,\"b\":1},{\"a\":2,\"b\":2}]\nCopy code \n\nOr use lines Parameters , also JSON Adjust to one data per row\n\n{\"a\":1,\"b\":1}\n{\"a\":2,\"b\":2}\nCopy code \n\nif JSON There is Chinese in the document , Advice and encoding Parameters , assignment 'utf-8', Otherwise, an error will be reported\n\nParameters Chinese meaning\nio Receiving site \u3001 file \u3001 character string . Url not accepted https, Try to get rid of s Climb to the top of the\nmatch Regular expressions , Returns a table that matches the regular expression\nflavor The parser defaults to \u2018lxml\u2019\nheader Specifies the row where the column header resides ,list For multiple indexes\nindex_col Specifies the column corresponding to the row header ,list For multiple indexes\nskiprows Skip the first n That's ok \uff08 The sequence labeling \uff09 Or skip n That's ok \uff08 Integer labeled \uff09\nattrs attribute , such as attrs = {'id': 'table'}\nparse_dates Parsing the date\n\nUsage method , Right click on the page if the table is found That is to say table You can use\n\nfor example \uff1a\u00a0data.stcn.com\/2019\/0304\/1\u2026\n\n<table class=\"...\" id=\"...\">\n<tr>\n<th>...<\/th>\n<\/tr>\n<tbody>\n<tr>\n<td>...<\/td>\n<\/tr>\n<tr>...<\/tr>\n<\/tbody>\n<\/table>\n\n<table> : Define the form\n<tbody> : Define the body of the table\n<tr> : Defines the rows of the table\n<th> : Defines the header of the table\n<td> : Define table cell\nCopy code \n\ncommon BUG\n\ninstall html5lib that will do , Or using parameters\n\nimport pandas as pd\nCopy code \n\nMore reference source code , You can refer to >\u00a0pandas.pydata.org\/pandas-docs\u2026\n\npandas Read and write function table of\n\nimport pandas as pd\n2\n4 print(csvframe, \"\\n-----*-----\")\n6 print(csvframe1, \"\\n-----*-----\")\n8 print(csvframe2, \"\\n-----*-----\")\n10 print(csvframe20, \"\\n-----*-----\")\n11\n13 print(csvframe30, \"\\n-----*-----\")\n15 print(csvframe31, \"\\n-----*-----\")\n16\n17 txtframe4 = pd.read_table('pandas_data_test\\ch05_04.txt',sep='\\s+') # According to regular analysis\n18 print(txtframe4, \"\\n-----*-----\")\n20 print(txtframe5, \"\\n-----*-----\")\n21 # Use skiprows Options , You can exclude redundant lines . Put the row number of the row to be excluded into the array , Assign to this option .\n23 print(txtframe6)\n24 Out[1]: 25 white red blue green animal\n26 0 1 5 2 3 cat\n27 1 2 7 8 5 dog\n28 2 3 3 6 7 horse\n29 3 2 2 8 3 duck\n30 4 4 4 2 1 mouse\n31 5 4 4 2 1 mou\n32 -----*-----\n33 white red blue green animal\n34 0 1 5 2 3 cat\n35 1 2 7 8 5 dog\n36 2 3 3 6 7 horse\n37 3 2 2 8 3 duck\n38 4 4 4 2 1 mouse\n39 5 4 4 2 1 mou\n40 -----*-----\n41 0 1 2 3 4\n42 0 1 5 2 3 cat\n43 1 2 7 8 5 dog\n44 2 3 3 6 7 horse\n45 3 2 2 8 3 duck\n46 4 4 4 2 1 mouse\n47 -----*-----\n48 white red blue green animal\n49 0 1 5 2 3 cat\n50 1 2 7 8 5 dog\n51 2 3 3 6 7 horse\n52 3 2 2 8 3 duck\n53 4 4 4 2 1 mouse\n54 -----*-----\n55 color status iteml item2 item3\n56 0 black up 3 4 6\n57 1 black down 2 6 7\n58 2 white up 5 5 5\n59 3 white down 3 3 2\n60 4 white left 1 2 1\n61 5 red up 2 2 2\n62 6 red down 1 1 4\n63 -----*-----\n64 iteml item2 item3\n65 color status\n66 black up 3 4 6\n67 down 2 6 7\n68 white up 5 5 5\n69 down 3 3 2\n70 left 1 2 1\n71 red up 2 2 2\n72 down 1 1 4\n73 -----*-----\n74 white red blue green\n75 0 1 5 2 3\n76 1 2 7 8 5\n77 2 3 3 6 7\n78 -----*-----\n79 0 1 2\n80 0 0 123 122\n81 1 1 124 321\n82 2 2 125 333\n83 -----*-----\n84 white red blue green animal\n85 0 1 5 2 3 cat\n86 1 2 7 8 5 dog\n87 2 3 3 6 7 horse\n88 3 2 2 8 3 duck\n89 4 4 4 2 1 mouse\nCopy code \n\nfrom TXT The file reads part of the data\n\n 1 print(csvframe2, \"\\n-----*-----\")\n2 # nrows=2 Specify the number of rows to get ,skiprows=[2] Delete the corresponding line\n4 print(csvfram20)\n5 Out[2]: 6 0 1 2 3 4\n7 0 1 5 2 3 cat\n8 1 2 7 8 5 dog\n9 2 3 3 6 7 horse\n10 3 2 2 8 3 duck\n11 4 4 4 2 1 mouse\n12 -----*-----\n13 0 1 2 3 4\n14 0 1 5 2 3 cat\n15 1 2 7 8 5 dog\nCopy code \n\nAnother interesting and common operation is to segment the text you want to parse , And then go through the parts , One by one A particular operation .\n\nfor example , For a column of numbers , Add one every two lines , Finally, put and plug into Series In the object \u201e This little example It's easy to understand ,\n\nIt has no practical value , But once you understand how it works , You can use it in more complex situations .\n\n 1 csvframe1 = pd.read_table('pandas_data_test\\myCSV_01.csv',sep=',')\n2 print(csvframe1, \"\\n-----*-----\")\n3 out = pd.Series()\n4 pieces = pd.read_csv('pandas_data_test\\myCSV_01.csv',chunksize=4) # chunksize The parameter determines the number of rows to split per part\n5 i = 0\n6 for piece in pieces:\n7 print(piece['white'])\n8 out.at[i] = piece['white'].sum()\n9 i += 1\n10 print(out, \"\\n-----*-----\")\n11 Out[3]: 12 white red blue green animal\n13 0 1 5 2 3 cat\n14 1 2 7 8 5 dog\n15 2 3 3 6 7 horse\n16 3 2 2 8 3 duck\n17 4 4 4 2 1 mouse\n18 5 4 4 2 1 mou\n19 -----*-----\n20 0 1\n21 1 2\n22 2 3\n23 3 2\n24 Name: white, dtype: int64\n25 4 4\n26 5 4\n27 Name: white, dtype: int64\n28 0 8\n29 1 8\n30 dtype: int64\nCopy code \n\nGo to CSV File write data\n\n 1 print(csvframe1)\n2 print(csvframe1.to_csv('pandas_data_test\\ch05_07.csv'))\n3 # Use index and header Options , Set their values to False, The default write can be cancelled index and header\n5 print(csvframe30.to_csv('pandas_data_test\\ch05_08.csv'))\n6 # It can be used to_csv() Functional na_rep Option to replace the empty field with the value you want . Common values are NULL\u30010 and NaN\n7 print(csvframe30.to_csv('pandas_data_test\\ch05_09.csv',na_rep=\" empty \"))\n8 Out[4]: 9 white red blue green animal\n10 0 1 5 2 3 cat\n11 1 2 7 8 5 dog\n12 2 3 3 6 7 horse\n13 3 2 2 8 3 duck\n14 4 4 4 2 1 mouse\n15 5 4 4 2 1 mou\n16 None\n17 None\n18 None\n19 None\nCopy code \n\nEnter the folder and we can see the corresponding file \uff1a\n\n 1 frame = pd.DataFrame(np.arange(4).reshape(2,2))\n2 print(frame.to_html())\n3 frame2 = pd.DataFrame( np.random.random((4,4)),index = ['white','black','red','blue1'],columns = ['up','down','right','left'])\n4 s = ['<HTML>']\n6 s.append(' <B0DY>')\n7 s.append(frame.to_html())\n8 s.append('<\/BODY><\/HTML>')\n9 html = ''.join(s)\n10 html_file = open('pandas_data_test\\myFrame.html','w')\n11 html_file.write(html)\n12 html_file.close()\n14 print(web_frames[0])\n16 print(ranking[0][1:10]) # Before outputting the content of a web page 10 That's ok\n17 Out[5]: 18 <table border=\"1\" class=\"dataframe\">\n20 <tr style=\"text-align: right;\">\n21 <th><\/th>\n22 <th>0<\/th>\n23 <th>1<\/th>\n24 <\/tr>\n26 <tbody>\n27 <tr>\n28 <th>0<\/th>\n29 <td>0<\/td>\n30 <td>1<\/td>\n31 <\/tr>\n32 <tr>\n33 <th>1<\/th>\n34 <td>2<\/td>\n35 <td>3<\/td>\n36 <\/tr>\n37 <\/tbody>\n38 <\/table>\n39 Unnamed: 0 0 1\n40 0 0 0 1\n41 1 1 2 3\n42 # Nome Exp Livelli\n43 1 2 admin 9029 NaN\n44 2 3 BrunoOrsini 2124 NaN\n45 3 4 Berserker 700 NaN\n46 4 5 Dnocioni 543 NaN\n47 5 6 albertosallusti 409 NaN\n48 6 7 Jon 233 NaN\n49 7 8 Mr.Y 180 NaN\n50 8 9 michele sisinni 157 NaN\n51 9 10 Selina 136 NaN\nCopy code \n\npandas All of the I\/O API Function , It's not specifically used to deal with XML( Extensible markup language \uff09 Format . Although there is no , But this format is actually\n\nVery important , Because a lot of structured data is based on XML Format stored .pandas There is no special processing letter It doesn't matter , because Python\n\nThere's a lot of reading and writing XML A library of formatted data \uff08 except pandas). One of the libraries is called lxml, It has excellent performance in large file processing , So from\n\nIt stands out from many of its kind . this This section describes how to use it to deal with XML file , And how to combine it with pandas integrated , With the most\n\nEventually XML In the document Get the data you need and convert it to DataFrame object .\n\nXML The source file is shown in the figure below\n\n 1 from lxml import objectify\n2\n3 xml = objectify.parse('pandas_data_test\\books.xml')\n4 root = xml.getroot() # Get root node\n5 print(root.Book.Author)\n6 mes1 = root.Book.getchildren()\n7 print(\"root.Book.getchildren() Get the sub tag content \uff1a\\n\", mes1)\n8 mes2 = root.Book[1].getchildren() # Take the second Book label\n9 print([child.tag for child in mes2]) # Get child tags\n10 print([child.text for child in mes2]) # Get the sub tag content\n11 Out[6]: 12 272103_l_EnRoss, Mark\n13 root.Book.getchildren() Get the sub tag content \uff1a\n14 [' 272103_l_EnRoss, Mark', 'XML Cookbook', 'Computer', 23.56, '2014-22-0l']\n15 ['Author', 'Title', 'Genre', 'Price', 'PublishDate']\n16 [' 272l03_l_EnBracket, Barbara', 'XML for Dummies', 'Computer', '35.95', '20l4-l2-l6']\nCopy code \n\nReading and writing Microsoft Excel file\n\nHDF5 Format\nthus , I have learned how to read and write the text format . To analyze a lot of data , Binary format is preferred .Python How much It's a kind of binary data processing\n\nTools .HDF5 Ku has had some success in this area .HDF Represents the hierarchy data format \uff08hierarchical data format ).HDF5\n\nKu is concerned about HDF5 Reading and writing of documents , The data structure of this file consists of nodes , The ability to store large data sets . The library uses all c Language\n\nDevelopment , Provides python\/matlab and Java Language interface . Its rapid expansion benefits from developers The widespread use of , And thanks to its effectiveness\n\nrate , In particular, using this format to store large amounts of data , It's very efficient . Compared to other formats that are simpler to process binary data ,HDF5\n\nSupport real time compression , Therefore, it is possible to compress files by using the repeating pattern in the data structure . at present ,Python There are two kinds of maneuvers HDF5 Format data\n\nMethods \uff1aPyTables and h5py. There are several differences between the two methods , Which one to choose depends largely on the specific needs .\n\nh5py by HDF5 Advanced API Provide the interface .PyTables It's packaged a lot of HDF5 details , Provide more flexible data containers \u3001 Index table \u3001 Search for\n\nFunctions and other computing related media .pandas Another one is called HDFStore\u3001 Be similar to diet Class , It USES PyTables Storage pandas\n\nobject . Use HDF5 Before the format , You have to guide people HDFStore class .\n\n 1 from pandas.io.pytables import HDFStore\n2 # Pay attention to the need tables This package , No, please install it yourself\n3 frame = pd.DataFrame(np.arange(16).reshape(4,4),index=['white','black1','red','blue'],columns=['up','down','right','left'])\n4 store = HDFStore('pandas_data_test\\mydata.h5')\n5 store['obj1'] = frame\n6 frame1 = pd.DataFrame(np.random.rand(16).reshape(4,4),index=['white','black1','red','blue'],columns=['up','down','right','left'])\n7 store['obj2'] = frame1\n8 print(store['obj1'])\n9 print(store['obj2'])\n10 Out[7]: 11 up down right left\n12 white 0 1 2 3\n13 black1 4 5 6 7\n14 red 8 9 10 11\n15 blue 12 13 14 15\n16 up down right left\n17 white 0.251269 0.422823 0.619294 0.273534\n18 black1 0.593960 0.353969 0.966026 0.104581\n19 red 0.964577 0.625644 0.342923 0.638627\n20 blue 0.246541 0.997952 0.414599 0.908750\n21 Closing remaining open files:pandas_data_test\\mydata.h5...done\nCopy code \n\nImplement object serialization\n\n****pickle The module implements a powerful algorithm , Be able to use Python Implement the data structure serialization \uff08pickling) And deserialization operations .\n\nSerialization is the process of transforming the hierarchy of an object into a byte stream . Serialization facilitates the transfer of objects \u3001 Storage and reconstruction , It's just a receiver that can rewind\n\nBuilding objects , It also retains all of its original features .\n\nuse pandas Library implements object serialization \uff08 Deserialization \uff09 Very convenient , All the tools are out of the box , No need to Python Import... In a conversation cPickle model\n\nblock , All operations are done implicitly . pandas The serialization format of is not fully used ASCII code .\n\n 1 import pickle\n2 data = { 'color': ['white','red'], 'value': [5, 7]}\n3 pickled_data = pickle.dumps(data)\n4 print(pickled_data)\n6 print(nframe)\n7\n8 # use pandas serialize\n9 frame = pd.DataFrame(np.arange(16).reshape(4,4), index = ['up','down','left','right'])\n10 frame.to_pickle('pandas_data_test\\frame.pkl') # Same as json The data is similar to\n12 Out[8]: 13 b'\\x80\\x03}q\\x00(X\\x05\\x00\\x00\\x00colorq\\x01]q\\x02(X\\x05\\x00\\x00\\x00whiteq\\x03X\\x03\\x00\\x00\\x00redq\\x04eX\\x05\\x00\\x00\\x00valueq\\x05]q\\x06(K\\x05K\\x07eu.'\n14 {'color': ['white', 'red'], 'value': [5, 7]}\n15 0 1 2 3\n16 up 0 1 2 3\n17 down 4 5 6 7\n18 left 8 9 10 11\n19 right 12 13 14 15\nCopy code \n\nDocking database\n**** In many applications , The data used comes from very few text files , Because text files are not the most efficient way to store data .\n\nData is often stored in SQL Class relational database , Supplement ,NoSQL Databases have become popular recently .\n\nfrom SQL Database loading data , Convert it to DataFrame The object is very simple pandas Several functions are provided to simplify the process .\n\npandas.io.sql Module provides database independent \u3001 called sqlalchemy The unified interface . This interface simplifies the connection mode , No matter for\n\nWhat kind of database , There is only one set of operation commands . Connected database usage create_engine() function , You can use it to configure the drive\n\nThe required user name \u3001 password \u3001 Port and database instance . database URL The typical form of this is \uff1a\n\nThe identifying name of the name , for example sqlite,mysql,postgresql,oracle, or mssql.drivername It's used to connect with all lowercase letters\n\nTo the database DBAPI The name of . If not specified , Then \u201c Default \u201dDBAPI\uff08 If available \uff09 - This default value is usually available for that backend\n\nThe most extensive driver .\n\n 1 from sqlalchemy import create_engine\n2\n3 # PostgreSQL database\n4 # default\n5 engine = create_engine('postgresql:\/\/scott:[email\u00a0protected]\/mydatabase')\n6 # pg8000 Driver\n7 engine = create_engine('postgresql+pg8000:\/\/scott:[email\u00a0protected]\/mydatabase')\n8 # psycopg2 Driver\n9 engine = create_engine('postgresql+psycopg2:\/\/scott:[email\u00a0protected]\/mydatabase')\n10 # MySql\n11 # default\n12 engine = create_engine('mysql:\/\/scott:[email\u00a0protected]\/foo')\n13\n14 # mysql-python Note the correspondence behind the drive\n15 engine = create_engine('mysql+mysqldb:\/\/scott:[email\u00a0protected]\/foo')\n16\n17 # MySQL-connector-python\n18 engine = create_engine('mysql+mysqlconnector:\/\/scott:[email\u00a0protected]\/foo')\n19 # OurSQL\n20 engine = create_engine('mysql+oursql:\/\/scott:[email\u00a0protected]\/foo')\n21 # Oracle\n22 engine = create_engine('oracle:\/\/scott:[email\u00a0protected]:1521\/sidname')\n23\n24 engine = create_engine('oracle+cx_oracle:\/\/scott:[email\u00a0protected]')\n25 # Microsoft SQL\n26 # pyodbc\n27 engine = create_engine('mssql+pyodbc:\/\/scott:[email\u00a0protected]')\n28\n29 # pymssql\n30 engine = create_engine('mssql+pymssql:\/\/scott:[email\u00a0protected]:port\/dbname')\nCopy code \n\nSQLite\uff1a\n\nbecause SQLite Connect to a local file , therefore URL Slightly different format .URL Of \u201c file \u201d Part of it is the file name of the database .\n\nFor relative file paths , It takes three slashes \uff1aengine = create_engine('sqlite:\/\/\/foo.db')\n\n**** For absolute file paths , After the three slashes is the absolute path \uff1a\n\nUnix\/Mac - 4 initial slashes in total\n\nengine = create_engine('sqlite:absolute\/path\/to\/foo.db')\n\n**Windows\n\nengine = create_engine('sqlite:\/\/\/C:\\path\\to\\foo.db')\n\nWindows alternative using raw string\n\nengine = create_engine(r'sqlite:\/\/\/C:\\path\\to\\foo.db')\n\n**** Learn to use Python Built in SQLite database sqlite3.SQLite3 The tool implements simplicity \u3001 Lightweight DBMS SQL,\n\nSo it can be built into Python Language implementation of any application . It's very practical , You can create an embedded database in a single file .\n\nIf you want to use all the functions of a database without installing a real database , This tool is the best choice . If you want to use real\n\nPractice database operation before using the database , Or use a database to store data in a single program without considering the interface , SQLite3 It's all No\n\nThe wrong choice .\n\n 1 from sqlalchemy import create_engine\n2 frame = pd.DataFrame( np.arange(20).reshape(4,5),columns=['white','red','blue','black','green'])\n3 # Connect SQLite3 database\n4 engine = create_engine('sqlite:\/\/\/pandas_data_test\/foo.db')\n5 # hold DataFrame Convert to database table .\n6 # to_sql(self, name, con, schema=None, if_exists='fail', index=True,\n7 # index_label=None, chunksize=None, dtype=None)\n8 frame.to_sql('colors',engine)\n9\nCopy code \n\nRunning results \uff1a","date":"2022-06-25 08:48:09","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.3547377586364746, \"perplexity\": 4043.8932667240556}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-27\/segments\/1656103034877.9\/warc\/CC-MAIN-20220625065404-20220625095404-00082.warc.gz\"}"} | null | null |
Гебрюдер Маузер () — немецкая компания, основанная братьями Петером-Паулем и Вильгельмом Маузерами для производства стрелкового оружия (преимущественно винтовок).
История
Оружейный завод в Оберндорфе-на-Неккаре, расположенном в 80 км к юго-западу от Штутгарта, был основан в 1811 году указом короля Вюртемберга Фридриха I.
Отец Петера Пауля и Вильгельма — Франц Андреас Маузер — большую часть жизни проработал на Королевском оружейном заводе в Оберндорфе-на-Неккаре. Петер Пауль Маузер начал свою работу на этом заводе в 12 лет и работал, пока в возрасте 19 лет не был призван в армию.
23 декабря 1872 года братья Петер Пауль и Вильгельм Маузер создали фирму Gebrüder Wilhelm und Paul Mauser.
В 1872 году братья Маузер купили у правительства Вюртемберга в Оберндорфе-на-Неккаре за 200 000 южно-германских гульденов (иногда именуемых «флоринами»).
В 1874 году — после преобразования — фирма стала именоваться Gebrüder Mauser und Cie («Братья Маузер и Компания»).
В 1922 году фирма преобразована в Waffenfabrik Mauser AG (акционерное общество Оружейный завод Маузер).
Впоследствии акционерному обществу Waffenfabrik Mauser также принадлежал оружейный завод в Борзигвальде, пригороде Берлина (Mauser AG Borsigwalde).
Завод Маузер в Оберндорфе-на-Неккаре разрушен авиацией союзников в 1945 году.
После Второй мировой войны инженерами компании Маузер в Оберндорфе-на-Неккаре была основана компания «Хеклер и Кох».
Продукция
Винтовки до 1945 года
Gewehr 71
В 1871 году братья Маузер создают однозарядную винтовку под 11-мм патрон, продемонстрированную в Прусской Королевской стрелковой школе в Шпандау, и она принимается на вооружение как Gewehr 1871.
Mauser 1889 «Бельгийский Маузер»
Первая винтовка фирмы Mauser M1889, изначально спроектированная под патрон уменьшенного калибра (7,65×53 мм) с бездымным порохом. Принята на вооружении в Бельгии, где и производилась. В следующем, 1890 году, принята на вооружении в Турции, причём винтовки для турецкой армии производились в Германии, на заводах и Mauser. Далее, в 1891 году, эта же модель принимается на вооружении в Аргентине. Поставки осуществлялись заводом Ludwig Loewe, и, также DWM. Такие же винтовки поставлялись в Перу, Эквадор и Колумбию.
Mauser 1894 и 1895
В 1894 году создана магазинная винтовка, в дальнейшем принятая на вооружение во многих странах мира (но не в Германии). Существовала модификация 1895 года. Винтовка модели 1894 года экспортировалась в Бразилию и Швецию.
Винтовка модели 1895 года под патрон 7×57 мм экспортировалась в Мексику, Чили, Уругвай, Китай, Иран и в бурские Южно-Африканские республики — Республику Трансвааль и Оранжевое Свободное государство.
В романе Луи Буссенара «Капитан Сорви-голова» (1901 год), описывающем события Второй англо-бурской войны 1899—1902 годов много раз упоминаются винтовки Маузер — по всей видимости имеется в виду именно модель 1895 года.
Mauser M96 «Шведский Маузер»
В 1896 году создана магазинная винтовка под патрон 6,5×55 мм, предназначенная для экспорта в Швецию, впоследствии получившая известность под неофициальным названием .
Mauser 98
Mauser Gewehr 98 (Маузер 98; Mauser 98, G98 или Gew.98) — созданная в 1898 году винтовка, принятая на вооружение объединённой германской армией вплоть до конца Второй мировой войны и получившая репутацию простого и надёжного оружия. По некоторым подсчётам в мире выпущено около 100 миллионов винтовок, которые можно считать разновидностями Mauser 98.
Mauser 98k
Винтовка принята на вооружение в 1935 году как Karabiner 98k (также Kar98k или K98k, то есть официально называлась «карабинер (способ крепления ремня) 98k»). В большинстве современных источников именуется Mauser 98k. Является модификацией винтовки Mauser 98.
Volkssturmkarabiner 98 (VK.98)
В дословном переводе с немецкого — «карабин фольксштурма». Является сильно упрощённой версией винтовки Mauser 98k.
Volkssturmkarabiner 98 выпускался в конце Второй мировой войны, как в однозарядном, так и в магазином варианте.
В конце Второй мировой войны другими немецкими производителями выпускались карабины Volkssturmkarabiner 1 (VK.1) и Volkssturmkarabiner 2 (VK.2), которые, несмотря на схожесть названий, имеют существенные отличия от Volkssturmkarabiner 98.
Mauser T-Gewehr — 1918 г.
Mauser T-Gewehr — первое в истории противотанковое ружьё, разработанное в 1918 году. Использовался патрон 13,25×92 мм SR. До конца Первой мировой войны применялось Германской армией на Западном фронте, а после войны состояло на вооружении некоторых европейских стран.
Gewehr 41 (Mauser)
Gewehr 41 (Mauser), она же G-41(M) — экспериментальная самозарядная винтовка под патрон 7,92×57 мм.
В 1940 году управлением вооружений сухопутных сил Германии были выдвинуты требования на новую самозарядную винтовку для вермахта. В конце 1941 года в части вермахта для войсковых испытаний начали поступать самозарядные винтовки двух типов: разработанная компанией Mauser винтовка G-41(M) и разработанная компанией Walther винтовка G-41(W). Общий выпуск винтовок G41(M) составил около 15 000 штук. В 1942 году, по итогам войсковых испытаний, на вооружение была принята винтовка G-41(W) системы Вальтера (часто именуется просто Gewehr 41 или G-41).
Sturmgewehr 45 (StG 45(M))
В 1944 году компания разработала «штурмовую винтовку» под промежуточный патрон 7,92×33 мм. В России и СССР оружие данного типа (автоматическая винтовка под промежуточный патрон) принято называть «автомат». Sturmgewehr 45 не была запущена в серийное производство в связи с разрушением в 1945 году авиацией союзников завода Маузер в Оберндорфе-на-Неккаре и окончанием Второй мировой войны. Впоследствии Sturmgewehr 45 послужила прототипом винтовки Heckler & Koch G3.
Винтовки после 1945 года
Mauser SP66
Mauser SP66 — снайперская винтовка, разработанная в 1976 году на основе спортивной винтовки М66 Супер Матч.
Mauser 86 SR
Mauser 86 SR — полицейская снайперская винтовка, разработанная для замены Mauser SP66. Отличиями от предшественника стали: модифицированная ложа из фибергласа или ламинированной фанеры, оснащённая регулируемым затыльником и гребнем приклада, отъёмный магазин на 9 патронов вместо постоянного на 3 и новая затворная группа.
Mauser SR 93
Mauser SR 93 — снайперская винтовка, разработанная в начале 1990-х годов. Выпускается под патроны калибра 7,62×51 mm NATO (.308Win), .300 Winchester Magnum, .338 Lapua Magnum . Технически представляет собой 5 зарядную винтовку с продольно-скользящим поворотным затвором. Подача патронов при стрельбе производится из отъемных коробчатых магазинов ёмкостью 5 патронов.
Mauser M 98
Современная охотничья винтовка. Конструктивно близка к Mauser 98/Mauser 98k (отличается формой и материалом ложи, креплением под оптический прицел, внешней отделкой). Производится с 1999 года по настоящее время (2011 год).
Mauser M 03
Современная охотничья винтовка.
Пистолеты и револьверы
Револьвер «Зиг-Заг»
В 1878 году разработан револьвер «Зиг-Заг». Существуют модификации 1886 г. и 1896 г.
Пистолет Mauser C96
В 1896 году компания разработала самозарядный пистолет модели Mauser C96, ставший, благодаря кино и литературе, неотъемлемой частью образа чекиста или комиссара эпохи гражданской войны в России.
Пистолет Mauser М1910
Самозарядный пистолет под патрон 6,35×15 мм Браунинг.
Пистолет Mauser М1914
Самозарядный пистолет под патрон 7,65×17 мм, модификация пистолета Mauser M1910.
Пистолет Mauser М1934
Модификация пистолета Mauser M1910.
Пистолет Mauser HSc — 1935 г.
Самозарядный пистолет, принят на вооружение Вермахта в 1941 году.
Пистолет Mauser V.7082 (Volkspistole) — 1944 г.
Создан как дешёвый и простой в производстве пистолет для вооружения фольксштурма. Имеет некоторое конструктивное сходство с Mauser HSc.
Пистолет-пулемёт MP-3008 — 1945 г.
Пистолет-пулемёт, немецкая копия английского пистолета-пулемёта STEN (отличается вертикально расположенным магазином, использовался коробчатый двухрядный магазин от пистолета-пулемёта MP-40). Создан как дешёвый и простой в производстве пистолет-пулемёт для вооружения фольксштурма.
Пулемёты
MG 18 TuF
Модификация пулемёта MG-08 под патрон 13,25×92 мм SR (данный патрон использовался в противотанковом ружье Mauser T-Gewehr). Пулемёт разработан компанией Mauser в 1918 году как средство борьбы с танками и самолётами (аббревиатура TuF расшифровывается как «Tank und Flieger», в переводе с немецкого «танк и самолёт»). В серийное производство запущен не был.
MG 81
Авиационный пулемёт под патрон 7,92 x 57 mm, разработан компанией Mauser в 1939 году.
MG 42
Единый пулемёт под патрон 7,92 x 57 mm, разработан компанией Metall und Lackierwarenfabrik Johannes Grossfuss AG в 1942 году. Выпускался несколькими производителями, в том числе компанией Mauser.
Автоматические орудия
2 cm FlaK 30/FlaK 38
Автоматическое зенитное орудие калибра 20 мм.
MG FF
Авиапушка калибра 20 мм.
MG 151
Авиапушка, выпускалась в вариантах калибра 15 мм (MG 151/15) и 20 мм (MG 151/20).
MG 213
Авиапушка, выпускалась в вариантах калибра 20 мм (MG 213) и 30 мм (MG 213C).
BK 27
Автоматическая пушка калибра 27 мм. Разработана в 1976 году . На сегодня её носителем является истребитель "Тайфун Еврофайтер" , принятый на вооружение Бундесвера в 2003 году .
Существуют корабельные модификации данной пушки — СMN 27 GS и MLG 27.
MK 30
Автоматическая пушка калибра 30 мм. Разработана в начале 1980-х годов.
Обозначение производителя на продукции 1934—1945 годов
С 1934 года по 1945 год на стрелковом оружии Германии наименования производителей не указывались — указывались только по классификации управления вооружений сухопутных сил. При этом на экспортном оружии (в том числе конструктивно идентичном производимому для внутренних нужд Германии) указывались традиционные наименования производителей.
Продукции компании Маузер соответствуют следующие коды:
завод в Оберндорфе-на-Неккаре
WaA63 S/42 (1935—1938 годы)
WaA63 42 (1938—1939 годы)
WaA108 S/42 (1935 год)
WaA135 S/42 (1935 год)
WaA135 byf (1941—1945 годы)
WaA211 S/42 (1935 год)
WaA241 S/42 (1935 год)
WaA655 42 (1938—1941 годы)
завод в Борзигвальде, пригороде Берлина
WaA26 S/243 (1938 год)
WaA26 243 (1938—1940 годы)
WaA26 ar (1941—1944 годы)
WaA49
WaA211
WaA217 S/243 (1935—1937 годы)
WaA280 S/243 (1938 год)
WaA280 243 (1938 год)
Примечания
Производители огнестрельного оружия Германии | {
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"redpajama_set_name": "RedPajamaC4"
} | 1,107 |
Q: How do I SUM between a date range but return blank if there is no data? I am trying to SUM column N (on sheet TRADE LOG) between a date range however if there is no data for that date range I would like the cells of column B on sheet STATISTICS to return a blank.
I have worked out how to sum within the date range however cannot work out how to return a blank if there is no data for that date range.
The date ranges I am using is months.
Current formula is below:
=SUMIFS('TRADE LOG'!N:N,'TRADE LOG'!B:B,">="&DATE(2021,2,1),'TRADE LOG'!N:N,"<="&DATE(2021,2,28))
Current worksheet(s) below:
A: You can do a number format that returns a null string if the formula returns 0:
$#,##0.00;-$#,##0.00;;
Or you can return a null string if it is 0 with the reciprocal of the reciprocal:
=IFERROR(1/(1/SUMIFS('TRADE LOG'!N:N,'TRADE LOG'!B:B,">="&DATE(2021,2,1),'TRADE LOG'!N:N,"<="&DATE(2021,2,28))),"")
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,662 |
\section{Introduction}\label{sect:intro}
A generalised polygon $\Gamma$ is a point-line incidence structure such that the incidence graph is
connected and bipartite with girth twice that of its diameter. If the valency of every vertex is at
least $3$, then we say that $\Gamma$ is \textit{thick}, and it turns out that the incidence graph is
then biregular\footnote{That is, there are two constants $k_1$ and $k_2$ such that the valency of
each vertex in one bipartition is $k_1$, and the valency of each vertex in the other bipartition
is $k_2$.}. By a famous result of Feit and Higman \cite{FeitHigman}, a finite \textit{thick
generalised polygon} is a complete bipartite graph, projective plane, generalised quadrangle,
generalised hexagon or generalised octagon. There are many known classes of finite projective
planes and finite generalised quadrangles but presently there are only two known families, up to
isomorphism and duality, of finite generalised hexagons; the \textit{split Cayley hexagons} and the
\textit{twisted triality hexagons}.
The split Cayley hexagons $\mathcal{H}(q)$ are the natural geometries for Dickson's group
$\mathsf{G}_2(q)$, and they were introduced by Tits \cite{Tits59} as the set of points of the
parabolic quadric $\Q(6,q)$ and an orbit of lines of $\Q(6,q)$ under $\mathsf{G}_2(q)$. If $q$ is
even, then the polar spaces $\mathsf{W}(5,q)$ and $\Q(6,q)$ are isomorphic geometries, and hence
$\mathcal{H}(q)$ can be embedded into a five-dimensional projective space.
Thas and Van Maldeghem \cite{ThasVM98} proved that if $\mathcal{H}$ is a finite thick generalised hexagon
embedded\footnote{We will not discuss the various meanings of ``embedding'' here, but
instead refer the interested reader to \cite{ThasVM96,ThasVM98}.} into the projective space $\PG(d,q)$, then $d\le 7$
and this embedding is equivalent to one of the standard models of the known generalised hexagons.
So in particular, it is impossible to embed the split Cayley hexagon $\mathcal{H}(q)$
into a three-dimensional projective space.
However, there is an elegant model of $\mathcal{H}(q)$ which begins with geometric structures lying
in $\PG(3,q)$, and it is equivalent to the model provided by Cameron and Kantor
\cite[Appendix]{CameronKantor79}:
\begin{theorem}[Cameron and Kantor (paraphrased) \cite{CameronKantor79}]\label{thm:constL3}\ \\
Let $(p,\sigma)$ be a point-plane anti-flag of $\PG(3,q)$ and let $\Omega$ be a set of $q
(q^2-1)(q^2+q+1)$ parabolic congruences\footnote{A \textit{pencil} of lines of $\PG(3,q)$ refers to
the set of lines passing through a point, lying on a plane. A set of $q^2+q+1$ lines concurrent
with a common line $\ell$, no two of which meet in a point not on $\ell$, is called a
\textit{parabolic congruence}, and the line $\ell$ is its \textit{axis}. The image of a parabolic congruence under the Klein
correspondence yields a $3$-dimensional quadratic cone of $\Q^+(5,q)$, and vice-versa (see
\cite[p. 30]{Hirschfeld85}), and so a parabolic congruence is a union of $q+1$ pencils sharing a
line.} each having axis not incident with $p$ or $\sigma$, but having a pencil of lines with one
line incident with $p$ and another incident with $\sigma$. Suppose that for each pencil
$\mathcal{L}$ with vertex not in $\sigma$ and plane not incident with $p$, there are precisely $q+1$ elements
of $\Omega$ containing $\mathcal{L}$, whose union are the lines of some linear complex (i.e., the
lines of a symplectic geometry $\mathsf{W}(3,q)$). Then the following incidence structure $\Gamma$
is isomorphic to the split Cayley hexagon $\mathcal{H}(q)$.
\begin{table}[H]
\begin{tabular}{ll}\hline
\textsc{Points:} & (a) Lines of $\mathsf{PG}(3,q)$.\\
& (b) Pencils with a vertex not in $\sigma$ and plane not incident with $p$.\\
\textsc{Lines:} & (i) Pencils with a vertex in $\sigma$ and plane through $p$.\\
& (ii) Elements of $\Omega$.\\
\hline \\
\end{tabular}
\begin{minipage}{16cm}
An element $\ell$ of type (a) is incident with an element $\mathcal{P}$ of type (i) if $\ell$ is an
element of $\mathcal{P}$. If $\mathcal{C}$ is an element of type (ii), then $\ell$ is incident with
$\mathcal{C}$ if $\ell$ is the axis of $\mathcal{C}$. Elements of type (i) and (b) are never
incident. The containment relation defines incidence between elements of type (b) and (ii).
\end{minipage}
\end{table}
\end{theorem}
The central result of this note is a unitary analogue of this model.
\begin{theorem}\label{thm:constU3}\ \\
Let $\mathcal{O}$ be a Hermitian curve of $\mathsf{H}(3,q^2)$ and let $\Omega$ be a set of Baer
subgenerators with a point in $\mathcal{O}$, such that every point of
$\mathsf{H}(3,q^2)\backslash\mathcal{O}$ is on $q+1$ elements of $\Omega$ spanning a Baer subplane.
Then the following incidence structure $\Gamma$ is a generalised hexagon of order
$(q,q)$. \begin{table}[H]
\begin{tabular}{ll}\hline
\textsc{Points:} & (a) Lines of $\mathsf{H}(3,q^2)$.\\
& (b) Affine points of $\mathsf{H}(3,q^2)\backslash\mathcal{O}$.\\
\textsc{Lines:} & (i) Points of $\mathcal{O}$.\\
& (ii) Elements of $\Omega$.\\
\textsc{Incidence:} & Inclusion or inherited incidence.\\
\hline
\end{tabular}
\end{table}
\noindent Moreover, $\Gamma$ is isomorphic to the split Cayley hexagon $\mathcal{H}(q)$.
\end{theorem}
The proof that $\Gamma$ is a generalised hexagon is presented in Section \ref{proofconstU3}.
Note that the lines of type (i) form a spread of $\mathcal{H}(q)$.
There exists a natural candidate for $\Omega$ which we explain in detail in Section \ref{sect:partition},
and it is essentially the only one (Theorem \ref{thm:omega}), and this implies the ultimate result
that $\Gamma$ is isomorphic to $\mathcal{H}(q)$.
By the deep results of Thas and Van Maldeghem \cite{ThasVM96,ThasVM98} and Cameron and Kantor
\cite{CameronKantor79}, if a set of points $\mathcal{P}$ and lines $\mathcal{L}$ of
$\mathsf{PG}(6,q)$ form a generalised hexagon, then it is isomorphic to the split Cayley
hexagon $\mathcal{H}(q)$ if $\mathcal{P}$ spans $\mathsf{PG}(6,q)$ and for any point
$x\in\mathcal{P}$, the points collinear to $x$ span a plane. A similar result was proved recently by
Thas and Van Maldeghem \cite{ThasVM08}, by foregoing the assumption that $\mathcal{P}$ and
$\mathcal{L}$ form a generalised hexagon, and instead instituting the following five axioms: (i) the
size of $\mathcal{L}$ is $(q^6-1)/(q-1)$, (ii) every point of $\PG(6,q)$ is incident with either $0$
or $q+1$ elements of $\mathcal{L}$, (iii) every plane of $\PG(6,q)$ is incident with $0$, $1$ or
$q+1$ elements of $\mathcal{L}$, (iv) every solid of $\PG(6,q)$ contains $0$, $1$, $q+1$ or $2q+1$
elements of $\mathcal{L}$, and (v) every hyperplane of $\PG(6, q)$ contains at most $q^3+3q^2+3q$
elements of $\mathcal{L}$.
One could instead characterise the split Cayley hexagon viewed as points and lines of the parabolic
quadric $\Q(6,q)$, and the best result we have to date follows from a result of Cuypers and
Steinbach \cite[Theorem 1.1]{CuypersSteinbach}:
\begin{theorem}[Cuypers and Steinbach \cite{CuypersSteinbach} (paraphrased)]\label{thm:Q6embedding}
Let $\mathcal{L}$ be a set of lines of $\Q(6,q)$ such that every point of $\Q(6,q)$ is incident with
$q+1$ lines of $\mathcal{L}$ spanning a plane, and such that the concurrency graph of $\mathcal{L}$
is connected. Then the points of $\Q(6,q)$ together with $\mathcal{L}$ define a generalised hexagon isomorphic to
the split Cayley hexagon $\mathcal{H}(q)$.
\end{theorem}
In Section \ref{sect:hexagon} we will give an elementary proof of Theorem \ref{thm:Q6embedding} by using Theorem \ref{thm:omega}.
\subsection*{Some remarks on notation:}
In this paper, the \textit{relative norm} and \textit{relative trace} maps will be defined for the quadratic extension
$\GF(q^2)$ over $\GF(q)$.
The relative norm $\mathsf{N}$ is the
multiplicative function which maps an element $x\in\GF(q^2)$ to the
product of its conjugates of $\GF(q^2)$ over $\GF(q)$. That is, $\mathsf{N}(x)=x^{q+1}$.
The relative trace is instead the sum of the conjugates, $\mathsf{T}(x):=x+x^q$.
\section{The 3-dimensional Hermitian surface and its Baer substructures}\label{sect:hermitian}
The two (\textit{classical}) generalised quadrangles of particular importance in this note are
$\mathsf{H}(3,q^2)$ and $\Q^-(5,q)$. First there is the incidence structure of all points and lines
of a non-singular Hermitian variety in $\PG(3,q^2)$, which forms the generalised quadrangle
$\mathsf{H}(3,q^2)$ of order $(q^2,q)$. Its point-line dual is isomorphic to the geometry of points
and lines of the elliptic quadric $\Q^-(5,q)$ in $\PG(5,q)$, which yields a generalised quadrangle
of order $(q,q^2)$ (see \cite[3.2.3]{FGQ}). To construct $\mathsf{H}(3,q^2)$ given a prime power
$q$, we take a non-degenerate Hermitian form such as
$$\langle X,Y\rangle = X_0Y_0^{q}+X_1Y_1^{q}+X_2Y_2^{q}+X_3Y_3^{q}$$ and the totally isotropic
subspaces of the ambient projective space, with respect to this form. Most of the material
contained in this section can be found in Barwick and Ebert's book \cite{BarwickEbert} and
Hirschfeld's book \cite[Chapter 7]{Hirschfeld98}.
Every line of $\mathsf{PG}(3,q^2)$ is (i) a \textit{generator} (i.e., totally isotropic line) of
$\mathsf{H}(3,q^2)$, (ii) meets $\mathsf{H}(3,q^2)$ in one point (i.e., a tangent line), or (iii) meets $\mathsf{H}(3,q^2)$
in a Baer subline (also called a \textit{hyperbolic} line). A \textit{Baer subline} of the projective line $\PG(1, q^2 )$ is a subset of
$q+1$ points in $\PG(1, q^2 )$ which form a $\GF(q)$-linear subspace. We may also speak of \textit{Baer subplanes} and \textit{Baer
subgeometries} of $\PG(3,q^2)$ as sets of points giving rise to projective subgeometries
isomorphic to $\PG(2,q)$ and $\PG(3,q)$ respectively. A \textit{Baer subgenerator} of
$\mathsf{H}(3,q^2)$ is a Baer subline of a generator of $\mathsf{H}(3,q^2)$. We will often use the
fact that three collinear points determine a unique Baer subline (\cite[Theorem 2.6]{BarwickEbert}) and
a planar quadrangle determines a unique Baer subplane (\cite[Theorem 2.8]{BarwickEbert}). In particular,
if $b$ and $b'$ are two Baer sublines of $\PG(2,q^2)$ sharing a point, but not spanning the same line,
then there is a unique Baer subplane containing both $b$ and $b'$. We say that it is the \textit{Baer subplane
spanned by $b$ and $b'$}.
One class of important objects for us in this paper will be the degenerate Hermitian curves of rank
2. Suppose we have a fixed hyperplane, $\pi: X_3=0$ say, meeting $\mathsf{H}(3,q^2)$ in a Hermitian
curve $\mathcal{O}$. Let $\ell$ be a generator of $\mathsf{H}(3,q^2)$. Then the polar planes of the
points on $\ell$ meet $\pi$ in the $q^2+1$ lines through $L:=\ell\cap\mathcal{O}$. Now suppose we have a
Baer subgenerator $b$ contained in $\ell$, and containing the point $L$. Then the polar planes of
the points of $b$ meet $\pi$ in $q+1$ lines through the point $L$ giving a \textit{dual Baer subline} of
$\pi$ with \textit{vertex} $L$. Moreover, the points lying on this dual Baer subline define a
variety with Gram matrix $U$; a Hermitian matrix of rank $2$. So they correspond to solutions of
$XU(X^q)^T=0$ where $U$ satisfies $U^q=U^T$. For example, if we consider a point $P$ in $\pi$,
say $(1,\omega,0,0)$ where $\mathsf{N}(\omega)=-1$, and two points $A:(a_0,a_0\omega,a_2,1)$,
$B:(b_0,b_0\omega,b_2,1)$
spanning a line with $P$, then $P, A, B$ determine a Baer subline.
In fact, if we suppose $B=P+\alpha A$ for some $\alpha\in\GF(q^2)^*$, then this
Baer subline is $\{A\}\cup \{\langle p+t\cdot\alpha a\rangle \mid t\in\GF(q)\}$ where $A=\langle a\rangle$
and $P=\langle p\rangle$.
Let $\mathfrak{u}$ be the polarity defining $\mathsf{H}(3,q^2)$.
Since $P$ is precisely the nullspace of $U$, and the tangent line $P^{\mathfrak{u}}\cap \pi$ is contained in
the dual Baer subline, it is not difficult to calculate that $U$ can be written explicitly as
$$
U:
\begin{pmatrix}
-\delta\omega^q & \delta & -\gamma\omega\\
\delta^q & \delta^q\omega & \gamma\\
-\gamma^q\omega^q & \gamma^q & 0\\
\end{pmatrix},\quad \delta\omega^q=\delta^q\omega.
$$
If we also suppose that the points of $A^{\mathfrak{u}}\cap \pi$ and $B^{\mathfrak{u}}\cap \pi$ are contained
in the dual Baer subline defined by $U$, then
we can solve for $\alpha$ and $\gamma$ (but the expressions might be ugly!). Here we explore a simple example
where $A:(0,0,1,\omega)$. Now $A^\mathfrak{u}\cap\pi$ has points of the form
$(r,s,0,0)$, $\mathsf{N}(r)+\mathsf{N}(s)=0$. So if $(r,s,0,0)$ also satisfies $(r,s,0)U(r^q,s^q,0)^T=0$, then
$$(r,s,0)U(r^q,s^q,0)^T=\mathsf{T}(rs^q\delta)+2\mathsf{N}(s)\delta^q\omega.$$
So $\mathsf{T}(rs^q\delta)+2\mathsf{N}(s)\delta^q\omega=0$ for every $(r,s,0,0)$ satisfying $\mathsf{N}(r)+\mathsf{N}(s)=0$.
In particular, $\delta$ is forced to be zero. Therefore, we can write
$$
U:
\begin{pmatrix}
0 & 0 & -\gamma\omega\\
0 & 0 & \gamma\\
-\gamma^q\omega^q & \gamma^q & 0\\
\end{pmatrix}.
$$
\subsection{Proof of the first part of Theorem \ref{thm:constU3}}\label{proofconstU3}
Here we prove that the incidence structure $\Gamma$ of Theorem \ref{thm:constU3} is a generalised
hexagon. Our approach is to use a definition of a generalised hexagon which is equivalent to the
one stated in the introduction: (i) it contains no ordinary $k$-gon for $k\in\{2,3,4,5\}$, (ii) any
pair of elements is contained in an ordinary hexagon, and (iii) there exists an ordinary heptagon
(see \cite[\S 1.3.1]{HvM}). A thick generalised polygon has \textit{order} $(s,t)$ if every line has $s+1$
points and every point is incident with $t+1$ lines. A counting argument shows that if we know that
the number of points and lines of a generalised hexagon are $(s+1)(1+st+s^2t^2)$ and
$(t+1)(1+st+s^2t^2)$, then the conditions (ii) and (iii) automatically follow from the first
condition.
\begin{proof}
First we show that $\Omega$ induces a point-partition of each generator (minus its point in the
Hermitian curve $\mathcal{O}$). Let $\ell$ be a generator of $\mathsf{H}(3,q^2)$ and let $P$ be a point of
$\ell\backslash\mathcal{O}$. For a point $X$, we will let $X^*$ be the $q+1$ elements of $\Omega$
which lie on $X$.
Consider the $q+1$ elements $P^*$ of $\Omega$ on $P$. Since $P^*$
covers the points of a Baer subplane, it follows that there is a unique element of $\Omega$
contained in $\ell$ and containing $P$. Therefore $\Omega$ induces a point-partition of each
generator minus its point in the Hermitian curve $\mathcal{O}$. It follows immediately that $\Gamma$ is a partial
linear space (i.e., every two points lie on at most one line).
Since $\mathsf{H}(3,q^2)$ is a generalised quadrangle, $\Gamma$ has no triangles. So suppose now
that we have a quadrangle $R$, $S$, $T$, $U$ of $\Gamma$. Then at least three of these points
are necessarily affine points. For example, if two of these points were of type (a), two points of type (b),
and with one line of type (i) and three of type (ii) making up the quadrangle,
the three lines of type (ii) would yield a triangle of generators. So this case is clearly impossible.
At least three points, $S$, $T$, $U$
say, are necessarily affine points and the lines of the quadrangle are elements of
$\Omega$. Moreover, $R$ is also an affine point, since if $R$ were a generator then $S$ and $U$
would lie on $R$ and $ST$, $TU$, $SU$ would then be a triangle in $\mathsf{H}(3,q^2)$; a
contradiction. So all the four points $R$, $S$, $T$, $U$ of a quadrangle must be affine.
Recall that $\mathfrak{u}$ is the polarity defining $\mathsf{H}(3,q^2)$. Note that
$R^\mathfrak{u} \cap T^\mathfrak{u}$ is equal to $SU$ and that $SU\cap
\mathsf{H}(3,q^2)$ is a Baer subline with a point on $\mathcal{O}$. Indeed $R^*$
spans a Baer subplane fully contained in $\mathsf{H}(3,q^2)$ and it meets $\mathcal{O}$
in a Baer subline and since $R^\mathfrak{u}\cap T^\mathfrak{u}\cap
\mathsf{H}(3,q^2)$ is a Baer subline contained in $R^*$ then $SU\cap \mathsf{H}(3,q^2)$
has a point in $\mathcal{O}$.
Likewise $S^\mathfrak{u}\cap U^\mathfrak{u}$ equal to $RT$ and $RT\cap
\mathsf{H}(3,q^2)$ is a Baer subline with a point in $\mathcal{O}$. So $SU$ and
$RT$ are polar to each other under $\mathfrak{u}$, but then each point
of $\mathsf{H}(3,q^2)$ on $SU$ is collinear with each point of $\mathsf{H}(3,q^2)$ on
$RT$, while the points of $\mathcal{O}$ are pairwise non-collinear, a
contradiction. Hence $\Gamma$ has no quadrangles.
Suppose we have a pentagon $R$, $S$, $T$, $U$, $W$ of $\Gamma$.
Now points of type (b), which are affine points, are collinear in $\Gamma$ if they are
incident with a common element of $\Omega$. Since each element of $\Omega$ spans a generator, points
of type (b) are also collinear in $\mathsf{H}(3,q^2)$. So since $\mathsf{H}(3,q^2)$ is a generalised
quadrangle, we see immediately that each point of our pentagon is an affine point.
Suppose, by way of contradiction, that
our pentagon has a point of type (a), that is, a generator $\ell$ of $\mathsf{H}(3,q^2)$.
Then we would have four generators of $\mathsf{H}(3,q^2)$ forming a quadrangle and we
obtain a similar ``forbidden'' quadrangle of affine points (i.e., $RSTU$) from the above argument.
So there are no
pentagons in $\Gamma$.
A trivial counting argument shows that $\mathcal{L}$ has size $(q^6-1)/(q-1)$, which is equal to the
sum of the number of affine points and the number of generators of $\mathsf{H}(3,q^2)$, and so it
follows that $\Gamma$ is a generalised hexagon (of order $(q,q)$). \end{proof}
\subsection{Exhibiting a suitable set of Baer subgenerators}\label{sect:partition}
In this section, we describe a natural candidate for a set $\Omega$ of Baer subgenerators satisfying
the hypotheses of Theorem \ref{thm:constU3}. Consider the stabiliser $G_\mathcal{O}$ in
$\mathsf{PGU}_4(q)$ of the Hermitian curve $\mathcal{O}=\pi \cap \mathsf{H}(3,q^2)$, where $\pi$ consists of the
elements whose last coordinate is zero. Then the elements of $G_\mathcal{O}$ can be thought of
(projectively) as matrices $M_A$ of the form
$$M_A:=\left(\begin{smallmatrix}
&&&0\\
&A&&0\\
&&&0\\
0&0&0&1
\end{smallmatrix}\right), \quad A\in\mathsf{GU}_3(q).$$
\begin{lemma}\label{lemma:transBaer}
The group $G_{\mathcal{O}}$ acts transitively on the set of Baer subgenerators which
have a point in $\mathcal{O}$.
\end{lemma}
\begin{proof}
Inside the group $\mathsf{PGU}_4(q)$, the stabiliser $J$ of a generator $\ell$ induces a $\mathsf{PGL}_2(q^2)$ acting $3$-transitively
on the points of $\ell$. So the stabiliser in $J$ of a point $P$ of $\ell$ acts transitively on the Baer sublines within $\ell$ which contain $P$.
Now $J$ meets $G_\mathcal{O}$ in the stabiliser of a point of $\ell$, and so $G_{\mathcal{O},\ell}$ acts transitively on Baer subgenerators
contained in $\ell$. Since $G_\mathcal{O}$ acts transitively on $\mathcal{O}$, the result follows.
\end{proof}
The key to this construction is the action of a particular subgroup of $G_{\mathcal{O}}$. We will
see later that this group naturally corresponds to the stabiliser in $\mathsf{G}_2(q)$ of a
non-degenerate hyperplane $\Q^-(5,q)$ of $\Q(6,q)$.
\begin{defn}[$\mathsf{SU}_3$]
Let $\mathsf{SU}_3$ be the group of collineations of $\mathsf{H}(3,q^2)$ obtained from the
matrices $M_A$ where $A\in\mathsf{SU}_3(q)$.
\end{defn}
In short, the orbits of $\mathsf{SU}_3$ on Baer subgenerators with a point in $\mathcal{O}$, each
form a suitable candidate for $\Omega$, as we will see.
\begin{lemma}\label{lem:stabbaer}
Let $\mathcal{O}=\pi\cap \mathsf{H}(3,q^2)$, where $\pi$ is the hyperplane $X_3=0$ of $\PG(3,q^2)$,
and let $G_\mathcal{O}$ be the stabiliser of $\mathcal{O}$ in $\mathsf{PGU}_4(q)$. Let $b$ be a Baer
subgenerator of $\mathsf{H}(3,q^2)$ with a point in $\mathcal{O}$. Then the stabiliser of $b$ in
$G_{\mathcal{O}}$ is contained in $\mathsf{SU}_3$.
\end{lemma}
\begin{proof}
Recall from the beginning of Section \ref{sect:hermitian} that given a Baer subgenerator $b$ of $\mathsf{H}(3,q^2)$ with a point
$B$ in $\mathcal{O}$, there is a dual Baer subline of $\pi$ with vertex $B$. So there is a set
of $3\times 3$ Hermitian matrices $U$ of rank $2$, which are equivalent up to scalar multiplication
in $\GF(q^2)^*$. Now $G_{\mathcal{O}}$ induces an action on the pairs $[U,\ell]$, where $U$ is a
Hermitian matrix of rank $2$ and $\ell$ is a generator containing the nullspace of $U$, which we can
write out explicitly by
$$[U,\ell]^{M_A}=[A^{-1}UA,\ell^{M_A}].$$
Let $\omega$ be an element of $\GF(q^2)$ satisfying $\mathsf{N}(\omega)=-1$, and let $U_0$ and
$\ell_0$ be
$$U_0:=\left(\begin{smallmatrix}
0&0&-\omega\\
0&0&1\\
-\omega^q&1&0\\
\end{smallmatrix}\right), \quad
\ell_0 := \langle (1,\omega,0,0), (0,0,1,\omega)\rangle.$$ Since $G_{\mathcal{O}}$ acts transitively
on Baer subgenerators with a point in $\mathcal{O}$ (Lemma \ref{lemma:transBaer}), we need only calculate the stabiliser of $[U_0,\ell_0]$. Now let $M_A$ be an
element of $G_ {\mathcal{O}}$ fixing $[U_0,\ell_0]$. Since $M_A$ fixes $\ell_0$, we can see by
direct calculation that $A$ is of the form
$$\left(\begin{smallmatrix}
a&b&-f\omega\\
d&e&f\\
g&g\omega&1\\
\end{smallmatrix}\right),$$
with $(a+d\omega)\omega =b+e\omega$.\\
Now we see what it means for $A$ to centralise $U_0$ up to a scalar $k$, that is,
$U_0A=kAU_0$. Hence
$$
\left(\begin{smallmatrix}
-g \omega& -g \omega^2& -\omega\\
g& g \omega& 1\\
d - a \omega^q& e - b \omega^q& 0
\end{smallmatrix}\right)
=
k\left(\begin{smallmatrix}
-f& -f \omega& b - a \omega\\
-f \omega^q& f& e - d \omega\\
-\omega^q& 1& 0
\end{smallmatrix}\right)$$
and we obtain
$$A=\left(\begin{smallmatrix}
k^{-1}-b\omega^q&b&-k^{-1}g\omega^2\\
(k^{-1}-k-b\omega^q)\omega^q&k+b\omega^q&k^{-1}g\omega\\
g&g\omega&1\\
\end{smallmatrix}\right)$$
where $k\in\GF(q)$, $\mathsf{N}(g)=k^2+\mathsf{T}(b^q\omega)-1$ and $\mathsf{T}(g\omega)=0$ (in order for this matrix to be unitary).
The determinant of $A$ is
$$1-g^2\omega(\mathsf{N}(\omega)+1) (\omega k^{-2} + b (\mathsf{N}(\omega)+1)k^{-1} +\omega)=1$$
and therefore, the stabiliser of $[U_0,\ell_0]$ in $G_{\mathcal{O}}$ is contained in $\mathsf{SU}_3$.
\end{proof}
The above lemma allows us to attach a value to a Baer subgenerator that is an invariant
for the action of $\mathsf{SU}_3$.
\begin{defn}[Norm of a Baer subgenerator]\label{norm}
Let $\mathcal{O}$ be the Hermitian curve $\mathsf{H}(3,q^2)\cap \pi$, where $\pi$ is the hyperplane
$X_3=0$ of $\PG(3,q^2)$ and let $G_\mathcal{O}$ be the stabiliser of $\mathcal{O}$ in
$\mathsf{PGU}_4(q)$. Fix a Baer subgenerator $b_0$ of $\mathsf{H}(3,q^2)$ with a point in
$\mathcal{O}$. Let $b$ be a Baer subgenerator of $\mathsf{H}(3,q^2)$ with a point in $\mathcal{O}$,
and suppose $M_A$ is an element of $G_\mathcal{O}$ such that $b=b_0^{M_A}$. Then the \textit{norm}
of $b$ is
$$\Vert b\Vert := \det(A).$$ Moreover (by Lemma \ref{lem:stabbaer}), the map $b\mapsto \Vert b\Vert$ induces a group
homomorphism $\phi$ from $G_{\mathcal{O}}$ to the multiplicative subgroup of elements of
$\GF(q^2)^*$ satisfying $\mathsf{N}(x)=1$.
\end{defn}
Note that the kernel of $\phi$ is $\mathsf{SU}_3$. The homomorphism $\phi$ is surjective and hence
there is a natural partition of Baer subgenerators with a point in $\mathcal{O}$ into $q+1$
classes. Each orbit of $\mathsf{SU}_3$ consists of Baer subgenerators with a common value for their norm.
\begin{lemma}\label{lemma:omega}
Let $\mu$ be an element of $\GF(q^2)$ such that $\mathsf{N}(\mu)=1$. Let $\mathcal{O}$ be a Hermitian
curve of $\mathsf{H}(3,q^2)$ defined by $X_3=0$, and let $\Omega$ be a set of Baer subgenerators
with a point in $\mathcal{O}$ which have norm equal to $\mu$. Then:
\begin{enumerate}
\item[(i)] Every affine point is on $q+1$ elements of $\Omega$ covering a Baer subplane.
\item[(ii)] For every point $X\in\mathcal{O}$ and for every affine point $Y$ in
$X^\mathfrak{u}$, there is a unique element of $\Omega$ through $X$ and $Y$.
\end{enumerate}
\end{lemma}
\begin{proof}
Recall that $\Omega$ is an orbit of $\mathsf{SU}_3$ on Baer subgenerators and $\mathsf{SU}_3$ acts
transitively on the affine points $\mathsf{H}(3,q^2)\backslash\mathcal{O}$, and so clearly every
affine point is on $q+1$ elements of $\Omega$. Moreover, such a set of $q+1$ elements of $\Omega$
will cover a Baer subplane, as we show now. Let $Y$ be an affine point, let $Y^*$ be the set
of $q+1$ elements of $\Omega$ through $Y$ and let $b_0$ be one particular
element of $Y^*$. Then every other element of $Y^*$ is in the orbit of $b_0$ under the stabiliser of $Y$
in $\mathsf{SU}_3$. Now for every $g\in (\mathsf{SU}_3)_Y$, we know that $\langle b_0^g\rangle=\langle b_0\rangle^g
\in Y^\perp$ and so every element of $Y^*$ lies in the plane $Y^\perp$. At infinity, $Y^\perp$ meets $\mathcal{O}$
in a Baer subline and so we have a triangle of Baer sublines spanning a Baer subplane of $Y^\perp$, and
it is covered completely by the elements of $Y^*$.
To complete the proof, we need only prove (ii).
Since the stabiliser of a point in $\mathcal{O}$ is transitive on the set of affine points in the perp of that point, we can
assume that $X=(1,\omega,0,0)$ and $Y=(0,0,1,\omega)$ for some $\omega$ satisfying
$\mathsf{N}(\omega)=-10$. We have already seen, in the proof of Lemma \ref{lem:stabbaer}, that $X$ and
$Y$ lie on a Baer subgenerator, which we can assume without loss of generality, is in $\Omega$.
This Baer subgenerator is uniquely defined by a $3\times 3$ Hermitian matrix $U$ of rank $2$ and the
generator $\ell$ spanning $X$ and $Y$, and we assume (as before) that $U$ has the form
$$U:=\left(\begin{smallmatrix}
0&0&-\omega\\
0&0&1\\
-\omega^q&1&0\\
\end{smallmatrix}\right).$$
Then the two-point stabiliser of $X$ and $Y$ inside $\mathsf{SU}_3$ consists of elements $M_A$ with $A$
of the form
$$A=\left(\begin{smallmatrix}
a&b&0\\
d&e&0\\
0&0&1\\
\end{smallmatrix}\right)$$
where $(a+d\omega)\omega=b+e\omega$ and
$\left(\begin{smallmatrix}
a&b\\
d&e
\end{smallmatrix}\right)\left(\begin{smallmatrix}
a^q&d^q\\
b^q&e^q
\end{smallmatrix}\right)=I$. Let's consider one of these elements $M_A$. Then
\begin{align*}
(A^q)^TUA
&=\left(\begin{smallmatrix}
0&0&-a^q\omega+d^q\\
0&0&-b^q\omega+e^q\\
-a\omega^q+d&-b\omega^q+e&0\\
\end{smallmatrix}\right)
\end{align*}
and we see that this matrix is a scalar multiple of $U$ (the scalar being $(-b\omega^q+e)$).
Therefore $M_A$ fixes the Baer subgenerator defined by $[U,\ell]$. Hence there is a unique element
of $\Omega$ on $X$ and $Y$. \end{proof}
\subsection{Classifying the suitable sets of Baer subgenerators}\label{sect:classify}
\begin{theorem}\label{thm:omega}
Suppose $\Omega$ is a set of Baer subgenerators of $\mathsf{H}(3,q^2)$ with a point in $\mathcal{O}$,
such that every affine point is on $q+1$ elements of $\Omega$ spanning a Baer subplane. Then
$\Omega$ is an orbit under $\mathsf{SU}_3$.
\end{theorem}
\begin{proof}
Let $b$ be a Baer subgenerator of $\mathsf{H}(3,q^2)$ with a point in $\mathcal{O}$. If $b'$ is
another Baer subgenerator of $\mathsf{H}(3,q^2)$ with a point in $\mathcal{O}$ such that $b$ and
$b'$ meet in an affine point and span a fully contained Baer subplane, then we will show that there
is some element of $\mathsf{SU}_3$ which maps $b$ to $b'$. Without loss of generality, we can
choose our favourite Baer subgenerator and our favourite affine point. Suppose we have a fixed Baer
subgenerator $b$ giving the dual Baer subline defined by
$$U=\left(\begin{smallmatrix}
0&0&-\omega&\\
0&0&1\\
-\omega^q&1&0\\
\end{smallmatrix}\right)$$
and on the generator $\ell=\langle (1,\omega,0,0),(0,0,1,\omega)\rangle$ where $\mathsf{N}(\omega)=-1$.
Let $P$ be the affine point $(0,0,1,\omega)$ and consider an arbitrary generator $\ell'$ on $P$
where $\ell':=\langle (0,0,1,\omega),(1,\nu,0,0)\rangle$ and $\mathsf{N}(\nu)=-1$. Suppose we have a Baer
subgenerator $b'$ on $P$, on the generator $\ell'$, defined by the matrix $U'$. Since every element
of $P^\mathfrak{u}\cap\mathcal{O}$ is in the dual Baer subline defined by $U'$, we have that $U'$ can be written as
$$\left(\begin{smallmatrix}
a&0&\beta\\
0&a&\gamma\\
\beta^q&\gamma^q&c\\
\end{smallmatrix}\right)$$
where $a\in\GF(q)$ and $\beta,\gamma\in\GF(q^2)$. For $(1,\nu,0,0)$ to be in the nullspace of
$U'$, we must have $a=0$ and $\beta=-\gamma\nu$. That is, $U'$ is just
$$\left(\begin{smallmatrix}
0&0&-\gamma\nu\\
0&0&\gamma\\
-\gamma^q\nu^q&\gamma^q&c\\
\end{smallmatrix}\right).$$
Now $b$ and $b'$ span a fully contained Baer subplane if and only if the dual Baer sublines defined by $U$ and $U'$ share
only the points of $P^\mathfrak{u} \cap \mathcal{O}$, on $\mathcal{O}$.
Indeed suppose, by way of contradiction, that there is a point $Z$ of
$\mathcal{O}$ in common between the dual
Baer sublines defined by $U$ and $U'$. Then $Z^\mathfrak{u}$ meets $b$
in a point $Q$, different from $L$ and $P$ and it meets $b'$ in a point
$Q'$ different from $L'$ ($L'=\pi\cap b'$) and $P$.
Thus $Z^\mathfrak{u} \cap P^\mathfrak{u}$ meets $\mathsf{H}(3,q^2)$ in a Baer
subline $b''$ containing $Q$ and $Q'$. Now the Baer subplane spanned
by $b$ and $b'$ is fully contained if and only if $b''$ has a point
$T$ in $\mathcal{O}$.
This implies that $T$ and $Z$ are points of $\mathcal{O}$ collinear on
$\mathsf{H}(3,q^2)$; a contradiction.
Note that $P^\mathfrak{u}\cap\mathcal{O}$ consists of the points of the form $(1,\delta,0,0)$ together with $(0,1,0,0)$.
Suppose $(1,\delta,\eta,0)$ is an element of both dual
Baer sublines. That is, $(1,\delta,\eta)U(1,\delta^q,\eta^q)^T=0$ and
$(1,\delta,\eta)U'(1,\delta^q,\eta^q)^T=0$. Now
\begin{align*}
(1,\delta,\eta)U(1,\delta^q,\eta^q)^T&= (1,\delta,\eta)\left(\begin{smallmatrix}
0&0&-\omega&\\
0&0&1\\
-\omega^q&1&0\\
\end{smallmatrix}\right)(1,\delta^q,\eta^q)^T\\
&=-\eta\omega^q+\eta\delta^q+(-\omega+\delta)\eta^q\\
&=T(\eta(\delta-\omega)^q),
\end{align*}
\begin{align*}
(1,\delta,\eta)U'(1,\delta^q,\eta^q)^T&= (1,\delta,\eta)\left(\begin{smallmatrix}
0&0&-\gamma\nu\\
0&0&\gamma\\
-\gamma^q\nu^q&\gamma^q&c\\
\end{smallmatrix}\right)(1,\delta^q,\eta^q)^T\\
&=-\eta\gamma^q\nu^q+\eta\gamma^q\delta^q+(-\gamma\nu+\delta\gamma+\eta c)\eta^q\\
&=-(\eta\gamma^q\nu^q+\eta^q\gamma\nu)+(\eta\gamma^q\delta^q+\eta^q\gamma\delta)+c\eta^{q+1}\\
&=T(\eta\gamma^q(\delta-\nu)^q)+c\mathsf{N}(\eta).
\end{align*}
Since $1+\mathsf{N}(\delta)+\mathsf{N}(\eta)=0$, we see that our equations become
\begin{equation*}\tag{*}\label{traceequation}
\mathsf{T}(\eta(\delta-\omega)^q)=0\text{ and }\mathsf{T}(\eta\gamma^q(\delta-\nu)^q)=c(1+\mathsf{N}(\delta))
\end{equation*}
So since the dual Baer sublines defined by $U$ and $U'$ share
only the points of $P^\mathfrak{u}\cap\mathcal{O}$, then whenever condition (\ref{traceequation}) holds for a choice of $\delta$, $\eta$,
we will have $\eta=0$. Therefore, we must have a priori that $c=0$ and $\gamma\notin\GF(q)$.
Let $\eta=(\gamma\nu-\gamma^q\omega)^q$ and
$$\delta=\frac{-\eta^q+\eta^{q-1}\gamma^q(\omega-\nu)^q}{\gamma^q-\gamma}.$$
Then a straightforward
calculation shows that $1+\mathsf{N}(\delta)+\mathsf{N}(\eta)=0$, $\mathsf{T}(\eta(\delta-\omega))= 0$ and $\mathsf{T}(\eta\gamma(\delta-\nu))= 0$, so condition (\ref{traceequation}) holds, and hence
$\eta=0$.
Therefore, $\nu = \omega\gamma^{q-1}$ and
$$U'=\left(\begin{smallmatrix}
0&0&-\gamma^q\omega\\
0&0&\gamma\\
-\gamma \omega^q&\gamma^q&0\\
\end{smallmatrix}\right).$$
We want to show that $U'$ is conjugate to $U$ under some element of $\mathsf{SU}_3(q)$. Now the
group $\mathsf{SU}_2(q)$ of invertible $2\times 2$ matrices with unit determinant, and fixing the
form $X_0Y_0^q+X_1Y_1^q=0$ on $\GF(q^2)^2$, has $q+1$ orbits on totally isotropic vectors of
$\GF(q^2)^2$. Each orbit consists of vectors $(x,y)$ where $y/x^q$ attains a common
value. Therefore, there exists some element $C_0$ of $\mathsf{SU}_2(q)$ such that
$C_0(-\gamma\nu,\gamma)^T=(-\omega,1)$. Let
$${\small C:=\left(\begin{array}{c|c}
C_0 &\begin{smallmatrix}0\\0\\ \end{smallmatrix} \\
\hline
\begin{matrix}0&0 \end{matrix}& 1\\
\end{array}\right).}$$
Then one can check easily that $C$ has determinant $1$ and $CU(C^q)^T=U'$. Therefore, there is some
element of $\mathsf{SU}_3$ which maps $b$ to $b'$.
For every affine point $P$, let $P^*$ be the set of $q+1$ elements of $\Omega$ incident with
$P$. Then by the above, every element of $P^*$ is contained in a common orbit of $\mathsf{SU}_3$.
Note that $\mathsf{SU}_3$ is transitive on generators of $\mathsf{H}(3,q^2)$, and the stabiliser of
a point $X$ of $\mathcal{O}$ in $\mathsf{SU}_3$ is transitive on the affine points of
$X^\mathfrak{u}$. Suppose now that $b$ and $b'$ do not meet in an affine point. Let $P\in b$. Then
$P^*\subset b^{\mathsf{SU}_3}$. Now there exists $g\in\mathsf{SU}_3$ such that $\langle
b\rangle^g=\langle b'\rangle$ and $P^g\in b'$. Thus $b'\in (P^g)^*\subset
(b')^{\mathsf{SU}_3}$. Note also that $P^g\in b^g$, and hence $b^g\in
(b')^{\mathsf{SU}_3}$. Therefore $b$ and $b'$ are in the same orbit under $\mathsf{SU}_3$.
\end{proof}
In Section \ref{sect:proofQ6embedding}, we will use the above result to prove Theorem
\ref{thm:Q6embedding}.
\section{The connection with the 6-dimensional parabolic quadric}\label{sect:parabolicquadric}
A non-degenerate hyperplane section of $\Q(6,q)$ can be of one of two types (up to isometry):
it could induce a hyperbolic quadric $\Q^+(5,q)$ or it could induce an elliptic
quadric $\Q^-(5,q)$. The stabiliser of a hyperbolic quadric section in $\mathsf{G}_2(q)$ is
isomorphic to $\mathsf{SL}_3(q):2$, whilst the stabiliser of an elliptic quadric section in
$\mathsf{G}_2(q)$ is isomorphic to $\mathsf{SU}_3(q):2$ (see \cite{Kle88a}). These two maximal
subgroups bring forth the two low-dimensional models of the Split Cayley hexagon that appear in this paper,
and a second way to explain the interplay between these `linear' and `unitary' models is
via Curtis-Tits and Phan systems; see Section \ref{sect:phan}.
We begin first with some
observations about the situation where we fix a $\Q^+(5,q)$ hyperplane section.
The stabiliser $\mathsf{SL}_3(q):2$ of $\Q^+(5,q)$ fixes two disjoint planes $p'$ and $\sigma'$ of
$\Q^+(5,q)$, and then the lines of $\mathcal{H}(q)$ contained in $\Q^+(5,q)$ are just the lines of
$\Q^+(5,q)$ which meet both $p'$ and $\sigma'$ in a point. It was noticed in \cite{CameronKantor79}
that we can reconstruct $\mathcal{H}(q)$ from these two fixed planes together with an orbit $\Omega$
of $\mathsf{SL}_3(q)$ on affine lines (of size $(q^3-q)(q^2+q+1)$). We can capture the affine
points by noticing that the $q+1$ hexagon-lines through an affine point span a totally isotropic plane
(sometimes known as an $\mathcal{H}(q)$-plane) meeting $\Q^+(5,q)$ in a line disjoint from both $p'$
and $\sigma'$. Similarly, we can take the polar image of an affine line and consider its
intersection with $\Q^+(5,q)$. This results in a $3$-dimensional quadratic cone of $\Q^+(5,q)$
meeting both $p'$ and $\sigma'$ in a point, but having vertex not in $p'$ nor $\sigma'$. We can
then employ the Klein correspondence to map our projection of $\mathcal{H}(q)$ on $\Q^+(5,q)$, to
$\mathsf{PG}(3,q)$ (see \cite[\S 15.4]{Hirschfeld85} for more on the Klein correspondence). We summarise this
correspondence below:
\begin{center}
\begin{table}[H]\footnotesize
\begin{tabular}{p{8cm}|p{8cm}}
$\PG(3,q)$&$\Q(6,q)$\\ \hline
\rowcolor[gray]{0.95}Point-plane anti-flag $(p,\sigma)$ &A latin $p'$ and greek plane $\sigma'$ defining a hyperbolic quadric $\Q^+(5,q)$\\
Pencils with vertex not in $\sigma$ and plane not through $p$&Affine points of $\Q(6,q)\backslash \Q^+(5,q)$\\
\rowcolor[gray]{0.95}Lines& Points of $\Q^+(5,q)$\\
Pencils with vertex in $\sigma$ and plane through $p$& Lines of $\Q^+(5,q)$ meeting $p'$ and $\sigma'$ in a point\\
\rowcolor[gray]{0.95}Parabolic congruences & Affine lines of $\Q(6,q)$, quadratic cones of $\Q^+(5,q)$\\
Parabolic congruences having axis not incident with $p$ or $\sigma$, but having a pencil of lines with one line incident with $p$
and another incident with $\sigma$ & Affine lines of $\mathcal{H}(q)$\\
\hline
\end{tabular}
\medskip
\caption{The extended Klein representation.}
\end{table}
\end{center}
Now we describe how we can view $\mathcal{H}(q)$ as substructures of the $3$-dimensional Hermitian
surface $\mathsf{H}(3,q^2)$. A \textit{$t$-spread} of $\PG(d,q)$ is a collection of $t$-dimensional
subspaces which partition the points of $\PG(d,q)$. So necessarily, $t+1$ must divide $d+1$ and the
size of a $t$-spread of $\PG(d,q)$ is $(q^{d+1}-1)/(q^{t+1}-1)$. If $t+1$ is half of $d+1$, we
usually call a $t$-spread just a \textit{spread} of $\PG(d,q)$. Suppose we have a $t$-spread
$\mathcal{S}$ of $\PG(d,q)$ and embed $\PG(d,q)$ as a hyperplane in $\PG(d+1,q)$. If we define the
\textit{blocks} to be the $(t+1)$-dimensional subspaces of $\PG(d+1,q)$ not contained in $\PG(d,q)$
incident with an element of the $t$-spread, then together with the affine points
$\PG(d+1,q)\backslash\PG(d,q)$, we obtain a linear space; in fact, a $2$--$(q^{d+1},q^{t+1},1)$
design. This linear representation of a $t$-spread is a generalisation of the commonly called
\textit{Andr\'e/Bruck-Bose construction} (where $t+1=(d+1)/2$), and is fully explained by Barlotti
and Cofman \cite{BaCo}. More generally, it is possible that this construction produces a
Desarguesian affine space and we then say that the given $t$-spread is \textit{Desarguesian}. It
turns out that a $t$-spread $\mathcal{S}$ is Desarguesian if and only if $\mathcal{S}$ induces a
spread in any subspace generated by two distinct elements of $\mathcal{S}$ (see \cite{Lun} and
\cite{Segre64}).
Now consider $\PG(3,q^2)$ and a hyperplane $\pi_\infty$ therein, and identify $\mathsf{AG}(3,q^2)$
with the affine geometry $\PG(3,q^2)\backslash \pi_\infty$.
We will be considering the correspondence between objects in $\mathsf{H}(3,q^2)$ and $\Q(6,q)$,
where $\mathcal{S}$ is a Hermitian spread of a non-degenerate hyperplane section $\Q^-(5,q)$ of
$\Q(6,q)$. One can also obtain this correspondence via field reduction from $\mathsf{H}(3,q^2)$ to
$\Q^+(7,q)$, and then slicing with a non-degenerate hyperplane section (see \cite{Lunardon06}). We
will call this correspondence the \textit{Barlotti-Cofman-Segre representation} of
$\mathsf{H}(3,q^2)$. Below we summarise the various correspondences between objects in
$\mathsf{H}(3,q^2)$ and objects in $\Q(6,q)$ obtained by the Barlotti-Cofman-Segre representation of
$\mathsf{H}(3,q^2)$. Throughout, we fix a hyperplane $\Sigma_\infty$ at infinity intersecting
$\Q(6,q)$ in a $\Q^-(5,q)$, which corresponds to a fixed non-degenerate hyperplane $\pi_\infty$ of
$\mathsf{H}(3,q^2)$, and we let $\mathcal{S}$ denote a Hermitian spread of $\Sigma_\infty$.
\begin{center}
\begin{table}[H]\footnotesize
\begin{tabular}{p{8cm}|p{8cm}}
$\mathsf{H}(3,q^2)$&$\Q(6,q)$\\ \hline
\rowcolor[gray]{0.95}Hermitian curve $\mathcal{O}$ of $\pi_\infty$ & Hermitian spread $\mathcal{S}$ of $\Q^-(5,q)$\\
Affine points $\mathsf{H}(3,q^2)\backslash \pi_\infty$ & Affine points of $\Q(6,q)\backslash \Q^-(5,q)$\\
\rowcolor[gray]{0.95}Generators of $\mathsf{H}(3,q^2)$&Generators of $\Q(6,q)$ incident with some element of $\mathcal{S}$\\
Baer subplane contained in $\mathsf{H}(3,q^2)$ meeting $\mathcal{O}$ in a Baer subline&Generators of $\Q(6,q)$ not incident with any element of $\mathcal{S}$\\
\rowcolor[gray]{0.95}Baer subgenerators with a point in $\mathcal{O}$& Affine lines of $\Q(6,q)$\\ \hline
\end{tabular}
\medskip
\caption{The Barlotti-Cofman-Segre representation.}\label{BCSrepresentation}
\end{table}
\end{center}
The table below shows how we can directly obtain the model for the split Cayley hexagon on the
$3$-dimensional Hermitian surface via the Barlotti-Cofman-Segre correspondence. We can recover the
affine points of $\Q(6,q)$ by noticing that a plane incident with a spread element will correspond
to a hexagon-plane; a point of $\mathcal{H}(q)$ together with its $q+1$ incident lines.
\begin{center}
\begin{table}[H]\footnotesize
\begin{tabular}{ll|l}
&In $\mathsf{H}(3,q^2)$& Barlotti-Cofman-Segre image in $\Q(6,q)$ \\
\hline
\rowcolor[gray]{0.95} \textsc{Points} & (a) Lines of $\mathsf{H}(3,q^2)$ & Planes of $\Q(6,q)$ containing a spread element.\\
& (b) Affine points of $\mathsf{H}(3,q^2)\backslash\mathcal{O}$& Affine points of $\Q(6,q)\backslash\Q^-(5,q)$.\\
\hline
\rowcolor[gray]{0.95} \textsc{Lines} & (i) Points of $\mathcal{O}$ & Lines of the Hermitian spread.\\
& (ii) Elements of $\Omega$ & Affine lines spanning a totally isotropic plane with a spread element.\\ \hline
\end{tabular}
\caption{The split Cayley hexagon in $\mathsf{H}(3,q^2)$.}
\end{table}
\end{center}
\section{Characterising the split Cayley hexagon in the 6-dimensional parabolic quadric}\label{sect:hexagon}
By the Barlotti-Cofman-Segre correspondence, we can translate Theorem \ref{thm:constU3} to a
statement about substructures of $\Q(6,q)$. However, the information that can be transferred via
this correspondence is not sufficient to characterise a set of lines $\Q(6,q)$ as the lines of a
generalised hexagon; there is an additional case. The natural model of the split Cayley hexagon was
revised in the introduction, and here we briefly point out a characterisation of it as a set of
lines of $\Q(6,q)$. It is a special case of a result of Cuypers and Steinbach \cite[Theorem
1.1]{CuypersSteinbach}, but we give a direct proof for completeness.
\begin{theorem}\label{thm:HqQuadric}
Let $\mathcal{L}$ be a set of lines of $\Q(6,q)$ such that every point of $\Q(6,q)$
is incident with $q+1$ lines of $\mathcal{L}$ spanning a plane. Then one of the following
occurs:
\begin{enumerate}
\item[(a)] There is a spread $\mathcal{S}$ of $\Q(6,q)$ such that
$\mathcal{L}$ is equal to the union of the lines contained in each generator of
$\mathcal{S}$.
\item[(b)]
The points of $\Q(6,q)$ together with $\mathcal{L}$ define the points and
lines of a generalised hexagon, and a plane of $\Q(6,q)$ contains $0$ or $q+1$ elements of
$\mathcal{L}$ in it.
\end{enumerate}
\end{theorem}
\begin{proof}
Let $\Gamma$ be the geometry having the points of $\Q(6,q)$ as its points, and having $\mathcal{L}$
as its set of lines.
Clearly $\Gamma$ is a partial linear space where there are $q+1$ lines through every point, and $q+1$
points through every line. We will write $P^*$ for the pencil of $q+1$ lines of $\mathcal{L}$ incident
with $P$.
Since every plane of $\PG(6,q)$ meets $\Q(6,q)$ in a full plane, a conic, a line, a pair of
concurrent lines or a point, it follows that every plane intersects $\mathcal{L}$ in $q^2+q+1$,
$q+1$, $1$ or $0$ lines. We will show now that the first possibility leads to case (a). Suppose
there is a plane $\pi$ with $q^2+q+1$ lines of $\mathcal{L}$. Let $\ell$ be an element of
$\mathcal{L}$ not contained in such a plane. Then the $q+1$ planes on the tangent quadric containing $\ell$ (i.e., the points collinear to all the points on $\ell$)
contain $q+1$ elements of $\mathcal{L}$. Since there is always at least one point $p$ of $\pi$
collinear with all points of $\ell$, we see that the point $p$ is now incident with at least $q+2$
elements of $\mathcal{L}$; a contradiction. Hence either every point is in a plane with $q^2+q+1$
elements of $\mathcal {L}$ (and we obtain the spread of $\Q(6,q)$), or no point is.
Suppose now that $\mathcal{L}$ is not partitioned by a spread of $\Q(6,q)$. So no plane of
$\PG(6,q)$ contains $q^2+q+1$ elements of $\mathcal{L}$, and therefore, every plane intersects
$\mathcal{L}$ in $0$, $1$ or $q+1$ lines. We continue now to prove that $\Gamma$ is a generalised
hexagon. Clearly there is no triangle formed by lines of $\mathcal{L}$, so suppose we have a
quadrangle $R$, $S$, $T$, $U$ in $\Gamma$. Note that these points do not lie in a common plane. The
planes spanning $T^*$, $U^*$ and $R^*$ are three totally singular planes contained in a common
3-space, which implies that this 3-space is also totally singular; a contradiction. Suppose now we
have a pentagon $R$, $S$, $T$, $U$, $W$ of $\Gamma$, (and the ordering of these points is
important). So $RSTU$ spans a 3-space intersecting $\Q(6,q)$ in two totally singular planes, namely
$S^*$ and $T^*$. Now $W$ is collinear (in $\mathcal{L}$) with $R$ and $U$, and therefore the line
$RU$ is totally singular; which implies that $RSTU$ is totally singular, a contradiction. So there
are no $k$-gons in $\Gamma$ with $k<6$. Since $\mathcal{L}$ has size equal to the number of points
of $\Q(6,q)$, it follows that $\Gamma$ is a generalised hexagon of order $q$.
Let $N_i$ be the number of planes of $\Q(6,q)$ containing $i$ elements of $\mathcal{L}$.
So $N_0+N_1+N_{q+1}=(q+1)(q^2+1)(q^3+1)$. Now each point is on a unique
plane containing $q+1$ elements of $\mathcal{L}$, and so $N_{q+1}=(q^3+1)(q^2+q+1)$.
Now for a given point $X$, all but one of the planes on $X$ would have no lines of $\mathcal{L}$ on it,
which accounts for $N_0=q^3(q^3+1)$ planes (n.b., there are $(q+1)(q^2+1)$ planes
on any point, and a plane contains $q^2+q+1$ points). So it follows that $N_1=0$.
\end{proof}
\begin{lemma}\label{lemma:hermitianspread}
Let $\mathcal{L}$ be a set of lines of $\Q(6,q)$ such that every point $X$ of $\Q(6,q)$ is incident with
$q+1$ lines of $\mathcal{L}$ spanning a plane $X^*$, and such that the concurrency graph of $\mathcal{L}$
is connected. Suppose $\Pi$ is a nondegenerate hyperplane meeting $\Q(6,q)$ in a $\Q^-(5,q)$-quadric.
Then the set $\mathcal{S}:=\{X^*\cap \Pi: X\in\Q^-(5,q)\}$ defines a Hermitian spread of $\Q^-(5,q)$.
\end{lemma}
\begin{proof}
Any pair of lines of $\mathcal{S}$ are disjoint since otherwise they would intersect
in a point $P$ and the plane $P^*$ spanned by the $q+1$ elements of $\mathcal{L}$ incident
with $P$ would then be contained in $\Q^-(5,q)$. Therefore, $\mathcal{S}$ forms a spread of
$\Q^-(5,q)$.
Now consider two elements $\ell$ and $m$ of $\mathcal{S}$. The solid $\langle \ell,m\rangle$ meets
$\Q^-(5,q)$ in a $\Q^+(3,q)$ section. The polar image of $\langle \ell,m\rangle$, within $\Q(6,q)$, is then a
plane meeting $\Q(6,q)$ in a non-degenerate conic $\mathcal{C}$.
Let $r$ be a line in the regulus determined by $\ell$ and $m$, and suppose for a proof by contradiction
that $r$ is not an element of $\mathcal{S}$. Then each of the $q+1$ points $Z_i$ on $r$ defines a different element
$\ell_i:=Z_i^*\cap \Pi$ of $\mathcal{S}$. Since the lines contained in $\langle \ell,m\rangle$ concurrent with $r$
form the opposite-regulus to that defined by $\ell$ and $m$, it follows that none of the $\ell_i$ are contained
in $\langle \ell, m \rangle$.
Since $\ell$ is a line of $\mathcal{L}$ and, by Theorem 4.1, a plane of $\Q(6,q)$ has $0$ or $q+1$ elements of
$\mathcal{L}$ contained in it, each of the $q+1$ planes $\langle \ell,X_i\rangle$ for
$X_i\in \mathcal{C}$ is a plane $Y^*$ for some $Y\in \ell$. Similarly, each of the $q+1$ planes $\langle m,X_i\rangle$
is a plane $Y^*$ for some $Y\in m$. Hence for each $X_i\in \mathcal{C}$ the plane $X_i^*$ meets $\langle l,m\rangle$ in a line of the opposite regulus. Therefore, there is a one-to-one correspondence between points
$X_i$ of $\mathcal{C}$ and points $Z_i$ of $r$. That is, the line $X_iZ_i$ is a line of $\mathcal{L}$ for every $i$.
Recall that the concurrency graph of $\mathcal{L}$ is connected, and so by Theorem \ref{thm:HqQuadric},
$\mathcal{L}$ forms the lines of a generalised hexagon. Let $Z_1$ and $Z_2$ be two elements on $r$.
Now $Z_1^\perp$ is a hyperplane and $Z_2^*$ is a plane, so we have two cases: (i) $Z_2^*$ is contained in $Z_1^\perp$, or (ii) $Z_2^*$ meets $Z_1^\perp$ in a line $n$.
The first case cannot arise as a plane of $\Q(6,q)$ contained in $Z_1^\perp$ must go through $Z_1$, and we have assumed that $r$ is not in $\mathcal{L}$.
Suppose we have the second case. Since $Z_2$ lies in $Z_1^\perp$, the line $n$ lies on $Z_2$ and so $n$ is a line of $\mathcal{L}$ in $Z_2^*$. Note that $\langle Z_1,n\rangle$ is a plane of $\Q(6,q)$ having
at least one element of $\mathcal{L}$ contained in it.
By Theorem 4.1, a plane of $\Q(6,q)$ has $0$ or $q+1$ elements of $\mathcal{L}$ contained
in it. Therefore, there is some point $V$ on $n$ such that $Z_1\in V^*$.
Hence we have a line of $\mathcal{L}$ going through $Z_1$ concurrent with $n$,
and $Z_1$ and $Z_2$ are at distance $4$.
This requirement then forces $r$ to lie in $V^*$.
and hence each $Z_i^*$ goes through $V$.
Now $\langle \mathcal{C}\rangle $ is a non-degenerate plane through $\Pi^\perp$ and
$\Pi^\perp \notin \langle Z_i^*: Z_i\in r\rangle$. Therefore, each $Z_i^*$ meets
the conic only in the point $X_i$.
The lines $X_1Z_1$ and $VZ_1$ are lines of $\mathcal{L}$ and since $X_1,V,Z_1\in r^\perp$,
we have that $Z_1^*$ is contained in $r^\perp$; a contradiction.
(Otherwise, $Z_1^*$ would be a plane through $Z_2$).
Hence $r\in \mathcal{S}$ and
$\mathcal{S}$ is closed under taking reguli.
By \cite[\S 3.1.2]{BTVM98} and \cite{Luyckx:2001fk}, such a spread of $\Q^-(5,q)$ is necessarily a Hermitian spread of $\Q^-(5,q)$.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:Q6embedding}]\label{sect:proofQ6embedding}
First we will translate the hypothesis to the 3-dimensional Hermitian variety $\mathsf{H}(3,q^2)$
via the Barlotti-Cofman-Segre correspondence. So let us fix a non-degenerate hyperplane section
$\Q^-(5,q)$ and consider the set $\mathcal{S}$ of lines of $\mathcal{L}$ that are contained in
$\Q^-(5,q)$. By Lemma \ref{lemma:hermitianspread}, $\mathcal{S}$ is a Hermitian spread of $\Q^-(5,q)$
and so we have the ingredients for the Barlotti-Cofman-Segre correspondence, whereby the spread
$\mathcal{S}$ corresponds to a fixed Hermitian curve $\mathcal{O}$ of $\mathsf{H}(3,q^2)$. Recall
that the elements of $\mathcal{L}$ not contained in $\Q^-(5,q)$ are mapped to a subset $\Omega$ of
the Baer subgenerators having a point in $\mathcal{O}$. Also, the affine points of $\Q(6,q)$ are
mapped to the affine points of $\mathsf{H}(3,q^2)\backslash\mathcal{O}$. We will show that $\Omega$
satisfies the hypotheses of Theorem \ref{thm:constU3}; that is, a generator spanned by $q+1$
elements of $\mathcal{L}$ corresponds to a Baer subplane of $\mathsf{H}(3,q^2)$. Now by Theorem
\ref{thm:HqQuadric}, we have either (a) $\mathcal{L}$ is the union of lines of the planes of a
spread $\mathcal{S}$ of $\Q(6,q)$, or (b) $\mathcal{L}$ forms the lines of a generalised
hexagon. Case (a) cannot occur as the concurrency graph of $\mathcal{L}$ is connected. So
$\mathcal{L}$ is the lines of a generalised hexagon embedded into $\Q(6,q)$. Let $P$ be an affine
point of $\Q(6,q)$ and let $P^*$ be the $q+1$ elements of $\mathcal{L}$ incident with $P$. By our
hypothesis, $P^*$ spans a plane $\pi_P$. If this plane were to be incident with an element of
$\mathcal{S}$, then $\pi_P$ would contain more than $q+1$ elements of $\mathcal{L}$ thus implying
that $\pi_P$ would have all of its lines in $\mathcal{L}$; this would then imply that the
concurrency graph of $\mathcal{L}$ is disconnected (see the proof of Lemma
\ref{thm:HqQuadric}). Therefore, $\pi_P$ is not incident with any element of $\mathcal{S}$, and
hence, $\pi_P$ meets $\Q^-(5,q)$ in a transversal line to $q+1$ elements of $\mathcal{S}$. By the
Barlotti-Cofman-Segre correspondence, $\pi_P$ corresponds to a Baer subplane of $\mathsf{H}(3,q^2)$,
as required.
By Theorem \ref{thm:omega}, $\Omega$ is an orbit of $\mathsf{SU}_3$. Moreover,
this group $\mathsf{SU}_3$ lies within the stabiliser in $\mathsf{PGU}_4(q)$ of a non-degenerate hyperplane,
and so corresponds to a subgroup $\overline{\mathsf{SU}_3}$ of the stabiliser of $\mathcal{S}$.
Now there are $q+1$ split Cayley hexagons whose lines not lying in $\Q^-(5,q)$ form
an orbit under $\overline{\mathsf{SU}_3}$, so it remains to
observe that $\overline{\mathsf{SU}_3}$ has only $q+1$ orbits of size $q(q+1)(q^3+1)$. Indeed, the orbits of
$\overline{\mathsf{SU}_3}$ on lines of $\Q(6,q)$ can be described completely geometrically from
the corresponding orbits of objects in $\mathsf{H}(3,q^2)$ (see Table \ref{BCSrepresentation}). Therefore,
$\Omega$ is the set of lines of some split Cayley hexagon (having a set of lines containing $\mathcal{S}$).
\begin{center}
\begin{table}[H]\footnotesize
\begin{tabular}{p{6cm}|p{6cm}|p{4cm}}
Orbits in $\mathsf{H}(3,q^2)$&Orbits on lines of $\Q(6,q)$&Size\\ \hline
\rowcolor[gray]{0.95}Hermitian curve $\mathcal{O}$ of $\pi_\infty$ & Hermitian spread $\mathcal{S}$ of $\Q^-(5,q)$&$q^3+1$\\
Affine points $\mathsf{H}(3,q^2)\backslash \pi_\infty$ & Lines of $\Q^-(5,q)$ not in $\mathcal{S}$&$q^2(q^3+1)$\\
\rowcolor[gray]{0.95} Baer subgenerators with no point in $\mathcal{O}$& Affine lines not meeting an element of $\mathcal{S}$ in a totally singular plane &
$q^2(q^2-1)(q^3+1)$\\
Baer subgenerators with a point in $\mathcal{O}$& Affine lines meeting an element of $\mathcal{S}$ in a totally singular plane & $(q+1)\times q(q+1)(q^3+1)$\\ \hline
\end{tabular}
\medskip
\caption{Orbits of $\overline{\mathsf{SU}_3}$ on lines of $\Q(6,q)$.}\label{BCSrepresentation2}
\end{table}
\end{center}
\end{proof}
\section{A connection with Phan theory}\label{sect:phan}
In the theory of linear algebraic groups, if a simply connected simple algebraic group $G$ of type $B_n$,
$C_n$, $D_{2n}$, $E_7$, $E_8$, $F_4$ or $G_2$ has a Curtis-Tits system for its extended Dynkin
diagram then there is a \textit{twisted} version known as a \textit{Phan system} for associated finite groups
corresponding to fixed points of so-called \textit{Frobenius maps} of $G$,
where the $\mathsf{SL}_2$-subgroups of the Curtis-Tits system are replaced with certain
$\mathsf{SU}_2$-subgroups. This phenomenon has been known since the 1970's to both group theorists
and those working in the theory of twin buildings. In a Curtis-Tits system for a
finite group $G$ (defined over $\GF(q)$), if $K_\alpha$ and $K_\beta$ are two
$\mathsf{SL}_2$-subgroups for two fundamental roots $\alpha$ and $\beta$ joined by a single bond,
then $\langle K_\alpha,K_\beta\rangle$ is isomorphic to $\mathsf{(P)SL}_3(q)$. Whereas in the
corresponding Phan system, a single bond represents an amalgam $\langle K_\alpha,K_\beta\rangle$
isomorphic to $\mathsf{(P)SU}_3(q)$. (See \cite{BGHS}, \cite{BY} and \cite{Gramlich09} for more on Phan systems).
The geometric model of the split Cayley hexagon that we presented
in this paper was inspired by a \textit{unitary} analogue of the $\mathsf{SL}_3$-model introduced by
Cameron and Kantor \cite{CameronKantor79}.
\begin{table}[H]
\begin{tabular}{c|c}
Curtis-Tits system & Phan system\\
\begin{tikzpicture}[scale=1.4]
\node (a) at (0,0) [circle, draw,inner sep=2pt] {};
\node (b) at (1,0) [circle, draw, inner sep=2pt] {};
\node (c) at (2,0) [circle, draw, inner sep=2pt] {};
\draw (a) to (b);
\draw (b) to (c);
\draw (a.30) to (b.150);
\draw (a.-30) to (b.-150);
\draw (.5,.1) to (.4,0);
\draw (.4,0) to (.5,-.1);
\node [below] at (a.south) {\footnotesize $\alpha$};
\node [below] at (b.south) {\footnotesize $\beta$};
\node [below] at (c.south) {\footnotesize ${-3\alpha-2\beta}$};
\node at (0.5,0.3) {\scriptsize $\mathsf{G}_2(q)$};
\node at (1.5,0.3) {\scriptsize $\mathsf{SL}_3(q)$};
\end{tikzpicture}
&
\begin{tikzpicture}[scale=1.4]
\node (a) at (0,0) [circle, draw,inner sep=2pt] {};
\node (b) at (1,0) [circle, draw, inner sep=2pt] {};
\node (c) at (2,0) [circle, draw, inner sep=2pt] {};
\draw (a) to (b);
\draw (b) to (c);
\draw (a.30) to (b.150);
\draw (a.-30) to (b.-150);
\draw (.5,.1) to (.4,0);
\draw (.4,0) to (.5,-.1);
\node [below] at (a.south) {\footnotesize $\alpha$};
\node [below] at (b.south) {\footnotesize $\beta$};
\node [below] at (c.south) {\footnotesize ${-3\alpha-2\beta}$};
\node at (0.5,0.3) {\scriptsize $\mathsf{G}_2(q)$};
\node at (1.5,0.3) {\scriptsize $\mathsf{SU}_3(q)$};
\end{tikzpicture}\\
\end{tabular}
\medskip
\caption{A summary of the Curtis-Tits and Phan systems for the extended Dynkin diagram of $\mathsf{G}_2(q)$.}
\end{table}
\section{Acknowledgements}
The authors thank Prof Frank De Clerck for various discussions concerning this work (and for
\textit{ontbijten}!), and they also thank Prof Guglielmo Lunardon for his comments on Theorem
\ref{thm:HqQuadric}. We thank Dr {\c{S}}{\"u}kr{\"u} Yal{\c{c}}{\i}nkaya for his expert advice on Phan
systems. This work was supported by the GOA-grant ``Incidence Geometry'' at Ghent University. The
first author acknowledges the support of a Marie Curie Incoming International Fellowship within the
6th European Community Framework Programme (MIIF1-CT-2006-040360), and the second author
acknowledges the support of a GNSAGA-grant.
We are especially grateful to the anonymous referees whose remarks have greatly improved the
exposition and clarity of this paper.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,793 |
Q: Как принять значение переменной из php c помощью json Вот такой кусочек кода из name.php отправляет строку к исполняемому файлу function.php
$.ajax({
type: "POST",
url: "function.php",
data: dataString,
cache: false,
success: function(html)
{
//Вот здесь я так понимаю должна быть функция принимающая данные
//Должно быть так
//var country = (country1,country2,country3)
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});
а вот php
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$spisok .= $value.',';
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Данные приходят (судя по firebug), но как их установить в переменную JS?
A: $.ajax({
type: "POST",
url: "function.php",
data: dataString,
cache: false,
dataType: 'json', // добавили тип принимаемых данных
success: function(html)
{
// теперь переменная html - это объект (распарсеный JSON)
}
});
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 474 |
\section{Introduction}
The notion of depth has historical roots in the theory of operator algebras, with its most natural definition being in the ring-theoretic terms of subrings, balanced tensors and bimodules (see the end of Section~2). Specializing to group algebra extensions, the minimum depth of a subgroup $K \leq G$ of a finite group $G$, over a field of characteristic zero, for example, is two if $K$ is normal in $G$,
or more strongly one if $G=KC_G(K)$, and the odd number $2n+1$ if all simples occur in $n$ applications of induction-restriction applied to the unit simple (or trivial module), $4$ and $6$ in certain exceptional cases, and it is not known if minimum depth $2n > 6$ can occur. Also, the (minimum) depth is less than or equal to twice the number of conjugates of $K$ that intersect to equal the core$_G(K)$ \cite[Theorem 6.9]{BKK}.
The subgroup depth, over $\CC$, of the series of permutation groups $S_n \subseteq S_{n+1}$ is computed
to be $2n - 1$ in \cite{BKK}, having the same value over any commutative base ring \cite{BDK}. With a twist automorphism introduced, the depth of the corresponding twisted group algebra series shrinks to $ 2\left(n - \left\lceil \frac{\sqrt{8n+1} - 1}{2} \right\rceil \right) + 1$, as computed in \cite{D}: a general reason for why depth should shrink when twisting is introduced is given in \cite{HKS}. The subgroup depth of the alternating group series $A_n \subset A_{n+1}$ is $2\left(n - \lceil \sqrt{n}\, \rceil \right)+1$ over the complex numbers \cite{BKK} (a precise value is not known in positive characteristic, although it is between the value just given and $2n - 3$ \cite{BDK}).
Other results on subgroup depth can be seen in \cite{F1, F2, FKR, HHP, HHP2}.
The paper \cite{K2014} shows how subgroup depth is determined by the tensor powers of the permutation module of cosets: this extends to determining subalgebra depth of a Hopf subalgebra $R \subseteq H$ from its quotient module $Q$ as an $R$-module (see Section~\ref{S:two} for the definition of $Q$). The depth of $Q_R$ is determined from the least $n$ for which $Q \otimes \cdots \otimes Q$ ($n$ times $Q$, denoted by $Q^{\otimes (n)}$) has the exact same indecomposable constituents as $Q^{\otimes (n+1)}$ (a chain of subsets increasing with $n$).
Drinfeld's quantum double $D(H)$ of a Hopf algebra $H$ is frequently applied to a group algebra. The depth of a finite group $G$ in its double $D(G)$ (over $\CC$) is studied in \cite{HKY}, where it is shown to be closely related to the tensor power of the adjoint representation of $G$ on itself at which it is faithful, a topic introduced and explored in \cite{P}. For example, in \cite{P} it is proved that the adjoint representation of the symmetric group on itself is faithful, which rephrased in the terms of $Q$ in this paper, shows that $Q$ and $Q^{\otimes (2)}$ have the same indecomposable constituents, or $Q_G$ has ``depth $1$'' for the group $\CC$-algebra of a symmetric group on $3$ or more letters (in its Drinfeld double with quotient module $Q$).
It is also noted in \cite[Lemma 1.3]{P} that for $G$ set equal to certain semidirect products of elementary abelian $p$- and $q$-groups, where $p,q$ are primes such that $p \| q-1$, the adjoint action of $G$ on itself is not faithful: from this recipe, a group $G$ of order $108$ with $Q$ of depth $2$ in $D(G)$ is constructed in \cite[Example 6.5]{HKY}. For any semisimple Hopf subalgebra pair $R \subseteq H$, the depth of $Q$ coincides with the length $n$ of the chain of annihilator ideals of tensor powers of $Q$, i.e., the least $n$ for which $\Ann Q^{\otimes (n)}$ is a Hopf ideal
\cite[Theorem 3.14]{K}, \cite{H}.
The half quantum groups, or Taft algebras $U_n(q)$, are $n^2$-dimensional algebras generated by a group-like element of order $n$ and a nilpotent element of order $n$, with an anti-commutation relation between these two generators involving a primitive $n$-th root-of-unity $q$ in the base field. Like group algebras, they are Hopf algebras, but unlike complex finite group algebras, they are not semisimple nor cocommutative. The noncocommutativity is explicit in the skew primitivity of the nilpotent element. The Drinfeld doubles of the half quantum groups are reduced to the transparent terms of generators and relations in \cite{C}. The Green rings (or representation rings) of the half quantum groups are determined in \cite{COZ} by means of a computation in \cite{G} of the tensor products of the $n^2$ isoclasses of indecomposables: the Green ring is shown to be commutative, although the half quantum groups are not quasitriangular (nor almost cocommutative \cite{CC}).
Two finite-dimensional modules over a finite-dimensional algebra are similar, denoted with a $\sim$, if they have the same nonzero indecomposable constituents.
In this paper, the quotient module $Q$ for the Taft algebra $U_n(q)$, for any $n \in \N$, $n \geq 2$ and $q$ a primitive $n$-th root of unity in the base field, is computed in Section~\ref{S:four}, decomposed in Theorem~\ref{T:KSdecQ}, and its second tensor power is shown in Theorem~\ref{T:decQQ} to contain all indecomposables of $U_n(q)$, a very strong form of faithfulness; the conclusion is that $Q \, {\not \sim} \, Q \otimes Q$, but $Q^{\otimes (2)} \sim
Q^{\otimes (3)}$ as $U_n(q)$-modules. Thus, the depth of $Q$ is $2$, which translates, using \cite[Theorem 5.1]{K2014}, into a minimum even depth $d_{ev}(U_n(q), D(U_n(q))) = 6$ for this Hopf subalgebra pair.
\section{The general quotient module $Q$ in more detail}\label{S:two}
We introduce the quotient module of certain Hopf algebra-subalgebra pairs, note a generalization of a relative Maschke's theorem \cite[Theorem 3.7]{K} and point out some differences between the quotient module and its restriction to the subalgebra. The material in this section is mostly of theoretical interest and some of it may be skipped in reading the main results in Sections~4 and~5.
Throughout the paper, let $H$ be a finite-dimensional Hopf algebra over a field $k$. Suppose $R$ is a left coideal subalgebra of $H$, i.e., $R$ is a subalgebra and the coproduct satisfies $\cop(R) \subseteq H \otimes R$. The subalgebra $R$ is augmented by the counit $\eps$ of $H$; let $R^+$ denote the augmentation ideal.
Define the quotient $H$-module \begin{equation}
Q = H/ R^+H.
\end{equation}
E.g., note that $\overline{r} = \overline{1_H} \eps(r)$
for all $r \in R$, with usual coset notation. In later sections, we will focus on the $R$-module $Q$ resulting from restriction, but we point out some differences between $Q_R$ and the cyclic module $Q_H$ in this section. After the next theorem our focus for $R$ falls back to the more restrictive notion of Hopf subalgebra, where $\cop(R) \subseteq R \otimes R$ and $S(R) = R$: thus, $R$ is a Hopf algebra with a restriction of the structure of $H$.
The quotient module $Q$ is
also a coalgebra, since $R^+H$ is a coideal, by elementary considerations.
It satisfies the identity of a right $H$-module coalgebra: $\cop(qh) = q\1 h\1 \otimes q\2 h\2$
and $\eps_Q(qh) = \eps_Q(q) \eps(h)$, for every $q \in Q, h \in H$. The canonical epimorphism of right $H$-module coalgebras $H \rightarrow Q$ is denoted by $h \mapsto \overline{h}$.
Let $A$ be any ring, and $B$ be a unital subring of $A$. Recall that $A$ is a right semisimple extension of $B$ if every right $A$-module $M$ splits in the kernel exact sequence of the canonical epimorphism $M \otimes_B A
\rightarrow M$; left semisimple extensions are similarly defined with left modules. Equivalently, short exact sequences of $A$-modules that are $B$-split, also split as $A$-modules. Note that the $A$-module $M$ is isomorphic to a direct summand of $M \otimes_B A$,
a fact which we denote by $M \| M \otimes_B A$, and $M$ is therefore $B$-relative projective. A third equivalent condition for semisimple extensions is that all modules are relative projective. (Note that a projective module $P_A$ has the property that $P \otimes_B A \rightarrow P$ splits for any subalgebra $B$ in $A$.)
A ring $A$ is a separable extension of a subring $B$ if for every $A$-module $M$ there is a splitting of the canonical epimorphism $M \otimes_B A \rightarrow M$, natural with respect to $A$-module mappings
$M \rightarrow N$. Equivalently, there is an element
$e = e^1 \otimes_B e^2 \in A \otimes_B A$ satisfying $e^1 e^2 = 1_A$ and $ae = ea$ for all $a \in A$ (suppressing a possible summation $\sum_{(e)} e^1 \otimes_B e^2$, using Sweedler-type notation).
We note the following theorem, which generalizes Maschke's theorem for group and Hopf algebras, and has the same proof as in \cite{K}, so the details are omitted.
\begin{theorem}\label{T:21}
A finite-dimensional Hopf algebra $H$ is a right semisimple extension of a left coideal subalgebra $R$ iff $k_H$ is isomorphic to a direct summand of $Q_H$ iff $k_H$ is $R$-relative projective
iff there is a $q \in Q$ such that $\eps_Q(q) = 1$ and $qh = \eps(h) q$ for each $h \in H$ iff there is $s \in H$ such that $\eps(s) = 1$ and $sH^+ \subseteq R^+H$ iff
$H$ is a separable extension of $R$.
\end{theorem}
\begin{proof}
The module structure on $k_H$ is of course $1 . h = \eps(h)$. Note that $Q \cong k \otimes_R H$ via $1 \otimes h \mapsto \overline{h}$; thus $Q$ is $R$-relative projective by a standard exercise. Moreover, for any right $H$-module $M$, the $H$-module $M \otimes Q$ is given by the diagonal action $(m \otimes q)h = mh\1 \otimes qh\2$, and we have the natural isomorphism of right $H$-modules,
\begin{equation}
\label{eq: induct}
M \otimes_R H \cong M \otimes Q
\end{equation}
given by $m \otimes_R h \mapsto mh\1 \otimes \overline{h\2}$ with inverse
$m \otimes \overline{h} \mapsto mS(h\1) \otimes_R h\2$.
The counit of $Q$ is $R$-split by $\lambda \mapsto \lambda \overline{1_H}$ as noted above. If $k_H$ is relative projective, the counit of $Q$ splits over $H$.
The element $q = \overline{s}$ is the image of a splitting $\sigma: k_H \rightarrow Q_H$.
It follows from the natural isomorphism \eqref{eq: induct} that every module is $R$-relative projective
and indeed that $S(s\1) \otimes_R s\2$ is a separability element.
\end{proof}
Note that Eq.~(\ref{eq: induct}) implies that all tensor powers of $Q$ are relative projective, and that the class of relative projective $H$-modules enjoys the ideal property that $M \otimes V$ is relative projective if $V$ is relative projective, and $M$ is any module. The property of separability, or its absence, is usually easy to detect in a subalgebra pair; e.g., global dimension satisfies $\operatorname{gldim}(B) \geq \operatorname{gldim}(A)$ for a projective separable extension $B \subseteq A$. Thus a nonsemisimple Hopf algebra is never a semisimple extension of a semisimple Hopf subalgebra. Furthermore, a separable extension of Hopf algebras is an ordinary Frobenius extension, where the modular functions are related by restrctiction \cite[Corollary 3.8]{K}. Thus, a non-unimodular Hopf algebra (like the Taft algebra) does not form a semisimple extension with its Drinfeld double (by \cite[Radford's Theorem 6.10]{NEFE}).
Below we use the notation $V_H \| W_H$ (suggested by the Krull-Schmidt Theorem) if $V$ is a direct summand of $W$ up to module isomorphism, sometimes also denoted by
$V_H \oplus * \cong W_H$.
\begin{cor}
Suppose a finite-dimensional Hopf algebra extension $H \supseteq R$ is not a semisimple extension. Then $k_H$ is not a direct summand of any tensor power of $Q_H$ up to isomorphism
(i.e., $k_H {\not |} Q^{\otimes (n)}$ for any $n \in \N$). If $W_H$ satisfies $k_H \| W^{\otimes (n)}$
for some $n \in \N$, then $W_H {\not |} Q^{\otimes (m)}$ for any $m \in \N$.
\end{cor}
\begin{proof}
A direct summand of a relative projective, such as $Q_H$, is relative projective. By Theorem~\ref{T:21}, the extension is semisimple if $k_H$ is relative projective.
\end{proof}
The restricted module $Q_R$ then is very different than $Q_H$, since $k_R \| Q_R$ always, as noted above. We will focus only on the restricted module $Q_R$ below.
\subsection{The quotient module and quantum subgroup depth}
For any ring $A$, and $A$-module $X$, let $1 \cdot X = X$, $2 \cdot X = X \oplus X$, etc. The similarity relation $\sim$ mentioned in the introducion may be defined on $A$-modules as follows. Two $A$-modules $M,N$ are similar, written $X \sim Y$, if $X \| n \cdot Y$ and $Y \| m \cdot X$ for some positive integers $m, n$. This is an equivalence relation, and carries over to isoclasses in the Grothendieck group of $A$, or the Green ring if $A$ is a (quasi-) Hopf algebra. If $M \sim N$ and $X$ is an $A$-module,
then we have $M \oplus X \sim N \oplus X$; if $\otimes$ is a tensor on mod-$A$, then also $M \otimes X \sim N \otimes X$. In case $A$ is a finite-dimensional algebra, $M \sim N$ if and only if $\Indec (M) = \Indec (N)$, where $\Indec (X)$ denotes the set of isoclasses of the indecomposable module constituents of $X$ in its Krull-Schmidt decomposition.
If $A $ is a finite-dimensional Hopf algebra, and $r(A)$ denotes the Green ring of $A$, then $r(A)$ has a preferred basis as a $\Z$-algebra given by the isoclasses of all indecomposable $A$-modules; the algebra is of finite rank iff
$A$ has finite representation type. The set of nonzero linear combinations of indecomposables with non-negative integer coefficients we call the \textit{positive quadrant}; these elements correspond to the (isoclasses of) actual $A$-modules. The
relations $\|$ and $\sim$ are meaningful in the positive quadrant: say $a,b,c$ are elements there, then $a \sim b$
(so $a \| nb$ and $b \| ma$ for some $m, n \in \N$)
implies $a + c \sim b + c$ (but not conversely unless $\Indec (c) \cap \Indec (a) = \emptyset = \Indec (c) \cap \Indec (b)$), $ac \sim bc$, $3a \sim 5 b$, and others. Also, $a \sim b$ and
$c \sim d$ implies $ac \sim bd$ and $a + c \sim b + d$.
Given an $a \in r(A)$ in the positive quadrant, define for each $m \geq 1$, the polynomial $p_m(a) = a + \cdots + a^m$, noting that
$p_m(a) \| p_{m+1}(a)$. Let $p_0(a) = 1$.
The \textit{depth} of $a$ (or any module in its isoclass) is the least $n \in \N$ for which
$p_n(a) \sim p_{n+1}(a)$ if such exists, denoted by $d(a) = n$; otherwise $d(a) = \infty$. Note that $p_n(a) \sim p_{n+1}(a)$ implies that $p_n(a) \sim p_{n+r}(a)$, for any $r \in \N$, by an exercise. Elements in $r(A)$ that represent algebras or coalgebras in the tensor category mod-$A$ of finite-dimensional $A$-modules, such as the quotient module $Q$ defined above for $A = H$ with Hopf subalgebra $R$, with isoclass denoted by $\tilde{q}$, satisfy $\tilde{q}^i\| \tilde{q}^{i+1}$ for all $i \in \N$, a fact
which follows from applying the counit and comultiplication, or the unit and multiplication. In this case,
$p_n(\tilde{q}) \sim \tilde{q}^n$, and the depth $n$ condition is replaceable by $\tilde{q}^n \sim \tilde{q}^{n+1}$. If $A$ has finite representation type, it is clear that an algebra or coalgebra such as $\tilde{q}$ has finite depth; and indeed any element $a$ in the positive quadrant has finite depth.
Note that depth $d(\tilde{q}) = 0$ in the Green ring of $R$ if and only if
$R$ is a normal Hopf subalgebra of $H$, i.e. $R^+H = HR^+$. Then $\tilde{q} \sim 1$ by recalling the right module $Q = H/R^+H$.
Suppose $r(A)$ is commutative, which is the case if $H$ is quasi-triangular or braided \cite{CK}. Let $a,b \in r(A)$ be two elements of finite depth in the positive quadrant. Then $ab$ and $a + b$ have finite depth. It is noted in \cite{F} that elements of finite depth are algebraic elements in the Green ring
of a cocommutative Hopf algebra,
and conversely by a paraphrasing; i.e., $d(a) < \infty$ $\Leftrightarrow$ $g(a) = 0$ for some nonzero polynomial $g(x) \in \Z[x]$.
Then $d(ab) < \infty$ and $d(a + b) < \infty$ follows in an exercise, or as standard textbook material on the set of algebraic elements in a commutative ring forming a subring.
Consider the special case of a subgroup pair $K \leq G$ where $G$ is a finite group. Let $R =kK$ and $H = kG$ be the group algebra extensions over a ground field $k$ of any characteristic; then $Q \cong k[K \setminus G]$, the right coset space, by noting that $(1 - k')g \in R^+H$ are the generators for each $k' \in K, g \in G$. It is noted in \cite[Chapter 9]{F} that $\tilde{q} \in r(H)$ has finite depth, i.e., the isoclass of any permutation module, such as $\tilde{q}$, is an algebraic element in the Green ring of $G$.
Based on simplifications made in the literature (see the preliminaries in \cite{K}), the \textit{subgroup depth} $d_k(K,G)$ is a positive integer equal to, or one less than, $d_{ev}(kK,kG) = 2d(Q_R) + 2$, the minimum even depth, and satisfying $$ d_h(kK,kG) - 2 \leq d_k(K,G) \leq d_h(kK,kG) + 1,$$
where the h-depth $d_h(kK,kG) = 2d(Q_H) + 1$. The two equalities of depth are explained and proven in \cite[Theorem 5.1]{K2014} for any Hopf subalgebra $R \subseteq H$:
\begin{eqnarray}
\label{eq: h}
d_h(R,H) & = & 2d(Q_H) + 1; \\
\label{eq: ev}
d_{ev}(R,H) & = & 2d(Q_R) + 2.
\end{eqnarray}
The minimum odd depth is not determined by this approach; in case $H$ is semisimple, there is a nice graphical technique described in \cite{BKK} for its determination.
The minimum h-depth, even depth and odd depth $d_{odd}(R,H)$ may be viewed as closely related (to within two) and on an equal footing: subalgebra depth $d(R,H)$ may be even and equal to $d_{ev}(R,H)$,
or odd and equal to $d_{odd}(R,H) = d_{ev}(R,H) - 1$.
From Feit's theorem in group representation theory, it follows that $d_k(K,G) < \infty$, a result with interesting upper bounds given in \cite{BDK}.
This paper will not make use of the most general and full definition of depth in \cite{BDK} and h-depth in \cite{K2012}. For the reader's convenience, we sketch the general definition of depth, the minimum depth $d(B,A)$,and h-depth $d_h(B,A)$ of a subring $B$ in a ring $A$: they are defined as follows by a positive integer, or infinity, in terms of the natural
$A$-$A$-bimodule structure on the tensor powers $A^{\otimes_B (n)}$ (where $n \in \N$ and $A^{\otimes_B (0)} = B$), and its restrictions to the three other combinations of $B$- and $A$-bimodules. If
\begin{equation}
\label{eq: finitedepth}
A^{\otimes_B (m)} \sim A^{\otimes_B (m+1)}
\end{equation}
for a nonnegative integer $m$ for which this similarity holds in one of the four bimodule structures mentioned above, we say the subring $B \subseteq A$, or the ring extension $A \supseteq B$, has h-depth $2m-1= 1,3,5,\ldots$ in
the $A$-$A$-bimodule structure, the ring extension has depth $2m =2,4,6,\ldots$ in the $A$-$B$- or $B$-$A$-bimodule structures, and the ring extension has depth
$2m+1=1,3,5,\ldots$ in the $B$-$B$-bimodule structure. Having depth $2m$ implies that the subring also has h-depth $2m+1$ by tensoring the similarity, and depth $2m+1$ by restriction of the similarity. Also, having h-depth $2m+1$ implies that the subring has depth $2m+2$, and having depth $2m+1$ implies having depth $2m+2$. Then define $d(B,A)$ as the minimum such natural number as well as the minimum h-depth $d_h(B,A)$, an odd positive integer, unless there is no such $m \in \N$ satisfying Eq.~(\ref{eq: finitedepth}), in which case we may write
$d(B,A) = \infty = d_h(B,A)$. Note then that \begin{equation}
\label{eq: ineq}
-2 \leq d_h(B,A) - d(B,A) \leq 1
\end{equation}
if one of these is finite (also noted above for group algebras).
For example, we note a shorter proof of \cite[Corollary 3.3]{K2014}, which states
that an h-separable Hopf algebra extension $H \supseteq R$ is a trivial extension.
For h-separability is equivalent to $d_h(R,H) = 1$, which is equivalent to the quotient module $Q_H$ having depth $0$, i.e., $Q \sim k_H$. Thus $\overline{h} = \overline{1_H}h = \eps(h)\overline{1_H}$ for all $h \in H$. But $Q$ is cyclic:
$Q = \{ \overline{h} \| h \in H \} = k\overline{1}$. Thus, $\dim Q = 1 = \dim H/ \dim R$. Hence, $R = H$.
The next lemma follows from the considerations about general depth above and the inequality~(\ref{eq: ineq}), Eqs.~(\ref{eq: h}) and~(\ref{eq: ev}), and is left as an exercise.
\begin{lemma}.
The depth of the $H$-module $Q$ and its restricted module satisfy
$$ d(Q_H) - d(Q_R) \in \{ 0,1 \}. $$
Both values are attained.
\end{lemma}
\begin{remark}
\begin{rm}
Restricting $Q^{\otimes (n)} \sim Q^{\otimes (n+1)}$ from $H$-modules to $R$-modules shows
that $d(Q_H) \geq d(Q_R)$. The value $d(Q_H) - d(Q_R) = 1$ is very usual when $H \neq R$; however, $d(Q_H) = d(Q_R)$ occurs when $R,H$ are the complex group algebras of the alternating groups, $A_4, A_5$
\cite{BKK, BDK, K2012}.
\end{rm}
\end{remark}
\section{Preliminaries on the half quantum group}
We keep our previous assumption that $k$ is any field, but assume further that $0\neq q\in k$ is a primitive $n$-th root of unity with $n \geq 2$. The latter implies, in particular, that the characteristic of $k$ does not divide $n$. Define the Taft algebra by
$$U_n(q) = k\bra G,X \, \| G^n = 1, X^n = 0, GX = qXG \ket.$$
This is an $n^2$-dimensional algebra with basis $\{ G^i X^j \| i,j=0,1,\ldots,n-1 \}$. It has a coalgebra structure given by $\cop(G) = G \otimes G$, $\cop(X) = X \otimes G + 1 \otimes X$, $\eps(G) = 1$, and $\eps(X) = 0$. There is also a Hopf algebra antipode given by $S(G) = G^{-1}$ and $S(X) = -XG^{-1}$ \cite{SY}.
The quantum double $D(U_n(q))$ is given in terms of generators and relations in \cite{C} by
\begin{eqnarray*}
D(U_n(q)) &=& k\bra a,b,c,d \, \| a^n = 0 = d^n, b^n=1 = c^n, ba = qab, \\
& & \hspace{-15pt} dc = qcd, db = qbd, bc = cb, ca = qac, da - qad = 1 - bc \ket.
\end{eqnarray*}
A basis for $D(U_n(q))$ is given by $\{ a^i b^j c^r d^s \| i,j,r,s = 0,1,\ldots, n-1 \}$.
The Hopf algebra structure in \cite{C} is not needed below, but shows that the Hopf subalgebra
generated by $a,b$ is isomorphic to $U_n(q)$. We view this as the embedding $G \mapsto b$
and $X \mapsto a$ for the purposes of computing depth (a Morita invariant in terms of Morita invariance of ring extensions \cite{K}). The Hopf algebra $D(U_n(q))$, and its quotient in the small quantum groups, is further discussed in \cite[Chapter 9]{CK}, and for $n=2$ below in Example~\ref{E:51}.
\subsection{Indecomposable modules of $U_n(q)$}
The principal modules, or projective indecomposables, of the half quantum group $U_n(q) = \bra a,b \, \| b^n = 1, a^n = 0, ba = qab \ket$ are determined from a basic set of primitive idempotents given by $e_i = \frac{1}{n}\sum_{j=0}^{n-1} q^{(i-1)j}b^j$, for $i = 1, \ldots, n$.
The projective indecomposables are thus the $n$-dimensional modules $P_1 = e_1U_n(q)$, $P_2 = e_2U_n(q), \ldots, P_n = e_n U_n(q)$.
Each $P_i$ is the projective cover of the one-dimensional simple module $S_i = P_i / P_i \operatorname{rad}(U_n(q))$, which has eigenvalue
$q^{n - i +1}$ from the action of the grouplike $b$ and is annihilated by $a$.
The radical ideal $J=\operatorname{rad}(U_n(q))$ is generated by the nilpotent element $a$: $J = aU_n(q)$ with
$J^n = 0$, but $J^{n-1} \neq 0$.
The Loewy length of each $P_i$ is $n$ and equal to its composition length: the algebra is Nakayama (or serial) \cite{SY}. The composition series of each $P_i$ is given by $$P_i \supset P_i J \supset P_i J^2 \supset \cdots \supset P_i J^{n-1} \supset \{ 0 \} .$$
The indecomposable module isoclasses of $U_n(q)$ are represented by
$P_iJ^{r-1}$, for $i,r = 1,\ldots,n$ \cite{SY}.
\subsection{Preliminaries on the $q$-binomial coefficients}
We recall the definition and some basic properties of the $q$-binomial coefficients, also known as Gauss polynomials. For any integer $j\geq 1$, set $(j)_q=1+q+\cdots +q^{j-1}$, $(j)!_q=(j)_q\cdots (1)_q$ and $(0)!_q=1$. Note that, for $q\neq 1$, $(j)_q=0$ if and only if $q^j=1$. Finally, define the $q$-binomial coefficients ${k\choose j}_q$ inductively as follows, for $k\geq j\geq 0$:
\begin{gather*}
{k\choose 0}_q=1={k\choose k}_q \quad \mbox{for $k\geq 0$,}\\[5pt]
{k\choose j}_q=q^j {k-1\choose j}_q+{k-1\choose j-1}_q \quad \mbox{for $k> j> 0$.}
\end{gather*}
These $q$-binomial coefficients are thus polynomials in $q$ with integer coefficients which agree with the corresponding binomial coefficients when $q=1$. Furthermore, whenever $k> j> 0$ and $(k-1)!_q\neq 0$ we have $\displaystyle {k\choose j}_q=\frac{(k)!_q}{(j)!_q (k-j)!_q}$.
It will be convenient to add the following notational conventions: for all $k\geq 0$ define $\displaystyle {k-1\choose -1}_q=0$, $\displaystyle {k\choose k+1}_q=0$ and $\displaystyle {-1\choose 0}_q=1$. With these conventions, the recurrence relation above can be extended to
\begin{equation}\label{E:recurrence}
{k\choose j}_q=q^j {k-1\choose j}_q+{k-1\choose j-1}_q \quad \mbox{for all $k\geq j\geq 0$.}
\end{equation}
We need a further fact concerning the $q$-binomial coefficients:
\begin{lemma}\label{L:qbinom}
Let $n\geq 2$ and assume $q$ is a primitive $n$-th root of unity. If $\alpha$ and $\beta$ are integers satisfying $\alpha\geq n$, $1\leq \beta\leq n-1$ and $\alpha-\beta<n$, then $\displaystyle {\alpha \choose \beta}_q=0$.
\end{lemma}
\begin{proof}
The proof is a straightforward induction on $\alpha\geq n$. Since $(n-1)!_q\neq 0$ we have $\displaystyle {n\choose \beta}_q=\frac{(n)!_q}{(\beta)!_q (n-\beta)!_q}=0$. So the case $\alpha=n$ holds and we can assume $\alpha\geq n+1$. Thus $\beta\geq 2$ is implied by the condition $\alpha-\beta<n$. The induction hypothesis thus yields $\displaystyle {\alpha-1\choose \beta}_q=0={\alpha-1\choose \beta-1}_q$, and the recurrence relation
$\displaystyle{\alpha\choose \beta}_q=q^\beta {\alpha-1\choose \beta}_q+{\alpha-1\choose \beta-1}_q$ proves the inductive step.
\end{proof}
\section{The $U_n(q)$-module $Q$}\label{S:four}
We view the Taft algebra $U_n(q)$ as the Hopf subalgebra of its quantum double
\begin{align*}
D(U_n(q))=k\langle\, a, b, c, d \mid a^n=0=d^n, b^n=1=c^n, ba=qab, dc=qcd, &\\ db=qbd, bc=cb, ca=qac, da-qad=1-bc \, \rangle &
\end{align*}
generated by $a$ and $b$, so that $U_n(q)=k\langle a, b \mid a^n=0, b^n=1, ba=qab \rangle$. Then for the Hopf subalgebra pair $U_n(q)\subseteq D(U_n(q))$ the corresponding quotient module is
\begin{equation*}
Q=D(R)/R^+ D(R),
\end{equation*}
with $R = U_n(q)$ and $R^+=\ker\,\epsilon=(1-b)R+aR$.
The right $R$-module $Q$ is $n^2$-dimensional and has basis $\{ \ee{i, j} \mid 0\leq i, j\leq n-1\}$, where $\ee{i, j}=c^i d^j+R^+ D(R)\in Q$ and $\ee{i+n, j}=\ee{i, j}$ for all $i\in\ZZ$, as $c^n=1$. The right action of $R$ on $Q$ is determined by:
\begin{align}\label{E:b_action}
\ee{i, j}.b &=q^j\ee{i, j},\\ \label{E:a_action}
\ee{i, j}.a &=(j)_q(\ee{i, j-1}-q^{j-1}\ee{i+1, j-1}),
\end{align}
for all $i\in\ZZ$ and all $0\leq j\leq n-1$, with the additional convention that $\ee{i, -1}=0$.
The following computation will be used frequently so we record it in a lemma. The proof is routine and is omitted.
\begin{lemma}\label{L:a_action}
Fix $i_0\in\ZZ$ and let $X=\sum_{i=i_0}^{i_0 +n-1} \lambda_i \ee{i, j}$ be an element of $Q$, with $\lambda_i\in k$ for all $i_0\leq i\leq i_0 +n-1$. Define $\lambda_{i_0 -1}=\lambda_{i_0 +n-1}$. Then
\begin{equation*}
X.a=(j)_q\sum_{i=i_0}^{i_0 +n-1} (\lambda_i -q^{j-1}\lambda_{i-1})\ee{i, j-1}.
\end{equation*}
\end{lemma}
In order to describe the Krull-Schmidt decomposition of the $R$-module $Q$, we define a new basis for $Q$. Let $\ell=1, \ldots, n$ and define
\begin{align*}
\uu{\ell}&=\sum_{i=1}^n q^{i(\ell-1)}{i+n-2\choose n-1}_q \ee{i, \ell-1},\\[5pt]
\ww{\ell}&=\sum_{i=1}^n q^{-i(\ell+1)}{i+\ell-2\choose \ell-1}_q \ee{i, n-1}.
\end{align*}
It follows from Lemma~\ref{L:qbinom} that $\uu{\ell}=q^{\ell-1}\ee{1, \ell-1}$, but it will be more convenient to use the defining expression for $\uu{\ell}$ given above. Also note that $\uu{n}=\ww{n}$.
\begin{lemma}\label{L:a_action_t_w}
For all $1\leq\ell\leq n$ and all $0\leq r\leq \ell-1$ the action of $a\in U_n(q)$ on the elements $\uu{\ell}$ and $\ww{\ell}$ is given by the following expressions:
\begin{enumerate}
\item[\textup{(a)}] $\displaystyle \uu{\ell}.a^r=\frac{(\ell-1)!_q}{(\ell-r-1)!_q}\sum_{i=1}^n q^{i(\ell-1)}{i+n-2-r\choose n-1-r}_q \ee{i, \ell-1-r}$,\\[5pt]
\item[\textup{(b)}] $\displaystyle \ww{\ell}.a^r=\frac{(n-1)!_q}{(n-r-1)!_q}\sum_{i=1}^n q^{-i(\ell+1)}{i+\ell-2-r\choose \ell-1-r}_q \ee{i, n-1-r}$.
\end{enumerate}
Moreover, the elements $\uu{\ell}.a^{\ell-1}$ with $1\leq \ell\leq n$ are a basis of $\bigoplus_{i=0}^{n-1} k \ee{i, 0}$,
$\uu{\ell}.a^r\neq0$, $\ww{\ell}.a^r\neq 0$ for all $0\leq r\leq \ell-1$ and $\uu{\ell}.a^\ell=0=\ww{\ell}.a^\ell$.
\end{lemma}
\begin{proof}
The proof is by induction on $r$, with the case $r=0$ being clear. So assume the result for $0\leq r\leq\ell-2$ with $\ell\geq 2$. Then, by Lemma~\ref{L:a_action},
\begin{align*}
\uu{\ell}.a^{r+1} &=\frac{(\ell-1)!_q}{(\ell-r-1)!_q}\sum_{i=1}^n q^{i(\ell-1)}{i+n-2-r\choose n-1-r}_q \ee{i, \ell-1-r}.a\\
&=\frac{(\ell-1)!_q}{(\ell-r-2)!_q}\sum_{i=1}^n q^{i(\ell-1)}\left({i+n-2-r\choose n-1-r}_q -q^{-r-1}{i+n-3-r\choose n-1-r}_q\right)\ee{i, \ell-2-r},
\end{align*}
adopting the convention that ${n-2-r\choose n-1-r}_q={2n-2-r\choose n-1-r}_q$, as required by Lemma~\ref{L:a_action}. But then by Lemma~\ref{L:qbinom} we conclude that ${n-2-r\choose n-1-r}_q={2n-2-r\choose n-1-r}_q=0$, which is consistent with our convention that ${k\choose k+1}_q=0$ for $k\geq 0$. So we can apply \eqref{E:recurrence} to obtain
\begin{align*}
{i+n-2-r\choose n-1-r}_q-q^{n-r-1} {i+n-r-3\choose n-1-r}_q={i+n-r-3\choose n-r-2}_q \quad \mbox{for all $1\leq i\leq n$,}
\end{align*}
and thus, using $q^n=1$, we establish the $r+1$ case of (a):
\begin{align*}
\uu{\ell}.a^{r+1} &=\frac{(\ell-1)!_q}{(\ell-r-2)!_q}\sum_{i=1}^n q^{i(\ell-1)} {i+n-r-3\choose n-r-2}_q \ee{i, \ell-2-r}.
\end{align*}
For (b) we have, as above,
\begin{align*}
\ww{\ell}.a^{r+1} &=\frac{(n-1)!_q}{(n-r-1)!_q}\sum_{i=1}^n q^{-i(\ell+1)}{i+\ell-2-r\choose \ell-1-r}_q \ee{i, n-1-r}.a\\
&=\frac{(n-1)!_q}{(n-r-2)!_q}\sum_{i=1}^n q^{-i(\ell+1)}\left({i+\ell-2-r\choose \ell-1-r}_q-q^{\ell-1-r}{i+\ell-3-r\choose \ell-1-r}_q\right)\ee{i, n-2-r},
\end{align*}
with the convention that ${\ell-2-r\choose \ell-1-r}_q={n+\ell-2-r\choose \ell-1-r}_q$, as indicated in Lemma~\ref{L:a_action}. Again by Lemma~\ref{L:qbinom} we conclude that ${\ell-2-r\choose \ell-1-r}_q={n+\ell-2-r\choose \ell-1-r}_q=0$, which is consistent with our convention that ${k\choose k+1}_q=0$ for $k\geq 0$. So we can apply \eqref{E:recurrence} to obtain
\begin{align*}
{i+\ell-2-r\choose \ell-1-r}_q-q^{\ell-1-r} {i+\ell-3-r\choose \ell-1-r}_q={i+\ell-3-r\choose \ell-2-r}_q \quad \mbox{for all $1\leq i\leq n$,}
\end{align*}
and conclude the inductive step:
\begin{align*}
\ww{\ell}.a^{r+1} &=\frac{(n-1)!_q}{(n-r-2)!_q}\sum_{i=1}^n q^{-i(\ell+1)}{i+\ell-3-r\choose \ell-2-r}_q \ee{i, n-2-r}.
\end{align*}
Note that
\begin{align*}
\uu{\ell}.a^{\ell-1} &=(\ell-1)!_q\sum_{i=1}^n q^{i(\ell-1)}{i+n-1-\ell \choose n-\ell}_q \ee{i,0}\\
&=(\ell-1)!_q\sum_{i=1}^\ell q^{i(\ell-1)}{i+n-1-\ell \choose n-\ell}_q \ee{i,0} \quad \mbox{by Lemma~\ref{L:qbinom},}\\
&=(\ell-1)!_q\left( q^{\ell(\ell-1)}{n-1 \choose n-\ell}_q \ee{\ell,0}+\mathsf{x}_\ell\right),
\end{align*}
with $\mathsf{x}_\ell$ in the $k$-span of $\{ \ee{i, 0} \mid 1\leq i\leq \ell-1\}$. In particular, as $(\ell-1)!_q q^{\ell(\ell-1)}{n-1 \choose n-\ell}_q\neq 0$ for all $1\leq \ell\leq n$ it is evident that the $n$ elements $\uu{\ell}.a^{\ell-1}$ with $1\leq \ell\leq n$ are linearly independent, and thus form a basis of $\bigoplus_{i=1}^n k \ee{i, 0}=\bigoplus_{i=0}^{n-1} k \ee{i, 0}$. Moreover, as $\uu{\ell}.a^{\ell-1}$ is in the $k$-span of $\{ \ee{i, 0} \mid 1\leq i\leq \ell\}$ and $\ee{i, 0}.a=0$ for all $i\in\ZZ$, we also have $\uu{\ell}.a^\ell=0$.
Now using (b) we obtain $\ww{\ell}.a^{\ell-1}=\frac{(n-1)!_q}{(n-\ell)!_q}\sum_{i=1}^n q^{-i(\ell+1)} \ee{i, n-\ell}\neq 0$ and another application of Lemma~\ref{L:a_action} yields $\ww{\ell}.a^{\ell}=0$.
\end{proof}
We are ready to introduce a new basis for $Q$ that leads to its decomposition as an $U_n(q)$-module.
\begin{prop}\label{P:basis_Q}
The elements
\begin{equation}\label{E:basis_t}
\uu{\ell}.a^r, \quad \mbox{with $\ell=1, \ldots, n$ and $r=0, \ldots, \ell-1$},
\end{equation}
and the elements
\begin{equation}\label{E:basis_w}
\ww{\ell'}.a^{r'}, \quad \mbox{with $\ell'=1, \ldots, n-1$ and $r'=0, \ldots, \ell'-1$},
\end{equation}
together form a basis of $Q$.
\end{prop}
\begin{proof}
Since $\dim_k Q=n^2$ and there are $n^2$ elements listed in \eqref{E:basis_t} and \eqref{E:basis_w}, it suffices to show that the elements in \eqref{E:basis_t} and \eqref{E:basis_w}, when taken together, are linearly independent.
Notice first that, by \eqref{E:b_action},
\begin{equation*}
(\uu{\ell} .a^r).b=q^{\ell-r-1}(\uu{\ell} .a^r) \quad \mbox{and}\quad (\ww{\ell'} .a^{r'}).b=q^{-r'-1}(\ww{\ell'} .a^{r'}),
\end{equation*}
and for $\ell, \ell', r, r'$ in the ranges specified in \eqref{E:basis_t} and \eqref{E:basis_w}, the elements $\uu{\ell} .a^r$ and $\ww{\ell'} .a^{r'}$ are nonzero, as observed in Lemma~\ref{L:a_action_t_w}, and furthermore $0\leq \ell-r-1\leq n-1$ and $-(n-1)\leq -r'-1\leq -1$. Thus these elements are eigenvectors for the action of $b$ and it suffices to show that elements from \eqref{E:basis_t} and \eqref{E:basis_w} associated to the same eigenvalue of $b$ are linearly independent.
So fix $0\leq j\leq n-1$. The eigenvectors for $b$ among \eqref{E:basis_t} and \eqref{E:basis_w} corresponding to the eigenvalue $q^j$ are the $n$ elements of the form $\uu{\alpha}.a^{\alpha-j-1}$ and $\ww{\beta}.a^{n-j-1}$, for $j+1\leq \alpha\leq n$ and $n-j\leq \beta\leq n-1$ (in case $j=0$ there are no elements of the form $\ww{\beta}.a^{k}$). Suppose, by way of contradiction, that there exists $0\leq j\leq n-1$ and scalars $\lambda_1, \ldots, \lambda_n\in k$, not all $0$, such that
\begin{equation}\label{E:dependence_rel}
\sum_{i=1}^j \lambda_i \ww{n-i}.a^{n-j-1} + \sum_{i=j+1}^n \lambda_i \uu{i}.a^{i-j-1}=0.
\end{equation}
We can choose $j$ minimal with this property. Then $j\geq 1$, because for $j=0$ we get the elements $\uu{\ell}.a^{\ell-1}$ with $1\leq \ell\leq n$, which by Lemma~\ref{L:a_action_t_w} are linearly independent. Applying $a$ to \eqref{E:dependence_rel} yields
\begin{align*}
0&=\sum_{i=1}^j \lambda_i \ww{n-i}.a^{n-j} + \sum_{i=j+1}^n \lambda_i \uu{i}.a^{i-j}=\sum_{i=1}^{j-1} \lambda_i \ww{n-i}.a^{n-(j-1)-1} + \sum_{i=j+1}^n \lambda_i \uu{i}.a^{i-(j-1)-1},
\end{align*}
since $\ww{n-j}.a^{n-j}=0$. The minimality assumption on $j$ implies that $\lambda_i=0$ for all $i\neq j$. Hence, from \eqref{E:dependence_rel} we get $\lambda_j \ww{n-j}.a^{n-j-1}=0$. Since $\ww{n-j}.a^{n-j-1}\neq0$, we deduce that also $\lambda_j=0$, contradicting the assumption that not all of the $\lambda_i$ are $0$. This established our claim.
\end{proof}
In our next result, we obtain indecomposable $U_n(q)$-submodules of $Q$ from the new basis elements introduced in Proposition~\ref{P:basis_Q}.
\begin{prop}\label{P:indec_from_basis}\hfill
\begin{enumerate}
\item[\textup{(a)}] For $\ell=1, \ldots, n$, $\displaystyle\bigoplus_{r=0}^{\ell-1} k\, \uu{\ell}.a^r$ is an $\ell$-dimensional right $U_n(q)$-submodule of $Q$ isomorphic to the indecomposable module $P_2 J^{n-\ell}$.
\item[\textup{(b)}] For $\ell'=1, \ldots, n-1$, $\displaystyle\bigoplus_{r'=0}^{\ell'-1} k\, \ww{\ell'}.a^{r'}$ is an $\ell'$-dimensional right $U_n(q)$-submodule of $Q$ isomorphic to the indecomposable module $P_{2+\ell'} J^{n-\ell'}$, where by convention $P_{n+1}=P_1$.
\end{enumerate}
\end{prop}
\begin{proof}
For the proof of (a), write $\displaystyle V_\ell=\bigoplus_{r=0}^{\ell-1} k\, \uu{\ell}.a^r$. By Proposition~\ref{P:basis_Q}, it is clear that this sum is indeed direct and thus $\dim_k V_\ell=\ell$. Moreover, $V_\ell$ is a right $U_n(q)$-submodule of $Q$ because $\uu{\ell}.a^r$ is an eigenvector for $b$ and $\left(\uu{\ell}.a^{\ell-1}\right).a=\uu{\ell}.a^{\ell}=0$.
Under the action of $b$ on $V_\ell$, the eigenvectors $\uu{\ell}.a^r$ have corresponding eigenvalue $q^{\ell-r-1}$, for $r=0, \ldots,\ell-1$. Thus the assignment $\uu{\ell}.a^r\mapsto e_2.a^{n-\ell+r}$, for $r=0, \ldots,\ell-1$, gives the required isomorphism $V_\ell\longrightarrow P_2 J^{n-\ell}$.
Part (b) is entirely analogous and we just describe the isomorphism $W_{\ell'}\longrightarrow P_{2+\ell'} J^{n-\ell'}$, where $\displaystyle W_{\ell'}=\bigoplus_{r'=0}^{\ell'-1} k\, \ww{\ell'}.a^{r'}$, which is $\ww{\ell'}.a^{r'}\mapsto e_{2+\ell'}.a^{n-\ell'+r'}$, for $r'=0, \ldots,\ell'-1$.
\end{proof}
The Krull-Schmidt decomposition of $Q$ now falls out easily.
\begin{theorem}\label{T:KSdecQ}
We have the following decomposition of $Q$ into indecomposables, as a right $U_n(q)$-module:
\begin{align*}
Q&\cong\bigoplus_{\ell=1}^{n} P_2 J^{n-\ell} \oplus \bigoplus_{\ell'=1}^{n-1} P_{2+\ell'} J^{n-\ell'}\\
&\cong P_2 \oplus P_2 J \oplus \cdots \oplus P_2 J^{n-1}\oplus P_3 J^{n-1}\oplus P_4 J^{n-2}\oplus\cdots\oplus P_n J^2\oplus P_1 J.
\end{align*}
\end{theorem}
\begin{proof}
The decomposition follows immediately from Proposition~\ref{P:basis_Q} and Proposition~\ref{P:indec_from_basis}.
\end{proof}
For the remainder of this section and the next, we will adopt the notation $M(\ell, i)$ used in~\cite{COZ} for the indecomposable $U_n(q)$-modules. This will make it more convenient for the reader to check any additional details in~\cite{COZ} concerning the Green ring $r(U_n(q))$ of the Taft algebra $U_n(q)$ and the tensor product of indecomposable $U_n(q)$-modules. The $P_iJ^r$ notation may be recovered through the isomorphism
\begin{equation*}
M(\ell, i)\cong P_{\ell-i+1}J^{n-\ell}, \quad \mbox{with indices mod $n$.}
\end{equation*}
Note that the authors in \cite{COZ} consider left modules in contrast to our right modules; one goes from one to the other by changing from $R$ to $R^{\textup{op}}$, and from $q$ to $q^{-1}$, so that the results are unchanged.
\begin{cor}
In the notation of \cite{COZ}, the isoclass of $Q$ satisfies the equation
\begin{equation}
[Q] = 1 + a + (a^{n-1} + a)x + (a^{n-2}+a)u_3 + \cdots + (a^2 +a)u_{n-1} + au_n
\end{equation}
in the Green ring $r(U_n(q))$.
\end{cor}
\begin{proof}
Recall from \cite{COZ} that $r(U_n(q))$ is generated by the simple $a = [M(1,n-1)] = [P_3J^{n-1}]$, and
the $2$-dimensional indecomposable, $x = [M(2,0)] = [P_3J^{n-2}]$. The Fibonacci-like polynomials $u_{\ell}(a,x)$ satisfy
$u_{\ell + 1} = xu_{\ell} - au_{\ell - 1}$, where $u_1 = 1$ and $u_2 = x$, and satisfy $u_{\ell} = [M(\ell,0)]$. Thus, $[M(\ell,r)] = a^{n-r}u_{\ell}$. The rest follows from the Theorem, reinterpreted as Eq.~(\ref{eq: coznotation}) below.
\end{proof}
\section{Decomposition of $Q\otimes Q$}
The aim of this section is to show that all indecomposable $U_n(q)$-modules occur in the Krull-Schmidt decomposition of $Q\otimes Q$. We use the notation in \cite{COZ} as mentioned above.
For the reader's convenience we record below the results from \cite[Section 3]{COZ} which are most relevant to us. Recall that $M(\ell, r)=M(\ell, r+n)$ for all $1\leq \ell\leq n$ and $r\in\mathbb{Z}$.
\begin{theorem}[{\cite[Section 3]{COZ}}]\label{T:COZ14}
\label{thm-COZ} Let $1\leq \ell, \ell'\leq n$, $r, r'\in\mathbb{Z}$, $\ell_0 = \min\{\ell, \ell'\}$ and $\ell_1 = \max\{\ell, \ell'\}$. The following hold:
\begin{itemize}
\item[\textup{(a)}] $\displaystyle M(\ell, r)\otimes M(\ell', r')\cong M(\ell', r')\otimes M(\ell, r)$.
\item[\textup{(b)}] $\displaystyle M(\ell, r)\otimes M(1, r')\cong M(\ell, r+r')$.
\item[\textup{(c)}] If $\ell \geq 2$, then $\displaystyle M(\ell, r)\otimes M(n, r')\cong \bigoplus_{i=1}^\ell M(n,r+r'+i-\ell)$.
\item[\textup{(d)}] If $\ell +\ell' \leq n$, then $\displaystyle M(\ell, r)\otimes M(\ell', r')\cong \bigoplus_{i=1}^{\ell_0} M(\ell_1-\ell_0-1+2i,r+r'+i-\ell_0)$.
\item[\textup{(e)}] If $\ell, \ell'< n$ and $\ell +\ell' \geq n$, then
\begin{align*}
M(\ell, r)\otimes M(\ell', r')\cong \left(\bigoplus_{i=1}^{n-\ell_1} M(\ell_1-\ell_0-1+2i,r+r'+i-\ell_0)\right) \ &\\
\qquad \oplus \left(\bigoplus_{j=1}^{\ell+\ell'-n} M(n,r+r'+1-j)\right).&
\end{align*}
\end{itemize}
\end{theorem}
Given the decomposition
\begin{equation}
\label{eq: coznotation}
Q\cong\bigoplus_{\ell=1}^{n} M(\ell, \ell-1) \oplus \bigoplus_{\ell'=1}^{n-1} M(\ell', n-1)
\end{equation}
from Theorem~\ref{T:KSdecQ} and the distributivity of $\oplus$ relative to $\otimes$, it is enough to show that for any integers $1\leq x\leq n$ and $0\leq y\leq n-1$, the module $M(x, y)$ is isomorphic to a direct summand of $M(i, j)\otimes M(i', j')$ for some $(i, j), (i', j')\in \mathcal{J}$, where
\begin{equation*}
\mathcal{J}=\{ (\ell, \ell-1) \mid 1\leq \ell\leq n\}\cup \{ (\ell, n-1) \mid 1\leq \ell\leq n-1\}.
\end{equation*}
We will consider four cases.
\medskip
\textbf{Case 1:} Assume that $y=n-1$ or $x=y+1$. Then $(x, y)\in\mathcal{J}$, $(1, 0)\in\mathcal{J}$ and $M(x, y)\otimes M(1, 0)\cong M(x, y)$, by Theorem \ref{T:COZ14}(b).
\medskip
\textbf{Case 2:} Assume that $x=n$. Then as $(n, n-1)\in\mathcal{J}$ and $M(n, n-1)\otimes M(n, n-1)\cong\bigoplus_{i=1}^n M(n, i-2)$, by Theorem \ref{T:COZ14}(c), we deduce that $M(n, y)$ is a direct summand of $M(n, n-1)\otimes M(n, n-1)$ for any $y$, since as $i$ varies from $1$ to $n$, $i-2$ runs through all residue classes modulo $n$.
\medskip
\textbf{Case 3:} Assume that $x\leq y\leq n-2$, with $n\geq 3$. Define $k=n+x-y-2$ and $\ell=n-y-1$. Then $1\leq x\leq k\leq n-2$ and $1\leq \ell\leq k$, so that $(k, n-1), (\ell, n-1)\in \mathcal{J}$. We will show, using Theorem \ref{T:COZ14}, that $M(x, y)$ is a direct summand of $M(k, n-1)\otimes M(\ell, n-1)$.
Indeed, we have $\ell_0 = \min\{k, \ell\}=\ell$ and $\ell_1 = \max\{k, \ell\}=k$; in particular, $\ell_0\geq 1$ and $n-\ell_1\geq 1$. If $k+\ell\leq n$ (respectively, $k+\ell> n$), Theorem \ref{T:COZ14}(d) (respectively, Theorem \ref{T:COZ14}(e)) with $i=1$ shows that $M(k-\ell+1, 2n-1-\ell)$ is a direct summand of $M(k, n-1)\otimes M(\ell, n-1)$. The claim follows since
$k-\ell+1=x$ and $2n-1-\ell=y+n$, which is congruent to $y$ modulo $n$.
\medskip
\textbf{Case 4:} Assume that $y+2\leq x\leq n-1$, with $n\geq 3$. Define $k=n-y-2$ and $\ell=n+y-x+1$. Then $1\leq n-x\leq k\leq n-2$ and $2\leq n-k\leq \ell\leq n-1$, so that $(k, n-1), (\ell, \ell-1)\in \mathcal{J}$. We will show that $M(x, y)$ is a direct summand of $M(k, n-1)\otimes M(\ell, \ell-1)$.
Since $k+\ell\geq n$, Theorem \ref{T:COZ14}(e) applies and by taking $i=n-\ell_1\geq 1$ in the formula, we conclude that
$M(-(\ell_0+\ell_1)+2n-1,2n-2+\ell-(\ell_0+\ell_1))$ is a direct summand of $M(k, n-1)\otimes M(\ell, \ell-1)$, where $\ell_0 = \min\{k, \ell\}$ and $\ell_1 = \max\{k, \ell\}$. Using $\ell_0+\ell_1=k+\ell$ we see that $-(\ell_0+\ell_1)+2n-1=-(2n-1-x)+2n-1=x$ and $2n-2+\ell-(\ell_0+\ell_1)=2n-2-k=y+n$, which is congruent to $y$ modulo $n$.
We have thus proved the following.
\begin{theorem}\label{T:decQQ}
All indecomposable $U_n(q)$-modules occur in the Krull-Schmidt decomposition of $Q\otimes Q$ as a right $U_n(q)$-module.
\end{theorem}
We remark that the endomorphism ring $\End Q^{\otimes 2}$ of the $U_n(q)$-module $Q \otimes Q$ is therefore
Morita equivalent to the Auslander algebra of $U_n(q)$ \cite{SY}.
\begin{cor}
For the Taft algebra $U_n(q)$ in its Drinfeld double, the depth of $Q$ is $d(Q_{U_n(q)}) = 2$ and the minimum even depth $$d_{ev}(U_n(q), D(U_n(q)) = 6.$$
\end{cor}
\medskip
\subsection{Example: the $n = 2$ Sweedler Hopf algebra case}\label{E:51}
We recapitulate the computations in the toy model case when $n = 2$, $k=\mathbb{C}$ and $q = -1$, the Sweedler Hopf algebra, with some additional remarks. The Hopf algebra structure of $R=H_4$ is then given by $H_4 = \mathbb{C} \bra a,b,\| a^2 = 0, b^2 = 1, ab = -ba \ket$
with coproduct \begin{equation}
\label{eq: cop}
\cop(b) = b \otimes b, \ \ \ \ \cop(a) = a \otimes b + 1 \otimes a,
\end{equation}
and counit $\eps(b) = 1$, $\eps(a) = 0$, and antipode $S(b) = b$, $S(a) = -ab$. The radical ideal
is $J = aH_4$.
In all, there are four indecomposables $P_1 = e_1H_4$, $P_2 = e_2 H_4$, where $e_1 = (1 + b)/2$, $e_2 = (1-b)/2$,
and $S_1 = P_1/P_1 a$, $S_2 = P_2 / P_2 a$, up to isomorphism.
The quantum double algebra is given in terms of generators and relations by
\begin{eqnarray*}
\label{eq: double}
D(H_4) & = & \mathbb{C} \bra a,b,c,d \| a^2 = 0 = d^2, b^2 = 1 = c^2, bc = cb, ab = -ba, \\
& & \ \ \ \ \ \ \ \ \ \ \ \ cd = -dc, ac = -ca, bd = -db, da + ad = 1 - bc \ket
\end{eqnarray*}
which is $16$-dimensional with basis $\{ a^ib^jc^rd^s \| i,j,r,s = 0,1 \}$. Note the obvious symmetry given by the involutory algebra automorphism exchanging $a$ with $d$ and $b$ with $c$.
The $\mathbb{C}$-algebra $A = D(H_4)$ has a basic set of two \textit{central} orthogonal idempotents $e_1 = \frac{1 - bc}{2}$ and
$e_2 = \frac{1 + bc}{2}$. Note that $ad + da = 2e_1$, $e_2 ad = -e_2da$ and that
$e_{11} := e_1ad/2$ is a noncentral idempotent. It follows that $A = e_1A \oplus e_2 A$,
where our computations give $e_1 A \cong M_2(\mathbb{C}) \otimes \mathbb{C} \Z_2$, where the other matrix units are given by
$e_{22} := e_1da/2$, $e_{12} := e_1 a/ \sqrt{2}$ and $e_{21} := e_1d/\sqrt{2}$. The block
$e_1A$ is in fact a bi-ideal in the Hopf algebra structure on $A$. The coproduct formulas and the representation theory of $D(H_4)$ may be found in \cite{HXC}, where it may also be noted that the symmetry is a Hopf algebra isomorphism of copies of $H_4$ within $D(H_4)$.
The block $e_2A$ is isomorphic as Hopf algebras to $H_8 := \overline{U_i}(sl_2(\mathbb{C}))$, the small quantum group at the fourth root of unity $i$ generated by a grouplike $K$, and two commuting nilpotent elements $E,F$
that anti-commute with $K$ (all of order $2$, see for example \cite[Example 4.9]{K2014}). The isomorphism is given by
$e_2 \mapsto 1$, $e_2 b \mapsto K$, $e_2 a \mapsto E$ and $e_2 bd \mapsto F$, the rest being an exercise; alternatively, it is part of a more general fact for the Drinfeld double of any half quantum group and any small quantum group sharing the same (odd or half-even) root-of-unity \cite[Chapter 9]{CK}.
The quotient module $Q = D(H_4)/ H_4^+ D(H_4)$ is computed by noting that $H_4^+ = aH_4 + (1 - b)H_4$, thus has basis $\{ \overline{1}, \overline{c}, \overline{d}, \overline{cd} \}$. The right $H_4$-module structure is given by
\begin{eqnarray*}
\overline{1}. a & = & \overline{0} = \overline{c}. a \\
\overline{1} . b & = & \overline{1}, \ \ \overline{c}.b = \overline{c} \\
\overline{d}.a & = & \overline{1}-\overline{c}, \ \ \overline{d}.b = -\overline{d} \\
\overline{cd}.a & = & \overline{c}-\overline{1}, \ \ \overline{cd}.b = -\overline{cd}
\end{eqnarray*}
which are short computations with the relations of $D(H_4)$, or follow from Eqs.~(\ref{E:b_action}) and~(\ref{E:a_action}).
Note the following submodules of $Q_{H_4}$. The $2$-dimensional submodule with basis $\{ \overline{d}, \overline{1}-\overline{c} \}$ isomorphic to $P_2$. The $1$-dimensional submodule spanned by $\overline{c}$, isomorphic to $S_1$. The $1$-dimensional submodule spanned by $\overline{d} + \overline{cd}$, isomorphic to $S_2$. Then it is easy to see that $Q_{H_4} \cong S_1 \oplus S_2 \oplus P_2$. Note that $P_1$ is not
a constituent of $Q$.
Since $S_1$ is the unit module, the tensor-square $Q \otimes Q$ has summand $P_2 \otimes P_2$, which may be computed directly or using Theorem~\ref{thm-COZ}, as
\begin{equation}
P_2 \otimes P_2 \cong P_1 \oplus P_2
\end{equation}
We see that $Q \otimes Q$, unlike $Q$, has all four indecomposables of $H_4$ as nonzero constituents, so its depth as an $H_4$-module is $2$.
\section*{Acknowledgements}
Samuel A.\ Lopes was partially supported by CMUP (UID/MAT/00144/2013), which is funded by FCT (Portugal) with national (MEC) and European structural funds (FEDER), under the partnership agreement PT2020.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 582 |
Q: Calculating a certain pushforward of a vector field So I'm new to differential geometry and this problem is giving me trouble, and more generally I'd just like to understand pushforwards of vector fields better. Let $\phi: \mathbb{S}^2\rightarrow\mathbb{R^2}$, $\phi(x,y,z) = (\frac{x}{1-z}, \frac{y}{1-z})$ and $\phi: \mathbb{S}^2\rightarrow\mathbb{R^2}$, $\psi(x,y,z) = (\frac{x}{1+z}, \frac{y}{1+z})$, and let $X$ be a vector field on $\mathbb{S^2}$ satisfying $\phi_*(X)(u,v) = (1,0)$ for all $(u,v) \in \mathbb{R}^2$. We want to find
$\psi_*(X)(u,v)$.
I really don't know how to start here. I understand the definition of the pushforward of a vector field, but it just seems so abstract and hard to do actual calculations with. Can anyone offer some insight?
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,013 |
Need A New Flavor? Try Some Red Wine With A Dash Of Eddie Vedder!
SAN FRANCISCO, CA - NOVEMBER 06: Musician Eddie Vedder performs during Citi Sound Vault Presents Eddie Vedder at The Chapel on November 6, 2017 in San Francisco, California. (Photo by Kevin Winter/Getty Images for Citi)
Pearl Jam has its fair share of covers, but how many of them include Eddie Vedder covered in wine?
Accompanied by Red Hot Chili Peppers' Chad Smith and Jeff Klinghoffer and many enthusiastic fans with tambourines, the band covered Neil Young's "Rockin' In The Free World" in Rio De Janeiro, Brazil.
What makes the performance lean toward the Flash Dance persona is that Vedder decides to pour a large bottle of Red Wine over his head, that was conveniently placed to his right on the stage.
No word if there will be a Vino a la Vedder on the shelves this year. I mean, Jon Bon Jovi did it, and Motorhead has their own wine… It's not like that would be COMPLETELY out of the question.
Pearl Jam - Rockin' In The Free World (With Chad Smith & Josh Klinghoffer) | Rio de Janeiro 2018
Uploaded by Jorge Cubas on 2018-03-22.
Red Hot Chili Peppers and Pearl Jam headlined Lollapalooza Brasil this weekend, in which Vedder beckoned Lollapalooza's founder Perry Farrell out on the stage, sang him Happy Birthday, and proceeded to slide into covering Jane's Addiction's "Mountain Song."
Watch the video here, which has a lot better sound quality than the wine shot from the nights prior:
Pearl Jam - Mountain Song with Perry Farrell (live @ Lollapalooza Brazil 2018)
Pearl Jam - Mountain Song with Perry Farrell (Jane's Addiction cover live @ Lollapalooza Brazil 2018)
Amy Cooper is a journalist with interests in Tech, Rock, Nerd Fandoms, and pop culture hilarity.
Chad Smith,Eddie Vedder,Flash Dance,Jane's Addiction,Jeff Klinghoffer,Jon Bon Jovi,Lollapalooza Brasil,Motorhead | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 8,216 |
Nilton Almeida, Senior Manager, Business & Application Development at Pulp & Paper, South America has 25 years' experience in the pulp and paper industry, including 18 with Kemira. He believes that as the number-one chemistry partner to the pulp and paper industry worldwide, Kemira's unique application expertise and Total Chemistry Management (TCM) partnership program is the key to helping customers in his region maximize and sustain their profitability.
South America has seen big investments in pulp plants over the last 15 years, and many are highly efficient state-of-the-art operations. However, with much of the pulp being sold overseas, local paper and board mills have often missed out on a similar level of investment. Our expertise in applying chemistry to the pulp and paper industry – which accounts for around 55% of our business – combined with our technologies for smart process management and strong local presence make us well placed to help any South American customer, whatever their level of technological development.
Our TCM concept is a partnership program tailored to take care of all a customer's chemistry requirements.
There's always room to improve the performance and profitability of both pulp mills and paper machines – this is the most interesting part of my job. Because no two customers or processes are quite the same, nothing is ever routine. The challenge is always to identify what approach will work best for each customer's unique challenges. Kemira has extensive expertise in the full range of papermaking chemistry – from pulping to coating, finishing and printing. We also have a tremendous knowledge base and portfolio for water treatment and reclaim. We combine our expertise with innovation to help our customers continuously optimize their processes, while reducing costs.
Our TCM concept is a partnership program tailored to take care of all a customer's chemistry requirements. At the moment it is common for a pulp or paper mill to have several chemical suppliers, one for each area of the process. This makes it difficult to optimize the process as a whole. With TCM we provide all the necessary facets to optimally manage the process including chemicals, industry and application expertise as well as state-of-the art, real-time smart process management technologies.
Customers in South America are already starting to see the benefits of this holistic approach. For example, Kemira supplies UPM's Fray Bentos pulp mill with several pulp chemicals from an onsite chemical production facility as part of a partnership that reduces the customer's total chemical spend and improves performance. In addition, Kemira has had a partnership with the Klabin Group for more than 20 years, supplying the necessary technologies for their pulp, paper and board machines.
These days the scope often includes technologies for smart process management and devices linked with online tools like Kemira KemConnect™ that provide full 24/7 visibility over all chemistry applications. Because KemConnect continuously gathers data, customers get the real-time insight needed for better decision-making, faster troubleshooting, smarter chemical management, and improved cost efficiency. KemConnect can help identify opportunities for future improvements as well.
Innovation to support the development of new kinds of stronger and lighter board utilizing low cost fibers. Developing Pulping Technologies to improve fiber quality and to reduce energy cost. | {
"redpajama_set_name": "RedPajamaC4"
} | 2,922 |
Choosing the right tempo and samples for house
Join Yeuda Ben-Atar for an in-depth discussion in this video Choosing the right tempo and samples for house, part of Ableton Live 9: Programming Beats.
Male 1: House music has been around for decades, and an old form of it can even be heard before the birth of the genre in various jazz, disco, African, and even rock tunes. First, let's listen to a couple of examples of house beats. In the electronic dance music world, also known as EDM, it's common to feature only the drums at the beginning and end of tracks. Producers do this to give DJs enough time to mix in and out their songs. There are many different types of house beats, progressive, electro, deep, tech, and even more. We're going to talk about the fundamentals, and make a basic house beat. That with some variation can fit all those styles of music.
The BPM range of most house music is between 110 to 140 BPM. Let's break down the BPM ranges for some house music sub genres. Disco, Soul, and Funk House, is usually between 110 to 118 BPM. Deep House is between 120 to 125 BPM. Electro and Progressive house is between 125 and 132 BPM. Techno is 125 to 140 BPM, Trance is 128 to 140 BPM, and Hardstyle is 145 to 150 BPM. It's important to note, house music is a very broad genre, that has many more subgenres than the ones I've listed here. For a house beat we're going to use a kick, a snare or clap, open hat, close hat, and various percussion sounds. Now the kick needs to be very punchy and heavy in bass and bottom end. The kick is the most important element in a house beat. Since it's the main element that makes people dance. And it's also the main characteristic drum part of the genre. Sometimes producers refer to the kick part as four on the floor. Let's listen to a few examples that might not work with the house beat.
And a few that might work. Nice. Now, after the kick we might add a clap or a snare. It's up to you to decide which one you want to use. This can be arranged to almost any type of clap or snare. Also feel free to layer different claps and snares together on top of each other. So a few examples And a layered one which is a clap and a snare playing together. Now next, we're gonna add the open hat, another essential part of house music, although it's not a must for any type of house. A few examples. Next we're also gonna add a closed hat.
You can choose almost any type of closed hat for a house beat. Try to use more than one kind for more variation and the ability to make interesting rhythms. Next, we're gonna add some various percussion sounds. This will be one of the key elements that will dictate the kind of house genre you are making. For example, for deep house, try to choose more organic sounds like bongos, African drums, and ride cymbals. For techno. Try to choose more industrial sounds like mechanical effects or robotic sounds. For our example we're going to choose two shakers, one ride, one low percussive sound, and two high percussive sounds, and finally a clash or white noise. Not a must for the fundamental house beat, but will help us indicate the listener, the start of the musical or rhythmical phrase. Now, let's listen to the beat I'll show you how to make in this chapter. Let's take a listen.
Yeuda Ben-Atar
The drum track is part of the hidden chemistry of a great song. This is your beat making lab. Yeuda Ben-Atar, educator and producer who performs as DJ Side Brain, shows you how to make beats in a variety of genres, from dubstep to hip-hop. Yeuda works with Ableton Live—but you can use these tutorials to make beats in whatever DAW you have accessible.
First get some basic rhythmic theory, including counting music and note subdivisions, and learn how elements like cymbals, percussive instruments like congas, and even homemade sounds from cans, bottles, and counters contribute to your beats. The following chapters tackle the particulars of house, dubstep, drum and bass, trap, juke, and hip-hop. In each of these chapters, Yeuda discusses how to choose the appropriate tempo and drum sounds for the style and how to sequence the kick, snare, and cymbals. The course closes with some pro mixing techniques that balance punch and presence, so your drums will cut through the mix and sound their best.
How to count music
Using a piano roll editor
Choosing the right tempo and samples for various genres
Sequencing your drum elements
How to program house, dubstep, drum and bass, trap, juke, and hip-hop beats
Adding extra percussion sounds
Adding breaks
How to mix your beat for presence and punch
Adjusting levels and panning
Adding reverb to your beat
Ableton Live 9 Tips and Tricks
with Michael Kiraly
Ableton Live 9 for Live Performance
with Yeuda Ben-Atar
Ableton Live 9 Essential Training
with Rick Schmunk
9h 4m Beginner
Ableton Push: Making Music
What you should know before watching this course
1. Basic Rhythmic Theory
Note divisions
Triplet notes and swing
Using a piano roll
Drum elements
2. House Beat
Sequencing the kick, snare, and hats for a house beat
Adding extra percussion sounds for house
3. Dubstep Beat
Choosing the right tempo and samples for dubstep
Sequencing the kick, snare, and rides
4. Drum-and-Bass Beat
Choosing the right tempo and samples for drum and bass
Sequencing the kick, snare, and hats
5. Trap Beat
Choosing the right tempo and samples for trap
Sequencing the kick, snare, and cymbals for a trap beat
Adding hats to a trap beat
6. Juke/Footwork Beat
Choosing the right tempo and samples for juke
Sequencing the kick, snare, and hats for juke beats
Adding sounds and extra percussion sounds to juke
7. Hip-Hop Beat
Choosing the right tempo and samples for hip-hop
Sequencing the kick, snare, and hats in hip-hop
Adding extra percussion sounds to a hip-hop beat
8. Mixing Beats
Using an EQ on the drums
Adding reverb
Parallel compression
Audio + Music DAWs Live Performance Music Production Virtual Instruments Ableton Live
Video: Choosing the right tempo and samples for house | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,638 |
import asyncio
import pickle
import time
from concurrent.futures import ProcessPoolExecutor
from thrift.perf.load.services import LoadTestInterface
from thrift.perf.load.types import LoadError
def us_to_sec(microseconds):
return microseconds / 1000000
def burn_in_executor(us):
start = time.time()
end = start + us_to_sec(us)
while time.time() < end:
pass
class LoadTestHandler(LoadTestInterface):
def __init__(self, loop=None):
super().__init__()
self.loop = loop or asyncio.get_event_loop()
self.pool = ProcessPoolExecutor()
pickle.DEFAULT_PROTOCOL = pickle.HIGHEST_PROTOCOL
async def noop(self):
pass
async def onewayNoop(self):
pass
async def asyncNoop(self):
pass
async def sleep(self, us):
await asyncio.sleep(us_to_sec(us))
async def onewaySleep(self, us):
await asyncio.sleep(us_to_sec(us))
async def burn(self, us):
return await self.loop.run_in_executor(self.pool, burn_in_executor, us)
async def onewayBurn(self, us):
return await self.loop.run_in_executor(self.pool, burn_in_executor, us)
async def badSleep(self, us):
# "bad" because it sleeps on the main thread
time.sleep(us_to_sec(us))
async def badBurn(self, us):
return burn_in_executor(us)
async def throwError(self, code):
raise LoadError(code=code)
async def throwUnexpected(self, code):
raise LoadError(code=code)
async def send(self, data):
pass
async def onewaySend(self, data):
pass
async def recv(self, bytes):
return "a" * bytes
async def sendrecv(self, data, recvBytes):
return "a" * recvBytes
async def echo(self, data):
return data
async def add(self, a, b):
return a + b
async def largeContainer(self, data):
pass
async def iterAllFields(self, data):
for item in data:
_ = item.stringField
for _ in item.stringList:
pass
return data
| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,973 |
<?php
namespace Devhelp\DatatablesBundle\Service;
/**
* Datatables interface
*
* @author <michal@devhelp.pl>
*/
interface DatatablesInterface
{
/**
* @param $grid
* @return mixed
*/
public function load($grid);
/**
* @return mixed
*/
public function getResult();
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,626 |
{"url":"http:\/\/math.stackexchange.com\/questions\/296977\/how-to-find-probability-relation","text":"# How to find probability relation\n\nI'm not really sure what the term would be for such a problem (is it probability distribution?) but here it is.\n\nI have a list divided into 4 sections and I am looking for a particular item.\n\nThe probability that the item is in the first fourth of the list is 3 times that of the second fourth of the list.\n\nThe probability that the item is in the last fourth of the list is twice that of the second fourth of the list.\n\nAnd the probability that the item is in the third fourth of the list is twice the probability it is in the last fourth of the list.\n\nGiving a relation like: [3x | x y | 2z | 2y z]\n\nHow can I find the probability that the item is in a particular part of the list? ie What is the probability it is in the first fourth of the list?\n\n-\n\nSince I am not fond of fractions, it is numerically convenient to let $x$ be the probability the item is in the second quartile.\n\nThe probability it is in the first quartile is $3x$.\n\nThe probability it is in the last is $2x$.\n\nThe probability it is in the third is $2(2x)$, that is, $4x$.\n\nSo $x+3x+2x+4x=1$. Now we know everything.\n\nRemark: If we like lots of variables, and subscripts, we can let $p_1$ be the probability the item is in the first quartile, $p_2$ the probability it is in the second, and so on.\n\nThe given information can be translated into equations, such as $p_1=3p_2$, and so on. We also have $p_1+p_2+p_3+p_4=1$. Solve.\n\n-\nI'm not quite following you, if the probability for the 3rd quartile is 4x and for the first is 3x, then isn't that saying there is a larger chance it is in the 3rd quartile? That may be correct, but the way the problem is worded it sounds like the largest chance is the first quartile. \u2013\u00a0schwiz Feb 7 '13 at 6:41\nIt says that probability of last fourth is twice the second, and that third fourth is twice last fourth, so third fourth is $2(2x)$. \u2013\u00a0Andr\u00e9 Nicolas Feb 7 '13 at 6:44\nOk this definitely helps, I think the wording of the question is bothering me but I get the concept. I'll accept your answer once I get it worked out in case I have more questions or others want to chime in. The X-route just confuses me and while working out with the Ps I didn't get the sum to equal 1 when I was finished so I think its the wording for sure that is throwing me off. \u2013\u00a0schwiz Feb 7 '13 at 7:11\nThe simple way is the one I described first. For the quartiles, we get in order $3\/10$, $1\/10$, $4.10$, and $2\/10$. If you want to use $p_i$, or $w,x,y,z$ we get $p_1=3p_2$ (I had a typo), $p_4=2p_2$, $p_3=2p_4$, $p_1+\\cdots+p_4=1$. \u2013\u00a0Andr\u00e9 Nicolas Feb 7 '13 at 7:15\nOk, yeah so that is where I was getting confused, when using x's q1 is x and q2 is 3x, but with p p1 = 3p2, it switches around and that threw me off. But I think I understand it better using the p even though the x notation is more desirable. \u2013\u00a0schwiz Feb 7 '13 at 7:22","date":"2015-11-27 14:05:56","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9169417023658752, \"perplexity\": 259.4665747233757}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2015-48\/segments\/1448398449160.83\/warc\/CC-MAIN-20151124205409-00162-ip-10-71-132-137.ec2.internal.warc.gz\"}"} | null | null |
<?php
namespace Vivo\Controller\CLI;
use Zend\Mvc\Controller\AbstractActionController;
/**
* Abstract controller for CLI controllers
*/
abstract class AbstractCliController extends AbstractActionController
{
const COMMAND = '';
public function notFoundAction()
{
$event = $this->getEvent();
$routeMatch = $event->getRouteMatch();
return sprintf("Unknown subcommand '%s'\nUsage:\n %s",
$routeMatch->getParam('action'),
$this->getConsoleUsage());
}
abstract public function getConsoleUsage();
public function helpAction()
{
return $this->getConsoleUsage();
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,537 |
ZERO closing costs on this home if you use a preferred lender and $5000 in upgrades.
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Directions: TAKE VAUGHN ROAD TOWARDS PIKE ROAD, CROSS OVER PIKE ROAD AND WOODLAND CREEK IS .5 MILE ON THE RIGHT. TURN IN THEN TAKE A RIGHT ONTO CRESCENT LODGE DRIVE. FOLLOW TO THE BACK AND TAKE A RIGHT ONTO CRESCENT LODGE CIRCLE. | {
"redpajama_set_name": "RedPajamaC4"
} | 764 |
Excerpt from difficulties in electric EngineeringThis selection of difficulties has been ready for using scholars on the Massachusetts Institute of know-how, yet because the booklet can be utilized in different technical colleges it kind of feels top to nation what floor the issues are meant to hide. on the Institute the booklet should be utilized by the 3rd 12 months scholars in electric Engineering, and by means of the 3rd and fourth yr scholars within the classes of Civil, Mechanical, Mining and Chem ical Engineering.
This textbook is meant to enrich present litterature through offering extra element on the various very fundamentals facets of ways turbomachinery operates.
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Ian Smith's ideas for profitable administration Buy-Outs identifies a number of the levels of administration buy-outs, highlighting advertisement calls to be made, mentioning benchmarks the place acceptable and supplying useful assistance on getting ready for a administration buy-out, sourcing funders and pricing MBOs, negotiating felony agreements and extra issues for public region offers.
In practical terms, this allowed Russell to devise a wave-line system for the design and construction of ships which would revolutionise naval architecture. ) earned him the Gold Medal of the Royal Society of Edinburgh. More importantly, vessels were built to his specifications: these were initially small iron steamers such as the Wave, but the years 1839-1840 saw the launch of larger ocean-going ships, the Flambeau and Fire-King. Eventually, Russell's ideas were used in the design of a fleet of transatlantic Royal Mail ships, plying the West Indies route.
14 wire gauge thickness. The span is 50 feet, the roofs being supported upon lattice girders of an average length of 45 feet. The position of the columns is shown on the ground plan, Fig. 11, Plate 123; and it will be observed that the entire mill floor is free from obstruction. The flooring will be of cast iron plates 1 inch thick. 41 DOWLAIS IRONWORKS ENGINES. 115 It has long been felt that the power of rolling wrought iron of large section and great lengths has not kept pace with the requirements of engineers, who are hampered in their designs by the impossibility of obtaining iron of sufficient dimensions.
The plot of land being bounded on the south by the Railway, on the east by the Peak Forest and Macclesfield Canal (also belonging to this 29 LOCOMOTIVE WORKSHOPS. 28 Company), and on the north adjacent to the Manchester and Ashtonunder-Lyne highroad. The Reservoirs are calculated to hold a month's consumption of water, and are supplied from the adjoining Canal, the water passing through filter beds in its course from the Canal to the Reservoirs. These Reservoirs from their elevated position supply the water directly into the Tenders upon the Railway and throughout the Workshops, their position being sufficiently high to do this, and the Canal high enough to supply the Reservoirs. | {
"redpajama_set_name": "RedPajamaC4"
} | 5,138 |
\section{Linear Bandits: Regret Upper Bound for ESCB} \label{sec:escb}
We first recall a regret upper bound for ESCB found in \cite{cuvelier2020}, based on the more general analysis of \cite{degenne2016}.
\begin{thm}\label{th:escb}
Consider a linear combinatorial bandit problem.
Then the regret of ESCB is upper bounded by:
$$
R(T,\theta) \le C(m) + {2 d m^3 \over \Delta_{\min}^2} + {24 d ( \ln T + 4 m \ln \ln T) \over \Delta_{\min}} \left\lceil {\ln m \over 1.61} \right\rceil^2
,$$
with $C(m)$ a positive number that depends solely on $m$.
\end{thm}
\section{Proofs}\label{sec:proofs}
\subsection{Technical Results}
We state a technical result about the product of i.i.d. random variables with Beta distribution.
\begin{lem}\label{lem:product_beta}
Let $V_1,...,V_m$ i.i.d. with distribution $V_i \sim \text{Beta}(\alpha,1)$. Then for all $\Delta \in [0,1]$:
\begin{equation*}
\mathbb{P}\left(\prod_{i=1}^m V_i \ge 1-\Delta \right) \leq {\alpha^m \over m (m !)} \left[\ln \left({1 \over 1 - \Delta} \right)\right]^{m}.
\end{equation*}
\end{lem}
\begin{proof}\label{proof:product_beta}
Taking logarithms:
\begin{align*}
\mathbb{P}\left(\prod_{i=1}^m V_i \ge 1-\Delta\right) = \mathbb{P}\left(\sum_{i=1}^m \ln {1 \over V_i} \le \ln \left({1 \over 1 - \Delta} \right) \right)
\end{align*}
Now if $V_i \sim \text{Beta}(\alpha,1)$ then $\ln {1 \over V_i} \sim \text{Exp}(\alpha)$ and since $V_i$ are i.i.d. we have
\begin{equation*} \sum_{i=1}^m \ln {1 \over V_i} \sim \text{Erlang}\left(m,\alpha \right).
\end{equation*}
Therefore:
\begin{align*} \mathbb{P}\left(\sum_{i=1}^m \ln {1 \over V_i} \le \ln \left({1 \over 1 - \Delta} \right) \right)
&= \frac{\alpha^m}{m!}\int_0^{\ln \left({1 \over 1 - \Delta} \right)}x^{m-1}e^{- \alpha x}dx\\
&\le \frac{\alpha^m}{m!}\int_0^{\ln \left({1 \over 1 - \Delta} \right)}x^{m-1}dx\\
\\
&= \frac{\alpha^m}{(m)!m}\left[\ln \left({1 \over 1 - \Delta} \right)\right]^{m}.
\end{align*}
which concludes the proof.
\end{proof}
We state another technical result about the Beta distribution near $1$.
\begin{lem}\label{lem:beta_tail_bound}
Consider $V \sim \text{Beta}(\alpha+1,\beta+1)$ with $\alpha, \beta > 0$. Define $T = \alpha + \beta$ and $M = {\alpha \over \alpha + \beta}$.
For all $c \in ]0,1[$
\begin{equation*}
\mathbb{P}(V \leq c) \leq \frac{e^{\frac{1}{12}}}{\sqrt{2 \pi T M(1-M)}} \int_{0}^{c} e^{-T D(M \mid x)} d x,
\end{equation*}
with $D$ the Kullback-Leibler divergence between Bernoulli distributions:
\begin{equation*}
D(M \mid x)=M \ln \frac{M}{x}+(1-M) \ln \frac{1-M}{1-x}.
\end{equation*}
We also have the simpler bound for $c \leq M$ :
\begin{equation*}
\mathbb{P}(V \leq c) \leq \frac{e^{\frac{1}{12}} \sqrt{T}}{\sqrt{2 \pi}} e^{ -T(M-c)^{2}}.
\end{equation*}
\end{lem}
\begin{rem}{\label{rem:concentration_beta_mode}} This proves that for any $\nu \in (0,1]$
\begin{equation*} \mathbb{P}\left(V \leq M-\sqrt{\frac{1}{T} \ln \left(\frac{e^{1 / 12} \sqrt{T}}{\nu \sqrt{2 \pi}}\right)}\right) \leq \nu . \end{equation*}
\end{rem}
\begin{rem}{\label{rem:concentration_multi_beta_mode}} Consider $V_{i} \sim \text{Beta}(\alpha_i+1, \beta_i+1)$ with $V_{1}, \ldots, V_{m}$ independent, by negation and union bound we have for all $c \in[0,1]:$
\begin{equation*}
\mathbb{P}\left(\sum_{i=1}^{m} V_{i} \leq \sum_{i=1}^m c_i \right) \leq \sum_{i=1}^{m} \mathbb{P}\left(V_{i} \leq c_i \right).
\end{equation*}
so that the above bound easily extends to the multidimensional case:
\begin{equation*}
\mathbb{P}\left(\sum_{i=1}^{m} V_{i} \leq \sum_i M_i-\sqrt{\frac{m^2}{T} \ln \left(\frac{e^{1 / 12} m \sqrt{T}}{\nu \sqrt{2 \pi}}\right)}\right) \leq \nu.
\end{equation*}
\end{rem}
\begin{proof}\label{proof:beta_tail_bound}
The density of $V$ is given by:
\begin{equation*}
f(x)=\frac{(\alpha+\beta) !}{\alpha ! \beta !} x^{\alpha}(1-x)^{\beta}.
\end{equation*}
The Stirling approximation yields for all $n$ (see \cite{robbins_remark_1955})
\begin{equation*}
\sqrt{2 \pi} n^{n+1 / 2} \leq \sqrt{2 \pi} n^{n+1 / 2} e^{\frac{1}{12 \pi+1}} \leq n ! \leq \sqrt{2 \pi} n^{n+1 / 2} e^{\frac{1}{5 \pi}} \leq \sqrt{2 \pi} n^{n+1 / 2} e^{\frac{1}{12}}.
\end{equation*}
Therefore:
\begin{align*}
\frac{(\alpha+\beta) !}{\alpha ! \beta !}&\leq \frac{\sqrt{2 \pi}(\alpha+\beta)^{\alpha+\beta+1 / 2} e^{\frac{1}{12}}}{(2 \pi)(\alpha)^{\alpha+1 / 2}(\beta)^{\beta+1 / 2}} \\\\
&= \frac{T^{T+1 / 2} e^{\frac{1}{12}}}{\sqrt{2 \pi}(T M)^{T M+1 / 2}(T(1-M))^{T(1-M)+1 / 2}} \\\\
&= \frac{e^{\frac{1}{12}}}{\sqrt{2 \pi T M(1-M)}\left[M^{M}(1-M)^{1-M}\right]^{T}}.
\end{align*}
Furthermore:
\begin{align*}
x^{\alpha}(1-x)^{\beta}&= e^{\alpha \ln (x)+\beta \ln (1-x)} \\\\
&= e^{T[M \ln (x)+(1-M) \ln (1-x)]} \\\\
&= M^{T M}(1-M)^{(1-M) T} e^{-T D(M \mid x)}.
\end{align*}
Replacing:
\begin{equation*}
f(x) \leq \frac{e^{\frac{1}{12}} e^{-T D(M \mid x)}}{\sqrt{2 \pi T M(1-M)}}.
\end{equation*}
Therefore:
\begin{equation*}
\mathbb{P}(V \leq c) = \int_{0}^c f(x) dx \leq \frac{e^{\frac{1}{12}}}{\sqrt{2 \pi T M(1-M)}} \int_{0}^{c} e^{-T D(M \mid x)} d x.
\end{equation*}
The simpler bound comes from $T M(1-M)=\frac{\alpha \beta}{T} \geq \frac{1}{T}$ and using Pinsker's inequality $D(x \mid M) \geq$ $2 (x-M)^{2},$ for $c \leq M$
\begin{equation*}
\mathbb{P}(V \leq c) \leq \frac{e^{\frac{1}{12}} \sqrt{T}}{\sqrt{2 \pi}} e^{ -2 T(M-c)^{2}}.
\end{equation*}
\end{proof}
We recall a result on the Irwin-Hall distribution.
\begin{rem}\label{rem:irwinhall}
Consider $U_1,...,U_m$ i.i.d. uniformly distributed in $[0,1]$. Then their sum follows the Irwin-Hall distribution and for any $\Delta \le 1$ we have that:
\begin{equation*}
\mathbb{P}\Big(\sum_{i=1}^m U_i \ge m - \Delta\Big) = \mathbb{P}\Big(\sum_{i=1}^m U_i \le \Delta\Big) = {\Delta^m \over m!}.
\end{equation*}
\end{rem}
We present a technical result on the tail behaviour of the sum of beta random variables.
\begin{lem}\label{lem:betatail}
Consider $V_1,...,V_m$ independent random variables following beta laws of parameters $(\alpha_1,\beta_1),...,(\alpha_m,\beta_m)$. For $\epsilon < 1$ we have that :
\begin{equation*}\mathbb{P}\Big(\sum_{i=1}^m V_i \geq m-\epsilon\Big) \leq
\frac{\epsilon^{\sum_{i=1}^m \beta_i}}{m! \prod_{i=1}^m B(\alpha_i,\beta_i)}.
\end{equation*}
where $B(\alpha,\beta) = \Gamma(\alpha) \Gamma(\beta) / \Gamma(\alpha+\beta)$ is the beta function.
\end{lem}
\begin{proof}\label{proof:betatail}
We define $A_\epsilon \triangleq \{(u_1,...,u_m)\in [0,1]^m, m-\epsilon \leq \sum_{i=1}^m u_i \leq m\}$. It is noted that if $(u_1,...,u_m) \in A_\epsilon$ we have that $u_i \geq 1- \epsilon$ for all $i$. We recall that the probability density of a Beta$(\alpha_i,\beta_i)$ law is $p_i(u) = u^{\alpha_i-1}(1-u)^{\beta_i-1} / B(\alpha_i,\beta_i)$.
We have
\begin{align*}
\mathbb{P}\Big(\sum_{i=1}^m V_i \geq m-\epsilon\Big)&= \int_{A_\epsilon} \Pi_{i=1}^m p_i(u_i) du_1...du_m
\\
\\
&= \int_{A_\epsilon} \Pi_{i=1}^m \frac{ u_i^{\alpha_i-1}(1-u_i)^{\beta_i-1}}{B(\alpha_i,\beta_i)} du_1...du_m
\\
\\
&\leq \int_{A_\epsilon} \Pi_{i=1}^m \frac{ \epsilon^{\beta_i-1}}{B(\alpha_i,\beta_i)} du_1...du_m
\\
\\
& = \frac{\epsilon^{\sum_{i=1}^m(\beta_i-1)}}{\prod_{i=1}^m B(\alpha_i,\beta_i)} \int_{A_\epsilon}1du_1...u_m.
\end{align*}
But we know that the integral $\int_{A_t}1du_1...u_m$ corresponds to the cumulative distribution function of the sum of $m$ uniform random variables in $[0,1]$. This is known as the Irving Hall distribution. So we have that $\int_{A_t}1du_1...u_m$ = $\frac{\epsilon^m}{m!}$
Which proves the announced result \begin{equation*}\mathbb{P}\Big(\sum_{i=1}^m V_i \geq m-\epsilon\Big) \leq
\frac{\epsilon^{\sum_{i=1}^m \beta_i}}{m! \prod_{i=1}^m B(\alpha_i,\beta_i)}.
\end{equation*}
\end{proof}
Finally we make an important remark about the link between regret and the first time the optimal decision is selected.
\begin{rem}\label{rem:regret_first_time}
Define $\tau$ the first time the optimal decision is selected. Then we have that:
\begin{equation*}
R(T,\theta) = \mathbb{E}(\sum_{t=1}^T \Delta_{x(t)}) \ge \Delta_{\min} \mathbb{E}(\sum_{t=1}^T {\bf 1}\{\Delta_{x(t)} \ne 0 \} ) \ge \Delta_{\min} \sum_{t=1}^T \mathbb{P}(\tau \ge t).
\end{equation*}
\end{rem}
\subsection{Proof of Theorem~\ref{th:linear_frequentist}}
Define $b = 1 - {\Delta \over m}$. Consider $\epsilon > 0$ such that $b -\epsilon \ge {1 \over 2}$ and denote the two decisions as $x^{1} = (1,...,1,0,...,0)$ and $x^{2} = (0,...,0,1,...,1)$. Consider the event where the empirical mean of decision $x^2$ does not deviate too much from its expectation when it is selected:
\begin{equation*}
{\cal A} = \left\{ \exists t \ge 0: x(t) = x^2 , \sum_{i=m+1}^d {A_i(t) \over N_i(t)} \le (b-\epsilon) m \right\}.
\end{equation*}
We decompose ${\cal A}$ as $\cup_{n \ge 1} {\cal A}_n$ where
\begin{equation*}
{\cal A}_n = \left\{ \exists t \ge 0: x(t) = x^2, N_i(t) = n, i=m+1,...,d, {1 \over n} \sum_{i=m+1}^d A_i(t) \le (b-\epsilon) m \right\}.
\end{equation*}
Using Hoeffding's inequality we have that:
\begin{equation*}
\mathbb{P}({\cal A}) \le \sum_{n \ge 1} \mathbb{P}({\cal A}_n) \le \sum_{n \ge 1} \exp(-2 mn \epsilon^2) = {\exp(-2 m \epsilon^2) \over 1 - \exp(-2 m \epsilon^2)}.
\end{equation*}
where we have used the fact that if $N_i(t) = n$ for $i=m+1,...,d$ then $\sum_{i=m+1}^d A_i(t)$ is a sum of $m n$ i.i.d. Bernoulli variables with parameter $b$. Let us control the probability that decision $x^1$ is never selected between time $0$ and time $t$, which is the probability of event:
\begin{equation*}
{\cal B}_t = \{ x(s) = x^2 : s=1,...,t \}.
\end{equation*}
Let us assume that ${\cal B}_{t}$ occurs and ${\cal A}$ does not occur. Since decisions $x^{1}$ and $x^2$ have been selected $0$ and $t$ times respectively, the probability of selecting $x^2$ is lower bounded by:
\begin{equation*}
\mathbb{P}( {\cal B}_{t+1} | {\cal B}_t , \bar{\cal A} ) \ge \mathbb{P}( \sum_{i=1}^m V_i(t) \le \sum_{i=m+1}^d V_i(t) | {\cal B}_t , \bar{\cal A} ).
\end{equation*}
where $V_1(t),...,V_d(t)$ are independent, distributed in $[0,1]$. For $i=1,...,m$, $V_i(t)$ is uniformly distributed in $[0,1]$ and has mean $1/2$. For $i=m+1,...,d$, $V_i(t)$ has Beta$(A_i(t) + 1,t - A_i(t) + 1)$ distribution with mean ${A_i(t) + 1 \over t + 2}$ so that expectations verify:
\begin{align*}
\sum_{i=m+1}^d \mathbb{E}( V_i(t) | {\cal B}_t , \bar{\cal A}) - \sum_{i=1}^m \mathbb{E}( V_i(t) | {\cal B}_t , \bar{\cal A}) &= \sum_{i=m+1}^d {A_i(t) + 1 \over t + 2} - \sum_{i=1}^m {1 \over 2}\\
&\ge {t m (b-\epsilon) + m \over t + 2} - {m \over 2} \\
&={m t (b-\epsilon -1/2) \over t + 2} \\
&\ge {m (b-\epsilon -1/2) \over 3},
\end{align*}
since $\sum_{i=m+1}^d A_i(t) \ge t m (b-\epsilon)$.
Using Hoeffding's inequality once again we have:
\begin{align*}
\mathbb{P}\Big( \sum_{i=1}^m V_i(t) \ge \sum_{i=m+1}^d V_i(t) | {\cal B}_t , \bar{\cal A} \Big) &= \mathbb{P}\Big( \sum_{i=1}^m V_i(t) - \sum_{i=m+1}^d V_i(t) \ge 0 | {\cal B}_t , \bar{\cal A} \Big) \\
&\le \exp\{- 2 m (b-\epsilon -1/2)^2 / 9 \} \\
&\equiv p_\Delta.
\end{align*}
We have proven that for all $t > 1$:
\begin{align*}
\mathbb{P}( {\cal B}_{t+1} | {\cal B}_t , \bar{\cal A} ) \ge 1 - p_\Delta,
\end{align*}
and since $\mathbb{P}({\cal B}_{1}|\bar{\cal A}) = 1/2$:
\begin{align*}
\mathbb{P}( {\cal B}_{t} ) \ge \mathbb{P}( {\cal B}_{t}, \bar{\cal A}) = \mathbb{P}(\bar{\cal A}) \mathbb{P}({\cal B}_{t} | \bar{\cal A}) \ge \mathbb{P}(\bar{\cal A})\mathbb{P}({\cal B}_{1}|\bar{\cal A}) (1 - p_\Delta)^{t-1} = \frac{\mathbb{P}(\bar{\cal A})}{2}(1 - p_\Delta)^{t-1}.
\end{align*}
Denote by $\tau$ the first time that $x^1$ is selected. If ${\cal B}_t$ occurs then $\tau \ge t$ and using Remark \ref{rem:regret_first_time} yields the lower bound:
\begin{align*}
R(T,\theta) \ge \Delta \sum_{t=1}^T \mathbb{P}(\tau \ge t) \ge \frac{\Delta \mathbb{P}(\bar{\cal A})}{2} \sum_{t=1}^T (1 - p_\Delta)^{t-1}.
\end{align*}
Setting $\epsilon = {1 \over \sqrt{m}}$ we get that \begin{equation*} \mathbb{P}({\cal A}) \le {e^{-2} \over 1 - e^{-2}} \le {1 \over 2},\end{equation*}
and we get the announced result:
\begin{align*}
R(T,\theta) \ge {\Delta \over 4} \sum_{t=1}^T (1 - p_\Delta)^t.
\end{align*}
\subsection{Proof of Theorem~\ref{th:linear_frequentist_bis}}
\label{proof:linear_frequentist_bis}
Consider $m \ge 5$. We denote by $N^1(t)$ and $N^2(t)$ the number of times that decisions $x^1$ and $x^2$ have been respectively selected, and it is noted that $N_i(t) = N^1(t)$ for $i=1,...,m$ and $N_i(t) = N^2(t)$ for $i=m+1,...,d$. Consider the event where the empirical mean of decision $x^2$ deviates significantly from its expectation when it is selected:
\begin{equation*}
{\cal A} = \left\{ \exists t \ge 0: x(t) = x^2 ,{1 \over N^2(t)} \sum_{i=m+1}^d A_i(t) \le m - \Delta -\sqrt{m \ln (2 N^2(t)) \over N^2(t)} \right\}.
\end{equation*}
We decompose ${\cal A}$ as $\cup_{n \ge 1} {\cal A}_n$ where
\begin{equation*}
{\cal A}_n = \left\{ \exists t \ge 0: x(t) = x^2, N^2(t) = n, {1 \over n} \sum_{i=m+1}^d A_i(t) \le m - \Delta - \sqrt{m \ln (2 n) \over n} \right\}.
\end{equation*}
Using Hoeffding's inequality we have that :
\begin{equation*}
\mathbb{P}({\cal A}) \le \sum_{n \ge 1} \mathbb{P}({\cal A}_n) \le \sum_{n \ge 1} {1 \over (2n)^{2}} = {\pi^2 \over 24} \le {1 \over 2},
\end{equation*}
where we have used the fact that if $N^2(t) = n$ then $\sum_{i=m+1}^d A_i(t)$ is a sum of $m n$ i.i.d. Bernoulli random variables with parameter $1-{\Delta \over m}$. Let us control the probability that decision $x^1$ is never selected between time $0$ and time $t$, which is the probability of event:
\begin{equation*}
{\cal B}_t = \{ x(s) = x^2 : s=1,...,t \}.
\end{equation*}
We have that:
\begin{align*}
\mathbb{P}( {\cal B}_{t+1} | {\cal B}_t , \bar{\cal A}) &\ge \mathbb{P}\Big( \sum_{i=1}^m V_i(t) \le \sum_{i=m+1}^d V_i(t) | {\cal B}_t , \bar{\cal A} \Big) \ge (1-p_{t,1}) (1- p_{t,2}),
\end{align*}
with
\begin{align*}
p_{t,1} &= \mathbb{P}\left( \sum_{i=1}^m V_i(t) \ge m - \Delta - h(m,t) | {\cal B}_t , \bar{\cal A} \right), \\
p_{t,2} &= \mathbb{P}\left( \sum_{i=m+1}^d V_i(t) \le m - \Delta - h(m,t) | {\cal B}_t , \bar{\cal A} \right), \\
h(m,t) &= \sqrt{m \ln (2 t) \over t} + \sqrt{\frac{m^2}{t} \ln \left(\frac{e^{1 / 12} m \sqrt{t}}{ \frac{1}{t^2}\sqrt{2 \pi}} \right)}.
\end{align*}
It is noted that there exists a universal constant $C_1 > 0$ such that
\begin{equation*}
h(m,t) \le \sqrt{C_1 m^2(\ln t + \ln m) \over t}.
\end{equation*}
Let us define $T_0 = C_0 m^2 \ln m$ with $C_0 > 0$ a universal constant such that the five following inequalities are true:
\begin{itemize}
\item $T_0 \ge m,$
\item $h(m,t) \le {1 \over 3}$ for all $t \ge T_0,$
\item ${C_1 8 e^2 \ln t \over t} \le 1$ for all $t \ge T_0,$
\item $\sum_{t = T_0}^{+\infty} \left({C_1 8 e^{2} \ln t \over t}\right)^{{3 \over 2}} \le {1 \over 3}.$
\item $\sum^{+\infty}_{t=T_0} {1 \over t^2} \le {1 \over 2}.$
\end{itemize}
Consider $p_{t,2}$, and recall that for $i=1,...,d$
\begin{equation*}
M_i(t) \triangleq \frac{A_i(t)}{A_i(t) +B_i(t) } = \frac{A_i(t)}{N_i(t) }.
\end{equation*}
is the mode of $V_i(t)$. If event $\bar{\cal A}$ occurs then
\begin{equation*}\sum_{i=m+1}^d M_i(t) > m - \Delta - \sqrt{m \ln (2 N^2(t)) \over N^2(t)}.\end{equation*}
So using lemma \ref{lem:beta_tail_bound} and remark \ref{rem:concentration_multi_beta_mode} we have that:
\begin{equation*}
p_{t,2} \le \frac{1}{t^2}.
\end{equation*}
Consider $p_{t,1}$. Since $\Delta \le \frac{1}{6}$ and $h(m,t) \le {1 \over 3}$ we have
\begin{equation*}m - \Delta - h(m,t) \ge m- {1 \over 2} \ge m-1.\end{equation*}
If event ${\cal B}_t$ occurs, then $A_i(t) = B_i(t) = 0$ for all $i=1,...,m$ therefore $\sum_{i=1}^{m} V_i(t)$ follows the Irwin-Hall distribution of size $m$, so from remark ~\ref{rem:irwinhall}, for $t \ge T_0$ we have:
\begin{align*}
p_{t,1} &= {1 \over m!}\left(\Delta + h(m,t)\right)^m \le {1 \over m!} \left((2\Delta)^m + (2h(m,t))^m\right),
\end{align*}
where we used the convexity inequality $({x+y \over 2})^m \le {x^m + y^m \over 2}$.
We have, for $T > T_0$ :
\begin{align*}
{\mathbb{P}( {\cal B}_{T} | \bar{\cal A}) \over \mathbb{P}( {\cal B}_{T_0} | \bar{\cal A})} & = \prod^{T-1}_{t=T_0} \mathbb{P}({\cal B}_{t+1}|{\cal B}_{t}, \bar{\cal A}) \geq \prod^{T-1}_{t=T_0} (1-p_{t,1})(1-p_{t,2}).\\
\end{align*}
Using the union bound and the definition of $T_0$:
\begin{align*}
\prod^{T-1}_{t=T_0} (1-p_{t,1}) \ge 1 - \sum^{T-1}_{t=T_0} p_{t,1} \ge 1 - \sum^{T-1}_{t=T_0} {1 \over t^2} \ge 1 - \sum^{+\infty}_{t=T_0} {1 \over t^2} \ge {1 \over 2}.
\end{align*}
Now:
\begin{align*}
1 - p_{t,2} &= 1 - {(2 \Delta)^m \over m!} - {(2 h(m,t))^m \over m!} \\
&= (1 - {(2 \Delta)^m \over m!}) {1 - {(2 \Delta)^m \over m!} - {(2 h(m,t))^m \over m!} \over 1 - {(2 \Delta)^m \over m!}} \\
&\ge (1 - {(2 \Delta)^m \over m!})(1 - {3 \over 2} {(2 h(m,t))^m \over m!} ),
\end{align*}
where we used the fact that $\Delta \le {1 \over 6}$ so that ${(2 \Delta)^m \over m!} \le {1 \over 3}$.
Using the union bound once more:
\begin{align*}
\prod^{T-1}_{t=T_0} (1-p_{t,2}) &\ge \prod^{T-1}_{t=T_0} (1 - {(2 \Delta)^m \over m!}) (1 - {3 \over 2} {(2 h(m,t))^m \over m!} ) \\
&\ge (1 - {(2 \Delta)^m \over m!})^{T - T_0} (1 - {3 \over 2} \sum^{T-1}_{t=T_0} {(2 h(m,t))^m \over m!} ) \\
&\ge (1 - {(2 \Delta)^m \over m!})^{T - T_0} (1 - {3 \over 2} \sum^{\infty}_{t=T_0} {(2 h(m,t))^m \over m!} ) .
\end{align*}
We turn to the last sum in the right hand side of the equation above. Since $t \ge T_0 \ge m$ we have
\begin{equation*}
h(m,t) \le \sqrt{C_1 m^2(\ln t + \ln m) \over t} \le \sqrt{C_1 2 m^2 \ln t \over t} .
\end{equation*}
Using Stirling's approximation we have $m! \ge (m/e)^m$ so that
\begin{equation*}
\sum^{\infty}_{t=T_0} {(2 h(m,t))^m \over m!} \le \sum^{\infty}_{t=T_0} \left({C_1 8 e^{2} \ln t \over t}\right)^{{m \over 2}} \le \sum^{\infty}_{t=T_0} \left({C_1 8 e^{2} \ln t \over t}\right)^{{3 \over 2}} \le {1 \over 3},
\end{equation*}
where we used twice the definition of $T_0$ and $m \ge 5 \ge 3$.
Putting things together we have proven that :
\begin{equation*}
{\mathbb{P}( {\cal B}_{T} | \bar{\cal A}) \over \mathbb{P}( {\cal B}_{T_0} | \bar{\cal A})} \ge {1 \over 4} \left(1 - {(2 \Delta)^m \over m!} \right)^{T-T_0} .
\end{equation*}
We showed previously with Theorem \ref{th:linear_frequentist} that :
\begin{equation*}
\mathbb{P}( {\cal B}_{T_0} | \bar{\cal A}) \ge {1 \over 2}(1-p_\Delta)^{T_0-1}
\end{equation*}
Let us lower bound the r.h.s. of this inequality. Since $m \ge 5$ and $\Delta \le 1/6$ we have, by definition
$$
p_{\Delta} = \exp\left\{- {2 m\over 9} \Big[{1 \over 2} - \Big({\Delta \over m} + {1 \over \sqrt{m}}\Big)\Big]^2 \right\} \le \exp\left\{ - \xi m \right\},
$$
with
$$\xi = {2 \over 9}\left[{1 \over 2} - \left({1 \over 30} + {1 \over \sqrt{5}}\right)\right]^2 > 0.
$$
Using the definition of $T_0$ this yields
$$
{1 \over 2} (1-p_\Delta)^{T_0-1} \ge {1 \over 2}(1 - e^{-\xi m} )^{C_0 m^2 \ln m - 1} \ge \min_{m \ge 5} \left\{ {1 \over 2}(1 - e^{-\xi m} )^{C_0 m^2 \ln m - 1} \right\} \equiv C_2,
$$
where $C_2$ is a universal constant and $C_2 > 0$ since
$$
\lim_{m \to \infty}\left\{ {1 \over 2}(1 - e^{-\xi m} )^{C_0 m^2 \ln m - 1} \right\} = {1 \over 2} > 0.
$$
We have proven that:
\begin{equation*}
\mathbb{P}( {\cal B}_{T_0} | \bar{\cal A}) \ge {1 \over 2}(1-p_\Delta)^{T_0-1} \ge C_2 .
\end{equation*}
which gives
\begin{equation*}
\mathbb{P}( {\cal B}_{T}) \ge C_2 \left(1-{(2\Delta)^m \over m !}\right)^{T-T_0},
\end{equation*}
and applying Remark~\ref{rem:regret_first_time} concludes the proof.
\subsection{Proof of Corollary \ref{co:linear_minimax}}
Using the same notation as above, we recall that
\begin{equation*}
\mathbb{P}({\cal B}_T) \geq C_3\left(1-{(2\Delta)^m \over m !}\right)^T.
\end{equation*}
If ${\cal B}_T$ occurs, decision $x^1$ is never played, resulting in a regret of $\Delta T$, therefore:
\begin{equation*}
R(T,\theta) \ge \Delta T \mathbb{P}({\cal B}_T) \ge C_3 \Delta T\left(1-\frac{\left(2\Delta\right)^m}{m!}\right)^T .
\end{equation*}
(i) If $T \ge 3^m m!$ let us set
\begin{equation*}
\Delta = \frac{1}{2} \left(\frac{m!}{T}\right)^{\frac{1}{m}} ,
\end{equation*}
so that we have $\Delta \le {1 \over 6}$ and, using Stirling's approximation $m! \ge (m/e)^m$ we get
\begin{align*}
\max_{\theta \in [0,1]^d} R(T,\theta) &\ge \frac{C_3}{3} \left(m!\right)^{\frac{1}{m}} T^{1 - \frac{1}{m}}\left(1- {1 \over T}\right)^T \\
&\ge {C_3 \over 3} {m \over e} (1- e^{-1}) T^{1 - \frac{1}{m}}.
\end{align*}
this yields
\begin{equation*}
\max_{\theta \in [0,1]^d} R(T,\theta) \geq \mathcal{O}(mT^{1-\frac{1}{m}}).
\end{equation*}
(ii) If $T \le 3^m m!$ let us set $\Delta = 1/6$, which yields
\begin{equation*}
\max_{\theta \in [0,1]^d} R(T,\theta) \geq \mathcal{O}(T).
\end{equation*}
and completes the proof.
\subsection{Proof of Theorem~\ref{th:linear_frequentist_forced}}
To simplify notation, we assume that the $\ell$ rounds of exploration are done before the algorithm starts, so that at time $t=0$ each decision has been explored $\ell / 2$ times and the TS algorithm starts.
We consider the following event :
\begin{equation*}
{\cal C} = \left\{\forall i \in [d] , A_i(0) = \frac{\ell}{2} \right\}.
\end{equation*}
We know that $A_i(0)$, $i=1,...,d$ are independent with a Binomial$(\ell/2,\theta_i)$ distribution so that
$$
\mathbb{P}({\cal C}) = \Big(1-{\Delta \over m}\Big)^\frac{\ell m}{2}.
$$
Define $\epsilon = {1 \over \sqrt{m}}$. We consider again the event where the empirical mean of decision $x^2$ deviates significantly from its expectation when it is selected, accounting for the rounds of forced exploration:
\begin{equation*}
{\cal A} = \left\{ \exists t \ge 0: x(t) = x^2 , \sum_{i=m+1}^d A_i(t) \le (1-{\Delta \over m}-\epsilon) (N^2(t)-{\ell \over 2}) m + {\ell \over 2} m \right\}.
\end{equation*}
We decompose ${\cal A}$ as $\cup_{n \ge 1} {\cal A}_n$ where
\begin{equation*}
{\cal A}_n = \left\{ \exists t \ge 0: x(t) = x^2, N^2(t) = n+{\ell \over 2}, \sum_{i=m+1}^d A_i(t) \le (1-{\Delta \over m}-\epsilon) n m + {\ell \over 2} m \right\}
\end{equation*}
Since $\epsilon = {1 \over \sqrt{m}}$, using Hoeffding's inequality we have that :
\begin{equation*}
\mathbb{P}({\cal A}|{\cal C}) \le \sum_{n \ge 1} \mathbb{P}({\cal A}_n|{\cal C}) \le \sum_{n \ge 1} \exp(-2 mn \epsilon^2) = {\exp(-2 m \epsilon^2) \over 1 - \exp(-2 m \epsilon^2)} \le {1 \over 2}.
\end{equation*}
where we have used the fact that if $N^2(t) = n+{\ell/2}$, then $\sum_{i=m+1}^d A_i(t)$ equals ${\ell \over 2}m$ plus the sum of $m n$ i.i.d Bernoulli random variables with parameter $1 - {\Delta \over m}$. Let us control the probability that decision $x^1$ is never selected between time $0$ and time $t$, which is the probability of event:
\begin{equation*}
{\cal B}_t = \{ x(s) = x^2 : s=1,...,t \}.
\end{equation*}
Let us assume that ${\cal B}_{t}$ and ${\cal C}$ occurs but ${\cal A}$ does not occur. Since decisions $x^{1}$ and $x^2$ have been selected $\ell/2$ and $\ell/2+t$ times respectively, the probability of selecting $x^2$ is lower bounded by:
\begin{equation*}
\mathbb{P}( {\cal B}_{t+1} | {\cal B}_t , \bar{\cal A},{\cal C} ) \ge \mathbb{P}\Big( \sum_{i=1}^m V_i(t) \le \sum_{i=m+1}^d V_i(t) | {\cal B}_t , \bar{\cal A}, {\cal C} \Big).
\end{equation*}
where $V_1(t),...,V_d(t)$ are independent, distributed in $[0,1]$. For $i=1,...,m$, $V_i(t)$ follows a Beta$(\frac{\ell}{2},1)$ law and has mean $\frac{{\ell \over 2} +1}{{\ell \over 2} + 2}$. For $i=m+1,...,d$, $V_i(t)$ follows a Beta$(A_i(t) + 1,t+{\ell \over 2} - A_i(t) + 1)$ distribution with mean ${A_i(t) + 1 \over t+{\ell \over 2} + 2}$ so that the expectations verify:
\begin{align*}
\sum_{i=m+1}^d \mathbb{E}( V_i(t) | {\cal B}_t , \bar{\cal A}, {\cal C}) - \sum_{i=1}^m \mathbb{E}( V_i(t) | {\cal B}_t , \bar{\cal A},{\cal C}) &\ge m {t (1-{\Delta \over m}-\epsilon) +\frac{\ell}{2} + 1 \over t +\frac{\ell}{2} + 2} - m\frac{{\ell \over 2} +1}{{\ell \over 2} + 2}\\
&\ge m {(1-{\Delta \over m}-\epsilon) +\frac{\ell}{2} + 1 \over \frac{\ell}{2} + 3} - m\frac{{\ell \over 2} +1}{{\ell \over 2} + 2}\\
&= m {\left({1 \over {\ell \over 2}+2} - ({\Delta \over m}+\epsilon)\right)\over ({\ell \over2}+3)},
\end{align*}
since $\sum_{i=m+1}^d A_i(t) \ge t m (1-{\Delta \over m}-\epsilon) + {m\ell \over 2}$. Recall that $\epsilon = {1 \over \sqrt{m}}$ so that ${1 \over {\ell \over 2}+2} - ({\Delta \over m}+\epsilon) \ge 0$. Using Hoeffding's inequality:
\begin{align*}
\mathbb{P}\Big( \sum_{i=1}^m V_i(t) \ge \sum_{i=m+1}^d V_i(t) | {\cal B}_t , \bar{\cal A} ,{\cal C}\Big) &= \mathbb{P}\Big( \sum_{i=1}^m V_i(t) - \sum_{i=m+1}^d V_i(t) \ge 0 \Big) \\
&\le \exp\left\{- 2 m {\left({1 \over {\ell \over 2}+2} - ({\Delta \over m}+\epsilon)\right)^2\over ({\ell \over2}+3)^2} \right\} \equiv p_{\Delta}^\ell.
\end{align*}
We have proven that for all $t > 1$:
\begin{equation*}
\mathbb{P}( {\cal B}_{t+1} | {\cal B}_t , \bar{\cal A} ) \ge 1 - p_{\Delta}^\ell.
\end{equation*}
so that:
\begin{align*}
\mathbb{P}( {\cal B}_{t} ) & \ge \mathbb{P}( {\cal B}_{t}, \bar{\cal A},{\cal C})\\
&= \mathbb{P}(\bar{\cal A}|{\cal C})\mathbb{P}({\cal C}) \mathbb{P}({\cal B}_{t} | \bar{\cal A},{\cal C}) \\
&\ge \mathbb{P}(\bar{\cal A}|{\cal C})\mathbb{P}({\cal B}_{1}|\bar{\cal A},{\cal C}) (1 - p_{\Delta}^\ell)^{t-1}(1-{\Delta \over m})^{\ell m\over 2} \\
&= \frac{\mathbb{P}(\bar{\cal A}|{\cal C})}{2}(1 - p_{\Delta}^\ell)^{t-1}(1-{\Delta \over m})^{\ell m\over 2}.
\end{align*}
Denote by $\tau$ the first time that $x^1$ is selected. If ${\cal B}_t$ occurs then $\tau \ge t$ and using Remark \ref{rem:regret_first_time} yields the lower bound:
\begin{equation*}
R(T,\theta) \ge \Delta \sum_{t=1}^T \mathbb{P}(\tau \ge t) \ge \Delta {\mathbb{P}(\bar{\cal A}|{\cal C}) \over 2}(1-{\Delta \over m})^{\ell m\over 2} \sum_{t=1}^T (1 - p_{\Delta}^\ell)^{t-1}.
\end{equation*}
From above, $\mathbb{P}({\cal A}|{\cal C}) \le {1 \over 2}$, and we get the announced result:
\begin{equation*}
R(T,\theta) \ge {\Delta \over 4}(1-{\Delta \over m})^{\ell m\over 2} \sum_{t=1}^T (1 - p_{\Delta}^\ell)^t.
\end{equation*}
\subsection{Proof of Theorem~\ref{th:linear_frequentist_forced_minimax}}
To simplify notation, we assume that the $\ell$ rounds of exploration are done before the algorithm starts, so that at time $t=0$ each decision has been explored $\ell / 2$ times and the TS algorithm starts.
We consider the following event :
\begin{equation*}
{\cal C} = \left\{\forall i \in [d] , A_i(0) = \frac{\ell}{2} \right\}.
\end{equation*}
We know that $A_i(0)$, $i=1,...,d$ are independent with a Binomial$(\ell/2,\theta_i)$ distribution so that
$$
\mathbb{P}({\cal C}) = \Big(1-{\Delta \over m}\Big)^\frac{\ell m}{2}.
$$
We consider the event where the empirical mean of decision $x^2$ deviates significantly from its expectation when it is selected, accounting for the rounds of forced exploration:
\begin{equation*}
{\cal A} = \left\{ \exists t \ge 0: x(t) = x^2 , \sum_{i=m+1}^d A_i(t) \le (m - \Delta)\Big(N^2(t) - {\ell \over 2}\Big) + {\ell m \over 2} - \sqrt{m \ln (2 (N^2(t)-{\ell \over 2}))} \right\} .
\end{equation*}
We decompose ${\cal A}$ as $\cup_{n \ge 1} {\cal A}_n$ where
\begin{equation*}
{\cal A}_n = \left\{ \exists t \ge 0: x(t) = x^2 ,N^2(t) = n+{\ell \over 2}, \sum_{i=m+1}^d A_i(t) \le (m - \Delta)n + {\ell m \over 2} - \sqrt{m \ln (2n)} \right\}.
\end{equation*}
Using Hoeffding's inequality we have that :
\begin{equation*}
\mathbb{P}({\cal A}| {\cal C_\ell}) \le \sum_{n \ge 1} \mathbb{P}({\cal A}_n| {\cal C_\ell}) \le \sum_{n \ge 1} {1 \over (2n)^{2}} = {\pi^2 \over 24} \le {1 \over 2},
\end{equation*}
where we have used the fact that if $N^2(t) = n+{\ell/2}$, then $\sum_{i=m+1}^d A_i(t)$ equals ${\ell \over 2}m$ plus the sum of $m n$ i.i.d Bernoulli random variables with parameter $1 - {\Delta \over m}$.
Let us control the probability that decision $x^1$ is never selected between time $0$ and time $t$, which is the probability of event:
\begin{equation*}
{\cal B}_t = \{ x(s) = x^2 : s=1,...,t \} ,
\end{equation*}
We have that:
\begin{align*}
\mathbb{P}( {\cal B}_{t+1} | {\cal B}_t , \bar{\cal A}) &\ge \mathbb{P}\left( \sum_{i=1}^m V_i(t) \le \sum_{i=m+1}^d V_i(t) | {\cal B}_t , \bar{\cal A} \right) \ge (1-p_{t,1}) (1- p_{t,2}),
\end{align*}
with
\begin{align*}
p_{t,1} &= \mathbb{P}\left( \sum_{i=1}^m V_i(t) \ge m - \Delta - h(m,\ell,t) | {\cal B}_t , \bar{\cal A},{\cal C_\ell} \right), \\
p_{t,2} &= \mathbb{P}\left( \sum_{i=m+1}^d V_i(t) \le m - \Delta - h(m,\ell,t) | {\cal B}_t , \bar{\cal A},{\cal C_\ell} \right), \\
h(m,\ell,t) &= - {m \ell \over 2 t} + \sqrt{m \ln (2 t) \over t} + \sqrt{\frac{m^2}{t+\ell} \ln \left(\frac{e^{1 / 12} m \sqrt{t+\ell}}{ \frac{1}{t^2}\sqrt{2 \pi}}\right)}.
\end{align*}
It is noted that there exists a constant $C_1 \ge 0$ such that
\begin{equation*}
h(m,\ell,t) \le \sqrt{ {C_1 m^2 ( \ln m + \ln (t+\ell)) \over t} }.
\end{equation*}
Let us define $T_0(m,\ell) = C_0 m^2 (\ln m)\ell^{1 \over {1 \over 4} - {1 \over m}}$ with $C_0$ a universal constant such that the following inequalities are true
\begin{itemize}
\item $T_0 \ge \max(m,\ell,7),$
\item $h(m,\ell,t) \le {1 \over 6 \ell},$ for all $t \ge T_0,$
\item $\sum^{+\infty}_{t=T_0} {1 \over t^2} \leq {1 \over 2},$
\item $4 \left( {\ell e \sqrt{2 C_1} \over (T_0-1)^{{1 \over 4} - {1 \over m}}}\right) \le {1 \over 3}.$
\end{itemize}
First consider $p_{t,2}$, and recall that $\forall i \in [d], \forall t, M_i(t) \triangleq \frac{A_i(t)}{A_i(t) +B_i(t) } = \frac{A_i(t)}{N_i(t) } $ is the mode of $V_i(t)$. If event $\bar{\cal A}$ occurs then
\begin{equation*}\sum_{i=m+1}^d M_i(t) > m - \Delta - \sqrt{m \ln (2 N^2(t)) \over N^2(t)} + \frac{m\ell}{2N^2(t)}. \end{equation*}
So using lemma \ref{lem:beta_tail_bound} and remark \ref{rem:concentration_multi_beta_mode} we have that:
$p_{t,2} \le \frac{1}{t^2}.$
Consider $p_{t,1}$. Since $\Delta < \frac{1}{6 \ell}$ and for $t \ge T_0$ we have $h(m,\ell,t) \le {1 \over 6 \ell}$, hence
\begin{equation*}
m - \Delta - h(m,\ell,t) \ge m-{1 \over 3\ell}.
\end{equation*}
If event ${\cal B}_t$ and ${\cal C}_t$ occurs then $A_i(t) = \frac{\ell}{2}$ and $B_i(t) = 0$ for all $i=1,...,m$ therefore we may control the tail behaviour of $\sum_{i=m+1}^d V_i(t)$ thanks to lemma \ref{lem:betatail}. So we have for $t \ge T_0$:
\begin{align*}
p_{t,1} & \leq {\ell^m \over 2^m m!}\left(\Delta + h(m,\ell,t) \right)^{m}
\le{1 \over 2 (m!)} \left( (\ell \Delta )^{m} + (\ell h(m,\ell,t))^m \right).
\end{align*}
where we used the convexity inequality $({x+y \over 2})^m \le {x^m + y^m \over 2}$.
We have, for $T > T_0$ :
\begin{align*}
{\mathbb{P}( {\cal B}_{T} | \bar{\cal A},{\cal C}) \over \mathbb{P}( {\cal B}_{T_0} | \bar{\cal A},{\cal C})} & = \prod^{T-1}_{t=T_0} \mathbb{P}({\cal B}_{t+1}|{\cal B}_{t}, \bar{\cal A},{\cal C}) \geq \prod^{T-1}_{t=T_0} (1-p_{t,1})(1-p_{t,2}).\\
\end{align*}
Using the union bound and the definition of $T_0$:
\begin{align*}
\prod^{T-1}_{t=T_0} (1-p_{t,2}) \ge 1 - \sum^{T-1}_{t=T_0} p_{t,2} \ge 1 - \sum^{T-1}_{t=T_0} {1 \over t^2} \ge 1 - \sum^{+\infty}_{t=T_0} {1 \over t^2} \ge {1 \over 2}.
\end{align*}
Now:
\begin{align*}
1 - p_{t,1} &= 1 - {(\ell \Delta)^m \over m!} - {( \ell h(m,\ell,t))^m \over m!} \\
&= (1 - {(\ell \Delta)^m \over m!}) {1 - {(\ell \Delta)^m \over m!} - {( \ell h(m,\ell,t))^m \over m!} \over 1 - {(\ell \Delta)^m \over m!}} \\
&\ge (1 - {(\ell \Delta)^m \over m!})(1 - {3 \over 2} {( \ell h(m,\ell,t))^m \over m!} ).
\end{align*}
where we used the fact that $\Delta \le {1 \over 6}$ so that ${(\ell \Delta)^m \over m!} \le {1 \over 3}.$
Using the union bound once more:
\begin{align*}
\prod^{T-1}_{t=T_0} (1-p_{t,2}) &\ge \prod^{T-1}_{t=T_0} (1 - {(\ell \Delta)^m \over m!}) (1 - {3 \over 2} {(\ell h(m,\ell,t))^m \over m!} ) \\
&\ge (1 - {(\ell \Delta)^m \over m!})^{T - T_0} (1 - {3 \over 2} \sum^{T-1}_{t=T_0} {(\ell h(m,\ell,t))^m \over m!} ) \\
&\ge (1 - {(\ell \Delta)^m \over m!})^{T - T_0} (1 - {3 \over 2} \sum^{\infty}_{t=T_0} {(\ell h(m,\ell,t))^m \over m!} ) .
\end{align*}
We turn to the last sum in the right hand side of the equation above. It is noted that $\log(2t) \le \sqrt{t}$ for all $t \ge 7$. Since $t \ge T_0 \ge \max(m,\ell,7)$ we have
\begin{equation*}
h(m,\ell,t) \le \sqrt{C_1 m^2(\ln(t+\ell) + \ln m) \over t} \le \sqrt{C_1 2 m^2 \ln(2t) \over t}
\le m \sqrt{2 C_1} \hspace{0.2cm} t^{-{1 \over 4}}.
\end{equation*}
We now upper bound the sum as follows:
\begin{align*}
\sum^{\infty}_{t=T_0} {(\ell h(m,t))^m \over m!}
& \overset{(i)}{\le} {(\ell m \sqrt{2 C_1} )^{m} \over m!}
\sum^{\infty}_{t=T_0} t^{-{m \over 4}}
\overset{(ii)}{\le} (\ell e \sqrt{2 C_1} )^{m}
\sum^{\infty}_{t=T_0} t^{-{m \over 4}} \\
& \overset{(iii)}{\le} 4 \left( {\ell e \sqrt{2 C_1} \over (T_0-1)^{{1 \over 4} - {1 \over m}}} \right)^{m}
\overset{(iv)}{\le} 4 \left( {\ell e \sqrt{2 C_1} \over (T_0-1)^{{1 \over 4} - {1 \over m}}} \right) \le {1 \over 3},
\end{align*}
where we used (i) the bound above, (ii) Stirling's approximation $m! \ge (m/e)^m$, (iii) the following sum-integral comparison, for any $m \ge 5$:
$$
\sum_{t=T_0}^{+\infty} t^{-{m \over 4}} \le \int_{T_0-1}^{+\infty} t^{-{m \over 4}} dt = {(T_0-1)^{1-{m \over 4}} \over {m \over 4} - 1} \le 4 (T_0-1)^{1-{m \over 4}},
$$
and (iv) the definition of $T_0$.
Putting things together we have proven that
\begin{equation*}
{\mathbb{P}( {\cal B}_{T} | \bar{\cal A},{\cal C}) \over \mathbb{P}( {\cal B}_{T_0} | \bar{\cal A})} \ge {1 \over 4} \left(1 - {(\ell \Delta)^m \over m!} \right)^{T-T_0}.
\end{equation*}
We showed previously that
\begin{equation*}
\mathbb{P}( {\cal B}_{T_0} | \bar{\cal A}) \ge {1 \over 2}(1-p_\Delta)^{T_0-1} \ge C_2(\ell,m),
\end{equation*}
with
\begin{equation*}
C_2(\ell,m) = {1 \over 2}(1-p_\Delta^\ell)^{T_0-1},
\end{equation*}
and it is noted that $\lim_{m \to \infty} C_2(\ell,m) = 1$ for any fixed $\ell \ge 0$.
Therefore
\begin{equation*}
\mathbb{P}( {\cal B}_{T} | \bar{\cal A},{\cal C}) \ge {C_2(\ell,m) \over 4} \left(1 - {(\ell \Delta)^m \over m!} \right)^{T-T_0}.
\end{equation*}
Since $\mathbb{P}({\cal C}) = (1 - {\Delta \over m})^{\ell m \over 2}$ we get
\begin{equation*}
\mathbb{P}( {\cal B}_{T} | \bar{\cal A}) \ge {C_2(\ell,m) \over 4} (1 - {\Delta \over m})^{\ell m \over 2} \left(1 - {(\ell \Delta)^m \over m!} \right)^{T-T_0},
\end{equation*}
and applying Remark~\ref{rem:regret_first_time} concludes the first part of the proof.
Now choose:
\begin{equation*}
\Delta =\frac{1}{\ell} \min\left( \left(\frac{m!}{T}\right)^{\frac{1}{m}} , {1 \over 6}\right),
\end{equation*}
and lower bound the regret by
$$
\max_{\theta \in [0,1]^d} R(T,\theta) \ge \Delta T \mathbb{P}( {\cal B}_{T}).
$$
If $\Delta = {1 \over \ell} \left(\frac{m!}{T}\right)^{\frac{1}{m}} $, then replacing
\begin{align*}
\max_{\theta \in [0,1]^d} R(T,\theta) &\ge \Delta T {C_2(\ell,m) \over 4} \left(1 - {\Delta \over m}\right)^{\ell m \over 2} \left(1 - {(\ell \Delta)^m \over m!} \right)^{T} \\
&= {(m!)^{1 \over m} T^{1 - {1 \over m}} \over \ell} {C_2(\ell,m) \over 4} \left(1 - {\Delta \over m} \right)^{\ell m \over 2} \left(1 - {1 \over T} \right)^T .
\end{align*}
Using the facts that (i) $\left(1 - {1 \over T} \right)^T \ge e$, that (ii) $(m!)^{1 \over m} \ge {m \over e}$ which follows from Stirling's approximation $m! \ge ({m \over e})^m$ and that (iii) since $\Delta \le {1 \over 6 \ell}$:
$$
\left(1 - {\Delta \over m} \right)^{\ell m \over 2} \ge \left(1 - {1 \over 6 m \ell} \right)^{\ell m \over 2} \ge e^{-{1 \over 12}},
$$
which yields the minimax regret bound
\begin{equation*}
\max_{\theta \in [0,1]^d} R(T,\theta) \geq \mathcal{O}\left(C_2(\ell,m) {m\over \ell}T^{1-\frac{1}{m}}\right).
\end{equation*}
Otherwise $\Delta = {1 \over 6 \ell}$ and we simply have
\begin{equation*}\max_{\theta \in [0,1]^d} R(T,\theta) \geq \mathcal{O}\left(C_2(\ell,m) {T\over \ell}\right).\end{equation*}
which completes the proof.
\subsection{Proof of Theorem~\ref{th:nonlinear_frequentist}}
\label{proof:nonlinear_frequentist}
At round $t \ge 1$, if the optimal decision has never been played then $A_i(t) = B_i(t) = 0$ for $i=1,...,m$. In turn the samples $V_i(t)$ are independent and uniformly distributed in $[0,1]$ for $i=1,...,m$.
Lemma \ref{lem:product_beta} shows that at time $t$ the optimal decision is played with probability
\begin{equation*}
\mathbb{P}\left( \prod_{i=1}^m V_i(t) \ge 1 - \Delta\right) \le p_{\Delta} \equiv {1 \over m m !} \left[\ln \left({1 \over 1 - \Delta} \right)\right]^{m}.
\end{equation*}
So the distribution of the first time the optimal decision is played
\begin{equation*}
\tau = \min \{ t \ge 1: x(t) = x^\star\},
\end{equation*}
is lower bounded by a geometric law
\begin{equation*}
\mathbb{P}( \tau \ge t) \ge (1-p_{\Delta})^{t-1}.
\end{equation*}
Combining this with Remark~\ref{rem:regret_first_time} yields the announced regret bound
\begin{equation*}
R(T,\theta) \ge \Delta \sum_{t=1}^T \mathbb{P}(\tau \ge t) \ge \sum_{t=1}^T (1-p_{\Delta})^{t-1} = \frac{\Delta}{p_\Delta}(1- (1-p_\Delta)^T).
\end{equation*}
\subsection{Proof of Theorem~\ref{th:nonlinear_frequentist_forced}}
\label{proof:nonlinear_frequentist_forced}
At round $t \ge 1$, if the optimal decision has been played ${\ell \over 2}$ times then $A_i(t) = {\ell \over 2}$ and $B_i(t) = 1$ for $i=1,...,m$. In turn the samples $V_i(t)$ are independent with distribution $\text{Beta}(1 + {\ell \over 2},1)$ for $i=1,...,m$.
Lemma \ref{lem:product_beta} shows that at time $t$ the optimal decision is played with probability
\begin{equation*}
\mathbb{P}( \prod_{i=1}^m V_i(t) \ge 1 - \Delta) \le p_{\Delta}^\ell \equiv {1 \over m m !} \left[\left(1+{\ell \over 2}\right) \ln \left({1 \over 1 - \Delta} \right)\right]^{m}.
\end{equation*}
So the distribution of the first time the optimal decision is played
\begin{equation*}
\tau = \min \{ t \ge 1: x(t) = x^\star\},
\end{equation*}
is lower bounded by a geometric law
\begin{equation*}
\mathbb{P}( \tau \ge t) \ge (1-p_{\Delta}^\ell)^{t-1}.
\end{equation*}
Combining this with remark~\ref{rem:regret_first_time} yields the announced regret bound
\begin{equation*}
R(T,\theta) \ge \Delta \sum_{t=1}^T \mathbb{P}(\tau \ge t) \ge \sum_{t=1}^T (1-p_{\Delta}^\ell)^{t-1} = \frac{\Delta}{p_{\Delta}^\ell}(1- (1-p_{\Delta}^\ell)^T).
\end{equation*}
\section{Conclusion}\label{sec:conclusion}
We have shown through both theoretical analysis as well as numerical experiments that TS can perform very poorly in high dimensions, both for both linear and non linear problems, and for various combinatorial structures such as sets of paths and matchings (one could consider more complex combinatorial set including multiple non disjoint paths). Introducing forced exploration does not alleviate the problem either. Therefore, this is not an artifact, but rather a general problem. In essence, Thompson performs poorly because it has a tendency to play much too greedily, and in high dimensions this sometimes leads to a complete lack of exploration and missing the optimal arm. Our work points towards a new challenging open problem which is to design better TS-like algorithms for regret minimization that can deal with high-dimensional problems, while retaining the computational efficiency of TS. Two reasonable ideas to explore would be (i) carefully designing the prior distribution (ii) enforcing forced explorations at regular intervals, possibly in an adaptive manner. Also, our work concerns the Bernoulli setting but we believe that our results can be generalized to bounded distributions. It is not obvious whether or not our results still hold for Gaussian distributions, and seems like an interesting open problem.
\section{Introduction}
\label{sec:introduction}
We consider the problem of combinatorial bandits with semi-bandit feedback. At time $t=1,,...,T$ a learner selects a decision $x(t) \in \mathcal{X}$ where $\mathcal{X} \subset \{0,1\}^d$ is the set of available decisions. The environment then draws a random vector $Z(t) \in \mathbb{R}^d$. The learner then observes $Y(t) = x(t) \odot Z(t)$, where $\odot$ denotes the Hadamard (elementwise) product. This setting is called semi bandit feedback. We assume that $(Z(t))_{t \ge 1}$ are i.i.d., and that $Z_1(t),...,Z_d(t)$ are independent and distributed as $Z_i(t) \sim $ Bernoulli$(\theta_i)$ for all $t$,$i$. Then the learner receives a reward $f(x(t),Z(t))$ where $f$ is a known function.
The goal is to minimize the regret:
\begin{equation*}
R(T,\theta) = T \max_{x \in \mathcal{X}} \Big\{ \mathbb{E} f(x,Z(t)) \Big\} - \sum_{t=1}^T \mathbb{E} f(x(t),Z(t)).
\end{equation*}
Initially $\theta$ is unknown to the learner and minimizing regret involves exploring suboptimal decisions just enough in order to identify the optimal decision. For any decision $x \in \mathcal{X}$, define the reward gap
\begin{equation*}
\Delta_x = \max_{x \in \mathcal{X}} \{ \mathbb{E} f(x,Z(t)) \} - \mathbb{E} f(x,Z(t)),
\end{equation*}
which is the amount of regret incurred by choosing $x$ instead of an optimal decision
\begin{equation*}
x^\star \in \arg \max_{x \in \mathcal{X}} \Big\{ \mathbb{E} f(x,Z(t)) \Big\},
\end{equation*}
and
$
\Delta_{\min} = \min_{x \in \mathcal{X}: \Delta_{x} > 0} \Delta_x
$
the minimal gap. We define $m \triangleq \max_{x \in \mathcal{X}} \sum_{i=1}^d |x_i| $ the size of the maximal decision.
For this problem, an algorithm which has attracted a lot of interest is Thompson Sampling (TS), which at time $t$ selects the decision maximizing $x \mapsto f(x,V(t)) $ where $V(t)$ is a random variable distributed as the posterior distribution of $\theta$ knowing the information available at time $t$, which is $Y(1),...,Y(t-1)$. The prior distribution of $\theta$ can be chosen in various ways, the most natural being a non-informative distribution such as the uniform distribution.
TS is usually computationally simple to implement, for instance when $f$ is linear, since it involves maximizing $f$ over $\mathcal{X}$. Also, for some problem instances it tends to perform well numerically. A particular case of interest is \emph{linear combinatorial semi-bandits} where $f(x,\theta) = \theta^\top x$ so that the reward is a linear function of the decision.
{\bf Our contribution.} We show that the regret of TS in general does not scale polynomially in the ambient dimension $d$.
(i) We provide several examples, both for linear and non-linear combinatorial bandits, where the regret of TS does not scale polynomially in the dimension $d$ (in fact in some cases it may scale even faster than exponentially in the dimension). In some cases, we show that one must wait for an amount of time greater than $\Omega(d^{d})$ for TS to perform at least as well as random choice where one simply chooses $x(t)$ uniformly distributed in ${\cal X}$ at every round. Therefore, in high dimensions, in some instances, TS in general can perform strictly worse than random choice for all practically relevant time horizons.
(ii) We show that the minimax regret of TS scales at least as $\Omega(T^{1 - {1 \over d}} )$ so that it is not minimax optimal, as there exists algorithms such as CUCB and ESCB with minimax regret $O(\textbf{poly}(d) \sqrt{T (\ln T)})$. In fact, in high dimensions, the minimax regret of TS is almost linear.
(iii) We further show that adding forced exploration as an initialization step to TS does not correct the minimax problem, so that this is not an artifact due to initialization.
(iv) Using numerical experiments, we show that indeed, for reasonable time horizons, TS performs very poorly in high dimensions in some instances.
We believe that our results highlight two general characteristics of TS. First, TS tends to be much more greedy than optimistic algorithms such as ESCB and CUCB. This greedy behavior explains why the regret of TS is, in some instances, much smaller than that of optimistic algorithms. In fact it is sometimes so greedy that it misses the optimal decision. Second, TS tends to be by nature a ''risky'' algorithm so that its regret exhibits very large fluctuations across runs. In some cases it finds the optimal arm very quickly and with little to no regret, while in other cases it simply misses the optimal decision and performs worse than random choice.
{\bf Related work.} Combinatorial bandits are a generalization of classical bandits studied in \cite{lai1985}. Several asymptotically optimal algorithms are known for classical bandits, including the algorithm of \cite{lai1987}, KL-UCB \cite{cappe2012}, DMED \cite{honda2010} and TS \cite{thompson1933,kaufmann12a}. Other algorithms include the celebrated UCB1 \cite{auer2002}. A large number of algorithms for combinatorial semi-bandits have been proposed, many of which naturally extend algorithms for classical bandits to the combinatorial setting. CUCB \cite{chen2013,kveton2014tight} is a natural extension of UCB1 to the combinatorial setting. ESCB \cite{combes2015,degenne2016} is an improvement of CUCB which leverages the independence of rewards between items. AESCB \cite{cuvelier2020} is an approximate version of ESCB with roughly the same performance guarantees and reduced computational complexity. TS for combinatorial bandits was considered in \cite{gopalan2014,wang2018,perrault2020}. Also, combinatorial semi bandits are a particular case of structured bandits, for which there exists asymptotically optimal algorithms such as OSSB \cite{combes2017}. Table~\ref{tab:perfs} presents the best known regret upper bounds for CUCB, ESCB and TS. For completeness, we also recall the complete regret upper bound for ESCB as Theorem~\ref{th:escb} (Appendix~\ref{sec:escb}).
We provide two types of bounds for TS: problem dependent bounds (sometimes called gap-dependent bounds) and minimax bounds (or gap-free bounds). The former involves $T$, $d$, $m$ and $\Delta_{\min}$, while the latter hold for any value of $\Delta_{\min}$.
\begin{table}
\caption{Algorithms and best known regret bounds.}
\label{tab:perfs}
\centering
\begin{tabular}{c|c}
Algo.& Regret \\
& ((i) problem dependent and (ii) minimax) \\ \hline
CUCB & (i) $O( d m (\ln T) / \Delta_{min})$
\cite{kveton2014tight}[Theorem 4] \\
& (ii) $O( \sqrt{d m T (\ln T)} + dm)$
\cite{kveton2014tight}[Theorem 6] \\ \hline
ESCB & (i) $O( d (\ln m)^2 (\ln T) / \Delta_{min} + d m^3/\Delta_{\min}^2)$ \cite{degenne2016}[Theorem 2] \\
& (ii) $O( \sqrt{d (\ln m)^2 T (\ln T)} + dm)$
\cite{degenne2016}[Corollary 1] \\ \hline
TS & (i) $O( d (\ln m)^2 \ln (|\mathcal{X}|T) / \Delta_{\min} + d m^3/\Delta_{\min}^2
+ m ((m^2 + 1)/\Delta_{min})^{2 + 4 m})$\\
&\cite{perrault2020}[Theorem 1] \\
& (ii) not available \\ \hline
\end{tabular}
\end{table}
An important observation is that all of the known regret upper bounds for TS \cite{gopalan2014}, \cite{wang2018}, \cite{perrault2020} feature at least one term that does not scale polynomially with the dimension.
In particular, the paper \cite{perrault2020} shows that there exists a universal constant $C \ge 0$ such that the regret of TS is upper bounded by
\begin{align*}
R(T) & \le C \Big[d (\ln m)^2 \ln (|\mathcal{X}|T) / \Delta_{\min} + d m^3/\Delta_{\min}^2
+ m ((m^2 + 1)/\Delta_{min})^{2 + 4 m}) \Big].
\end{align*}
This is a general bound for all combinatorial sets of interest. The bound has a super exponential term (that does not depend on $T$) : $m ((m^2 + 1)/\Delta_{min})^{2 + 4 m})$. Regret bounds for CUCB and ESCB do not feature this exponential dependency in the dimension. However, since TS tends to perform very well in all of the numerical experiments presented in the literature, a natural intuition would be that all known upper bounds are simply not sharp, and that the true regret of TS does not really grow exponentially with the dimension. We show in this work that this intuition is incorrect, and that the regret of TS really does scale (at least) exponentially in the dimension i.e. it suffers from the ''curse of dimensionality''. This directly implies that, in high dimensions, and for some combinatorial sets, CUCB and ESCB perform much better than TS.
In order to alleviate this problem, it would be natural to attempt to modify the prior distribution used by TS, since it is known to have a strong influence on its performance \cite{Bubeck14,Liu16}. Similarly, in a related problem, \cite{Agrawal17} suggests to use correlated Thompson samples, and \cite{Grant20} studies the influence of the prior on numerical performance. We believe that this is an interesting open problem.
\begin{figure}[ht]
\begin{mdframed}
Prior knowledge to the learner: combinatorial set ${\cal X} \subset \{0,1\}^d$, function $f:{\cal X} \times [0,1]^d \to \mathbb{R}$
For $t=1,...,T$:
\hspace{0.5cm} 1. The learner computes statistics
$A(t) = \sum_{s=1}^{t-1} Z(s) \odot x(s)$ and $B(t) = \sum_{s=1}^{t-1} x(s) - Z(s) \odot x(s)$
\hspace{0.5cm} 2. The learner draws $V(t)$ with $V_1(t),...,V_d(t)$ independent and $V_i(t) \sim \text{Beta}(A_i(t)+1,B_i(t)+1)$
\hspace{0.5cm} 3. The learner chooses decision $x(t) \in \arg\max_{x \in \mathcal{X}} \{ f(x,V(t))\}$
\hspace{0.5cm} 4. The environment draws $Z(t)$ with $Z_1(t),...,Z_d(t)$ independent and $Z_i(t) \sim \text{Ber}(\theta_i)$
\hspace{0.5cm} 5. The learner observes $Z(t) \odot x(t)$ and receives the reward $f(x(t),Z(t))$
Performance metric: expected regret $R(T,\theta) = T (\max_{x \in \mathcal{X}} \{ \mathbb{E} f(x,Z(t)) \} ) - \sum_{t=1}^{T} \mathbb{E} f(x(t),Z(t))$
\end{mdframed}
\caption{\label{myfig} TS for combinatorial semi-bandits with Bernoulli rewards and uniform prior.}
\end{figure}
\section{Main Results}\label{sec:mainresult}
We now state our main theoretical results. All proofs are found in the appendix.
\subsection{Some Combinatorial Sets of Interest} \label{subsec : combiset}
We will provide several examples of combinatorial sets where the regret of TS indeed scales exponentially with the dimension, so that this phenomenon is quite general and is not an artifact that only occurs for one particular family of combinatorial structures. We define $\mathcal{X}^{p}$ the set of paths of the directed acyclic graph depicted in figure~\ref{fig:directed_graph_matching}. This set has two disjoint decisions $(1,...,1,0,...,0)$ and $(0,...,0,1,...,1)$ of equal size $m = {d \over 2}$. We define $\mathcal{X}^{m}$ the set of matchings of the bipartite graph depicted in figure~\ref{fig:directed_graph_matching}. This graph has $d$ vertices and $d$ edges.
\begin{figure}[h]
\centering
\includegraphics[width=0.45\linewidth]{directedgraph.pdf} \hfill
\includegraphics[width=0.25\linewidth]{matchinggraph.pdf}
\caption{Paths in a directed acyclic graph (left) and matchings of the Z graph (right)}
\label{fig:directed_graph_matching}
\end{figure}
The combinatorial sets presented are very simple. However our results can by generalised for more complex set of interest without losing the exponential nature of the regret. For example the two path environment can be generalized to $k > 2$ paths. It can also be generalized for non disjoint paths if the optimal path does not share "a lot" of edges with all the other paths. This could be the case for real life applications like shortest path routing or in medical trials where treatments cannot be associated with each other.
With those simple examples in mind many other more complex sets that exhibit exponential regret can be found. However we do not provide formal proof for those more complex examples as the simple example of paths is sufficient to prove the suboptimality of TS here.
\subsection{Linear Combinatorial Bandits}
We focus on linear bandits, where the expected reward function is linear i.e. $f(x,\theta) = x^\top \theta$. In the example of~\cite{wang2018}[Theorem 3], the Thompson sample of sub-optimal decisions has no variance. One could be lead to think that the exponential dependency of the regret on the dimension could be caused by this feature, and it is hence natural to investigate the linear case, which is not only more common, but also where the Thompson sample of any decision always has a non-null variance.
In Theorem~\ref{th:linear_frequentist} we consider a linear problem over the combinatorial set ${\cal X}^p$ which is formed of two disjoint paths. We show that the regret of TS does scale exponentially in the dimension for this problem. Therefore this phenomenon is not linked to a particular, well chosen, non-linear reward function, but also occurs for the classical case of linear reward functions.
\begin{thm}\label{th:linear_frequentist}
Consider a linear combinatorial bandit problem over combinatorial set $\mathcal{X}^{p}$ and parameter $\theta_i = 1$ if $1 \le i \le d/2$ and $\theta_i = 1 - {\Delta \over m}$ otherwise. Assume that ${\Delta \over m}+{1\over \sqrt{m}} < {1\over2}$.
Then the regret of TS is lower bounded by
\begin{equation*}
R(T,\theta) \ge {\Delta \over 4 p_\Delta}(1 - (1-p_{\Delta})^{T-1}), \text{ with }
p_\Delta = \exp\left\{- {2 m\over 9} \left( {1 \over 2} - ({\Delta \over m} + {1 \over \sqrt{m}})\right)^2 \right\} .
\end{equation*}
\end{thm}
Theorem~\ref{th:linear_frequentist} is proven by showing that the first time that the optimal decision is selected is exponentially large in general. The central argument can be summarized as follows. Consider $t$ such that at times $1,...,t$ only the suboptimal decision has been selected.
The probability of selecting the optimal decision is
$$
\mathbb{P}\left( \sum_{i=1}^m V_i(t) \ge \sum_{i=m+1}^d V_i(t) | A(t),B(t) \right)
$$
where $V_1(t),...,V_d(t)$ are independent, distributed in $[0,1]$, and their respective expectations are
$$
\mathbb{E}\Big(V_i(t) | A(t),B(t) \Big) = \begin{cases} {1 \over 2}, & \text{ if } 1 \le i \le m ,\\ {A_i(t) + 1 \over t + 2}, &\text{ if } m+1 \le i \le d .
\end{cases}
$$
Furthermore, from the law of large numbers, when $t$ is large, $$\sum_{i=m+1}^d A_i(t) \approx (1 - {\Delta \over m}) t$$ since we have sampled the sub-optimal decision $t$ times. Therefore
$$
\sum_{i=1}^m \mathbb{E}(V_i(t)|A(t),B(t)) = {m \over 2}
$$
and again because t is large,
$$
\sum_{i=m+1}^{d} \mathbb{E}(V_i(t)|A(t),B(t)) \approx m - \Delta.
$$ Since $V_1(t),...,V_d(t)$ are independent and distributed in $[0,1]$, their sums must concentrate around their expectation, and from Hoeffding's inequality:
$$
\mathbb{P} \Big( \sum_{i=1}^m V_i(t) \ge \sum_{i=m+1}^d V_i(t) | A(t),B(t) \Big) \le O\Big( e^{-u m(({1 \over 2} - {\Delta \over m}))^2 }\Big),
$$
where $u > 0$ is some positive exponent related to how concentrated the Thompson samples are. This implies that, for large $t$, the probability of selecting the optimal decision is exponentially small if it has never been selected previously.
Also, we see that this phenomenon of lack of exploration by TS is a typically high dimensional phenomenon. In short, the Thompson samples of decisions will tend to concentrate around their expectation, so that TS will, most of the time, act greedily and simply select the decision maximizing the empirical reward. We can also emphasize the fact that when $m$ grows, we can have an arbitrary large gap $\Delta$ and still have exponential regret. This is unexpected, since the difficulty of a bandit problem is usually a decreasing function of the gap $\Delta$. Furthermore another version of Theorem \ref{th:linear_frequentist} which exhibits exponential behavior can be shown with parameters $\theta_i = u$ if $1 \le i \le d/2$ and $\theta_i = u - {\Delta \over m}$ otherwise, under the condition ${\Delta \over m}+{1\over \sqrt{m}} < u-{1\over2}$ where $u \in ]{1\over2},1]$. We chose the parameters of Theorem \ref{th:linear_frequentist} for the sake of clarity and being at the edge of the parameter space is not a necessary condition to have exponential regret.
\subsection{Linear Combinatorial Bandits: Small Gap Regime}
Theorem~\ref{th:linear_frequentist_bis} is another regret bound for TS which is more accurate in the regime where $\Delta$ is small, it allows to deduce a lower bound for the minimax regret as well. The proof is a more intricate version of the proof of Theorem~\ref{th:linear_frequentist} highlighted above. Corollary~\ref{co:linear_minimax} states that the minimax regret of TS scales as $O(T)$ for $T \le m!$, and at least as $\Omega(T^{1-{2 \over d}})$ for $T \ge m!$. Of course, in practice, when $m$ is large we have $T \le m!$ for any reasonable time horizon, so that the regret of TS is linear in this regime. Also, as stated by Corollary~\ref{co:linear_minimax}, TS is provably not minimax optimal, as there exist algorithms with minimax regret scaling at most as $O(d^{1/4} \sqrt{T \ln T} )$.
\begin{thm}\label{th:linear_frequentist_bis}
Consider a linear combinatorial bandit problem over combinatorial set $\mathcal{X}^{p}$ and parameter $\theta_i = 1$ if $1 \le i \le d/2$ and $\theta_i = 1 - {\Delta \over m}$ otherwise.
Then there exists universal constants $C_2,C_3$ such that for all $T \ge T_0(m) \equiv C_2 m^2\ln m$, and $\Delta \le 1/6$ and $m \geq 5$, the regret of TS is lower bounded by
\begin{align*}
R(T,\theta) &\ge \frac{\Delta}{4p_\Delta}(1- (1-p_\Delta)^{T_0-1})
+ C_3 \Delta \frac{\left(1- g_\Delta \right)^{T_0}-\left(1-g_\Delta \right)^{T-T_0+1}}{g_\Delta}
\end{align*}
with
\begin{align*}
g_\Delta = {\left(2\Delta\right)^m \over m!} \text{ and }
p_\Delta = \exp\left\{- {2 m\over 9} \left[ {1 \over 2} - \left({\Delta \over m} + {1 \over \sqrt{m}}\right)\right]^2 \right\}
\end{align*}
\end{thm}
\begin{cor}\label{co:linear_minimax}
Consider a linear combinatorial bandit problem over combinatorial set $\mathcal{X}^{p}$ with $m \geq 5$. If $T > 2^mm!$ then the minimax regret of TS is lower bounded by
\begin{equation*}
\max_{\theta \in [0,1]^d} R(T,\theta) \ge C_4 m T^{1 - {2 \over d}}
\end{equation*}
Otherwise it is lower bounded by
\begin{equation*}
\max_{\theta \in [0,1]^d} R(T,\theta) \ge C'_4 T
\end{equation*}
with $C_4,C'_4 > 0$ universal constants.
The minimax regret of ESCB for this set $\mathcal{X}^p$ is upper bounded by:
\begin{equation*}
\max_{\theta \in [0,1]^d} R(T,\theta) \le C_5 d^{1 \over 4} \sqrt{ T \ln T}.
\end{equation*}
with $C_5 \ge 0$ a universal constant. Therefore TS is \emph{not} minimax optimal.
\end{cor}
\subsection{Linear Combinatorial Bandits with Forced Exploration}
We finally extend our results to show that, even when forced exploration is added, TS still provably incurs a regret growing exponentially with the dimension, as stated by Theorem~\ref{th:linear_frequentist_forced} and Theorem~\ref{th:linear_frequentist_forced_minimax}. In particular, if the number of forced exploration rounds $\ell$ satisfies
\begin{equation*}
\ln \left( {1 \over 1-{\Delta \over m}}\right) \le \frac{2\left(\frac{1}{\frac{\ell}{2}+2} -({\Delta \over m} + { 1 \over \sqrt{m}}) \right)^2}{(\frac{\ell}{2}+3)^2 \frac{\ell}{2}}
\end{equation*}
then Theorem~\ref{th:linear_frequentist_forced} implies that the regret of TS still increases exponentially in $m$, in spite of the forced exploration added to the algorithm. In fact it is impossible to set $\ell$ to prevent exponential regret from happening, unless the learner knows the value of the gap $\Delta$ in advance. Indeed, for any fixed $\ell$, the above inequality always holds providing that $\Delta$ is small enough.
\begin{thm}\label{th:linear_frequentist_forced}
Consider a linear combinatorial bandit problem over combinatorial set $\mathcal{X}^{p}$ and parameter $\theta_i = 1$ if $1 \le i \le d/2$ and $\theta_i = 1 - {\Delta \over m}$ otherwise.
Then if ${\Delta \over m} + { 1 \over \sqrt{m} } < \frac{1}{\frac{\ell}{2}+2}$ the regret of TS with $\ell$ forced exploration rounds is lower bounded by
\begin{equation*}
R(T,\theta) \ge
(1-{\Delta \over m})^{m\ell \over 2 }{\Delta \over 4p^{\ell}_\Delta}(1 - (1-p^{\ell}_\Delta)^{T-1}), \text{ with }
\end{equation*}
\begin{equation*}
p^{\ell}_\Delta = \exp\Big\{- 2 m \Big(\frac{1}{\frac{\ell}{2}+2} -({\Delta \over m} + { 1 \over \sqrt{m} }) \Big)^2 / ({\ell \over2}+3)^2 \Big\}.
\end{equation*}
\end{thm}
\begin{thm}\label{th:linear_frequentist_forced_minimax}
Consider a linear combinatorial bandit problem over combinatorial set $\mathcal{X}^{p}$ with $m \geq 5$ and parameter $\theta_i = 1$ if $1 \le i \le d/2$ and $\theta_i = 1 - {\Delta \over m}$ otherwise.
Then if $T>4^m m!$ and $T > T_0(m,l) \equiv C_6 m^2 (\ln m)\ell^{1 \over {1 \over 4} - {1 \over m}}$ the minimax regret of TS with $\ell$ forced exploration rounds is lower bounded by
\begin{equation*}
\max_{\theta \in [0,1]^d} R(T,\theta) \ge C_7 C(\ell,m) {m\over \ell} T^{1 - {1 \over m}}.
\end{equation*}
Otherwise it is bounded by
\begin{equation*}
\max_{\theta \in [0,1]^d} R(T,\theta) \ge C'_7 C(\ell,m) \frac{T}{\ell}.
\end{equation*}
with $C_6, C_7,C_7' > 0$ universal constants and $C$ such that $\forall \ell \in \mathbb{N}, \lim_{m \rightarrow \infty} C(\ell,m) = 1$
\end{thm}
\subsection{Non-Linear Combinatorial Bandits}
We here provide a non-linear combinatorial bandits example. The example is inspired by \cite{wang2018}: there are two decisions, the optimal decision has an expected reward of $1$ and the other one an expected reward of $1-\Delta$. Theorem \ref{th:nonlinear_frequentist} shows that the regret of TS for this problem scales super-exponentially with the dimension $d$ which is an improvement over \cite{wang2018}[Theorem 3]. By corollary, we prove that TS does not outperform random choice (i.e. a trivial algorithm which chooses one of the two decisions uniformly at random at each time) until $t \ge T_0(m)$, where $T_0(m)$ grows super-exponentially with $m$, As an illustration of how large this number might be, for $\Delta = {1 \over 2}$, the value of $T_0(9)$ is greater than a million, and the value of $T_0(20)$ is greater than the estimated age of the universe in seconds. Therefore, in practice as well as in theory, TS does not outperform random choice in high dimensions which is perhaps even more surprising.
The proof of Theorem~\ref{th:nonlinear_frequentist} is based on the fact that there exists a non zero probability that the optimal decision will never be selected for an exponentially large amount of time. Indeed, if the optimal decision has never been selected, it is chosen with a probability equal to $\mathbb{P}( \prod_{i=1}^m U_i \ge 1 - \Delta)$ where $U_1,...,U_m$ are i.i.d. uniformly distributed on $[0,1]$, and since this probability is exponentially small in $d$, one must wait for an exponentially large time before selecting the optimal decision and the regret must scale accordingly. It is noted that this proof technique of lower bounding the expected value of the first time the optimal decision is ever selected is very powerful and will be used many times to prove our results.
\begin{thm}\label{th:nonlinear_frequentist}
Consider a non-linear combinatorial bandit problem over combinatorial set
$
\mathcal{X} = \left\{ \sum_{i=1}^{m} e_i , e_{m+1} \right\}
$
where $(e_i)_{i\in [m+1]}$ is the canonical base of $\mathbb{R}^{m+1} $ with parameter $\theta = (1,...,1)$ and reward function $f(x,\theta) = \prod_{i=1}^{m} \theta_i$ if $x = \sum_{i=1}^{m} e_i$ and $f(x,\theta) = 1 - \Delta$ otherwise.
Then the regret of TS is lower bounded by
\begin{align*}
R(T,\theta) &\ge {\Delta \over p_\Delta}(1 - (1-p_\Delta)^{T}) \text{ with }
p_{\Delta} = {1 \over m m !} \left[\ln \left({1 \over 1 - \Delta} \right)\right]^{m}.
\end{align*}
\end{thm}
\begin{cor}
For any $T \le T_0(m) \equiv {1 \over p_\Delta}$ TS performs strictly worse than random choice in the sense that
\begin{align*}
R(T,\theta) &\ge T \Delta \left(1 - {1 \over e}\right) > {T \Delta \over 2}.
\end{align*}
\end{cor}
It is noted that Theorem~\ref{th:nonlinear_frequentist} is a parameter-dependent lower bound, where we consider a fixed parameter $\theta$ and we let the time horizon $T$ grow. From Theorem~\ref{th:nonlinear_frequentist} we deduce Corollary \ref{cor:nonlinear_minimax} which is a lower bound on the minimax regret of TS. The minimax regret of TS scales at least as $\Omega(T^{1 - {1 \over d}})$, so that it is almost linear in high dimensions when $d$ is large. This also proves that, as long as the dimension $d$ is strictly greater than $2$, TS is not minimax optimal, since there exists algorithms such as CUCB whose minimax regret scales at most as $O({\bf poly}(d) \sqrt{T \ln T})$. This demonstrates that TS has a tendency to be too "greedy" which prevents it from exploring enough, and while this is not a problem in low dimensions, in high dimensions this matters a great deal, and causes it to perform much worse than optimistic algorithms. Corollary \ref{cor:nonlinear_minimax} is proven simply by letting $\Delta = {T^{-{1 \over d}}}$ in Theorem~\ref{th:nonlinear_frequentist} and the regret upper bound for CUCB follows directly from \cite{kveton2014tight}.
\begin{cor}\label{cor:nonlinear_minimax}
Consider ${\cal F}$ the class of $1$-Lipschitz functions.
The minimax regret of TS is lower bounded by:
\begin{equation*}
\max_{\theta \in [0,1]^d, f \in {\cal F}} R(T,\theta,f) \ge C_1 T^{1 - {1 \over d}},
\end{equation*}
with $C_1 > 0$ a universal constant, the minimax regret of CUCB is upper bounded by
\begin{equation*}
\max_{\theta \in [0,1]^d, f \in {\cal F}} R(T,\theta,f) \le C_1' d \sqrt{T \ln T},
\end{equation*}
where $C_1'$ is a universal constant. Hence TS is \emph{not} minimax optimal.
\end{cor}
\subsection{Non-Linear Combinatorial Bandits with Forced Exploration}
Our results above show that the regret of TS scales exponentially in the dimension since the expectation of the first time at which the optimal decision is selected can grow exponentially in the dimension. Therefore it is natural to assume that forcing some exploration initially would alleviate the problem. Theorem~\ref{th:nonlinear_frequentist_forced} considers the same non linear bandit problem as that considered in Theorem~\ref{th:nonlinear_frequentist}, and shows that, while forced exploration does bring some improvement, for any fixed value of $\ell > 0$, the regret of TS with $\ell$ forced exploration rounds still scales exponentially in the dimension. The reason for this is that once again the first time at which the optimal decision is selected can be exponentially large, even with forced exploration. Upon closer inspection of Theorem~\ref{th:nonlinear_frequentist_forced}, one can see that, in order for the regret lower bound not to scale exponentially in the dimension one would require $1/ p^{\ell}_{\Delta}$ to grow at most polynomially in $d$, which in turn would require
$
{\ell \over 2} \ln \left({1 \over 1 - \Delta} \right) \ge 1.
$
This indicates that, unless the gap $\Delta$ is known in advance (and in general $\Delta$ is of course unknown), it is not possible to select a value of $\ell$ that prevents the regret from scaling exponentially in the dimension. This suggests that some more complex modifications need to be made to TS in order to "fix" this exponential dependency on the dimension.
\begin{thm}\label{th:nonlinear_frequentist_forced}
Consider a non-linear combinatorial bandit problem with over combinatorial set
$
\mathcal{X} = \left\{ \sum_{i=1}^{m} e_i , e_{m+1} \right\}
$
with parameter $\theta = (1,...,1)$ and reward function $f(x,\theta) = \prod_{i=1}^{m} \theta_i$ if $x = \sum_{i=1}^{m} e_i$ and $f(x,\theta) = 1 - \Delta$ otherwise.
Then the regret of TS with $\ell$ forced exploration rounds is lower bounded by
\begin{equation*}
R(T,\theta) \ge {\Delta \over p^\ell_\Delta}(1 - (1-p^\ell_\Delta)^{T}) \text{ with }
p^\ell_{\Delta} = {1 \over m m !} \left[\left(1+{\ell \over 2}\right) \ln \left({1 \over 1 - \Delta} \right)\right]^{m}.
\end{equation*}
\end{thm}
\section{Model}\label{sec:model}
\subsection{Problem Dependent Regret and Minimax Regret}
In order to evaluate the performance of an algorithm over the set of instances $\theta \in [0,1]^d$, there are two main figures of merit that we study in this paper. The first is the problem-dependent regret which is $R(T,\theta)$ when $\theta \in [0,1]^d$ is fixed. The second is the minimax regret which is the worse case over $\theta$ for $T$ fixed:
$
\max_{\theta \in [0,1]^d} R(T,\theta).
$
\subsection{TS}
The basic TS algorithm works as follows. For $i=1,...,d$, define
$
A_i(t) = \sum_{s=1}^{t-1} Z_i(s) x_i(s)
$
and
$
B_i(t) = \sum_{s=1}^{t-1} (1-Z_i(s)) x_i(s)
$
which represent the number of successes and failures observed when getting a sample to estimate $\theta_i$. We define
$
N_i(t) = A_i(t) + B_i(t) = \sum_{s=1}^{t-1} x_i(s)
$
the number of samples available at time $t$ to estimate $\theta_i$, and
\begin{equation*}
\hat\theta_i(t) = {A_i(t) \over \max(N_i(t),1)},
\end{equation*}
the corresponding estimate of $\theta_i$ which is simply the empirical mean.
The TS algorithm selects decision
\begin{equation*}
x(t) \in \arg\max_{x \in \mathcal{X}} f(x,V(t)) \text{ where } V_i(t) \sim \text{Beta}( A_i(t) + 1,B_i(t) + 1),
\end{equation*}
and $V_1(t),...,V_d(t)$ are independent. Vector $V(t)$ is called the \emph{Thompson sample} at time $t$. TS is based on a Bayesian argument, $V(t)$ is drawn according to the posterior distribution of $\theta$ knowing the information available at time $t$, where the prior distribution for $\theta$ is uniform over $[0,1]^d$. Choosing decision $x(t)$ as done above should ensure that one explores just enough to find the optimal decision. In the linear case, the decision can be computed by linear maximization over $\mathcal{X}$ :
\begin{equation*}
x(t) \in \arg\max_{x \in \mathcal{X}} \{ V(t)^\top x \}.
\end{equation*}
This explains the practical appeal of TS, since whenever linear maximization over $\mathcal{X}$ can be implemented efficiently, the algorithm has low computational complexity.
\subsection{TS with Forced Exploration}
A natural extension of TS is to add $\ell$ forced exploration rounds, where $\ell$ is a fixed number, in order to avoid some artifacts that could possibly occur due to the prior distribution. The algorithm operates as follows. At time $1 \le t \le \ell$, one selects $x(t) \in \mathcal{X}$ such that $x_{i(t)}(t) = 1$ with
\begin{equation*}
i(t) \in \min_{i=1,...,d} N_i(t).
\end{equation*}
Otherwise for $t \ge \ell+1$ one selects
\begin{equation*}
x(t) \in \arg\max_{x \in \mathcal{X}} f(x,V(t)),
\end{equation*}
with $V(t)$ the Thompson sample defined above. Namely, one first performs a forced exploration during $\ell$ rounds then apply TS. This guarantees that $N_i(t) \ge \lfloor \ell/d \rfloor$ samples are available to estimate $\theta_i$ for all $i=1,...,d$, then one subsequently applies TS. We call this variant TS with $\ell$ forced exploration rounds.
\section{Numerical Experiments}\label{sec:numericalexperiments}
We now illustrate the exponential regret of TS in practical settings using numerical experiments. Due to this exponential nature, some of those experiments involve a significant amount of computing time in high dimensions. Due to limited space, we solely consider the linear case, which is the most often considered in the literature. Unless specified otherwise we use $1000$ independent sample paths for averaging, and $95\%$ confidence intervals are presented on the plots.
\paragraph{First selection of the optimal decision} As shown by our theoretical results, the first time that the optimal decision is selected
$
\tau = \min \{ t \ge 1: x(t) = x^\star \}
$
can be exponentially large, and this is what causes exponential regret.
On Figure~\ref{fig:2dz_ECDF}, we present the c.d.f. (cumulative distribution function) of $\tau$ as a function of $m$ for combinatorial sets ${\cal X}^p$ and ${\cal X}^m$ introduced above. The parameter values are chosen as in the previous sections $\theta_i = 1$ if $1 \le i \le d/2$ and $\theta_i = {\Delta \over m}$ otherwise. For each sample path we generate $\tau$ by simulating TS until the optimal decision is played for the first time.
Some quantiles of $\tau$ indeed seem to increase exponentially as $m$ grows and for reasonable values of $m$, $\tau$ can be very large with appreciable probability, for instance on Fig. \ref{fig:2dz_ECDF} for $m = 14$, $\tau \ge 5.10^{4}$ with probability greater than $0.1$. Clearly, on sample paths where this happens, TS performs worse than random choice for the first $5.10^{4}$ time steps which is a surprisingly poor behaviour, especially on such a simple problem. This also showcases the fact that those sample paths happen relatively often. Thus the regret of TS is not only due to very rare occasions with high regret but also because of those poor behavior that can happen quite often.
\begin{figure}[ht]
\centering
\begin{tabular}{cc}
\includegraphics[width = 0.4\linewidth]{2paths_ECDF_delta02_m6_l0.png}
&\includegraphics[width = 0.4\linewidth]{2paths_ECDF_delta02_m14_l0.png}\\
$m = 6,{\Delta \over m} = {1 \over 5}$ on ${\cal X}^p$ & $m = 14,{\Delta \over m} = {1 \over 5}$ on ${\cal X}^p$ \\
\includegraphics[width = 0.4\linewidth]{zgraph_ECDF_delta0125_m4_l0.png} &\includegraphics[width = 0.4\linewidth]{zgraph_ECDF_delta0125_m6_l0.png} \\
$m=4, {\Delta \over m} = {1 \over 8}$ on ${\cal X}^m$ & $m=6,{\Delta \over m} = {1 \over 8}$ on ${\cal X}^m$
\end{tabular}
\caption{C.d.f. of the first time the optimal decision is played $\tau$ as a function of $m$ for set of paths and matchings ${\cal X}^p$, ${\cal X}^m$}
\label{fig:2dz_ECDF}
\end{figure}
To investigate the impact of the gap $\Delta$, on Figure~\ref{fig:TS_first_opti_2decision} and Figure~ \ref{fig:TS_first_opti_zgraph}, we plot the expected first time the optimal decision is selected $\mathbb{E}(\tau)$ as a function of $m$ for various $\delta = {\Delta \over m}$, for the sets of paths ${\cal X}^p$ and matchings ${\cal X}^m$. Once again we observe an exponential growth in both figures, and this growth is particularly fast for small values of $\delta$. When the gap gets small, the exponential growth of regret is exacerbated, leading to an even worse performance.
\begin{figure}[ht]%
\centering
\subfloat[Set of paths ${\cal X}^p$ combinatorial set]{\label{fig:TS_first_opti_2decision} \includegraphics[width = 0.45\linewidth]{2paths_First_opti_diffdelta.png}}%
\subfloat[Set of matchings ${\cal X}^m$ combinatorial set]{\label{fig:TS_first_opti_zgraph}\includegraphics[width = 0.45\linewidth]{zgraph_First_opti_differentdelta.png} }%
\caption{Expectation of the first time the optimal decision is played $\mathbb{E}(\tau)$ as a function of $m$ and various values of ${\Delta \over m} = \delta$}
\end{figure}
\paragraph{Impact of forced exploration} We now investigate if forced exploration alleviates the problem in practice, and consider $\ell$ forced exploration rounds. On figures \ref{fig:TS_first_opti_2decision_fe} and \ref{fig:TS_first_opti_2decision_fe_varl} we plot the expected first time the optimal decision is selected $\mathbb{E}(\tau)$ as a function of $m$ for various $\delta = {\Delta \over m}$ and $\ell$, for the sets of paths ${\cal X}^p$. As predicted by Theorem \ref{th:linear_frequentist_forced}, $\mathbb{E}(\tau)$ still seems to scale exponentially in $m$ which causes exponential regret.
Theorem~\ref{th:linear_frequentist_forced} states that if $\ell$ is chosen such that ${\Delta \over m} < {2 \over \ell + 1}$ then regret scales exponentially. On the other hand one could think that when $\ell$ is chosen large enough to violate this condition then regret does not grow as rapidly. Figure~\ref{fig:TS_first_opti_2decision_fe_varl} shows that, at least numerically, this does not appear to be the case, indeed, we choose $\ell = m$, and there are values of $\delta$ such that the regret still seem to scale exponentially in $m$.
\begin{figure}[ht]%
\centering
\subfloat[Set of paths ${\cal X}^p$ combinatorial set]{\label{fig:TS_first_opti_2decision_fe}\includegraphics[width = 0.45\linewidth]{2paths_First_opti_diffdelta_diffl.png} }%
\subfloat[Set of matchings ${\cal X}^m$ combinatorial set ${\Delta \over m} = {1\over100}$ and $\ell=m$]{\label{fig:TS_first_opti_2decision_fe_varl}\includegraphics[width = 0.45\linewidth]{2paths_First_opti_varl.png} }%
\caption{Expectation of the first time the optimal decision is played $\mathbb{E}(\tau)$ as a function of $m$ and various values of ${\Delta \over m} = \delta$ and $\ell$}
\label{fig:MFL}%
\end{figure}
\paragraph{Comparison with optimistic algorithms} We now compare TS with the state-of-the-art frequentist algorithms ESCB and/or CUCB. These experiments are averaged over $40$ sample paths due to computational limits. On figures~\ref{fig:Regret_comparison_2d} and~\ref{fig:Regret_comparison_z} we present the regret as a function of $m$ for the set of paths ${\cal X}^p$, ${\Delta \over m} = 0.1$, $T = 4.10^4$ and the set of matchings ${\cal X}^m$, ${\Delta \over m} = 0.05$, $T = 4.10^4$ respectively. The results show that the regret of TS is larger than that of CUCB and/or ESCB by several orders of magnitude in high dimensions, as predicted by our theoretical results. In fact the regret of TS is so overwhelmingly large that, due to the scale of the figure, it looks like the regret of ESCB and/or CUCB does not increase with the dimension (this is of course not the case). On figures \ref{fig:TS_regret_2decision_delta_var} and \ref{fig:TS_regret_zgraph_delta_var} we perform similar experiments but with smaller gaps, ${\Delta \over m} = {1 \over m}$ and the same behaviour arises.
\begin{figure}[ht]%
\centering
\subfloat[$\delta = 0.1$ and $T = 4.10^4$, Set of paths ${\cal X}^p$]{\label{fig:Regret_comparison_2d}\includegraphics[width = 0.45\linewidth]{2paths_Regret_delta01.png} }%
\subfloat[$\delta = 0.05$ and $T = 10^5$, set of matchings ${\cal X}^m$]{\label{fig:Regret_comparison_z}\includegraphics[width = 0.45\linewidth]{zgraph_Regret_delta005.png} }%
\caption{Regret comparison between ESCB and TS for set of paths ${\cal X}^p$ and matching ${\cal X}^m$ (Averaged over 40 experiences)}
\end{figure}
\begin{figure}[h!]%
\centering
\subfloat[$\Delta = 1$ and $T = 4.10^5$, set of paths ${\cal X}^p$]{\label{fig:TS_regret_2decision_delta_var}\includegraphics[width = 0.45\linewidth]{2paths_Regret_vardelta.png} }%
\subfloat[$\Delta = 1$ and $T = 10^5$, set of matchings ${\cal X}^m$]{\label{fig:TS_regret_zgraph_delta_var}\includegraphics[width = 0.45\linewidth]{zgraph_Regret_Vardelta.png} }%
\caption{Regret comparison between ESCB and TS for set of paths ${\cal X}^p$ and matching ${\cal X}^m$ For a fixed $\Delta$ (Averaged over 20 and 40 experiences)}
\end{figure}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,381 |
WASHINGTON - As open enrollment begins this weekend for the third year of Obamacare, insurance carriers in Arizona are "still trying to figure ... out" which types of plans work best for them and for their customers.
At the root of the Phoenix Suns new "WeArePHX" marketing campaign is the team's nearly 50-year connection to the Valley, a connection that goes straight to Phoenix City Hall.
More than 2,000 people attended the inaugural Southwest Cannabis Conference and Expo at the Phoenix Convention Center this week.
SCOTTSDALE — From biology to bowling, officials with Scottsdale-based tutoring company AvidBrain said they offer lessons for a variety of subjects.
The historic Welnick Bros. Marketplace building in downtown Phoenix has sat vacant for the past decade. But now, developers plan to restore it as an adaptive reuse project with restaurant, retail and entertainment space.
Editor's note: FYI is an ongoing series of multimedia storytelling that explain various issues and topics that impact the Valley. Halloween is just around the corner and Americans are busting out their scary decorations and jack-o-lanterns. But how much does it all cost? National Retail Federation and U.S. Census data have some answers. This graphic displays Halloween facts about money, activities and the industry. And be sure to take the quiz at the end of the graphic to test your knowledge on Halloween business.
Three of the Valley's most-accomplished businesswomen were awarded the Athena Award by the Greater Phoenix Chamber of Commerce on Thursday in its annual program to support women in leadership.
More than 170 times last year, someone pointed a laser at an aircraft in Arizona, according to data compiled by the Federal Aviation Administration, and pilots and passengers are in agreement that the perpetrators need to be punished.
The Verizon IndyCar Series will return to Phoenix International Raceway (PIR) for the first time in over a decade.
The clothes on his back and crumbled documents were the only two possessions Ethiopian refugee Anduale Hassan had to his name after stepping off of the plane at Sky Harbor International Airport. A government official stared when he first saw Hassan.
WASHINGTON - With more than a year until the election, congressional candidates in Arizona have already spent $5.5 million on their campaigns - for everything from consultants to coffee mugs, according to the latest Federal Election Commission filings.
ParkX has won Phoenix's Smart City App Hack competition and will represent the city at the global competition in Barcelona, Spain, in November. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,689 |
Q: Null Pointer Exception occurs while starting fragment I am facing this exception while starting the app on tablet. It works perfectly fine on phone.
> 05-06 00:34:48.213: E/AndroidRuntime(545): Caused by:
> java.lang.NullPointerException: name == null 05-06 00:34:48.213:
> E/AndroidRuntime(545): at
> java.lang.VMClassLoader.findLoadedClass(Native Method) 05-06
> 00:34:48.213: E/AndroidRuntime(545): at
> java.lang.ClassLoader.findLoadedClass(ClassLoader.java:354) 05-06
> 00:34:48.213: E/AndroidRuntime(545): at
> java.lang.ClassLoader.loadClass(ClassLoader.java:491) 05-06
> 00:34:48.213: E/AndroidRuntime(545): at
> java.lang.ClassLoader.loadClass(ClassLoader.java:461) 05-06
> 00:34:48.213: E/AndroidRuntime(545): at
> android.app.Fragment.instantiate(Fragment.java:562) 05-06
> 00:34:48.213: E/AndroidRuntime(545): at
> android.preference.PreferenceActivity.switchToHeaderInner(PreferenceActivity.java:1117)
> 05-06 00:34:48.213: E/AndroidRuntime(545): at
> android.preference.PreferenceActivity.switchToHeader(PreferenceActivity.java:1150)
> 05-06 00:34:48.213: E/AndroidRuntime(545): at
> android.preference.PreferenceActivity.onCreate(PreferenceActivity.java:551)
It looks like if the headers don't have a fragment defined, it fails on tablet as it tries to focus on the first header by default. My first header is not having a fragment, hence, the installation is failing probably. Is there a way to get around this ?
Thanks!
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,777 |
Ilba – wieś w Rumunii, w okręgu Marmarosz, w gminie Cicârlău. W 2011 roku liczyła 1220 mieszkańców.
Przypisy
Wsie w okręgu Marmarosz | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,840 |
package com.vladsch.md.nav.psi.element;
import com.intellij.lang.ASTNode;
import org.jetbrains.annotations.NotNull;
import org.jetbrains.annotations.Nullable;
public interface MdAdmonition extends MdIndentingComposite, MdBlockElementWithChildren {
@NotNull
String getMarker();
@Nullable
ASTNode getInfoNode();
@NotNull
String getInfo();
@Nullable
ASTNode getTitleElement();
@Nullable
String getTitle();
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,434 |
WINRT_EXPORT namespace winrt {
namespace ABI::Windows::Foundation {
#ifndef WINRT_GENERIC_7b44e581_cfa9_5763_bed7_6a65739f0dbf
#define WINRT_GENERIC_7b44e581_cfa9_5763_bed7_6a65739f0dbf
template <> struct __declspec(uuid("7b44e581-cfa9-5763-bed7-6a65739f0dbf")) __declspec(novtable) IAsyncOperation<winrt::Windows::ApplicationModel::Background::BackgroundAccessStatus> : impl_IAsyncOperation<winrt::Windows::ApplicationModel::Background::BackgroundAccessStatus> {};
#endif
#ifndef WINRT_GENERIC_47cbd985_0f08_5a3d_92cf_b27960506ed6
#define WINRT_GENERIC_47cbd985_0f08_5a3d_92cf_b27960506ed6
template <> struct __declspec(uuid("47cbd985-0f08-5a3d-92cf-b27960506ed6")) __declspec(novtable) IAsyncOperation<winrt::Windows::ApplicationModel::Background::ApplicationTriggerResult> : impl_IAsyncOperation<winrt::Windows::ApplicationModel::Background::ApplicationTriggerResult> {};
#endif
#ifndef WINRT_GENERIC_2595482c_1cbf_5691_a30d_2164909c6712
#define WINRT_GENERIC_2595482c_1cbf_5691_a30d_2164909c6712
template <> struct __declspec(uuid("2595482c-1cbf-5691-a30d-2164909c6712")) __declspec(novtable) IAsyncOperation<winrt::Windows::ApplicationModel::Background::MediaProcessingTriggerResult> : impl_IAsyncOperation<winrt::Windows::ApplicationModel::Background::MediaProcessingTriggerResult> {};
#endif
}
namespace ABI::Windows::Foundation::Collections {
#ifndef WINRT_GENERIC_78c880f6_a7dc_5172_89da_7749fc82aa82
#define WINRT_GENERIC_78c880f6_a7dc_5172_89da_7749fc82aa82
template <> struct __declspec(uuid("78c880f6-a7dc-5172-89da-7749fc82aa82")) __declspec(novtable) IMapView<GUID, Windows::ApplicationModel::Background::IBackgroundTaskRegistration> : impl_IMapView<GUID, Windows::ApplicationModel::Background::IBackgroundTaskRegistration> {};
#endif
#ifndef WINRT_GENERIC_851e3cfd_306b_5c8e_ae3c_a8d83c623604
#define WINRT_GENERIC_851e3cfd_306b_5c8e_ae3c_a8d83c623604
template <> struct __declspec(uuid("851e3cfd-306b-5c8e-ae3c-a8d83c623604")) __declspec(novtable) IIterable<Windows::Storage::StorageLibrary> : impl_IIterable<Windows::Storage::StorageLibrary> {};
#endif
}
namespace ABI::Windows::Foundation {
#ifndef WINRT_GENERIC_b5136c46_2f2e_511d_9e8e_5ef4decb1da7
#define WINRT_GENERIC_b5136c46_2f2e_511d_9e8e_5ef4decb1da7
template <> struct __declspec(uuid("b5136c46-2f2e-511d-9e8e-5ef4decb1da7")) __declspec(novtable) IAsyncOperation<winrt::Windows::ApplicationModel::Background::DeviceTriggerResult> : impl_IAsyncOperation<winrt::Windows::ApplicationModel::Background::DeviceTriggerResult> {};
#endif
#ifndef WINRT_GENERIC_9d01424d_5653_59f8_ba6b_d0c077346d2d
#define WINRT_GENERIC_9d01424d_5653_59f8_ba6b_d0c077346d2d
template <> struct __declspec(uuid("9d01424d-5653-59f8-ba6b-d0c077346d2d")) __declspec(novtable) IAsyncOperation<Windows::ApplicationModel::Background::DeviceConnectionChangeTrigger> : impl_IAsyncOperation<Windows::ApplicationModel::Background::DeviceConnectionChangeTrigger> {};
#endif
}
namespace ABI::Windows::Foundation::Collections {
#ifndef WINRT_GENERIC_e3e660d6_d041_5ecd_b18b_fa254e4a860f
#define WINRT_GENERIC_e3e660d6_d041_5ecd_b18b_fa254e4a860f
template <> struct __declspec(uuid("e3e660d6-d041-5ecd-b18b-fa254e4a860f")) __declspec(novtable) IVector<winrt::Windows::Devices::Sensors::ActivityType> : impl_IVector<winrt::Windows::Devices::Sensors::ActivityType> {};
#endif
#ifndef WINRT_GENERIC_fc7a0488_2803_505c_9e62_9200afe416c6
#define WINRT_GENERIC_fc7a0488_2803_505c_9e62_9200afe416c6
template <> struct __declspec(uuid("fc7a0488-2803-505c-9e62-9200afe416c6")) __declspec(novtable) IVectorView<winrt::Windows::Devices::Sensors::ActivityType> : impl_IVectorView<winrt::Windows::Devices::Sensors::ActivityType> {};
#endif
}
namespace ABI::Windows::Foundation {
#ifndef WINRT_GENERIC_a55a747d_59f6_5cb6_b439_c8aad670905c
#define WINRT_GENERIC_a55a747d_59f6_5cb6_b439_c8aad670905c
template <> struct __declspec(uuid("a55a747d-59f6-5cb6-b439-c8aad670905c")) __declspec(novtable) IAsyncOperation<winrt::Windows::ApplicationModel::Background::AlarmAccessStatus> : impl_IAsyncOperation<winrt::Windows::ApplicationModel::Background::AlarmAccessStatus> {};
#endif
#ifndef WINRT_GENERIC_26dd26e3_3f47_5709_b2f2_d6d0ad3288f0
#define WINRT_GENERIC_26dd26e3_3f47_5709_b2f2_d6d0ad3288f0
template <> struct __declspec(uuid("26dd26e3-3f47-5709-b2f2-d6d0ad3288f0")) __declspec(novtable) AsyncOperationCompletedHandler<winrt::Windows::ApplicationModel::Background::BackgroundAccessStatus> : impl_AsyncOperationCompletedHandler<winrt::Windows::ApplicationModel::Background::BackgroundAccessStatus> {};
#endif
#ifndef WINRT_GENERIC_d0065ef6_ee9d_55f8_ac2b_53a91ff96d2e
#define WINRT_GENERIC_d0065ef6_ee9d_55f8_ac2b_53a91ff96d2e
template <> struct __declspec(uuid("d0065ef6-ee9d-55f8-ac2b-53a91ff96d2e")) __declspec(novtable) AsyncOperationCompletedHandler<winrt::Windows::ApplicationModel::Background::ApplicationTriggerResult> : impl_AsyncOperationCompletedHandler<winrt::Windows::ApplicationModel::Background::ApplicationTriggerResult> {};
#endif
#ifndef WINRT_GENERIC_3814c6a5_2ad1_5875_bed5_5031cd1f50a2
#define WINRT_GENERIC_3814c6a5_2ad1_5875_bed5_5031cd1f50a2
template <> struct __declspec(uuid("3814c6a5-2ad1-5875-bed5-5031cd1f50a2")) __declspec(novtable) AsyncOperationCompletedHandler<winrt::Windows::ApplicationModel::Background::MediaProcessingTriggerResult> : impl_AsyncOperationCompletedHandler<winrt::Windows::ApplicationModel::Background::MediaProcessingTriggerResult> {};
#endif
}
namespace ABI::Windows::Foundation::Collections {
#ifndef WINRT_GENERIC_5a1f6d75_8678_547c_8fd7_fbceb6ebf968
#define WINRT_GENERIC_5a1f6d75_8678_547c_8fd7_fbceb6ebf968
template <> struct __declspec(uuid("5a1f6d75-8678-547c-8fd7-fbceb6ebf968")) __declspec(novtable) IKeyValuePair<GUID, Windows::ApplicationModel::Background::IBackgroundTaskRegistration> : impl_IKeyValuePair<GUID, Windows::ApplicationModel::Background::IBackgroundTaskRegistration> {};
#endif
#ifndef WINRT_GENERIC_0a1c6409_fbd3_58c8_9af3_6262cc56e5b3
#define WINRT_GENERIC_0a1c6409_fbd3_58c8_9af3_6262cc56e5b3
template <> struct __declspec(uuid("0a1c6409-fbd3-58c8-9af3-6262cc56e5b3")) __declspec(novtable) IIterator<Windows::Storage::StorageLibrary> : impl_IIterator<Windows::Storage::StorageLibrary> {};
#endif
}
namespace ABI::Windows::Foundation {
#ifndef WINRT_GENERIC_d5aa9506_1464_57d4_859d_7ee9b26cb1f9
#define WINRT_GENERIC_d5aa9506_1464_57d4_859d_7ee9b26cb1f9
template <> struct __declspec(uuid("d5aa9506-1464-57d4-859d-7ee9b26cb1f9")) __declspec(novtable) AsyncOperationCompletedHandler<winrt::Windows::ApplicationModel::Background::DeviceTriggerResult> : impl_AsyncOperationCompletedHandler<winrt::Windows::ApplicationModel::Background::DeviceTriggerResult> {};
#endif
#ifndef WINRT_GENERIC_3fd5a57e_47e4_5921_b148_5cb586166ca8
#define WINRT_GENERIC_3fd5a57e_47e4_5921_b148_5cb586166ca8
template <> struct __declspec(uuid("3fd5a57e-47e4-5921-b148-5cb586166ca8")) __declspec(novtable) AsyncOperationCompletedHandler<Windows::ApplicationModel::Background::DeviceConnectionChangeTrigger> : impl_AsyncOperationCompletedHandler<Windows::ApplicationModel::Background::DeviceConnectionChangeTrigger> {};
#endif
}
namespace ABI::Windows::Foundation::Collections {
#ifndef WINRT_GENERIC_40524281_a7c6_50b1_b6f5_0baa95d902c2
#define WINRT_GENERIC_40524281_a7c6_50b1_b6f5_0baa95d902c2
template <> struct __declspec(uuid("40524281-a7c6-50b1-b6f5-0baa95d902c2")) __declspec(novtable) IIterator<winrt::Windows::Devices::Sensors::ActivityType> : impl_IIterator<winrt::Windows::Devices::Sensors::ActivityType> {};
#endif
#ifndef WINRT_GENERIC_2a04cdfa_5dfd_5178_8731_ade998e4a7f6
#define WINRT_GENERIC_2a04cdfa_5dfd_5178_8731_ade998e4a7f6
template <> struct __declspec(uuid("2a04cdfa-5dfd-5178-8731-ade998e4a7f6")) __declspec(novtable) IIterable<winrt::Windows::Devices::Sensors::ActivityType> : impl_IIterable<winrt::Windows::Devices::Sensors::ActivityType> {};
#endif
}
namespace ABI::Windows::Foundation {
#ifndef WINRT_GENERIC_84108017_a8e7_5449_b713_df48503a953e
#define WINRT_GENERIC_84108017_a8e7_5449_b713_df48503a953e
template <> struct __declspec(uuid("84108017-a8e7-5449-b713-df48503a953e")) __declspec(novtable) AsyncOperationCompletedHandler<winrt::Windows::ApplicationModel::Background::AlarmAccessStatus> : impl_AsyncOperationCompletedHandler<winrt::Windows::ApplicationModel::Background::AlarmAccessStatus> {};
#endif
}
namespace ABI::Windows::Foundation::Collections {
#ifndef WINRT_GENERIC_2001aea5_1a86_517e_8be5_11d7fb5935b2
#define WINRT_GENERIC_2001aea5_1a86_517e_8be5_11d7fb5935b2
template <> struct __declspec(uuid("2001aea5-1a86-517e-8be5-11d7fb5935b2")) __declspec(novtable) IIterator<Windows::Foundation::Collections::IKeyValuePair<GUID, Windows::ApplicationModel::Background::IBackgroundTaskRegistration>> : impl_IIterator<Windows::Foundation::Collections::IKeyValuePair<GUID, Windows::ApplicationModel::Background::IBackgroundTaskRegistration>> {};
#endif
#ifndef WINRT_GENERIC_80fb0327_5a00_55cc_85db_a852719981b9
#define WINRT_GENERIC_80fb0327_5a00_55cc_85db_a852719981b9
template <> struct __declspec(uuid("80fb0327-5a00-55cc-85db-a852719981b9")) __declspec(novtable) IIterable<Windows::Foundation::Collections::IKeyValuePair<GUID, Windows::ApplicationModel::Background::IBackgroundTaskRegistration>> : impl_IIterable<Windows::Foundation::Collections::IKeyValuePair<GUID, Windows::ApplicationModel::Background::IBackgroundTaskRegistration>> {};
#endif
}
namespace Windows::ApplicationModel::Background {
template <typename D>
struct WINRT_EBO impl_IActivitySensorTrigger
{
Windows::Foundation::Collections::IVector<winrt::Windows::Devices::Sensors::ActivityType> SubscribedActivities() const;
uint32_t ReportInterval() const;
Windows::Foundation::Collections::IVectorView<winrt::Windows::Devices::Sensors::ActivityType> SupportedActivities() const;
uint32_t MinimumReportInterval() const;
};
template <typename D>
struct WINRT_EBO impl_IActivitySensorTriggerFactory
{
Windows::ApplicationModel::Background::ActivitySensorTrigger Create(uint32_t reportIntervalInMilliseconds) const;
};
template <typename D>
struct WINRT_EBO impl_IAlarmApplicationManagerStatics
{
Windows::Foundation::IAsyncOperation<winrt::Windows::ApplicationModel::Background::AlarmAccessStatus> RequestAccessAsync() const;
Windows::ApplicationModel::Background::AlarmAccessStatus GetAccessStatus() const;
};
template <typename D>
struct WINRT_EBO impl_IApplicationTrigger
{
Windows::Foundation::IAsyncOperation<winrt::Windows::ApplicationModel::Background::ApplicationTriggerResult> RequestAsync() const;
Windows::Foundation::IAsyncOperation<winrt::Windows::ApplicationModel::Background::ApplicationTriggerResult> RequestAsync(const Windows::Foundation::Collections::ValueSet & arguments) const;
};
template <typename D>
struct WINRT_EBO impl_IApplicationTriggerDetails
{
Windows::Foundation::Collections::ValueSet Arguments() const;
};
template <typename D>
struct WINRT_EBO impl_IAppointmentStoreNotificationTrigger
{
};
template <typename D>
struct WINRT_EBO impl_IBackgroundCondition
{
};
template <typename D>
struct WINRT_EBO impl_IBackgroundExecutionManagerStatics
{
Windows::Foundation::IAsyncOperation<winrt::Windows::ApplicationModel::Background::BackgroundAccessStatus> RequestAccessAsync() const;
Windows::Foundation::IAsyncOperation<winrt::Windows::ApplicationModel::Background::BackgroundAccessStatus> RequestAccessAsync(hstring_ref applicationId) const;
void RemoveAccess() const;
void RemoveAccess(hstring_ref applicationId) const;
Windows::ApplicationModel::Background::BackgroundAccessStatus GetAccessStatus() const;
Windows::ApplicationModel::Background::BackgroundAccessStatus GetAccessStatus(hstring_ref applicationId) const;
};
template <typename D>
struct WINRT_EBO impl_IBackgroundTask
{
void Run(const Windows::ApplicationModel::Background::IBackgroundTaskInstance & taskInstance) const;
};
template <typename D>
struct WINRT_EBO impl_IBackgroundTaskBuilder
{
void TaskEntryPoint(hstring_ref value) const;
hstring TaskEntryPoint() const;
void SetTrigger(const Windows::ApplicationModel::Background::IBackgroundTrigger & trigger) const;
void AddCondition(const Windows::ApplicationModel::Background::IBackgroundCondition & condition) const;
void Name(hstring_ref value) const;
hstring Name() const;
Windows::ApplicationModel::Background::BackgroundTaskRegistration Register() const;
};
template <typename D>
struct WINRT_EBO impl_IBackgroundTaskBuilder2
{
void CancelOnConditionLoss(bool value) const;
bool CancelOnConditionLoss() const;
};
template <typename D>
struct WINRT_EBO impl_IBackgroundTaskBuilder3
{
void IsNetworkRequested(bool value) const;
bool IsNetworkRequested() const;
};
template <typename D>
struct WINRT_EBO impl_IBackgroundTaskCompletedEventArgs
{
GUID InstanceId() const;
void CheckResult() const;
};
template <typename D>
struct WINRT_EBO impl_IBackgroundTaskDeferral
{
void Complete() const;
};
template <typename D>
struct WINRT_EBO impl_IBackgroundTaskInstance
{
GUID InstanceId() const;
Windows::ApplicationModel::Background::BackgroundTaskRegistration Task() const;
uint32_t Progress() const;
void Progress(uint32_t value) const;
Windows::IInspectable TriggerDetails() const;
event_token Canceled(const Windows::ApplicationModel::Background::BackgroundTaskCanceledEventHandler & cancelHandler) const;
using Canceled_revoker = event_revoker<IBackgroundTaskInstance>;
Canceled_revoker Canceled(auto_revoke_t, const Windows::ApplicationModel::Background::BackgroundTaskCanceledEventHandler & cancelHandler) const;
void Canceled(event_token cookie) const;
uint32_t SuspendedCount() const;
Windows::ApplicationModel::Background::BackgroundTaskDeferral GetDeferral() const;
};
template <typename D>
struct WINRT_EBO impl_IBackgroundTaskInstance2
{
uint32_t GetThrottleCount(Windows::ApplicationModel::Background::BackgroundTaskThrottleCounter counter) const;
};
template <typename D>
struct WINRT_EBO impl_IBackgroundTaskInstance4
{
Windows::System::User User() const;
};
template <typename D>
struct WINRT_EBO impl_IBackgroundTaskProgressEventArgs
{
GUID InstanceId() const;
uint32_t Progress() const;
};
template <typename D>
struct WINRT_EBO impl_IBackgroundTaskRegistration
{
GUID TaskId() const;
hstring Name() const;
event_token Progress(const Windows::ApplicationModel::Background::BackgroundTaskProgressEventHandler & handler) const;
using Progress_revoker = event_revoker<IBackgroundTaskRegistration>;
Progress_revoker Progress(auto_revoke_t, const Windows::ApplicationModel::Background::BackgroundTaskProgressEventHandler & handler) const;
void Progress(event_token cookie) const;
event_token Completed(const Windows::ApplicationModel::Background::BackgroundTaskCompletedEventHandler & handler) const;
using Completed_revoker = event_revoker<IBackgroundTaskRegistration>;
Completed_revoker Completed(auto_revoke_t, const Windows::ApplicationModel::Background::BackgroundTaskCompletedEventHandler & handler) const;
void Completed(event_token cookie) const;
void Unregister(bool cancelTask) const;
};
template <typename D>
struct WINRT_EBO impl_IBackgroundTaskRegistration2
{
Windows::ApplicationModel::Background::IBackgroundTrigger Trigger() const;
};
template <typename D>
struct WINRT_EBO impl_IBackgroundTaskRegistrationStatics
{
Windows::Foundation::Collections::IMapView<GUID, Windows::ApplicationModel::Background::IBackgroundTaskRegistration> AllTasks() const;
};
template <typename D>
struct WINRT_EBO impl_IBackgroundTrigger
{
};
template <typename D>
struct WINRT_EBO impl_IBackgroundWorkCostStatics
{
Windows::ApplicationModel::Background::BackgroundWorkCostValue CurrentBackgroundWorkCost() const;
};
template <typename D>
struct WINRT_EBO impl_IBluetoothLEAdvertisementPublisherTrigger
{
Windows::Devices::Bluetooth::Advertisement::BluetoothLEAdvertisement Advertisement() const;
};
template <typename D>
struct WINRT_EBO impl_IBluetoothLEAdvertisementWatcherTrigger
{
Windows::Foundation::TimeSpan MinSamplingInterval() const;
Windows::Foundation::TimeSpan MaxSamplingInterval() const;
Windows::Foundation::TimeSpan MinOutOfRangeTimeout() const;
Windows::Foundation::TimeSpan MaxOutOfRangeTimeout() const;
Windows::Devices::Bluetooth::BluetoothSignalStrengthFilter SignalStrengthFilter() const;
void SignalStrengthFilter(const Windows::Devices::Bluetooth::BluetoothSignalStrengthFilter & value) const;
Windows::Devices::Bluetooth::Advertisement::BluetoothLEAdvertisementFilter AdvertisementFilter() const;
void AdvertisementFilter(const Windows::Devices::Bluetooth::Advertisement::BluetoothLEAdvertisementFilter & value) const;
};
template <typename D>
struct WINRT_EBO impl_ICachedFileUpdaterTrigger
{
};
template <typename D>
struct WINRT_EBO impl_ICachedFileUpdaterTriggerDetails
{
Windows::Storage::Provider::CachedFileTarget UpdateTarget() const;
Windows::Storage::Provider::FileUpdateRequest UpdateRequest() const;
bool CanRequestUserInput() const;
};
template <typename D>
struct WINRT_EBO impl_IChatMessageNotificationTrigger
{
};
template <typename D>
struct WINRT_EBO impl_IChatMessageReceivedNotificationTrigger
{
};
template <typename D>
struct WINRT_EBO impl_ICommunicationBlockingAppSetAsActiveTrigger
{
};
template <typename D>
struct WINRT_EBO impl_IContactStoreNotificationTrigger
{
};
template <typename D>
struct WINRT_EBO impl_IContentPrefetchTrigger
{
Windows::Foundation::TimeSpan WaitInterval() const;
};
template <typename D>
struct WINRT_EBO impl_IContentPrefetchTriggerFactory
{
Windows::ApplicationModel::Background::ContentPrefetchTrigger Create(const Windows::Foundation::TimeSpan & waitInterval) const;
};
template <typename D>
struct WINRT_EBO impl_IDeviceConnectionChangeTrigger
{
hstring DeviceId() const;
bool CanMaintainConnection() const;
bool MaintainConnection() const;
void MaintainConnection(bool value) const;
};
template <typename D>
struct WINRT_EBO impl_IDeviceConnectionChangeTriggerStatics
{
Windows::Foundation::IAsyncOperation<Windows::ApplicationModel::Background::DeviceConnectionChangeTrigger> FromIdAsync(hstring_ref deviceId) const;
};
template <typename D>
struct WINRT_EBO impl_IDeviceManufacturerNotificationTrigger
{
hstring TriggerQualifier() const;
bool OneShot() const;
};
template <typename D>
struct WINRT_EBO impl_IDeviceManufacturerNotificationTriggerFactory
{
Windows::ApplicationModel::Background::DeviceManufacturerNotificationTrigger Create(hstring_ref triggerQualifier, bool oneShot) const;
};
template <typename D>
struct WINRT_EBO impl_IDeviceServicingTrigger
{
Windows::Foundation::IAsyncOperation<winrt::Windows::ApplicationModel::Background::DeviceTriggerResult> RequestAsync(hstring_ref deviceId, const Windows::Foundation::TimeSpan & expectedDuration) const;
Windows::Foundation::IAsyncOperation<winrt::Windows::ApplicationModel::Background::DeviceTriggerResult> RequestAsync(hstring_ref deviceId, const Windows::Foundation::TimeSpan & expectedDuration, hstring_ref arguments) const;
};
template <typename D>
struct WINRT_EBO impl_IDeviceUseTrigger
{
Windows::Foundation::IAsyncOperation<winrt::Windows::ApplicationModel::Background::DeviceTriggerResult> RequestAsync(hstring_ref deviceId) const;
Windows::Foundation::IAsyncOperation<winrt::Windows::ApplicationModel::Background::DeviceTriggerResult> RequestAsync(hstring_ref deviceId, hstring_ref arguments) const;
};
template <typename D>
struct WINRT_EBO impl_IDeviceWatcherTrigger
{
};
template <typename D>
struct WINRT_EBO impl_IEmailStoreNotificationTrigger
{
};
template <typename D>
struct WINRT_EBO impl_IGattCharacteristicNotificationTrigger
{
Windows::Devices::Bluetooth::GenericAttributeProfile::GattCharacteristic Characteristic() const;
};
template <typename D>
struct WINRT_EBO impl_IGattCharacteristicNotificationTriggerFactory
{
Windows::ApplicationModel::Background::GattCharacteristicNotificationTrigger Create(const Windows::Devices::Bluetooth::GenericAttributeProfile::GattCharacteristic & characteristic) const;
};
template <typename D>
struct WINRT_EBO impl_ILocationTrigger
{
Windows::ApplicationModel::Background::LocationTriggerType TriggerType() const;
};
template <typename D>
struct WINRT_EBO impl_ILocationTriggerFactory
{
Windows::ApplicationModel::Background::LocationTrigger Create(Windows::ApplicationModel::Background::LocationTriggerType triggerType) const;
};
template <typename D>
struct WINRT_EBO impl_IMaintenanceTrigger
{
uint32_t FreshnessTime() const;
bool OneShot() const;
};
template <typename D>
struct WINRT_EBO impl_IMaintenanceTriggerFactory
{
Windows::ApplicationModel::Background::MaintenanceTrigger Create(uint32_t freshnessTime, bool oneShot) const;
};
template <typename D>
struct WINRT_EBO impl_IMediaProcessingTrigger
{
Windows::Foundation::IAsyncOperation<winrt::Windows::ApplicationModel::Background::MediaProcessingTriggerResult> RequestAsync() const;
Windows::Foundation::IAsyncOperation<winrt::Windows::ApplicationModel::Background::MediaProcessingTriggerResult> RequestAsync(const Windows::Foundation::Collections::ValueSet & arguments) const;
};
template <typename D>
struct WINRT_EBO impl_INetworkOperatorHotspotAuthenticationTrigger
{
};
template <typename D>
struct WINRT_EBO impl_INetworkOperatorNotificationTrigger
{
hstring NetworkAccountId() const;
};
template <typename D>
struct WINRT_EBO impl_INetworkOperatorNotificationTriggerFactory
{
Windows::ApplicationModel::Background::NetworkOperatorNotificationTrigger Create(hstring_ref networkAccountId) const;
};
template <typename D>
struct WINRT_EBO impl_IPhoneTrigger
{
bool OneShot() const;
Windows::ApplicationModel::Calls::Background::PhoneTriggerType TriggerType() const;
};
template <typename D>
struct WINRT_EBO impl_IPhoneTriggerFactory
{
Windows::ApplicationModel::Background::PhoneTrigger Create(Windows::ApplicationModel::Calls::Background::PhoneTriggerType type, bool oneShot) const;
};
template <typename D>
struct WINRT_EBO impl_IPushNotificationTriggerFactory
{
Windows::ApplicationModel::Background::PushNotificationTrigger Create(hstring_ref applicationId) const;
};
template <typename D>
struct WINRT_EBO impl_IRcsEndUserMessageAvailableTrigger
{
};
template <typename D>
struct WINRT_EBO impl_IRfcommConnectionTrigger
{
Windows::Devices::Bluetooth::Background::RfcommInboundConnectionInformation InboundConnection() const;
Windows::Devices::Bluetooth::Background::RfcommOutboundConnectionInformation OutboundConnection() const;
bool AllowMultipleConnections() const;
void AllowMultipleConnections(bool value) const;
Windows::Networking::Sockets::SocketProtectionLevel ProtectionLevel() const;
void ProtectionLevel(Windows::Networking::Sockets::SocketProtectionLevel value) const;
Windows::Networking::HostName RemoteHostName() const;
void RemoteHostName(const Windows::Networking::HostName & value) const;
};
template <typename D>
struct WINRT_EBO impl_ISecondaryAuthenticationFactorAuthenticationTrigger
{
};
template <typename D>
struct WINRT_EBO impl_ISensorDataThresholdTrigger
{
};
template <typename D>
struct WINRT_EBO impl_ISensorDataThresholdTriggerFactory
{
Windows::ApplicationModel::Background::SensorDataThresholdTrigger Create(const Windows::Devices::Sensors::ISensorDataThreshold & threshold) const;
};
template <typename D>
struct WINRT_EBO impl_ISmartCardTrigger
{
Windows::Devices::SmartCards::SmartCardTriggerType TriggerType() const;
};
template <typename D>
struct WINRT_EBO impl_ISmartCardTriggerFactory
{
Windows::ApplicationModel::Background::SmartCardTrigger Create(Windows::Devices::SmartCards::SmartCardTriggerType triggerType) const;
};
template <typename D>
struct WINRT_EBO impl_ISmsMessageReceivedTriggerFactory
{
Windows::ApplicationModel::Background::SmsMessageReceivedTrigger Create(const Windows::Devices::Sms::SmsFilterRules & filterRules) const;
};
template <typename D>
struct WINRT_EBO impl_ISocketActivityTrigger
{
bool IsWakeFromLowPowerSupported() const;
};
template <typename D>
struct WINRT_EBO impl_IStorageLibraryContentChangedTrigger
{
};
template <typename D>
struct WINRT_EBO impl_IStorageLibraryContentChangedTriggerStatics
{
Windows::ApplicationModel::Background::StorageLibraryContentChangedTrigger Create(const Windows::Storage::StorageLibrary & storageLibrary) const;
Windows::ApplicationModel::Background::StorageLibraryContentChangedTrigger CreateFromLibraries(const Windows::Foundation::Collections::IIterable<Windows::Storage::StorageLibrary> & storageLibraries) const;
};
template <typename D>
struct WINRT_EBO impl_ISystemCondition
{
Windows::ApplicationModel::Background::SystemConditionType ConditionType() const;
};
template <typename D>
struct WINRT_EBO impl_ISystemConditionFactory
{
Windows::ApplicationModel::Background::SystemCondition Create(Windows::ApplicationModel::Background::SystemConditionType conditionType) const;
};
template <typename D>
struct WINRT_EBO impl_ISystemTrigger
{
bool OneShot() const;
Windows::ApplicationModel::Background::SystemTriggerType TriggerType() const;
};
template <typename D>
struct WINRT_EBO impl_ISystemTriggerFactory
{
Windows::ApplicationModel::Background::SystemTrigger Create(Windows::ApplicationModel::Background::SystemTriggerType triggerType, bool oneShot) const;
};
template <typename D>
struct WINRT_EBO impl_ITimeTrigger
{
uint32_t FreshnessTime() const;
bool OneShot() const;
};
template <typename D>
struct WINRT_EBO impl_ITimeTriggerFactory
{
Windows::ApplicationModel::Background::TimeTrigger Create(uint32_t freshnessTime, bool oneShot) const;
};
template <typename D>
struct WINRT_EBO impl_IToastNotificationActionTriggerFactory
{
Windows::ApplicationModel::Background::ToastNotificationActionTrigger Create(hstring_ref applicationId) const;
};
template <typename D>
struct WINRT_EBO impl_IToastNotificationHistoryChangedTriggerFactory
{
Windows::ApplicationModel::Background::ToastNotificationHistoryChangedTrigger Create(hstring_ref applicationId) const;
};
template <typename D>
struct WINRT_EBO impl_IUserNotificationChangedTriggerFactory
{
Windows::ApplicationModel::Background::UserNotificationChangedTrigger Create(Windows::UI::Notifications::NotificationKinds notificationKinds) const;
};
struct BackgroundTaskCanceledEventHandler : Windows::IUnknown
{
BackgroundTaskCanceledEventHandler(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<BackgroundTaskCanceledEventHandler>(m_ptr); }
template <typename L> BackgroundTaskCanceledEventHandler(L lambda);
template <typename F> BackgroundTaskCanceledEventHandler (F * function);
template <typename O, typename M> BackgroundTaskCanceledEventHandler(O * object, M method);
void operator()(const Windows::ApplicationModel::Background::IBackgroundTaskInstance & sender, Windows::ApplicationModel::Background::BackgroundTaskCancellationReason reason) const;
};
struct BackgroundTaskCompletedEventHandler : Windows::IUnknown
{
BackgroundTaskCompletedEventHandler(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<BackgroundTaskCompletedEventHandler>(m_ptr); }
template <typename L> BackgroundTaskCompletedEventHandler(L lambda);
template <typename F> BackgroundTaskCompletedEventHandler (F * function);
template <typename O, typename M> BackgroundTaskCompletedEventHandler(O * object, M method);
void operator()(const Windows::ApplicationModel::Background::BackgroundTaskRegistration & sender, const Windows::ApplicationModel::Background::BackgroundTaskCompletedEventArgs & args) const;
};
struct BackgroundTaskProgressEventHandler : Windows::IUnknown
{
BackgroundTaskProgressEventHandler(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<BackgroundTaskProgressEventHandler>(m_ptr); }
template <typename L> BackgroundTaskProgressEventHandler(L lambda);
template <typename F> BackgroundTaskProgressEventHandler (F * function);
template <typename O, typename M> BackgroundTaskProgressEventHandler(O * object, M method);
void operator()(const Windows::ApplicationModel::Background::BackgroundTaskRegistration & sender, const Windows::ApplicationModel::Background::BackgroundTaskProgressEventArgs & args) const;
};
struct IActivitySensorTrigger :
Windows::IInspectable,
impl::consume<IActivitySensorTrigger>,
impl::require<IActivitySensorTrigger, Windows::ApplicationModel::Background::IBackgroundTrigger>
{
IActivitySensorTrigger(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<IActivitySensorTrigger>(m_ptr); }
};
struct IActivitySensorTriggerFactory :
Windows::IInspectable,
impl::consume<IActivitySensorTriggerFactory>
{
IActivitySensorTriggerFactory(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<IActivitySensorTriggerFactory>(m_ptr); }
};
struct IAlarmApplicationManagerStatics :
Windows::IInspectable,
impl::consume<IAlarmApplicationManagerStatics>
{
IAlarmApplicationManagerStatics(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<IAlarmApplicationManagerStatics>(m_ptr); }
};
struct IApplicationTrigger :
Windows::IInspectable,
impl::consume<IApplicationTrigger>,
impl::require<IApplicationTrigger, Windows::ApplicationModel::Background::IBackgroundTrigger>
{
IApplicationTrigger(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<IApplicationTrigger>(m_ptr); }
};
struct IApplicationTriggerDetails :
Windows::IInspectable,
impl::consume<IApplicationTriggerDetails>
{
IApplicationTriggerDetails(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<IApplicationTriggerDetails>(m_ptr); }
};
struct IAppointmentStoreNotificationTrigger :
Windows::IInspectable,
impl::consume<IAppointmentStoreNotificationTrigger>,
impl::require<IAppointmentStoreNotificationTrigger, Windows::ApplicationModel::Background::IBackgroundTrigger>
{
IAppointmentStoreNotificationTrigger(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<IAppointmentStoreNotificationTrigger>(m_ptr); }
};
struct IBackgroundCondition :
Windows::IInspectable,
impl::consume<IBackgroundCondition>
{
IBackgroundCondition(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<IBackgroundCondition>(m_ptr); }
};
struct IBackgroundExecutionManagerStatics :
Windows::IInspectable,
impl::consume<IBackgroundExecutionManagerStatics>
{
IBackgroundExecutionManagerStatics(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<IBackgroundExecutionManagerStatics>(m_ptr); }
};
struct IBackgroundTask :
Windows::IInspectable,
impl::consume<IBackgroundTask>
{
IBackgroundTask(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<IBackgroundTask>(m_ptr); }
};
struct IBackgroundTaskBuilder :
Windows::IInspectable,
impl::consume<IBackgroundTaskBuilder>
{
IBackgroundTaskBuilder(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<IBackgroundTaskBuilder>(m_ptr); }
};
struct IBackgroundTaskBuilder2 :
Windows::IInspectable,
impl::consume<IBackgroundTaskBuilder2>,
impl::require<IBackgroundTaskBuilder2, Windows::ApplicationModel::Background::IBackgroundTaskBuilder>
{
IBackgroundTaskBuilder2(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<IBackgroundTaskBuilder2>(m_ptr); }
};
struct IBackgroundTaskBuilder3 :
Windows::IInspectable,
impl::consume<IBackgroundTaskBuilder3>,
impl::require<IBackgroundTaskBuilder3, Windows::ApplicationModel::Background::IBackgroundTaskBuilder>
{
IBackgroundTaskBuilder3(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<IBackgroundTaskBuilder3>(m_ptr); }
};
struct IBackgroundTaskCompletedEventArgs :
Windows::IInspectable,
impl::consume<IBackgroundTaskCompletedEventArgs>
{
IBackgroundTaskCompletedEventArgs(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<IBackgroundTaskCompletedEventArgs>(m_ptr); }
};
struct IBackgroundTaskDeferral :
Windows::IInspectable,
impl::consume<IBackgroundTaskDeferral>
{
IBackgroundTaskDeferral(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<IBackgroundTaskDeferral>(m_ptr); }
};
struct IBackgroundTaskInstance :
Windows::IInspectable,
impl::consume<IBackgroundTaskInstance>
{
IBackgroundTaskInstance(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<IBackgroundTaskInstance>(m_ptr); }
};
struct IBackgroundTaskInstance2 :
Windows::IInspectable,
impl::consume<IBackgroundTaskInstance2>,
impl::require<IBackgroundTaskInstance2, Windows::ApplicationModel::Background::IBackgroundTaskInstance>
{
IBackgroundTaskInstance2(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<IBackgroundTaskInstance2>(m_ptr); }
};
struct IBackgroundTaskInstance4 :
Windows::IInspectable,
impl::consume<IBackgroundTaskInstance4>,
impl::require<IBackgroundTaskInstance4, Windows::ApplicationModel::Background::IBackgroundTaskInstance>
{
IBackgroundTaskInstance4(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<IBackgroundTaskInstance4>(m_ptr); }
};
struct IBackgroundTaskProgressEventArgs :
Windows::IInspectable,
impl::consume<IBackgroundTaskProgressEventArgs>
{
IBackgroundTaskProgressEventArgs(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<IBackgroundTaskProgressEventArgs>(m_ptr); }
};
struct IBackgroundTaskRegistration :
Windows::IInspectable,
impl::consume<IBackgroundTaskRegistration>
{
IBackgroundTaskRegistration(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<IBackgroundTaskRegistration>(m_ptr); }
};
struct IBackgroundTaskRegistration2 :
Windows::IInspectable,
impl::consume<IBackgroundTaskRegistration2>,
impl::require<IBackgroundTaskRegistration2, Windows::ApplicationModel::Background::IBackgroundTaskRegistration>
{
IBackgroundTaskRegistration2(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<IBackgroundTaskRegistration2>(m_ptr); }
};
struct IBackgroundTaskRegistrationStatics :
Windows::IInspectable,
impl::consume<IBackgroundTaskRegistrationStatics>
{
IBackgroundTaskRegistrationStatics(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<IBackgroundTaskRegistrationStatics>(m_ptr); }
};
struct IBackgroundTrigger :
Windows::IInspectable,
impl::consume<IBackgroundTrigger>
{
IBackgroundTrigger(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<IBackgroundTrigger>(m_ptr); }
};
struct IBackgroundWorkCostStatics :
Windows::IInspectable,
impl::consume<IBackgroundWorkCostStatics>
{
IBackgroundWorkCostStatics(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<IBackgroundWorkCostStatics>(m_ptr); }
};
struct IBluetoothLEAdvertisementPublisherTrigger :
Windows::IInspectable,
impl::consume<IBluetoothLEAdvertisementPublisherTrigger>,
impl::require<IBluetoothLEAdvertisementPublisherTrigger, Windows::ApplicationModel::Background::IBackgroundTrigger>
{
IBluetoothLEAdvertisementPublisherTrigger(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<IBluetoothLEAdvertisementPublisherTrigger>(m_ptr); }
};
struct IBluetoothLEAdvertisementWatcherTrigger :
Windows::IInspectable,
impl::consume<IBluetoothLEAdvertisementWatcherTrigger>,
impl::require<IBluetoothLEAdvertisementWatcherTrigger, Windows::ApplicationModel::Background::IBackgroundTrigger>
{
IBluetoothLEAdvertisementWatcherTrigger(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<IBluetoothLEAdvertisementWatcherTrigger>(m_ptr); }
};
struct ICachedFileUpdaterTrigger :
Windows::IInspectable,
impl::consume<ICachedFileUpdaterTrigger>,
impl::require<ICachedFileUpdaterTrigger, Windows::ApplicationModel::Background::IBackgroundTrigger>
{
ICachedFileUpdaterTrigger(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<ICachedFileUpdaterTrigger>(m_ptr); }
};
struct ICachedFileUpdaterTriggerDetails :
Windows::IInspectable,
impl::consume<ICachedFileUpdaterTriggerDetails>
{
ICachedFileUpdaterTriggerDetails(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<ICachedFileUpdaterTriggerDetails>(m_ptr); }
};
struct IChatMessageNotificationTrigger :
Windows::IInspectable,
impl::consume<IChatMessageNotificationTrigger>,
impl::require<IChatMessageNotificationTrigger, Windows::ApplicationModel::Background::IBackgroundTrigger>
{
IChatMessageNotificationTrigger(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<IChatMessageNotificationTrigger>(m_ptr); }
};
struct IChatMessageReceivedNotificationTrigger :
Windows::IInspectable,
impl::consume<IChatMessageReceivedNotificationTrigger>,
impl::require<IChatMessageReceivedNotificationTrigger, Windows::ApplicationModel::Background::IBackgroundTrigger>
{
IChatMessageReceivedNotificationTrigger(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<IChatMessageReceivedNotificationTrigger>(m_ptr); }
};
struct ICommunicationBlockingAppSetAsActiveTrigger :
Windows::IInspectable,
impl::consume<ICommunicationBlockingAppSetAsActiveTrigger>,
impl::require<ICommunicationBlockingAppSetAsActiveTrigger, Windows::ApplicationModel::Background::IBackgroundTrigger>
{
ICommunicationBlockingAppSetAsActiveTrigger(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<ICommunicationBlockingAppSetAsActiveTrigger>(m_ptr); }
};
struct IContactStoreNotificationTrigger :
Windows::IInspectable,
impl::consume<IContactStoreNotificationTrigger>,
impl::require<IContactStoreNotificationTrigger, Windows::ApplicationModel::Background::IBackgroundTrigger>
{
IContactStoreNotificationTrigger(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<IContactStoreNotificationTrigger>(m_ptr); }
};
struct IContentPrefetchTrigger :
Windows::IInspectable,
impl::consume<IContentPrefetchTrigger>,
impl::require<IContentPrefetchTrigger, Windows::ApplicationModel::Background::IBackgroundTrigger>
{
IContentPrefetchTrigger(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<IContentPrefetchTrigger>(m_ptr); }
};
struct IContentPrefetchTriggerFactory :
Windows::IInspectable,
impl::consume<IContentPrefetchTriggerFactory>
{
IContentPrefetchTriggerFactory(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<IContentPrefetchTriggerFactory>(m_ptr); }
};
struct IDeviceConnectionChangeTrigger :
Windows::IInspectable,
impl::consume<IDeviceConnectionChangeTrigger>,
impl::require<IDeviceConnectionChangeTrigger, Windows::ApplicationModel::Background::IBackgroundTrigger>
{
IDeviceConnectionChangeTrigger(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<IDeviceConnectionChangeTrigger>(m_ptr); }
};
struct IDeviceConnectionChangeTriggerStatics :
Windows::IInspectable,
impl::consume<IDeviceConnectionChangeTriggerStatics>
{
IDeviceConnectionChangeTriggerStatics(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<IDeviceConnectionChangeTriggerStatics>(m_ptr); }
};
struct IDeviceManufacturerNotificationTrigger :
Windows::IInspectable,
impl::consume<IDeviceManufacturerNotificationTrigger>,
impl::require<IDeviceManufacturerNotificationTrigger, Windows::ApplicationModel::Background::IBackgroundTrigger>
{
IDeviceManufacturerNotificationTrigger(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<IDeviceManufacturerNotificationTrigger>(m_ptr); }
};
struct IDeviceManufacturerNotificationTriggerFactory :
Windows::IInspectable,
impl::consume<IDeviceManufacturerNotificationTriggerFactory>
{
IDeviceManufacturerNotificationTriggerFactory(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<IDeviceManufacturerNotificationTriggerFactory>(m_ptr); }
};
struct IDeviceServicingTrigger :
Windows::IInspectable,
impl::consume<IDeviceServicingTrigger>,
impl::require<IDeviceServicingTrigger, Windows::ApplicationModel::Background::IBackgroundTrigger>
{
IDeviceServicingTrigger(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<IDeviceServicingTrigger>(m_ptr); }
};
struct IDeviceUseTrigger :
Windows::IInspectable,
impl::consume<IDeviceUseTrigger>,
impl::require<IDeviceUseTrigger, Windows::ApplicationModel::Background::IBackgroundTrigger>
{
IDeviceUseTrigger(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<IDeviceUseTrigger>(m_ptr); }
};
struct IDeviceWatcherTrigger :
Windows::IInspectable,
impl::consume<IDeviceWatcherTrigger>,
impl::require<IDeviceWatcherTrigger, Windows::ApplicationModel::Background::IBackgroundTrigger>
{
IDeviceWatcherTrigger(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<IDeviceWatcherTrigger>(m_ptr); }
};
struct IEmailStoreNotificationTrigger :
Windows::IInspectable,
impl::consume<IEmailStoreNotificationTrigger>,
impl::require<IEmailStoreNotificationTrigger, Windows::ApplicationModel::Background::IBackgroundTrigger>
{
IEmailStoreNotificationTrigger(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<IEmailStoreNotificationTrigger>(m_ptr); }
};
struct IGattCharacteristicNotificationTrigger :
Windows::IInspectable,
impl::consume<IGattCharacteristicNotificationTrigger>,
impl::require<IGattCharacteristicNotificationTrigger, Windows::ApplicationModel::Background::IBackgroundTrigger>
{
IGattCharacteristicNotificationTrigger(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<IGattCharacteristicNotificationTrigger>(m_ptr); }
};
struct IGattCharacteristicNotificationTriggerFactory :
Windows::IInspectable,
impl::consume<IGattCharacteristicNotificationTriggerFactory>
{
IGattCharacteristicNotificationTriggerFactory(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<IGattCharacteristicNotificationTriggerFactory>(m_ptr); }
};
struct ILocationTrigger :
Windows::IInspectable,
impl::consume<ILocationTrigger>,
impl::require<ILocationTrigger, Windows::ApplicationModel::Background::IBackgroundTrigger>
{
ILocationTrigger(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<ILocationTrigger>(m_ptr); }
};
struct ILocationTriggerFactory :
Windows::IInspectable,
impl::consume<ILocationTriggerFactory>
{
ILocationTriggerFactory(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<ILocationTriggerFactory>(m_ptr); }
};
struct IMaintenanceTrigger :
Windows::IInspectable,
impl::consume<IMaintenanceTrigger>,
impl::require<IMaintenanceTrigger, Windows::ApplicationModel::Background::IBackgroundTrigger>
{
IMaintenanceTrigger(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<IMaintenanceTrigger>(m_ptr); }
};
struct IMaintenanceTriggerFactory :
Windows::IInspectable,
impl::consume<IMaintenanceTriggerFactory>
{
IMaintenanceTriggerFactory(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<IMaintenanceTriggerFactory>(m_ptr); }
};
struct IMediaProcessingTrigger :
Windows::IInspectable,
impl::consume<IMediaProcessingTrigger>,
impl::require<IMediaProcessingTrigger, Windows::ApplicationModel::Background::IBackgroundTrigger>
{
IMediaProcessingTrigger(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<IMediaProcessingTrigger>(m_ptr); }
};
struct INetworkOperatorHotspotAuthenticationTrigger :
Windows::IInspectable,
impl::consume<INetworkOperatorHotspotAuthenticationTrigger>,
impl::require<INetworkOperatorHotspotAuthenticationTrigger, Windows::ApplicationModel::Background::IBackgroundTrigger>
{
INetworkOperatorHotspotAuthenticationTrigger(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<INetworkOperatorHotspotAuthenticationTrigger>(m_ptr); }
};
struct INetworkOperatorNotificationTrigger :
Windows::IInspectable,
impl::consume<INetworkOperatorNotificationTrigger>,
impl::require<INetworkOperatorNotificationTrigger, Windows::ApplicationModel::Background::IBackgroundTrigger>
{
INetworkOperatorNotificationTrigger(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<INetworkOperatorNotificationTrigger>(m_ptr); }
};
struct INetworkOperatorNotificationTriggerFactory :
Windows::IInspectable,
impl::consume<INetworkOperatorNotificationTriggerFactory>
{
INetworkOperatorNotificationTriggerFactory(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<INetworkOperatorNotificationTriggerFactory>(m_ptr); }
};
struct IPhoneTrigger :
Windows::IInspectable,
impl::consume<IPhoneTrigger>,
impl::require<IPhoneTrigger, Windows::ApplicationModel::Background::IBackgroundTrigger>
{
IPhoneTrigger(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<IPhoneTrigger>(m_ptr); }
};
struct IPhoneTriggerFactory :
Windows::IInspectable,
impl::consume<IPhoneTriggerFactory>
{
IPhoneTriggerFactory(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<IPhoneTriggerFactory>(m_ptr); }
};
struct IPushNotificationTriggerFactory :
Windows::IInspectable,
impl::consume<IPushNotificationTriggerFactory>
{
IPushNotificationTriggerFactory(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<IPushNotificationTriggerFactory>(m_ptr); }
};
struct IRcsEndUserMessageAvailableTrigger :
Windows::IInspectable,
impl::consume<IRcsEndUserMessageAvailableTrigger>,
impl::require<IRcsEndUserMessageAvailableTrigger, Windows::ApplicationModel::Background::IBackgroundTrigger>
{
IRcsEndUserMessageAvailableTrigger(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<IRcsEndUserMessageAvailableTrigger>(m_ptr); }
};
struct IRfcommConnectionTrigger :
Windows::IInspectable,
impl::consume<IRfcommConnectionTrigger>,
impl::require<IRfcommConnectionTrigger, Windows::ApplicationModel::Background::IBackgroundTrigger>
{
IRfcommConnectionTrigger(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<IRfcommConnectionTrigger>(m_ptr); }
};
struct ISecondaryAuthenticationFactorAuthenticationTrigger :
Windows::IInspectable,
impl::consume<ISecondaryAuthenticationFactorAuthenticationTrigger>,
impl::require<ISecondaryAuthenticationFactorAuthenticationTrigger, Windows::ApplicationModel::Background::IBackgroundTrigger>
{
ISecondaryAuthenticationFactorAuthenticationTrigger(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<ISecondaryAuthenticationFactorAuthenticationTrigger>(m_ptr); }
};
struct ISensorDataThresholdTrigger :
Windows::IInspectable,
impl::consume<ISensorDataThresholdTrigger>,
impl::require<ISensorDataThresholdTrigger, Windows::ApplicationModel::Background::IBackgroundTrigger>
{
ISensorDataThresholdTrigger(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<ISensorDataThresholdTrigger>(m_ptr); }
};
struct ISensorDataThresholdTriggerFactory :
Windows::IInspectable,
impl::consume<ISensorDataThresholdTriggerFactory>
{
ISensorDataThresholdTriggerFactory(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<ISensorDataThresholdTriggerFactory>(m_ptr); }
};
struct ISmartCardTrigger :
Windows::IInspectable,
impl::consume<ISmartCardTrigger>,
impl::require<ISmartCardTrigger, Windows::ApplicationModel::Background::IBackgroundTrigger>
{
ISmartCardTrigger(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<ISmartCardTrigger>(m_ptr); }
};
struct ISmartCardTriggerFactory :
Windows::IInspectable,
impl::consume<ISmartCardTriggerFactory>
{
ISmartCardTriggerFactory(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<ISmartCardTriggerFactory>(m_ptr); }
};
struct ISmsMessageReceivedTriggerFactory :
Windows::IInspectable,
impl::consume<ISmsMessageReceivedTriggerFactory>
{
ISmsMessageReceivedTriggerFactory(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<ISmsMessageReceivedTriggerFactory>(m_ptr); }
};
struct ISocketActivityTrigger :
Windows::IInspectable,
impl::consume<ISocketActivityTrigger>
{
ISocketActivityTrigger(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<ISocketActivityTrigger>(m_ptr); }
};
struct IStorageLibraryContentChangedTrigger :
Windows::IInspectable,
impl::consume<IStorageLibraryContentChangedTrigger>,
impl::require<IStorageLibraryContentChangedTrigger, Windows::ApplicationModel::Background::IBackgroundTrigger>
{
IStorageLibraryContentChangedTrigger(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<IStorageLibraryContentChangedTrigger>(m_ptr); }
};
struct IStorageLibraryContentChangedTriggerStatics :
Windows::IInspectable,
impl::consume<IStorageLibraryContentChangedTriggerStatics>
{
IStorageLibraryContentChangedTriggerStatics(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<IStorageLibraryContentChangedTriggerStatics>(m_ptr); }
};
struct ISystemCondition :
Windows::IInspectable,
impl::consume<ISystemCondition>,
impl::require<ISystemCondition, Windows::ApplicationModel::Background::IBackgroundCondition>
{
ISystemCondition(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<ISystemCondition>(m_ptr); }
};
struct ISystemConditionFactory :
Windows::IInspectable,
impl::consume<ISystemConditionFactory>
{
ISystemConditionFactory(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<ISystemConditionFactory>(m_ptr); }
};
struct ISystemTrigger :
Windows::IInspectable,
impl::consume<ISystemTrigger>,
impl::require<ISystemTrigger, Windows::ApplicationModel::Background::IBackgroundTrigger>
{
ISystemTrigger(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<ISystemTrigger>(m_ptr); }
};
struct ISystemTriggerFactory :
Windows::IInspectable,
impl::consume<ISystemTriggerFactory>
{
ISystemTriggerFactory(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<ISystemTriggerFactory>(m_ptr); }
};
struct ITimeTrigger :
Windows::IInspectable,
impl::consume<ITimeTrigger>,
impl::require<ITimeTrigger, Windows::ApplicationModel::Background::IBackgroundTrigger>
{
ITimeTrigger(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<ITimeTrigger>(m_ptr); }
};
struct ITimeTriggerFactory :
Windows::IInspectable,
impl::consume<ITimeTriggerFactory>
{
ITimeTriggerFactory(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<ITimeTriggerFactory>(m_ptr); }
};
struct IToastNotificationActionTriggerFactory :
Windows::IInspectable,
impl::consume<IToastNotificationActionTriggerFactory>
{
IToastNotificationActionTriggerFactory(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<IToastNotificationActionTriggerFactory>(m_ptr); }
};
struct IToastNotificationHistoryChangedTriggerFactory :
Windows::IInspectable,
impl::consume<IToastNotificationHistoryChangedTriggerFactory>
{
IToastNotificationHistoryChangedTriggerFactory(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<IToastNotificationHistoryChangedTriggerFactory>(m_ptr); }
};
struct IUserNotificationChangedTriggerFactory :
Windows::IInspectable,
impl::consume<IUserNotificationChangedTriggerFactory>
{
IUserNotificationChangedTriggerFactory(std::nullptr_t = nullptr) noexcept {}
auto operator->() const noexcept { return ptr<IUserNotificationChangedTriggerFactory>(m_ptr); }
};
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,205 |
Q: Dependency Injection with inheritance in .NET Core I have the following model:
public class BaseClass
{
public int id {get; set;}
}
and two child classes:
public class ChildA : BaseClass {}
public class ChildB : BaseClass {}
I need to implement a DI like this, because of specific app logic.
services.AddScoped<BaseClass, ChildA>();
But I am getting an error:
There is no implicit conversion from ChildA to BaseClass
Is there a way to accomplish this?
Thanks
A: It was a namespace naming problem as @Panagiotis Kanavos suggested.
Nevertheless, I am going to create a new question since this solve this particular simple example, but It's not working for the more complex real scenario I have.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,886 |
Best Friends with a Frequent Flyer Card? WhenTheMeetingsOver.com is Here to Help Make Business Travel Better
Adding a new and unique dimension to business travel, WhenTheMeetingsOver.com is a networking site aimed at making business travel more fun by connecting people who are in the same city at the same time.
Sarah Russell and Caroline Purdon, Communications and Marketing Directors of WTMO
'I love being able to explore a new city with someone of similar interests' Amy Galloway, Director and Lawyer, Fort Lauderdale, USA
WhenTheMeetingsOver.com Launches --...
(PRWEB) January 22, 2010
Business travel has a glamorous image, and often it is glamorous. (think George Clooney spending most of his time in business class seats and deluxe hotels in his new movie 'Up in the Air'). But the flip side is that it can be boring, and a trip can leave the corporate traveller not knowing what to do or where to go.
http://www.whenthemeetingsover.com is a new network for business travellers, it connects people who are in the same city at the same time and allows members to link up for email exchange about the best places to go, or, they can meet up 'when the meeting's over', to explore the city, have a game of golf/tennis, go for coffee, develop business contacts… whatever - the site helps members make contact with others who are travelling.
Most people who have been on a business trip flick through the hotel's 'Where' magazine, to find out what to do/where to go. But then it is tedious to go out alone, or, just wanting to get an endorsement of the places to go. http://www.whenthemeetingsover.com is perfect site to help. Locating people of similar interests and destinations, it can link like-minded people together.
Entrepreneur, Brian Catton, says 'On a recent trip to Phuket, and after my meeting, I wanted to play tennis, but who with? I waited ages at the hotel's courts, eventually someone showed up. He was Swedish, we were of a similar standard and we had some great matches. He had interests in a Pro Am tennis event in Sweden and his company was building a resort in Phuket. Coincidentally I also have property interests and I promote professional tennis. A lucky chance...
Then it hit me. There must be over one million people travelling on business every day, who might want to do something when the meeting's over. Or, to find out about a city from somebody else who is also travelling. London, Paris, Rome, Hong Kong, Shanghai, Sydney, New York, Los Angeles... for a site like ours the possibilities are endless! Poignantly there's a kinship amongst travellers, even strangers, and our site has the capacity to bring people together. Safely and at arm's length.'
The site is live, it has a forum and is constantly being updated and improved, and best of all it's free. Check out http://whenthemeetingsover.com
brian(at)whenthemeetingsover(dot)com
Brian +1 954 612 1540
Brian +44 7968 055497
Caroline Purdon
WhenTheMeetingsOver.com
Sarah Russell
Make business travel better...'WhenTheMeetingsOver.com captures the essence of a mere 6 degrees separating us! WTMO focuses on a concept thatⳠgoing to be welcome to many of us who spend a significant time of our lives on the road ... Connecting ᗨen The Meetings Over⠠will bring everything thatⳠgood out of international business travel!' Micky Lawler, Managing Director, Tennis, Octagon Athletes and Personalities, Virginia, USA | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,764 |
I have heard Christians say they had more difficulty with temptation after Christ than they had before Christ. That may be. Before Christ you did what pleased you. After Christ you desire to be pleasing to the Lord. To no longer share in the world's rebellion against God is to experience the meaning of being faithful. We need Lent because we are not always faithful to the God who loves us. We need confession and penance. We need to hear again, "Your sins are absolved."
This first Sunday of Lent highlights baptism and temptation. Baptism separates us from the world. Just as the eight souls aboard the Ark were saved through water, baptism which corresponds to this now saves you. (1 Peter 3:20, 21) Temptation is that which we grapple with in the lust of the flesh, the lust of the eyes and the pride of life. As long as we are in the flesh and there is a devil there will be temptation. So we will always need Lent!
We cannot overcome evil with evil. We overcome evil with good.
The message of Lent is clear. A deeper conversion is needed if we are to overcome the world; if we are to convince this world of a kingdom that is not of this world. As it is, we do not belong to the world. Being almost persuaded of this will not result in a life truly overcoming the world. As we embrace fasting or less feasting as unto the Lord we advance in the things of the Lord. Excelling in the virtues of faith, hope and charity, helps us overcome the deadly sins of pride, greed, lust, anger, gluttony, envy, and sloth.
We need Lent as kind of a spiritual spring cleaning. Returning to things that nourish the soul and revive the heart will bring us closer to him whose heart burst with love for the world on an old rugged cross. Make no mistake about it, God is not willing that any should perish, (meaning you and me), but that all should come to repentance, (meaning you and me).
Let us come to this season of repentance, a time that promises newness of life and joy and receive the Spirit's power that we might be more than conquerors through him who loved us and gave himself for us.
Let us pray: Dear Jesus, restore to me the joy of your salvation. Ever let me be filled with you Spirit. Help me this Lent to be willing to enter into a deeper conversion. Amen. | {
"redpajama_set_name": "RedPajamaC4"
} | 2,199 |
\section{Introduction}
The goal of statistical inference from data
can be stated as follows: given some data determine
the posteriors of some parameters, while marginalizing over other parameters. The
posteriors are best parametrized in terms of 1D probability density
distributions, but alternative
descriptions such as the mean and the variance can sometimes be used, especially in the asymptotic
regime of large data where they fully describe the posterior. Occasionally we also want
to examine higher dimensional posteriors, such as 2D probability density
plots, but we rarely
go to higher dimensions due to the difficulty of visualizing it. While we want to
summarize the results in a series of 1D and 2D plots, the actual problem can have a
large number of dimensions, most of which we may not care about, but which
are correlated with the ones we do.
The main difficulty in obtaining reliable lower dimensional posteriors
lies in the marginalization part: marginals, i.e., averaging over the probability
distribution of certain parameters, can change the answer significantly
relative to the answer for the unmarginalized
posterior, where those parameters are assumed to
be known.
A standard approach to posteriors is Monte Carlo Markov Chain (MCMC) sampling.
In this approach we sample all the parameters according to their probability
density. After the samples are created marginalization is trivial, as one
can simply count the 1D and 2D posterior density
distribution of the parameter of interest. MCMC is
argued to be exact, in the sense that in the limit of
large samples it converges to the true answer. But in practice
this limit may never be reached. For example,
doing Metropolis sampling without knowing the covariance structure
of the variables suffers from the curse of dimensionality.
In very high dimensions this is practically impossible.
One avoids the curse of
dimensionality by having access to the gradient of the loss
function, as in Hamiltonian
Monte Carlo (HMC).
However, in high dimensions
thousands of model
evaluations may be needed to produce a single independent sample (e.g. \cite{JascheWandelt13}),
which often makes it prohibitively expensive, specially if model evaluation
is costly.
Alternatives to MCMC are approximate methods such as Maximum Likelihood Estimation (MLE)
or its Bayesian version, Maximum A Posteriori (MAP) estimation,
or KL divergence minimization based variational inference (VI). These can be less expensive,
but can also give inaccurate results and must be used with care. MAP can give incorrect
estimators in many different situations, even in the limit of large data, and
is thus an inconsistent estimator, a result known
since \cite{NeymanScott1948}. Similarly, mean field VI can give an incorrect
mean in
certain situations. Full rank VI (FRVI) typically gives
the posterior mean close to the correct value, but not always
(we present one example
in section \ref{sec4}). It often does not give correct variance.
Quantifying the error of the approximation is difficult \citep{YaoVSG18}.
A related method is Population Monte Carlo (PMC)
\citep{CappeDoucEtAl08,WraithKilbingerEtAl09}, which uses sampling
from a proposal distribution to obtain new samples, and the posterior at the
sample to improve upon the proposal sampling density. Both of these methods
use KL divergence to quantify the agreement between the proposal
distribution and the true distribution.
In many scientific applications the cost of evaluating the model and
the likelihood
can be very high. In these situations MCMC becomes prohibitively
expensive.
The goal of this paper is to develop a method that is optimization
based and
extends MAP and stochastic VI methods such as ADVI \citep{KucukelbirTRGB17}, such that it requires a
low number
of likelihood evaluations, while striving to be as accurate as MCMC.\footnote{ In general exact inference is impossible because global optimization of non-convex surfaces is an unsolved problem in high dimensions.} We would like to
avoid some of the main pitfalls of the approximate
methods like MAP or VI.
Our goal is to have a method that works for both
convex and non-convex problems, and works for moderately high
dimensions, where a full rank matrix inversion is not a computational
bottleneck: this can be a dozen or up to a few thousand dimensions,
depending on the computational cost of the likelihood
and the complexity of posterior surface.
The outline of the paper is as follows.
In section \ref{sec2} we compare traditional stochastic KL divergence
approaches to our new proposal of using L$_2$ distance on a
toy 1D Gaussian example.
In section \ref{sec3} we develop the method further by
incorporating higher derivative information, and increasing
the expressivity of approximate posteriors, while still allowing
for analytic marginalization. In section \ref{sec4} we show
several examples, including a realistic data analysis example
from our research area of cosmology. We conclude by discussion
and conclusions in section \ref{sec5}.
\section{Stochastic KL divergence minimization versus ${\bf EL_2O}$}
\label{sec2}
A general problem of statistical inference is how to infer parameters from the data:
we have some data $\bi{x}= \{x_i\}_{i=1}^N$ and some parameters
the data depend on, $\bi{z}= \{z_j\}_{j=1}^M$.
We want to
describe the posterior of $\bi{z}$ given data $\bi{x}$.
We can define the posterior $p(\bi{z}|\bi{x})$ as
\begin{equation}
p(\bi{z}|\bi{x})=\frac{p(\bi{x}|\bi{z})p(\bi{z})}{p(\bi{x})}=\frac{p(\bi{x},\bi{z})}{p(\bi{x})},
\label{loss}
\end{equation}
where $p(\bi{x}|\bi{z})$ is the likelihood of the data, $p(\bi{z})$ is the prior of $\bi{z}$
and $p(\bi{x})=\int p(\bi{x}|\bi{z})p(\bi{z})d\bi{z}$ is the normalization.
In general we have access to the prior and likelihood, but not the normalization.
We can define the negative log of posterior in terms of what we have access to, which is negative log joint distribution $\tilde{\mathcal{L}}_p$, defined as
\begin{equation}
\tilde{\mathcal{L}}_p=-\ln p(\bi{x},\bi{z})=-\ln p(\bi{x}|\bi{z})-\ln p(\bi{z})= -\ln p(\bi{z}|\bi{x})-\ln p(\bi{x})\equiv \mathcal{L}_p-\ln p(\bi{x}).
\label{loss1}
\end{equation}
For flat prior this is simply the negative log likelihood of the data.
Note
that the difference between $\tilde{\mathcal{L}}_p$ and $\mathcal{L}_p$
is $\ln p(\bi{x})$, which is independent of $\bi{z}$, so in terms
of gradients with respect to $\bi{z}$ there is no difference between
the two and we will not distinguish between them.
We would like to have accurate posteriors, but we would also like to avoid
the computational cost of MCMC.
Our goal is to describe the posteriors of parameters, and our approach will be rooted in
optimization methods such as MAP or VI, where we assume a simple analytic form for the posterior, and try
to fit its parameters to the information we have.
To explain the motivation behind our
approach we will for simplicity in this section
assume we only have a single parameter
$z$ given the data $\bi{x}$,
$\tilde{\mathcal{L}}_p=-\ln p(z|\bi{x})+\ln p(\bi{x})$.
We would like to fit the posterior of $z$ to a simple form, and the Gaussian ansatz is the
simplest,
\begin{equation}
\mathcal{L}_q=-\ln q(z), \, q(z)=N(z;\mu,\Sigma).
\label{1drank}
\end{equation}
We will also assume the posterior is Gaussian for the purpose
of expectations, but since this is something we do not know a
priori we will perform the estimation of parameters of $q$.
\subsection{Stochastic KL divergence minimization}
Many of the most popular statistical inference methods are rooted in the minimization of
KL divergence, defined as
\begin{equation}
{\rm KL}(q||p)=\E_q(\mathcal{L}_p-\mathcal{L}_q), \, {\rm KL}(p||q)= \E_p(\mathcal{L}_q-\mathcal{L}_p).
\end{equation}
Here $\E_{q}$, $\E_{p}$ is the expectation over $q$ and $p$, respectively.
For intractable posteriors this cannot be evaluated exactly, and one tries to minimize
KL divergence sampled over the corresponding probability distributions. Deterministic evaluation using quadratures
is possible in
very low dimensions or under the mean field assumption, but this
becomes impossible in more than a few dimensions and we will not
consider it further here.
In this case stochastic sampling is the only practical
method: we will thus consider stochastic minimization of KL divergence.
We first briefly show that stochastic
minimization of ${\rm KL}(p||q)$ is noisy (this is a known result and
not required for the rest of the paper).
Let's assume
we have
generated $N_k$ samples $z_k$ from $p$, using for example MCMC, with which we evaluate $\mathcal{L}_p(z_k)$ for $k=1,..N_k$.
We have
\begin{equation}
{\rm KL}(p||q)=N_k^{-1}\left[\sum_k -\mathcal{L}_p(z_k) + \frac{(z_k-\mu)^2}{2\Sigma}+\frac{\ln (2\pi \Sigma)}{2}\right].
\end{equation}
Let us minimize
${\rm KL}(p||q)$ with respect to $\mu$ and $\Sigma$. We find that $\mathcal{L}_p(z_k)$ do not
enter into the answer at all, and we get
\begin{equation}
\mu=N_k^{-1}\sum_k z_k,\, \Sigma=N_k^{-1}\sum_k (z_k-\mu)^2.
\end{equation}
As expected
this is the standard Monte Carlo (MC) result for the
first two moments of the posterior given the MCMC samples
from $p$. The answer converges to the true value as
$N_k^{-1/2}$: one requires many samples for convergence.
MCMC sampling usually requires many calls to $\tilde{\mathcal{L}}_p(z)$ before an independent sample
of $z_k$ is generated, with the correlation length strongly dependent on the nature of the
problem and the sampling method.
If one instead tries to approximate the
moments of $p$ one obtains Expectation
Propagation method \citep{Minka01}.
Now let us look at stochastic minimization of ${\rm KL}(q||p)$. This corresponds to Variational Inference (VI)
\citep{WainwrightJordan08,BleiEtAl16}, which is argued to be significantly faster than MCMC.
Here we assume $q$ is approximate posterior with a known
analytic form, but since the posterior $p$
is not analytically
tractable we will create samples from $q$ (an example of this
procedure is ADVI, \cite{KucukelbirTRGB17}).
Let us define the samples as
\begin{equation}
z_k=\Sigma^{1/2}\epsilon_k+\mu,
\label{zk}
\end{equation}
where $\epsilon_k$ is a random number drawn from a unit variance zero mean
Gaussian $N(\epsilon_k;0,1)$.
With this we find
\begin{equation}
{\rm KL}(q||p)=N_k^{-1}\sum_k\left[ -\frac{\epsilon_k^2}{2}-\frac{\ln (2\pi \Sigma)}{2}+\mathcal{L}_p(z_k)\right].
\end{equation}
We want to use $\mathcal{L}_p(z_k)$ to update information on
the mean $\mu$, but it only enters via $z_k$ inside
$\mathcal{L}_p(z_k)$.
So if we want to minimize KL divergence with respect to $\mu$ we have to propagate its
derivative through $z_k$, the so called reparametrization trick \citep{KingmaWelling13,RezendeMW14}.
To proceed let us assume that the posterior is given by a Gaussian
\begin{equation}
p(z|\bi{x})=N(z;\mu_t,\Sigma_t),
\end{equation}
where the subscripts $t$ denote true value. Since we are for now assuming that
we do not have access to the analytic
gradient, we will envision that the gradient with
respect to the $\mu$ and $\Sigma$ parameters inside $z_k$ can be evaluated via a finite
difference, evaluating $\nabla_{z}\mathcal{L}_p(z_k)=[\mathcal{L}_p(z_k+\delta z_k)-\mathcal{L}_p(z_k)]/\delta z_k)$. With $\nabla_{\mu}\mathcal{L}_p=\nabla_{z}\mathcal{L}_p (dz/d\mu)=\nabla_{z}\mathcal{L}_p$ we find that the gradient of KL divergence with
respect to $\mu$ equal zero gives
\begin{equation}
\nabla_{\mu} {\rm KL}(q||p)=\sum_k \frac{(z_k-\mu_t)}{\Sigma_t}=0,\, \mu=\mu_t-N_k^{-1} \sum_k\Sigma^{1/2}\epsilon_k.
\end{equation}
Since the mean of $\epsilon_k$ is zero this will converge to the correct answer, but
will be noisy and the
convergence to the true value will be as $N_k^{-1/2}$.
To solve for the variance we similarly take a gradient with respect to $\Sigma$ and set it to zero,
\begin{equation}
\nabla_{\Sigma} {\rm KL}(q||p)=-\frac{1}{2\Sigma}+\sum_k \frac{\Sigma^{-1/2}\epsilon_k(z_k-\mu_t)}{2\Sigma_t}=0,
\end{equation}
with solution
\begin{equation}
\Sigma=\frac{N_k\Sigma_t}{\sum_k \left[\epsilon_k^2+(\mu-\mu_t)\Sigma^{-1/2}\epsilon_k \right]}.
\end{equation}
Note that this is really a quadratic equation in $\Sigma^{1/2}$, which may have multiple roots: minimizing
stochastic ${\rm KL}(q||p)$ is not a convex optimization problem.
Even if we have converged on $\mu=\mu_t$ rapidly so that we can drop the last term in the
denominator and avoid solving this as a quadratic equation, we are still left with a fluctuating term $(\sum_k\epsilon_k^2)^{-1}$. This expression
also converges as $N_k^{-1/2}$ to the true value. As we iterate
towards the correct solution we also have to vary $q$, so the overall number of calls
to $\tilde{\mathcal{L}}_p(z_k)$ will be larger. Results are shown in figure \ref{fig:ADVI}.
In summary, minimizing ${\rm KL}(p||q)$ and ${\rm KL}(q||p)$ with sampling is a noisy
process, converging to the true answer with $N_k$ samples as $N_k^{-1/2}$. We will argue below this is a consequence of KL
divergence integrand not being positive definite.
In this context it is not immediately obvious
why should stochastic VI be faster than MCMC, except that the prefactor for MCMC
is typically larger, because the MCMC
samples of $p$ are correlated, while VI samples drawn from $q$ are not (this is
however somewhat
offset by the fact that in stochastic VI one must also iterate on $q$).
\subsection{EL$_2$O: Optimizing the expectation of L$_2$ distance squared of log posteriors}
Minimizing KL divergence is not the
only way to match two probability distributions.
Recent work has argued that
KL divergence objective function may not be optimal,
and that other
objective functions may have better
convergence properties \citep{RanganathTAB16}. In this
work we will also modify the objective function, but with the goal
of preserving the expectation of
KL divergence minimization in appropriate limits.
A conceptually simple approach is to
minimize the Euclidean distance squared between the true and approximate log posterior
averaged over the samples drawn from some fiducial posterior $\tilde p$
close to the posterior $p$:
this too will be zero when the two
are equal, and will be averaged over $\tilde p$: as long as $\tilde p$ is close to $p$ this will
provide approximately correct weighting for the samples.
If the distance is not zero it will also
provide an estimate of the error generated by $q$ not being equal to $p$, which can be
reduced by improving on $q$. While current
estimate for $q$ can be used for $\tilde p$, and iterate on it,
we wish to separate its role in terms
of sampling versus evaluating its log posterior, so we will always denote
the sampling from $\tilde p$, even when this will mean sampling from the
current estimate of $q$.
The proposal of this paper is to replace the stochastic
KL divergence
minimization with a simpler and
(as we will show) less noisy ${\rm L_2}$ optimization.
Since KL divergence enjoys many information theory based
guarantees, we would also like this ${\rm L_2}$ optimization
to reduce to KL divergence minimization
in the high sampling limit, if $\tilde{p}=q$. We will
show later that this is indeed the case. To be slightly
more general, we can introduce
expectation of ${\rm L_n}$ distance (to the power $n$) of log posterior between the two
distributions,
\begin{equation}
{\rm EL_n}(\tilde{p})= \E_{\tilde p}\left(|\mathcal{L}_q-\mathcal{L}_p|^n \right),
\end{equation}
where $\tilde p$ denotes some approximation to $p$.
These belong to a larger class of f-divergences, $D_f(p,q)=\E_q f(p/q)$, such that KL($p|q$) is for $f(t)=t\ln t$, while for
${\rm EL_n}(p)$ we have $f(t)=t|\ln t|^n$.
In this paper we will
focus on $n=2$. We cannot directly minimize the
$L_2$ distance because we do not know the
normalization $\ln p(x)$,
so instead
we will minimize it up to the unknown
normalization,
\begin{equation}
{\rm EL_2O}(\tilde{p})=\argmin_{\mu,\Sigma,\ln \bar{p}}
\E_{\tilde p}[(\mathcal{L}_q-\tilde{\mathcal{L}}_p-\ln \bar{p})^2 ],
\label{EL2O}
\end{equation}
where $\ln \bar{p}$
is an
approximation to $\ln p(x)$ and is a free parameter to be minimized together with $\mu,\, \Sigma$.
Later we will generalize this to higher order derivatives of $\mathcal{L}_p$, for which
we do not need to distinguish between $\tilde{\mathcal{L}}_p$ and ${\mathcal{L}}_p$.
The choice of $\tilde{p}$ defines the distance.
We present first the version where $\tilde p=q$, since we know how to sample from it, later we will generalize it to other sampling proposals. However,
we view the sampling distribution as unrelated to the
hyper-parameters of $q$
we optimize for, even when $\tilde{p}=q$,
so unlike ADVI we will not be propagating the gradients with respect to the samples $z_k$
inside $\mathcal{L}_p$. For the 1D Gaussian case
we have
\begin{equation}
{\rm EL_2O}(\tilde{p})=\argmin_{\mu,\Sigma,\ln \bar{p}}N_k^{-1}\sum_k \left[\frac{(z_k-\mu)^2}{2\Sigma}-\tilde{\mathcal{L}}_p(z_k)-c\right]^2.
\label{elo1}
\end{equation}
where $c=\ln \bar{p}-(\ln 2\pi \Sigma)/2$ is a constant to be optimized together with $\mu$ and $\Sigma$. Note that we are not using equation
\ref{zk} to simplify the first term: we are separating the role of
$\tilde{p}$ as a sampling proposal, from $q$ as an approximation to
$p$, even when $\tilde p=q$.
We see that equation \ref{elo1} is a standard linear regression problem
with polynomial basis up to quadratic order in $z$, and
the linear parameters to solve are
$c-\mu^2/2\Sigma$ for $z^0$, $\mu/\Sigma$ for $z$
and $-1/2\Sigma$ for $z^2$. If we have $N_k=3$
one can obtain the complete solution via normal equations of linear algebra (or a single Newton update if using optimization), and then transform these to determine
$\mu$,
$\Sigma$ and $\ln \bar{p}$, which are uniquely determined, and if
$p$ is Gaussian ($\mathcal{L}_p$ quadratic) ${\rm EL_2O}$ is zero.
If $N_k>3$ the problem is over-constrained: if $\mathcal{L}_p$ is quadratic in
$z$
we are not gaining any additional information and ${\rm EL_2O}$ is still zero.
In fact, in this case the three samples could
have been drawn from any distribution. There is no sampling noise in minimizing ${\rm EL_2O}$ if $p$ is covered by $q$.
The results of KL divergence minimization implemented by ADVI and EL$_2$O minimization are shown in figure \ref{fig:ADVI}.
\begin{figure}[t!]
\centering \includegraphics[height=0.27\textwidth]{ADVI_fig.pdf}
\caption{Relative errors on the mean $\mu$ and variance $\Sigma$ for the Gaussian ansatz of $q$ in a setting where $p$ is Gaussian. We find that the ADVI solution is noisy and only slowly converges to the correct answer, while EL$_2$O gives the exact solution after 3 evaluations, since there are 3 parameters to determine, after which the problem is over-determined.}
\label{fig:ADVI}
\end{figure}
If $\mathcal{L}_p$ is not quadratic then minimizing ${\rm EL_2O}$ finds the solution that
depends on the
class of functions $q$ and also on the
samples drawn from $\tilde{p}$. In this case it is useful that the samples
are close to the true $p$, since that means that we are weighting ${\rm EL_2O}$ according to
the true sampling density (we will discuss the optimal choice of $\tilde{p}$ in section \ref{sec4}).
Even in this situation however minimizing ${\rm EL_2O}$ has
advantages. We will show below that the optimization is fast, more so
if $q$ comes close to covering $p$.
Moreover, the value of ${\rm EL_2O}$ at the minimum is informing us of the quality of
the fit: if the fit is good and ${\rm EL_2O}$ is low we can be more confident of the resulting $q$ being a good approximation. If the fit is poor we may
want to look for improvements in $q$. We will discuss several types of these
that can reduce ${\rm EL_2O}$ beyond the full rank Gaussian approximation for $q$.
Even though we focus on ${\rm L_2}$ distance in this paper it is worth
commenting on other distances. Of particular interest is ${\rm L_1}$
optimization defined as
\begin{equation}
{\rm EL_1O}(\tilde{p})=\argmin_{\mu,\Sigma,\ln \bar{p}}
\E_{\tilde p}(|\mathcal{L}_q-\tilde{\mathcal{L}}_p-\ln \bar{p}| ).
\end{equation}
If we use $\tilde{p}=q$ then this differs from ${\rm KL}(q||p)$ minimization only
in taking the absolute value $|\mathcal{L}_p-\mathcal{L}_q|$
(in terms of f-divergence $t\ln t$ is replaced with $t|\ln t|$).
The difference is that while minimizing ${\rm KL}(q||p)$
minimizes $\mathcal{L}_p$
regardless of the $z_k$ dependent part of $\mathcal{L}_q$, ${\rm EL_1O}$ tries to set it to
$\mathcal{L}_p=\mathcal{L}_q$, up to the normalization $\ln \bar{p}$.
For a finite number of samples the two solutions differ and the
latter enforces the sampling variance cancellation, since the
values of $\mathcal{L}_q$ and $\tilde{\mathcal{L}}_p$ are both evaluated
at the same sample $z_k$. We see from this example that the noise in
KL divergence can be traced to the fact that its integrand is not required to be
positive, while it is for ${\rm EL_1O}$ and ${\rm EL_2O}$. While there are
many f-divergences that have this property,
${\rm EL_2O}$ leads to
identical equations as KL divergence minimization
in the high sampling limit, so for the
rest of the paper we will focus on ${\rm EL_2O}$ only.
While $t\ln t$ and $t\ln^2t$ f-divergence minimization seems very similar, they are fundamentally different
optimization procedures. Minimization of
KL divergence only makes sense in the context of the KL divergence integral $\int dz q (\ln q -\ln p)$
: it is only positive after the integration and one cannot minimize the integrand
alone. Deterministic integration is only feasible in very low dimensions, and stochastic integration via Monte Carlo converges slowly, as $N_k^{-1/2}$. In contrast, minimizing EL$_2$O is based on comparing $\ln q(z_k)$ and $\ln p(z_k)$ at the same sampling
points $z_k$: if the two distributions are to be equal they should agree at every sampling point, up to the
normalization constant. There is no need
to perform an integral and instead it should be viewed as a loss function
minimization procedure: there is no stochastic integration noise.
\section{Expectation with $\bf L_2$ optimization ({\bf EL$_2$O}) method}
\label{sec3}
In this section we generalize the EL$_2$O concept in several directions.
An important trend in the modern statistics and machine learning (ML) applications
in recent years has been
the development of automatic differentiation (and Hessians) of the loss function $\mathcal{L}_p$.
Gradients enable us to do gradient based optimization and sampling, which is the
basis of recent successes in statistics and ML, from HMC to neural networks.
Codes such as Tensorflow, PyTorch and Stan have been developed to obtain analytic gradients using backpropagation method.
The key property of
these tools is that often the calculational cost of the analytic gradient
is comparable to the cost of evaluating the function itself: this is because the
calculation of the function and its gradient share many components, such that
the additional operations of the gradient do not significantly increase the overall
cost. In some cases, such as the nonlinear least squares to be discussed below,
a good approximation of the
Hessian can be obtained using function gradients alone (Gauss-Newton approximation),
so if one has the function gradient one also gets an approximation to the Hessian for free.
When one has access to the
gradients and Hessian it is worth taking advantage of this information for
posterior inference.
Assume we have a general gradient expansion of log posterior around a
sample $\bi{z}_k$,
\begin{equation}
\mathcal{L}_p(\bi{z}_k+\Delta\bi{z}_k)=
\sum_{n=0}^{\infty} \frac{1}{n!}\nabla^n_{\bi{z}}\mathcal{L}_p(\bi{z}_k)(\Delta\bi{z}_k)^n,
\end{equation}
where $\nabla^n_{\bi{z}}\mathcal{L}$ is $M^n$ dimensional tensor of higher order derivatives.
For $n=0$ this the log posterior value, for $n=1$ this is its gradient vector and for $n=2$ this is its Hessian matrix.
We perform the same expansion for our approximate posterior $q(\bi{z},\bi{\theta})$,
\begin{equation}
\mathcal{L}_q
(\bi{z}_k+\Delta\bi{z})=
-\ln q(\bi{z}_k+\Delta\bi{z})= \sum_{n=0}^{\infty}
\frac{1}{n!}\nabla^n_{\bi{z}}\mathcal{L}_q(\bi{z}_k)(\Delta\bi{z})^n.
\end{equation}
We assume that $q(\bi{z},\bi{\theta})$ has a simple form so that these gradients can be
evaluated analytically, and that it depends on hyper-parameters $\bi{\theta}$ we wish to
determine.
We will begin with
a multivariate Gaussian assumption for $q(\bi{z})$
with mean $\bi{\mu}$ and covariance $\bi{\Sigma}$,
\begin{equation}
q(\bi{z})=N(\bi{z};\bi{\mu},\bi{\Sigma})=(2\pi)^{-N/2}\det \bi{\Sigma}^{-1/2}e^{-\frac{1}{2}(\bi{z}-\bi{\mu})^T\bi{\Sigma}^{-1}(\bi{z}-\bi{\mu})},
\label{fullrank}
\end{equation}
\begin{equation}
\mathcal{L}_q =\frac{1}{2}\left[\ln \det \bi{\Sigma}+(\bi{z}-\bi{\mu})^T\bi{\Sigma}^{-1}(\bi{z}-\bi{\mu})+N\ln(2\pi) \right].
\label{lnq}
\end{equation}
To make the expansion coefficients dimensionless we
can first scale $z_i$,
\begin{equation}
z_i \rightarrow \frac{z_i-\mu_i}{\Sigma_{ii}^{1/2}}.
\label{scale}
\end{equation}
While centering (subtracting $\mu_i$) is not required at this stage,
we will use it when we discuss non-Gaussian posteriors later.
This scaling can be done using some approximate $\Sigma_{ii}$ from $q$
that need not be iterated upon, so for this reason
we will in general separate it from
the iterative method of determining $q$. In practice we
set the scaling $\Sigma_{ii}$ after the burn-in
phase of iterations, when $q$ is
typically determined by MAP and
Laplace approximation.
The considerations in previous section, combined
with the availability of analytic
gradient expansion terms, suggest to generalize ${\rm EL_2O}$ to the form that is the
foundation of this paper,
\begin{equation}
{\rm EL_2O}=\argmin_{\bi{ \theta}} \E_{\tilde p} \left\{N_{\rm der}^{-1}\sum_{n=0}^{n_{\rm max}} \sum_{i_1,..i_n}\alpha_n\left[
\nabla^n_{\bi{z}}\mathcal{L}_q(\bi{z})-\nabla^n_{\bi{z}}\mathcal{L}_p(\bi{z},\bi{\theta})
\right]^2\right\},
\label{argmin}
\end{equation}
where the averaging is done over the samples $\bi{z}_k$ drawn from $\tilde p$, and
where we define, for $n=0$, $\mathcal{L}_p=\tilde{\mathcal{L}}_p+\ln \bar{p}$. The sum over
index $i_1..i_n$ should be symmetrized, so for example for $n=2$ it is over
$i_1, i_2\ge i_1$. Here $N_{\rm der}$ is the total
number of all the terms, while $n_{\rm max}$ is the largest order of the
derivatives we wish to include.
This
equation combines three main elements: sampling from
$\tilde p$, evaluation of the log posterior $\mathcal{L}_p$
and its derivatives at samples, analytic evaluation of the same for $\mathcal{L}_q$,
and finally $L_2$ objective optimization to find the
best fit parameters $\bi{\theta}$. The latter involves evaluating
another sequence of first (and second, if second order optimization is used) order gradients, this time with respect to $\bi{\theta}$.
In equation \ref{argmin} we have introduced the
weight $\alpha_n$, which accounts for different weighting of different derivative order, so that for example
each tensor element of a Hessian $\nabla_{z_i}\nabla_{z_j}\mathcal{L}_p$ can have a different weight to each
vector element of a gradient $\nabla_{z_i}\mathcal{L}_p$, which can have different weight as each $\mathcal{L}_p$.
Because of scaling in equation \ref{scale} the
expression is invariant under reparametrization of $\bi{z}$, and one can view each element of gradient and
Hessian as one additional evaluation of $\mathcal{L}_p$, which
should have equal weight. For example, one can view $\mathcal{L}_p$ and $\nabla_{z_i}\mathcal{L}_p$
using a finite difference expression of the gradient into an evaluation of two $\mathcal{L}_p$
at two independent samples. In this view equation \ref{EL2O} turns into equation \ref{argmin}. However,
because we want the samples to be spread over sampling proposal $\tilde{p}$, it is suboptimal to have $N$ "samples" at
nearly the
same point gives by the gradient information, and for this reason $\alpha_0>\alpha_1>\alpha_2$.
Equal weight can also be justified using the fact that different gradient order terms determine
different components of $\bi{\theta}$.
In the
context of a full rank Gaussian for $q$ we
can think of the log likelihood determining an
approximation to the normalization constant $\ln p(x)$,
the gradient determining the means $\bi{\mu}$, and the
Hessian determining the covariance $\bi{\Sigma}$, of
which the
latter two
are needed for the posterior. We will in fact write the
optimization equations below explicitly in the form where these terms are
separated.
Due to the sample variance cancellation
a single sample evaluation suffices, but
each additional evaluation of these variables
can be used to improve the proposed $q$. For example, in a Gaussian mixture model for
$q$, at each sample evaluation of the log posterior, its gradient and Hessian gives enough
information to fit another full rank Gaussian mixture component.
However, other
weights may be worth exploring, such as downweighting the Hessian
in situations where it is only
approximate, as in the Gauss-Newton approximation discussed further below.
We will generally stop at $n_{\rm max}=2$, but
in some circumstances
having access
to analytic gradients beyond the Hessian could be beneficial. We will see below
that one of the main problems of full rank Gaussian variational methods is its inability to
model the change of sign of Hessian off-diagonal elements, which can be
described with the third order expansion ($n_{\rm max}=3$) terms.
However,
doing gradient expansion around a single point has its limitations: the shape of
the posterior could be very different just a short distance away:
there is no substitute for
sampling over the entire probability distribution. Moreover, evaluations beyond the
Hessian may be costly even if analytic derivatives are used.
For this reason
we will develop here three different versions depending on whether we have
access to only $n_{\rm max}=0$ information, $n_{\rm max}=1$, or $n_{\rm max}=2$.
\subsection{Gradient and Hessian version}
For $n_{\rm max}=2$ we optimize
\begin{equation}
{\rm EL_2O}=N_M^{-1}\argmin_{\ln \bar{p},\bi{ \mu},\bi{\Sigma}^{-1}}\E_{\tilde p}\left\{ \sum_{i,j\le i}^M\left[\nabla_{z_i}\nabla_{z_j}\mathcal{L}_q-\nabla_{z_i}\nabla_{z_j}\mathcal{L}_p\right]^2+\sum_{i=1}^M \left[\nabla_{z_i}\mathcal{L}_q-\nabla_{z_i}\mathcal{L}_p\right]^2
+[\mathcal{L}_q-\tilde{\mathcal{L}}_p-\ln \bar{p}]^2
\right\},
\label{argmin1}
\end{equation}
where $N_M=M(M+3)/2+1$.
In this section
we will drop for simplicity the last term
and optimization
over $\ln \bar{p}$ since
we do not need it
($q$ is already normalized). For a more
general $q$ such as a Gaussian mixture model this term needs to be included, as discussed
further below.
There is additional flexibility in terms of how much weight to give to Hessian versus
gradient information, which we will not explore in this
paper.
We first need analytic gradient and Hessian information of $q(\bi{z})$. Taking the gradient of $\mathcal{L}_q$ in
equation \ref{lnq} with respect to $\bi{z}$ gives
\begin{equation}
\nabla_{\bi{z}}\mathcal{L}_q=\bi{\Sigma}^{-1}(\bi{z}-\bi{\mu}).
\label{qgradz}
\end{equation}
The Hessian is obtained as a second derivative of $\mathcal{L}_q(\bi{z})$ with respect to $\bi{z}$,
\begin{equation}
\nabla_{\bi{z}}\nabla_{\bi{z}}\mathcal{L}_q(\bi{z})=\bi{\Sigma}^{-1}.
\label{qgrad2z}
\end{equation}
Even if we have only a single sample we can expect the Hessian evaluated at the
sample to determine $\bi{\Sigma}$
(since it has no dependence on $\bi{\mu}$), and with $\bi{\Sigma}$ determined we can use
its gradient to determine $\bi{\mu}$ from equation \ref{qgradz}. In this approach we can thus write
the optimization solution of equation \ref{argmin1} as
\begin{equation}
\bi{\Sigma}^{-1}=\E_{\tilde p} \left[ \nabla_{\bi{z}} \nabla_{\bi{z}} \mathcal{L}_p \right]
\approx
N_{k}^{-1} \sum_{k=1}^{\rm N_k}\nabla_{{\bi z}} \nabla_{\bi{z}} \mathcal{L}_p (\bi{z}_{k}).
\label{Laplace}
\end{equation}
Applying the optimization of equation \ref{argmin} with respect to $\bi{\mu}$ and
keeping the gradient terms only (i.e. dropping $n=0$ term, since $n=2$ term has no $\bi{\mu}$
dependence) we find,
\begin{equation}
\E_{\tilde p}[\nabla_{\bi{z}}\mathcal{L}_p(\bi{z})]=\bi{\Sigma}^{-1}(\E_{\tilde p} [\bi{z}] -\bi{\mu}),
\end{equation}
\begin{equation}
\bi{\mu}=-\bi{\Sigma}\E_{\tilde p} [ \nabla_{\bi{z}}\mathcal{L}_p(\bi{z})+\bi{z}]\approx
N_{\rm k}^{-1}\sum_{k=1}^{\rm N_k}\left[-\bi{\Sigma}\nabla_{\bi{z}} \mathcal{L}_p (\bi{z}_{k})+\bi{z}_{k}\right].
\label{mu2}
\end{equation}
Expectation of these equations has been derived in the context of variational methods
\citep{OpperArchambeau09}, showing that the solution to ${\rm EL_2O}$ is the same as VI minimization
of ${\rm KL}(q||p)$ in the high sample limit, if $\tilde p=q$. But there is a difference in the
sampling noise if the number of samples is low: the presence of $\bi{z}_{k}$ at the
end of equation \ref{mu2} guarantees there is no sampling noise, and no such term
appears in stochastic KL divergence based minimization.
As we argued above, stochastic minimization of KL divergence has sampling
noise, while minimizing ${\rm EL_2O}$
gives estimators in equations \ref{Laplace}, \ref{mu2} that
are exact even for a single sample, under the
assumption of the posterior belonging to the family of model posteriors
$q(\bi{z})$: it does not even matter where we draw the sample.
If the posterior
does not belong to this family we need to perform the expectation in equations \ref{Laplace}, \ref{mu2},
by averaging over more than one sample. There will be sampling noise, but the closer $q$ family is to the posterior $p$
the lower the noise.
This will be shown explicitly in examples of section \ref{sec4},
where we test the method on $q$'s generalized beyond the full
rank Gaussian.
The residual ${\rm EL_2O}$ informs us when this is needed: if it
is large it
indicates the need to improve $q$, by going beyond the full rank Gaussian.
The approach of this paper is to generalize $q$ until we
find a solution with low residual
${\rm EL_2O}$ so that the posterior is reliable: in examples of section \ref{sec4}
this is reached approximately when ${\rm EL_2O}< 0.2$.
Since the gradient and Hessian gives enough information to determine $\bi{\mu}$ and
$\bi{\Sigma}$ we can convert this into an iterative process where we draw a few
samples, even as low as a single sample only.
Assume that at the current iteration the estimate is $\bi{\mu}_t$ and
$\bi{\Sigma}_t$
and that we have drawn a single sample $\bi{z}_1$ from $q=N(\bi{z};\bi{\mu}_t,\bi{\Sigma}_t)$.
We also evaluate the gradient $\nabla_{\bi{z}}\mathcal{L}(\bi{z}_1)$
and Hessian, giving the following updates
\begin{equation}
\bi{\Sigma}^{-1}_{t+1}=\nabla_{\bi{z}}\nabla_{\bi{z}}\mathcal{L}(\bi{z}_1),
\end{equation}
\begin{equation}
\bi{\mu}_{t+1}=-\bi{\Sigma}_{t+1}\nabla_{\bi{z}}\mathcal{L}(\bi{z}_1)+\bi{z}_1.
\end{equation}
With an update $\bi{\mu}_{t+1}$ we can draw a new sample and repeat the process
until convergence. In this paper we are assuming
that the Hessian inversion to get the covariance
matrix and sampling from it via Cholesky decomposition
is not a computational bottleneck. This would limit the method
to thousands of dimensions if full rank description of all
the variables is needed, and if the cost
of evaluating $\mathcal{L}_p$ is moderate.
Note that if we were doing MAP we would have assumed $q$ is a delta function with
mean $\bi{\mu}_t$, in which case there is only one sample at $\bi{z}_1=\bi{\mu}_t$:
the equation above becomes equivalent to a
second order Newton's method for MAP optimization. One obtains a full distribution in the
full rank Gaussian approximation
by evaluating the Hessian at the MAP estimate (Laplace approximation).
The ${\rm EL_2O}$
method is a very simple generalization of the Laplace approximation
and for a single sample
has equal cost, as long as
the cost of matrix inversion is low.
Once we have approximately converged on $\bi{\mu}$ (burn-in phase) we can average over more
samples to obtain a more reliable approximation for both $\bi{\mu}$ and
$\bi{\Sigma}$. This often gives a more reliable estimate of $\bi{\mu}$ since
it smooths out any small scale ruggedness in the posterior. Typically we find a few samples suffice
for simple problems.
When problems are not simple (and ${\rm EL_2O}$ remains high)
it is better to increase the
expressiveness of $q$ beyond the full rank Gaussian, as discussed below.
Only the full rank Gaussian
allows analytic marginalization of correlated variables:
one inverts the Hessian matrix $\bi{\Sigma}^{-1}$ to obtain the
covariance matrix $\bi{\Sigma}$, and marginalization is
simply eliminatation of the rows
and columns of $\bi{\Sigma}$
for the parameters we want to marginalize over (for proper normalization one also needs to
evaluate the determinant of the remaining sub-matrix).
We want to
preserve this property for more general $q$ as well, and given the above
stated property of full rank Gaussian two ways to do so are one-dimensional
transforms and Gaussian mixtures, both of which will be discussed below.
\begin{algorithm}[tb]
\caption{Full rank Gradient and Hessian EL$_2$O}
\label{alg1}
\begin{algorithmic}
\STATE {\bfseries Input:} data $x_{i}$, size $N$
\STATE Initialize parameters $z_i$: random sample from prior, size $M$
\STATE Find MAP using optimization for initial $\bi{\mu}$. Use Laplace for initial $q(\bi{z})=N(\bi{\mu},\bi{\Sigma})$.
\WHILE{EL$_2$O value has not converged}
\STATE Draw a new sample $z_{N_k+1}$. Increase $N_k$ by 1.
\IF{Hessian available}
\STATE $\bi{\Sigma}^{-1}=N_{k}^{-1} \sum_{k=1}^{\rm N_k}\nabla_{{\bi z}} \nabla_{\bi{z}} \mathcal{L}_p (\bi{z}_{k})$
\ELSE{}
\STATE $\bi{\mathcal{H}}=\sum_{k=1}^{\rm N_k}(\bi{z}_k-\bi{\mu})(\bi{z}_k-\bi{\mu})$
\STATE $\bi{\Sigma}^{-1}= \bi{\mathcal{H}}^{-1}\sum_{k=1}^{\rm N_k} (\bi{z}_k-\bi{\mu})\nabla_{\bi{z}}\mathcal{L}(\bi{z}_k) $
\ENDIF
\STATE $\bi{\mu}=N_{\rm k}^{-1}\sum_{k=1}^{\rm N_k}\left[-\bi{\Sigma}\nabla_{\bi{z}}
\mathcal{L}_p (\bi{z}_{k})+\bi{z}_{k}\right]$
\STATE Compute EL$_2$O
\ENDWHILE
\end{algorithmic}
\end{algorithm}
\subsection{Gradient only and gradient free versions}
We argued above that it is always beneficial to evaluate the Hessian, since together
with the gradient this gives us
an immediate estimate of $M(M+3)/2$ parameters, which can be chosen to be
$\bi{\Sigma}^{-1}$ and $\bi{\mu}$, and so we get a full rank $q$
with a single sample.
Moreover,
for nonlinear least squares and related problems evaluating the Hessian in Gauss-Newton
approximation is no more expensive than evaluating the gradient. Suppose however that
the Hessian is not available and we only have access to the gradients.
This is for example the ADVI strategy \citep{KucukelbirTRGB17}, but let us look
at what our approach gives. Specifically, we want to minimize
\begin{equation}
{\rm EL_2O}=(M+1)^{-1}\argmin_{\ln \bar{p},\bi{ \mu},\bi{\Sigma}^{-1}} \E_{\tilde p}\left[
\sum_{i=1}^M \left\{\nabla_{z_i}\mathcal{L}_q-\nabla_{z_i}\mathcal{L}_p\right\}^2+
\left\{\mathcal{L}_q-\tilde{\mathcal{L}}_p-\ln \bar{p}\right\}^2
\right],
\label{grad}
\end{equation}
where again for simplicity of this section we will drop the last term and not optimize
over $\ln p(\bi{x})$, since we do not need
it. The first term on the RHS is called the Fisher divergence $F(q,p)$ if sampled
from $q$ and $F(p,q)$ if sampled from $p$ \citep{Hammad78}, and Jensen-Fisher
divergence if averaged over the two \citep{SanchezZarzo12}.
This shows the connection of the gradient
part of ${\rm EL_2O}$ to the Fisher information.
First derivatives with respect to $\bi{\mu}$ give equation \ref{mu2}. To evaluate it
we need to determine $\bi{\Sigma}$.
To get the equation for $\bi{\Sigma}$ we first derive its gradient
$ \nabla_{\bi{\Sigma}^{-1}}{\rm EL_2O}$, and its
Hessian, $\bi{\mathcal{H}}=\nabla_{\bi{\Sigma}^{-1}} \nabla_{\bi{\Sigma^{-1}}}{\rm EL_2O}$,
\begin{equation}
\bi{\mathcal{H}}=\E_{\tilde p} [ (\bi{z}-\bi{\mu})(\bi{z}-\bi{\mu})]
\approx \sum_{k=1}^{\rm N_k}(\bi{z}_k-\bi{\mu})(\bi{z}_k-\bi{\mu}) .
\end{equation}
We need $N_k=M+1$ gradients sampled at $\bi{z}_k$ for this matrix to be non-singular if $\bi{\mu}$
is also determined from the same samples. Taking the
first derivative of equation \ref{grad} with respect to $\bi{\Sigma}^{-1}$ gives
\begin{equation}
\bi{\Sigma}^{-1}=\bi{\mathcal{H}}^{-1}
\E_{\tilde p} \left[
(\bi{z}-\bi{\mu})\nabla_{\bi{z}}\mathcal{L} \right]
\approx \bi{\mathcal{H}}^{-1}\sum_{k=1}^{\rm N_k} (\bi{z}_k-\bi{\mu})\nabla_{\bi{z}}\mathcal{L}(\bi{z}_k) .
\label{gradsig}
\end{equation}
We obtained a set of equations \ref{mu2} and \ref{gradsig} that only use gradient information, but these are
different from the ADVI equations \citep{KucukelbirTRGB17}. In particular, our equations
have sampling variance cancellation built in, and if $q$ covers $p$ they
give zero error once we have drawn
enough samples so that the system is not under-constrained.
For $\tilde{p}=p$
the $L_2$ norm of equation \ref{grad} has also been proposed by \cite{Hyvarinen05} as a
score matching statistic, but was rewritten
through integration by parts into a form that does not cancel sampling variance,
similar to ${\rm KL}(p||q)$.
Finally, if we have no access to gradients we can still apply the ${\rm EL_2O}$ method:
to get all the
full rank parameters we need to evaluate the loss function in $M(M+3)/2+1$ points, and then
optimize
\begin{equation}
{\rm EL_2O}=\argmin_{\ln \bar{p},\bi{ \mu},\bi{\Sigma}^{-1}}\E_{\tilde p}\left[ \left\{\tilde{\mathcal{L}}_q-\mathcal{L}_p-\ln \bar{p}\right\}^2\right],
\label{gradfree}
\end{equation}
where this time we also need to optimize for $\ln \bar{p}$
together with $\bi{\mu}$ and $\bi{\Sigma}$.
These equations also incorporate the sampling variance cancellation.
Hybrid approaches are also possible: for example, we may have access to analytic first or
second derivatives
for some parameters, but not for others. In this case, one can design an optimization
process that uses analytic gradients and Hessian components for some of the parameters, while relying on
either numerical finite differences or gradient free approaches
for the other parameters. More generally, some parameters may require
expensive and slow evaluations (slow parameters) while others can be inexpensive (fast parameters).
In this case, we can afford to do numerical gradients with respect to fast parameters and
focus on development of analytic gradients for slow parameters. Another hybrid
approach will be discussed in the context of Gauss-Newton approximation below,
where we use the Hessian in the Gauss-Newton approximation for the covariance
matrix $\bi{\Sigma}$, and only the gradient for the remaining hyper-parameters of $q$.
\subsection{Posterior expansion beyond the full rank Gaussian: bijective 1D transforms}
So far we obtained the full rank VI solution with an iterative process which should
converge nearly as rapidly as MAP. If the posterior is close to the assumed multi-variate
Gaussian then this process converges fast, and only a few samples are needed. If there is
strong variation between the Hessian elements evaluated at different sampling points then we know
the posterior is not well described by a multi-variate Gaussian. In this case we may
want to consider proposal functions beyond equation \ref{fullrank}.
However, a multi-variate Gaussian is the only
correlated multi-variate distribution where analytic marginalization can be done by simply
inverting the Hessian matrix. This is a property that we do not want to abandon.
For this reason we will first consider one-dimensional transformations of the original
variables $\bi{z}$ in this subsection, and Gaussian mixtures in the next.
Variable transformations need to
be bijective so that we can easily go from one set of the variables to the other and back \citep{RezendeMohamed15}.
Here we will
use a very simple family of models that give rise to
skewness and curtosis, which are the one-dimensional versions of the gradient
expansion at third and fourth order.
Specifically, we will consider bijective transformations of the form $y_i(z_i)$ such that
\begin{equation}
q(\bi{z})=N({\bi{y}};\bi{\mu}, \bi{\Sigma})\Pi_i |J_i|,\,\,J_i=\frac{dy_i}{dz_i},
\label{multiy}
\end{equation}
with $N({\bi{y}};\bi{\mu}, \bi{\Sigma})$ given by equation \ref{fullrank}. Under this form the marginalization
over the variables is trivial. For example,
marginalized posterior distribution of $z_i$ is \newline $q[z_i(y_i)]=N({y_i};\mu_i,\Sigma_{ii})|dy_i/dz_i|$,
where $\Sigma_{ii}$ is the diagonal component of the covariance matrix, obtained by
inverting the Hessian matrix $\bi{\Sigma}^{-1}$.
We would like to modify the variables $z_i$ such that
the resulting posteriors can accommodate more of the variation of $\tilde{\mathcal{L}}_p$.
In one dimension this would be their skewness and curtosis, which are indicated for
example by the variation of the Hessian with the sample, but in higher dimensions
we also want to accommodate variation of off-diagonal terms of the Hessian.
In typical situations given the full rank
solution and the scaling of equation \ref{scale}
the posterior mass
will be concentrated around $-1<z_i<1$, but the distribution may be skewed, or have
more or less posterior mass outside this interval.
A very simple change of variable is
$
y_i=z_i+\frac{1}{2}\epsilon_i z_i^2+\frac{1}{6}\eta_i z_i^3$,
where we assume $\epsilon_i$ and $\eta_i$ are both parameters that can be either
positive or negative, but small such that the relation is invertible. In one dimension
the log of
posterior is, keeping the terms at the lowest order in $\epsilon_i$ and $\eta_i$,
$2\mathcal{L}_q \approx c+z_i^2+\epsilon_i z_i^3+\frac{1}{3}\eta_i z_i^4...$
Viewed as a Taylor expansion
we see that $\epsilon_i$ term determines the third order gradient expansion
and $\eta_i$ the fourth order, both around $z_i=0$.
To make these expressions valid for larger values of $\epsilon_i$ and $\eta_i$ we promote
the transformation into
\begin{equation}
y_i(z_i)={\rm sinh}_{\eta}\left[ \frac{\exp(\epsilon_iz_i)-1}{\epsilon_i}\right],
\label{sk}
\end{equation}
where for $\epsilon_i=0$ the above is just $y_i(z_i)={\rm sinh}_{\eta} z_i$ \citep{SchuhmannJoachimiEtAl16}
\begin{equation}
{\rm sinh}_{\eta}(x) =
\begin{cases}
\eta^{-1}{\rm sinh}(\eta x) & (\eta>0) \\ x & (\eta=0) \\ \eta^{-1}{\rm arcsinh}(\eta x) & (\eta<0).
\end{cases}
\end{equation}
These are bijective, but not guaranteed to give the required posteriors.
We can however apply the transformations multiple times for a more expressive
family of models. For small values of
$\epsilon$ and $\eta$ equation \ref{sk} reduces to the Taylor expansion above.
If the posteriors are multi-peaked then these transformations may not be sufficient,
and
Gaussian mixture models can be used instead, discussed below.
The gradient of $\mathcal{L}_q $ is
\begin{equation}
\nabla_{z_i}\mathcal{L}_q=\sum_j(\Sigma^{-1})_{ij}(y_j-\mu_j) J_i-\frac{\nabla_{z_i}|J_i|}{|J_i|},
\end{equation}
while the Hessian is
\begin{equation}
\nabla_{z_i}\nabla_{z_j}\mathcal{L}_q=(\Sigma^{-1})_{ij}J_iJ_j+\left[\sum_k (\Sigma^{-1})_{ik}(y_k-\mu_k) \nabla_{z_i}J_i -\frac{\nabla_{z_i}\nabla_{z_i}|J_i|}{|J_i|}+\left(\frac{\nabla_{z_i}|J_i|}{|J_i|}\right)^2\right]\delta_{ij}.
\label{hnl}
\end{equation}
If the Hessian is varying with the samples $\bi{z}_k$ we have an indication
that we need higher order corrections.
With the gradient and Hessian at a single sample $\bi{z}_1$ we
have the sufficient number of constraints, $M(M+3)/2$, to determine $\bi{\mu}$ and $\bi{\Sigma}^{-1}$. If we evaluate these variables at another sample $\bi{z}_2$ we
already have too many constraints to determine the additional $2M$ nonlinear transform variables, so the problem
is overconstrained even with two drawn samples. This is the power of having access to the gradient and
Hessian information: we converge fast both because we can use Newton's
method to find the solutions and because a few samples give us enough
constraints.
\subsection{Posterior expansion beyond the full rank Gaussian: Gaussian mixtures}
A second non-bijective way that can extend the expressivity of posteriors while still allowing for analytic
marginalizations is a Gaussian mixture model \citep{BishopLawrenceEtAl97}. Here we model the posterior as a weighted sum
of several multi-variate Gaussians, each of which can have an additional 1D NL transform, as
in equation \ref{multiy},
\begin{equation}
q(\bi{z})=\sum_j w_j N({\bi{y}^j};\bi{\mu}^j, \bi{\Sigma}^j)\Pi_i \left|\frac{dy_i^j}{dz_i}\right|
\equiv \sum_j w_j q^j(\bi{z}),
\label{gm}
\end{equation}
where $\sum_j w_j=1$.
We can introduce the position dependent weights
\begin{equation}
w_j(\bi{z})=\frac{w_jq^j(\bi{z})}{q(\bi{z})}.
\end{equation}
We can now derive the corresponding $\nabla_{\bi{z}}\mathcal{L}_q$ and $\nabla_{\bi{z}}\nabla_{\bi{z}}\mathcal{L}_q$. For example, if $\bi{y}=\bi{z}$ the gradient is
\begin{equation}
\nabla_{\bi{z}}\mathcal{L}_q=\sum_j w_j(\bi{z})\nabla_{\bi{z}} \mathcal{L}_q=\sum_j w_j(\bi{z})(\bi{\Sigma}^j)^{-1}(\bi{z}-\bi{\mu}^j),
\end{equation}
and is simply a weighted gradient of each of the Gaussian mixture components. The Hessian is
\begin{eqnarray}
&&\nabla_{\bi{z}}\nabla_{\bi{z}}\mathcal{L}_q=\sum_j \left[\nabla_{\bi{z}}w_j(\bi{z})\nabla_{\bi{z}}
\mathcal{L}_q+w_j(\bi{z})\nabla_{\bi{z}}\nabla_{\bi{z}}\mathcal{L}_q\right]\nonumber \\
&=&\sum_j w_j(\bi{z})
(\bi{\Sigma}^j)^{-1}-\sum_i\sum_{j \ne i} \frac{w_i(\bi{z})w_j(\bi{z})}{w_i}
\left[(\bi{\Sigma}^j)^{-1}(\bi{z}-\bi{\mu}^j)(\bi{\Sigma}^i)^{-1}(\bi{z}-\bi{\mu}^i)\right].
\label{hessgm}
\end{eqnarray}
Once we have these
expressions we can insert them into equation \ref{argmin} and optimize against the parameters of $q(\bi{z})$.
This is an optimization problem and requires iterative method to find the solution, but
no additional evaluations of $\mathcal{L}_p$. While previously we did not use
$\mathcal{L}_p$ itself since we did not need $c$, now we need to use this value as
well, at it determines the weights $w_j$.
With $\mathcal{L}_p$, its gradient and its
Hessian we have enough data to determine one Gaussian mixture component per sample
$\bi{z}_k$.
Once we have constructed the full $q$,
to analytically marginalize over some of $\bi{z}$ we need to invert separately each of the matrices $(\bi{\Sigma}^j)^{-1}$.
To summarize, there exist expressive posterior parametrizations beyond the
full rank Gaussian that allow for
analytic marginalizations and that can fit a broad range of posteriors.
Gaussian mixture model can for example be used for a multi-modal posterior distribution, and a
single evaluation with a Hessian can fit one component of the multi-variate Gaussian mixture model.
One dimensional nonlinear transforms can be used to give skewness, curtosis and even multi-modality
to each dimension.
\subsection{Sampling proposals}
\label{sampl}
We argued above that KL$(p||q)$ minimization leads to standard MC method, while
minimizing KL$(q||p)$ gives VI method. ${\rm EL_2O}$ has more flexibility in terms of the choice of the sampling proposal $\tilde{p}$. We list some of these below, with some
specific examples presented in section \ref{sec4}.
{\it Sampling from $q$}: If we sample from $\tilde{p}=q$ and minimize
${\rm EL_2O}$ we get results equivalent to VI in the large sample limit.
While sampling from $q$, and iterating on it, is the simplest choice,
it is not the only choice, and may not be the best choice either.
One disadvantage is that the samples change as we vary $q$ during optimization, increasing the
number of calls to $\mathcal{L}_p$. A second disadvantage is that in high
dimensions sampling from the full rank Gaussian becomes impossible since the
cost of Cholesky decomposition becomes prohibitive. We will address this
problem elsewhere.
{\it Sampling from $p$}: one
alternative is to sample
from $p$ itself. This has the advantage that the samples do not need to change
as we iterate on $q$, and if the cost of $\mathcal{L}_p$ is dominant this
can be an attractive possibility. There are inference problems where sampling from $p$ is
easy. An example are forward inference problems: suppose we know
the prior distribution of $\bi{x}$ and we would like to
know the posterior of $\bi{z}=f(\bi{x})+\bi{n}$, where $f$ is some
function and $\bi{n}$ is noise. In this case we can create samples of $\bi{z}$ drawn from $p$ simply
by drawing samples of $\bi{x}$ from its prior and $\bi{n}$ from noise distribution
and evaluating $\bi{z}=f(\bi{x})+\bi{n}$.
In most cases however sampling from $p$ is hard.
One possibility is to create a number of true samples from $p$ using
MCMC. This may be expensive, since there is a burn-in period that one
needs to overcome first. We can use ${\rm EL_2O}$
optimization with $\tilde{p}=q$
for the burn-in phase to get to the minimum, and from there sampling can be
almost immediate, but samples will be correlated, and often
the correlation length can be hundreds or more. For modern methods like
HMC sampling can be more efficient in traversing the
posterior with a lower number of $\tilde{\mathcal{L}}_p$
calls, so this approach is worth exploring further.
{\it Sampling from approximate $p$}: since MCMC is expensive
one can try cheaper alternatives. One possibility is to sample from an approximate posterior generated with the help of
simulations. Suppose one generates a simulation where one knows the answer.
One then performs the analysis as on the data, obtaining the point estimate
on the parameters in terms of their best fit mean
or mode (MAP). Since we know the truth for simulation we can create a data sample
by adding the difference between the truth and the point estimate of the
simulation to the point estimate of the data. This will give an approximation
to the posterior distribution $p$ where each sample is completely independent.
It will not be exact because of realization dependence of the posterior,
and some of the samples may end up being very unlikely in the sense of
having a very high $\tilde{\mathcal{L}}_p$. Additional importance sampling
may be needed to improve this posterior further.
Another strategy is not to sample at all, but devise a deterministic algorithm to select the points where to evaluate
$\mathcal{L}_p$.
For example,
given a MAP+Laplace solution one could select the points that
are exactly a fixed fraction of standard deviation away from MAP for every parameter.
This strategy has had some success in filtering applications, where it is called
unscented Kalman filter (UKF, \cite{JulierUhlman04}). Another option is
deterministic quadratures, such as Gauss-Hermite integration. These
deterministic approaches become very expensive in high dimensions.
{\it Sampling from $q+p$}: symmetric KL divergence is called Jensen-Shannon divergence, and we can similarly do the same for
EL$_2$O. Since L$_2$ norm is already symmetric we just need
to sample from $p+q$.
This may have some benefits: if one samples from $p$ then
there is no penalty for posterior densities $q$ which do not vanish
where $p=0$. Conversely, if one samples from $q$ there is no penalty
for situations where $p$ does not vanish while $q=0$.
For example, there may be
multiple posterior peaks in true $p$, but if we only found one we would never know
the existence of the others. In both cases
the difficulty arises because the normalization of $p$ is not accessible to us.
The latter case is often argued to be more
problematic suggesting sampling from $p$ should be used if possible.
However, note that in the case of widely separated
posterior peaks sampling from true $p$ using MCMC may not be possible either, as MCMC
may not find all of the posterior peaks. In this case
sampling from $q$ using
multiple starting points with
a Gaussian mixture for $q$ is a
better alternative.
If both of these issues are a concern one can try to sample from both
$q$ and from $p$, mixing the two types of samples.
This will give us
samples where $p=0$ but $q>0$, and where $q=0$ but $p>0$, if
such regions exist.
Another
hybrid sampling $p+q$ method is, after the burn in, to iterate on $q$,
sample from it, use it as a starting point for
a MCMC sampling method with fast mixing properties (such as HMC)
to move to another point, record the sample, update $q$, and repeat the process.
The samples from $q$ may suffer from having too large values of $\tilde{\mathcal{L}}_p$,
so a Metropolis style acceptance rate can be added to prevent this.
It is worth emphasizing that the philosophy
of this paper is to find $q$ that fits the posterior everywhere: both
samples from $q$ and samples from $p$ should lead to ${\rm EL_2O}$ close to zero.
and if $q$ covers $p$ we should always find this solution in the limit of
large sampling density. These issues will be explored further in
next section, where we show for specific examples
that more expressive $q$ reduce the
differences between sampling from $q$ versus sampling from $p$ or $q+p$.
\subsection{Hessian for nonlinear least squares and related loss functions}
One of the most common statistical analyses in science is a nonlinear
least squares problem, which is a simple acyclic graphical
model. One has some data vector $\bi{x}$ and some
model for the data $\bi{f}(\bi{z})$, which is nonlinear in terms of
the parameters $\bi{z}$. We also assume a known data measurement noise covariance matrix $\bi{N}$,
which can be dependent of the parameters $\bi{z}$.
One may add a prior for the latent variables in the form,
\begin{equation}
\tilde{\mathcal{L}}_p=\frac {1}{2} \left\{\bi{z}^T\bi{Z}^{-1}\bi{z}+[\bi{x}-\bi{f}(\bi{z})]^T\bi{N}^{-1}[\bi{x}-\bi{f}(\bi{z})]+\ln \det \bi{ZN}\right\},
\label{ls}
\end{equation}
where $\bi{Z}$ is the prior covariance matrix of $\bi{z}$ and we assumed the prior
mean is zero (otherwise we also need to subtract out the prior mean from $\bi{z}$
in the first term).
The Hessian in the Gauss-Newton approximation is
\begin{equation}
\E_{\tilde{p}}[\nabla_{\bi{z}} \nabla_{\bi{z}}\mathcal{L}_p] \approx
\bi{Z}^{-1}+ \E_{\tilde{p}}\left\{ 2{\rm tr} \left[\bi{N}^{-1} (\nabla_{\bi{z}}\bi{N})
\bi{N}^{-1}(\nabla_{\bi{z}}\bi{N})\right]+(\nabla_{\bi{z}}\bi{f})^T\bi{N}^{-1}(\nabla_{\bi{z}}\bi{f})
\right\},
\end{equation}
where we dropped the second derivative terms of $\bi{f}$ and $\bi{N}$. The former
term multiplies the residuals
$\bi{x}-\bi{f}(\bi{z})$, which close to the best fit (i.e. where the posterior mass
is concentrated) are oscillating around zero
if the model is a good fit to the data. This
suppresses this term relative to the first derivative
term, which is always positive:
in the Gauss-Newton approximation the curvature matrix is explicitly positive definite,
and so is its expectation value over the samples. Clearly this is no longer valid if we move
away from the peak, or
we have a multi-modal posterior, since we have extrema that are saddle points or local minima
and Gauss-Newton is a poor approximation there,
so care must be exercised when using Gauss-Newton away
from the global minimum. In our applications we use the Hessian to determine
$\bi{\Sigma}^{-1}$, but we ignore it when evaluating nonlinear parameters $\epsilon$ and $\eta$.
The cost of evaluating the Hessian under the Gauss-Newton approximation
equals the cost of evaluating the gradient $\nabla_{\bi{z}}\mathcal{L}_p$, since it simply involves
first derivatives of $\bi{f}$ or $\bi{N}$.
\subsection{Range constraints}
If a variable has a boundary then finding a function extremum
may not be obtained by finding where its gradient is zero, but may instead be
found at the boundary. In this case the posterior distribution is abruptly
changed at the boundary, which is difficult to handle with
Gaussians.
The most common case is that a given variable is bounded to a one sided
interval, or sometimes to a two-sided interval.
There are two methods one can adopt, first one is a transformation to
an unconstrained variable and second one is a reflective boundary condition.
Suppose for example that we have a constraint
$z_i'>a_i$, and we would like to have an
unconstrained optimization that also transitions to the $z'$ prior on a scale $\xi_i$ away from $a_i$.
We can use
\begin{equation}
z_i=\xi_i\ln\left(e^{\frac{z_i'-a_i}{\xi_i}}-1\right)
\label{uncon}
\end{equation}
as our new variable \citep{KucukelbirTRGB17}.
This variable becomes $z_i'$ for $z_i'\gg a_i+\xi_i$ and $\xi_i\ln (z_i'-a_i)/\xi_i$ for
$a_i<z_i'\ll a_i+\xi_i$, so $z_i$ is now defined on the
entire real interval with no constraint. If we want to preserve the probability and use
the prior on original $z'$ we must also
include the Jacobian $J_i=|dz_i/dz_i'|$, $p(z_i)=p(z_i')J_i^{-1}$.
The presence of the Jacobian modifies the loss
function.
In our examples below we are including the Jacobian and we treat $\xi_i$
as another nonlinear parameter attached to parameter $z_i$ with a
constraint, so that we optimize EL$_2$O with respect to it.
A problem with this method is that posterior will always go to zero
at $z_i' \rightarrow a_i$, since the Gaussian goes to zero
at large values. So even though this method can be quite
successful in getting most of the posterior correct, it
will artificially turn down to zero
at the boundary. This is problematic,
since it suggests the data exclude the boundary even if they
do not.
Second approach to a boundary $z_i'>a_i$ is to extend the
range to $z_i'<a_i$ using a reflective (or mirror) boundary condition across
$z_i'=a_i$, such
that if $z_i'<a_i$ then $\tilde{\mathcal{L}}_p(z_i'-a_i)=\tilde{\mathcal{L}}_p(a_i-z_i')$.
This leads to the non-bijective transformation of
appendix A with $b_i=0$: we have $z_i'$ defined on
entire range and we model it with a sum of two mirrored Gaussians.
Effectively this is
equivalent to an unconstrained posterior analysis, where
we take the posterior
at $z_i'<a_i$ and add it to $z_i'>a_i$.
It solves the problem of the unconstrained
transformation
method above, as the posterior at the boundary is not forced to zero,
since it can be continuous and non-zero across the boundary $a_i$. The marginalization
over this parameter remains trivial, since it is as if the parameter is not constrained at all.
For the purpose of the marginalized posterior for the
parameter itself, we must add the $z_i'<a_i$
posterior to $z_i'>a_i$ posterior. If the posterior mass is non-zero at
$z_i'=a_i$ then this will result in the posterior abruptly transitioning
from a finite value to 0 at the boundary.
This method can be generalized to a two sided boundary.
In section \ref{sec4} we will show an example of both
methods.
\subsection{Related Work}
Our proposed divergence is in the family of f-divergences. Recently, several divergences have been introduced
(e.g. \citet{RanganathTAB16,DiengTRPB17})
to counter the claimed problems of KL divergence such as its asymmetry and exclusivity of $q$, but here
we argue that with
sufficiently expressive $q$ these problems may not be fundamental
for EL$_2$O method. Expectations of EL$_2$O equations agree with VI
expressions of
\citep{OpperArchambeau09}.
Stochastic VI has been explored for posteriors in several papers,
including ADVI \cite{KucukelbirTRGB17}. In direct comparison test we
find it has a slower convergence than EL$_2$O.
For $n=1$ the Fisher divergence minimization has been proposed by \citet{Hyvarinen05} as a
score matching statistic, but was rewritten
through integration by parts into a form that does not cancel sampling variance and has similar convergence properties as stochastic VI.
Reducing sampling noise has also been explored more recently in
\citet{RoederWD17} in a different context and with a different approach.
Quantifying the error of the VI approximation has been explored in \citet{YaoVSG18}, but
using EL$_2$O value
is simpler to evaluate.
NL transformations have been explored in terms of boundary effects in \citet{KucukelbirTRGB17}. Our NL transformations
correspond more explicitly to generalized skewness and curtosis parameters,
and as such are useful for general description of probability distributions. We employ
analytic marginals to obtain posteriors and for this reason we only employ a single layer point-wise NL transformations, instead of the more powerful
normalizing flows \cite{RezendeMohamed15}. More recently, \cite{LinKS19}
also adopt GM and NL for similar purposes, also using Hessian based
second order optimization (called natural gradient in recent ML literature).
\section{Numerical experiments}
\label{sec4}
In this section we look at several examples in increasing order of complexity.
\subsection{Non-Gaussian correlated 2D posterior}
\begin{figure}[t!]
\centering \includegraphics[height=0.34\textwidth]{Fig2_3.pdf} \hspace{0cm} \includegraphics[height=0.34\textwidth]{Fig2_1.pdf}
\caption{Example of a correlated non-Gaussian posterior problem, where one of the two Gaussian correlated variables $y_2$ is preceded by a nonlinear transformation (mapped by the exponential function, $z_2 = \mathrm{exp}(y_2)$). \textit{Left}: The 2D posterior and the means estimated by various methods. \textit{Right}: 1D marginalized posterior of $z_2$, with the black vertical line marking its true mean. MAP (blue) finds the mode and MFVI (green)
FRVI (yellow) estimate the mean relatively well, but all of them fail to capture the correct shape of the posterior and its variance. Fitting for the skewness and curtosis parameters, EL$_2$O with the NL transform (NL-EL$_2$O, red) accurately models the posterior. All curves have been normalized to the same value at the peak to reduce their
dynamical range.}
\label{banana}
\end{figure}
\begin{figure}[b!]
\centering \includegraphics[height=0.34\textwidth]{Fig2_2.pdf} \includegraphics[height=0.34\textwidth]{Fig1_3.pdf}
\caption{EL$_2$O values provide an estimate of the quality of the fit. Here we show
them as a function of the number of iterations for the correlated non-Gaussian posterior example (\textit{Left panel}) and for the forward model posterior example (\textit{Right panel}). Typically values of EL$_2$O $\lesssim 0.2$ indicate that we have obtained a
satisfactory posterior. The convergence is faster for NL-EL$_2$O despite
having more parameters, a consequence of sampling noise free
nature of EL$_2$O. In these examples each iteration draws 5 samples and we average
over the past samples after the burn in. \textit{Right}: Sampling from $p$ (red dashed) gives better EL$_2$O that sampling from $q$ (green dashed) for full rank GM. As we increase the expressivity of $q$, by applying the NL transform to better match the posterior, this makes EL$_2$O values for sampling from $p$ and $q$ more similar, and lower, improving the overall quality of the posterior.}
\label{ELO}
\end{figure}
In the first example we have a 2-dimensional problem modeled as two Gaussian distributed
and correlated variables $z_1$ and $z_2$, but second one is
nonlinear transformed using $\exp(z_2)$ mapping.
This transformation is not in the family of
skewness and curtosis transformations proposed in section \ref{sec3}. Here we will try to model the
posterior using $\epsilon$ and $\eta$ in addition to $\bi{\mu}$ and $\bi{\Sigma}$. The question
is how well can our method handle the posterior of $z_2$, as well as the joint posterior
of $z_1$ and $z_2$, and how does it compare to MAP, MF and FR VI or EL$_2$O.
The results are shown in figure \ref{banana}. Left panel shows the 2D contours,
which open up towards larger values of $z_2$ and as a result the MAP is away from the
mean. Right hand panel shows the resulting posterior of MAP+Laplace, MF and FR EL$_2$O
(which equals MFVI and FRVI in large sampling limit), and
NL EL$_2$O. MAP gets the peak posterior correct but not the mean, MF improves on the
mean and FR improves it further, but none of these get the full posterior. NL EL$_2$O
gets the full posterior in nearly perfect agreement with the correct distribution, which is , 0.13 versus 0.5 or 0.7,
respectively. What is interesting that the convergence of NL EL$_2$O is faster,
despite having more parameters: the convergence has been reached
after 8 iterations. We started with $N_k=1$ and ended with $N_k=5$ for this example. Note that we can reuse
samples from previous iterations.
\subsection{Forward model posterior}
A very simple state evolution model
is where we know
the prior distribution of $x$, assumed to be a Gaussian with zero mean and
variance $\Sigma$, and we would like to
know the posterior of $z=x^2+n$, where
$n$ is Gaussian noise with zero mean and variance $Q$.
The loss function $\tilde{\mathcal{L}}_p$ is
\begin{equation}
\tilde{\mathcal{L}}_p=\frac{1}{2}\left[x\Sigma^{-1}x
+(z-x^2)Q^{-1}(z-x^2)\right],
\label{likelihood}
\end{equation}
where we dropped all irrelevant constants.
We would like to find the posterior of $z$ marginalized over $x$.
In the absence of noise the problem can be solved using the Jacobian between $x$ and $z$,
but addition of noise requires an additional convolution. In higher dimensions evaluating
the Jacobian quickly becomes very expensive, so we will instead solve the problem by
approximating the joint probability distribution of $x$ and $z$, and then
marginalizing over $x$. This is a hard problem because the joint
distribution is very non-Gaussian, as seen in figure \ref{pq}.
We can first attempt to solve with MAP. The MAP solution is at $\hat{x}=\hat{z}=0$,
and at the MAP Laplace approximation gives a diagonal Hessian between $x$ and
$z$, so the two are uncorrelated. The variance on $z$ is $Q$, which
vanishes in no noise $Q \rightarrow 0$ limit. MAP+Laplace for $z$ is thus
a narrow distribution at zero, which is clearly a very
poor approximation to the correct posterior.
The full rank VI or ${\rm EL_2O}$ approach is to sample from full rank Gaussian
$q$ and iterate
until convergence. The off-diagonal elements of the Hessian are given by
$\nabla_z\nabla_x\mathcal{L}_p=-2xQ^{-1}$, which vanishes upon
averaging over $x$, so full rank and mean field solutions are equal
and FRVI assumes the two variables are uncorrelated. The variance on
$z$ is again given by $Q$.
The inverse variance of $x$ is $\nabla_x\nabla_x\mathcal{L}_p=\Sigma^{-1}+4x^2Q^{-1}$,
which in ${\rm EL_2O}$ or VI we need to average over $q$.
The stationary point is reached when $\E[ \nabla_x\nabla_x\mathcal{L}_p]^{-1}=\E[ x^2 ]=\sigma_x^2$, which in the low $Q$ limit
gives $\sigma_x^2=Q^{1/2}/2$. Once we have the full rank Hessian we can also determine
the means from the gradient of equation \ref{likelihood}, finding a solution $\mu_x=\mu_z=0$.
So we find a somewhat absurd result that even though
$x$ is not affected by $z$ and its posterior should equal its prior,
FRVI gives a different solution, one of a delta function at zero in the $Q \rightarrow 0$ limit,
which is identical to the MAP solution.
However, with this posterior the value of ${\rm EL_2O}$ is large, because the Hessian
is fluctuating across the posterior and is not well represented with a single average.
This is specially clear for the off-diagonal elements, whose average is zero, but the actual
values fluctuate with rms of order $Q^{-3/4}$, very large
fluctuations if $Q \rightarrow 0$.
\begin{figure}[t!]
\centering \includegraphics[height=0.34\textwidth]{Fig1_1.pdf} \hspace{0cm} \includegraphics[height=0.34\textwidth]{Fig1_2.pdf}
\caption{Example of a forward inference problem.
\textit{Left}: Contours of two symmetric Gaussian components (GM), with NL transform applied, together with samples from the posterior. The elliptical contours are warped by NL transform to better match the posterior. The total
posterior is the sum of the two, which enhances the posterior density at $x\sim 0$.
\textit{Right}: 1D marginalized posterior of $z$ as approximated by different methods. MAP and FRVI (blue, normalized to the same peak value to reduce the dynamic range of the plot) give a poor estimate of posterior compared to MCMC (histogram). For EL$_2$O, we evaluated GM+NL with the following sampling proposals $\tilde{p}$: sampling from $p$ (red), $q$ (green), and $p+q$ (black). Note that sampling
from $p$ is narrower than sampling from $q$.
Vertical bars indicate the means, including MCMC (light blue dashed).
}
\label{pq}
\end{figure}
One must improve the model by going beyond a single full rank Gaussian.
Here we will do so with a non-bijective transformation of
appendix A, using $b_x=0$ and $a_x=0$, i.e. we model it as
two Gaussian components
mirror symmetric across $x=0$ axis. We use equation \ref{hessgm}, which says that if the two Gaussian
components are well separated
the local Hessian can be used to determine $\bi{\Sigma}^{-1}$ of the local
Gaussian component (which then also determines the covariance of the other component
due to the symmetry).
The local Hessian
is given by $\nabla_x\nabla_x\mathcal{L}_p=\Sigma^{-1}+4\mu_x^2Q^{-1}$,
$\nabla_x\nabla_z\mathcal{L}_p=-2\mu_xQ^{-1}$ and $\nabla_z\nabla_z\mathcal{L}_p=Q^{-1}$.
To further improve the model we consider bijective
nonlinear transforms (NL).
These are useful as they warp the
ellipses, which allows to match $q$ closer to the true
posterior $p$.
The results are shown in figure \ref{pq}. We see from the figure that MAP or FRVI=FR-EL$_2$O
fail to give the correct posterior, while the Gaussian mixture with NL gives
a very good posterior of $z$, in good agreement with MCMC.
In this example we can sample from $p$ directly, so we do not need to iterate
on samples from $q$.
${\rm EL_2O}$ has flexibility to use samples from either of the two, which is distinctly
different from KL based VI. This forward model problem
gives us the opportunity to compare
the results between the two.
We would like to know if
sampling from $q$ versus $p$ gives different
answers, and if sampling from both further improves
the results. This is also shown in figure \ref{pq}. We see that there are some
small differences in the posteriors, and that sampling from $q$ is slightly worse:
in terms of ${\rm EL_2O}$ value, we get 0.20 for sampling from $p$ and
$p+q$ and 0.23 for sampling from $q$. Somewhat surprising, we find that
sampling from $q$ gives a broader approximation
that sampling from $p$, contrary to KL divergence based FRVI
\citep{Bishop07}.
Sampling from $p+q$ does
not further improve the results over sampling from $p$.
The difference between
$p$ and $q$ sampling is
larger if we restrict to the full rank Gaussian without
NL, and the ${\rm EL_2O}$ values are also larger: 0.4 for $p$ versus 0.5 for $q$.
This suggests that while for simple $q$ the results may
be biased and sampling from $p$ is preferred, more expressive
$q$ reduces the difference between the two. This is not
surprising: if $p$ is in the family of $q$ then we should be
able to recover the exact solution with
optimization, finding ${\rm EL_2O}=0$ upon convergence.
If we want to improve the $q$ sampling
results of figure \ref{pq} we can do so
by adding additional Gaussian mixture components, or
additional NL transforms, but we have not attempted to do so here.
A potential concern is that the exclusive nature of $q$ may lead to a situation
where EL$_2$O sampled from $q$ is low, but the quality is poor, because $p>0$
where $q=0$. If this happens because there is another posterior maximum elsewhere
far away then the only way to address it is using global optimization techniques
like multiple starting points. All methods, including MCMC, have difficulties in these
situations and require specialized methods (see below). If, on the other hand, there is excess
posterior mass that is smoothly attached to the bulk of $q$ then EL$_2$O method
should be able to detect it, specially with gradient and Hessian information.
We can test it on this example
by comparing EL$_2$O values on samples evaluated from $p$ versus samples evaluated
from $q$, while using the same $\mathcal{L}_q$ to evaluate EL$_2$O.
We find very little difference between the two, 0.24 versus 0.23, and so
${\rm EL_2O}$ evaluated on samples from $q$ gives a
reliable estimate of the quality of solution.
In these examples we used up to
15 iterations with $N_k=5$, and
averaging over all past iterations after the burn-in.
\subsection{Multi-modal posterior}
Multi-modal posteriors are very challenging for any method. If the
modes are widely separated then standard MCMC methods will fail, and specialized
techniques, such as annealing or nested sampling \citep{HandleyHobsonEtAl15} are required.
If the modes are closer to each other so that their posteriors overlap
then MCMC will be able to find them, and
this is considered to be a strength of MCMC as compared to MAP or VI, which
in the simplest implementations find only one of the modes. Here we
use EL$_2$O with a Gaussian mixture (GM) on a simple bimodal posterior, which is a sum of two
full rank Gaussians in 2 dimensions.
\begin{figure}[t!]
\centering \includegraphics[height=0.32\textwidth]{Fig3_1.pdf} \includegraphics[height=0.32\textwidth]{Fig3_2.pdf}
\caption{Application of a Gaussian mixture (GM) model to the multi-modal posterior problem.
\textit{Left}: Modeling the posterior as a weighted sum of two bivariate Gaussians, we demonstrate that the EL$_2$O method identifies both peaks, with means and covariances accurately estimated. A single starting point with 2 GM components converges to this solution after 15 iterations. For multiple starting points, each one converges within a few iterations to one of the two local minima, and EL$_2$O properly normalizes the two GM components. The two final solutions are identical (we show the multiple starting point method).
\textit{Right}: 1D marginalized posterior predicted by the EL$_2$O (blue dotted line), which closely matches the posterior from samples (red).}
\label{fig:multimodal}
\end{figure}
We show the results in figure \ref{fig:multimodal}.
For this example we consider two optimization strategies. The
first one is to first iterate on a full rank Gaussian,
and since the residuals are large, we add a nonlinear transform.
Since even after this residuals remain large we add a second
Gaussian component. After a total of 13 iterations we converge to the correct
posterior. This can be compared to Stein discrepancy
method of
\cite{LiuWang18}, where 500 iterations with 100 particles
were used to converge.
The convergence to the correct
result is possible because the two
modes overlap in their posterior density.
The second strategy for these problems is to have multiple starting points. We will not discuss strategies how to choose
the starting points, and we will adopt a simple random starting
point method.
In figure \ref{fig:multimodal} we show results with
several different starting points, each converging within a few iterations to one of two two modes (about half of the time onto each). How many starting points we need to choose depends
on how many modes we discover: if after a few starting points we
do not discover new modes we may stop the procedure.
We construct the initial solution as the sum of the two
Gaussians as found at each mode, using the gradient and Hessian
to determine the full rank Gaussian, with the relative normalization determined by equation \ref{gradfree}.
If the two modes are widely separated this is already
the correct solution, but in this case they are not and
we
use optimization to further improve
on the initial parameters. The end result
was identical to the above strategy, but the multiple starting points strategy is more robust, as it will
find a solution even for the case of widely separated modes. In general,
if multimodal posteriors are suspected (and even if not), multiple starting points
are always recommended as a way to verify that optimization found all the relevant
posterior peaks.
\subsection{A science graphical model example: galaxy clustering analysis}
\begin{figure}[t!]
\centering \includegraphics[height=0.47\textwidth]{Fig4_5.pdf}
\label{fig:Pk1}
\caption{The power spectrum multipoles ($l = 0, 2, 4$) from the best-fit theory model and measurements from the BOSS DR12 LOWZ+CMASS NGC data, with $0.4 < z < 0.6$. Fitting the model to data over the wavenumber range $k = 0.02 - 0.4 h$Mpc$^{-1}$, we find a good agreement between the model and the measurements.}
\end{figure}
\begin{figure}[t!]
\centering
\includegraphics[height=0.45\textwidth]{Fig4_1_ADVI.pdf} \includegraphics[height=0.45\textwidth]{Fig4_2_ADVI.pdf} \includegraphics[height=0.25\textwidth]{Fig4_3_mirror_ADVI.pdf}
\caption{\textit{Top}: 1D and 2D posterior distributions of four selected RSD model parameters whose posteriors are close to Gaussian.
\textit{Top left panel}: MAP+Laplace gives inaccurate 2D posterior relative to EL$_2$O, even if 1D projections
are accurate. \textit{Top right panel}: MAP can be displaced in the mean, while EL$_2$O and ADVI
results agree very well with MCMC samples. \textit{Bottom}: 1D posteriors for parameters which are most non-Gaussian.
Together with the NL transform (blue solid curves), EL$_2$O results closely match the MCMC posterior (red solid). Also shown are 2.5\%, 50\%, and 97.5\% intervals (dotted lines), for MCMC and EL$_2$O. 125 likelihood evaluations
were used for EL$_2$O, compared to $10^5$ for MCMC, and $2.3\times 10^4$ for ADVI. Despite taking 200 times more steps than ${\rm EL_2O}$, ADVI posteriors are
considerably worse.
For
$f_{sB}$ parameter we have a boundary $f_{sB}>0$, and we
model it with the unconstrained transformation method (green solid) and
adding the
reflective boundary method to it, the latter allowing
the posterior density at the boundary to be non-zero (blue solid).}
\label{fig:Pk2}
\end{figure}
Our main goal is a fast determination of posterior inference in a typical scientific
analysis, where the model
is expensive to evaluate, is nonlinear in its model parameters, and we have numerous nuisance
parameters we want to marginalize over. Here we give
an example from our own research in cosmology, which was the original motivation for
this work, because MCMC was failing to converge for this problem. We observe about
$10^6$ galaxy positions, measured out to about half of the lookback time of the universe and distributed over a
quarter of the sky, with the radial position determined by their redshift extracted from galaxy spectroscopic emission lines.
Galaxy clustering is anisotropic because of the
redshift space distortions (RSD),
generated by the Doppler shifts proportional to the galaxy velocities.
We can summarize the anisotropic clustering by measuring the power spectrum
as a function of the angle $\mu$ between the line of sight direction and the wavevector of the Fourier mode.
In this specific case we are given measured summary statistics of galaxy clustering $\hat{P}_{l}(k)$, where $l=0,2,4$
are the angular multipoles (Legendre transforming the angular dependence on $\mu$) of the power spectrum and $k$ is the wavevector amplitude. We have a model prediction
for the summary statistic $P_{l}(k)$ that depends on 13 different parameters, of which 3 are of
cosmological interest, since they inform us of the content of the universe, including
dark matter and dark energy.
Others can be viewed as nuisance parameters, although they can also be of
interest on their own \citep{HandSeljakEtAl17}.
We are also given the
covariance matrix of the summary statistics (generalized
noise matrix).
The covariance matrix depends on the signal $P_{l}(k)$, so the derivative of the noise matrix with respect to the
parameters needs to be included in the analysis.
We assume flat prior on the parameters and we use Gauss-Newton approximation for
the Hessian, so in terms of equation \ref{ls} we ignore the prior term with $\bi{S}$,
while the parameter dependence is both in $\bi{f}$ and in $\bi{N}$.
A common complication for scientific analyses is that the gradients are often not available in an analytic form:
the models are evaluated as a numerical evaluation of ODEs or PDEs with many
time steps to evolve the system from its initial conditions to the final output.
Doing back-propagation on ODEs or PDEs with a large number of steps
can be expensive and requires dedicated codes,
which are often not available. In our application, we were able to do analytic
derivatives for 9 parameters, leaving 4 to numerical finite difference method.
We have implemented these with one sided derivatives (step size of 5\%),
meaning that we need 5 function calls to get the function and the
gradient. Due to the use of
Gauss-Newton approximation we also get an approximate Hessian at no extra cost. This
finite difference approach could be improved, for example by using some more global interpolation schemes, but for this paper we will not attempt to do so.
Our specific optimization approach was to use L-BFGS (with L=5) for initial steps,
switching to Gauss-Newton optimization at later steps closer to the solution.
We started with assuming $q$ is a delta function (MAP approach), switching to
sampling from $q$ as we approach the minimum, and gradually increasing the number
of samples $N_k$ once we are past the burn-in, reusing samples from the previous iterations after the burn-in. Here
burn-in is defined in terms of EL$_2$O not rapidly
changing anymore.
Overall it took 25 iteration steps to converge
to the full non-Gaussian posterior solution. Number of samples and iteration steps used in all numerical examples are outlined in Table \ref{table1}.
Results are shown in figure \ref{fig:Pk2}. In the top
panel the parameters are $f \sigma_8$ (product of the growth rate $f$ and the amplitude of matter fluctuations $\sigma_8$), $b_1$ (linear bias), $\sigma_c$ (velocity dispersion for central galaxies), and $f_{1h,s_Bs_B}$ (normalization parameter of the 1-halo amplitude).
It is
of interest to explore how it compares to MAP+Laplace (using Hessian of $\mathcal{L}_p$ at MAP
to determine the inverse covariance matrix) in situations where the posterior is
approximately Gaussian, and we show these results as well
in the
top panel of figure \ref{fig:Pk2}. We see that MAP+Laplace can fail in the mean, or in the
covariance matrix. This could be caused by
the marginalization over non-Gaussian probability
distributions of other parameters, or caused by small scale noise in the log
posterior close to the minimum, which
EL$_2$O improves on by averaging over several samples.
The results have converged
to the correct posterior after 25 iterations, at which point the
EL$_2$O value is stable and around
0.18, which we have argued is low enough for the posteriors to be accurate.
Here we compare to MCMC emcee package \citep{ForemanHoggEtAl13}, which initially did not converge, so we restarted
it at the EL$_2$O best fit parameters (results are shown
with $10^5$ samples after burn-in).
In the bottom
panel we explore parameters that have the most non-Gaussian posteriors;
these parameters are $f_s$ (satellite fraction), $f_{sB}$ (type B satellite fraction), both of which have positivity constraint, $b_{1,\ c_A}$ (linear bias of the type A central galaxies), and $\gamma_{b_1 s_A}$ (slope parameter in the relation between $b_{1,\ s_A}$ and $b_{1,\ c_A}$). In all cases the EL$_2$O posteriors agree
remarkably well with MCMC. This is even the case
for the parameter $f_{sB}$, which has a positivity constraint $f_{sB}>0$, but
is poorly constrained, with a very non-Gaussian posterior that peaks at 0. Even for
this parameter the median and 2.5\%, 97.5\% lower and upper limits agree with MCMC. When we model this parameter
with the unconstrained transformation method, we see that the probability rapidly
descends to zero at the boundary $f_{sB}>0$. The
reflective boundary method corrects this and gives a
better result at the boundary,
as also shown in the same figure. In this example, with reflective boundary,
we allowed the posterior to
go to -0.2 on this parameter.
We do not show MAP+Laplace results since
they poorly match these non-Gaussian
posteriors.
\setlength\extrarowheight{3pt}
\begin{table}[t]
\centering
\begin{tabular}{l | c c c c}
\hline
\hline
& \phantom{...} Ex. 4.1\phantom{...} & \phantom{...}4.2\phantom{...} & \phantom{.....}4.3 \phantom{...}& \phantom{...}4.4\phantom{...} \\
\hline
$N_k$ & 1-10 & 1-10 & 1-10 & 1-10 \\
$N_{\mathrm{iteration}}$ & 10 & 15 & 15 & 25 \\
$N_{\rm tot}$ & 25 & 25 & 25 & 125 \\
\hline
\end{tabular}
\caption{Number of samples per iteration $N_k$, number of iteration steps $N_{\mathrm{iteration}}$,
and total number of $\tilde{\mathcal{L}}_p$ evaluations (incldding
burn-in) for 4 different numerical examples presented in this work. Example 4.4 does not have analytic gradients for 4 parameters, which are evaluated with a finite difference instead.}
\label{table1}
\end{table}
\begin{figure}[t!]
\centering \includegraphics[height=0.39\textwidth]{timingcomparison.pdf}
\label{fig:timing}
\caption{Timing results for 13-dimensional example of section 4.4. EL$_2$O is about 1000 times faster than MCMC and 200 times
faster than ADVI using the same parametrization (but which did not converge to exact posteriors, as seen in figure \ref{fig:Pk2}).}
\end{figure}
\section{Discussion and conclusions}
\label{sec5}
The main goal of this work is to develop a method that
gives reliable and smooth parameter posteriors,
while also
minimizing the number of
calls to log joint probability $\tilde{\mathcal{L}}_p$. In many settings, specially for scientific
applications, an
evaluation of $\tilde{\mathcal{L}}_p$ can be extremely costly.
The current gold standard are MCMC methods,
which asymptotically converge to the correct answer, but require a very large
number of likelihood evaluations, often exceeding $10^5$ or more.
Scientific models are becoming more and more
sophisticated, which comes at a heavy computational cost in terms
of evaluation of $\tilde{\mathcal{L}}_p$.
Using brute force MCMC sampling methods in these situations
is practically impossible.
In this paper we follow the optimization approaches of MAP+Laplace and stochastic VI \citep{KucukelbirTRGB17},
but we modify and extend these in several directions.
The focus of this paper is
on low dimensionality problems, where doing matrix inversion and Cholesky
decomposition of the Hessian is not costly compared to evaluating
$\tilde{\mathcal{L}}_p$. In practice this limits the method to of order thousands of
parameters, if they are correlated so that the full rank matrix description is
needed. If one can adopt sparse matrix approximations one can increase the
dimensionality of the problem.
Both VI and MCMC methods rely
on minimizing KL divergence.
When the problem is not
tractable using deterministic methods this minimization uses sampling, and
this leads to a sampling noise in optimization
that is only reduced as inverse square root
of the number of samples. This can be traced to the feature of KL divergence that
its integrand does not have
to be positive, even if KL divergence is. minimization of
KL divergence only makes sense in the context of the KL divergence integral $\int dz q (\ln q -\ln p)$
: it is only positive after the integration. Deterministic integration is only feasible in very low dimensions, and stochastic integration via Monte Carlo converges slowly, as $N_k^{-1/2}$.
In this
paper we propose instead to minimize ${\rm EL_2O}$,
Euclidean L$_2$ distance squared between the log posterior $\mathcal{L}_p$, which
we evaluate as $\tilde{\mathcal{L}}_p+\ln \bar{p}$, where $\ln \bar{p}$ is an
unknown constant,
and the equivalent terms of its
approximation $\mathcal{L}_q$. EL$_2$O is based on comparing $\ln q(z_k)$ and $\ln p(z_k)$ at the same sampling
points $z_k$: if the two distributions are to be equal they should agree at every sampling point, up to the
normalization constant. There is no need
to perform the integral to obtain a useful minimization procedure and there is no stochastic integration noise,
only one extra optimization parameter due to the unknown normalization.
When available
we add its higher order
derivatives, where we do not have to distinguish between
$\mathcal{L}_p$ and $\tilde{\mathcal{L}}_p$. This is
evaluated as expectation over some approximate probability distribution $\tilde{p}$ close to $p$.
While one can construct many different f-divergences,
${\rm EL_2O}$ optimization
agrees with KL divergence minimization based VI
in the high sampling limit, if samples are generated from $\tilde{p}=q$.
However, for a finite number of samples
the resulting algorithm differs from
recent VI methods such as ADVI \citep{KucukelbirTRGB17}, or the response method \citep{GiordanoBJ18}.
While $t\ln t$ (KL divergence) and $t\ln^2t$ f-divergence minimization (EL$_2$O) seem very similar, they are fundamentally
different, the former more related to stochastic integration.
A first advantage of ${\rm EL_2O}$ is that it
has no sampling noise if the family of models $q$ covers $p$, in contrast to the stochastic minimization of KL divergence, as we demonstrate
in section \ref{sec2}.
In this case
the method gives exact solution and ${\rm EL_2O}=0$,
as long as we have enough constraints as the parameters,
and it does not even matter where the samples are drawn from. In this limit additional samples make the problem over-constrained,
which does not improve the result.
If the family of $q$ is too simple to cover $p$
then the
results fluctuate depending on the drawn samples and the
convergence is slower. Having more
expressive $q$ so that it is closer to $p$
makes the convergence faster even if there are
more parameters to be determined.
In practical examples
we have observed this behavior once ${\rm EL_2O}$ dropped
below 0.2 (e.g. figure \ref{ELO}), where we approach
exact inference. This property of
${\rm EL_2O}$ is different from a stochastic KL divergence minimization,
which is noisy and will typically take longer to converge
as the number of paramaters increases.
Moreover, as we argue in section \ref{sec2}, even in the
simplest setting of Gaussian posterior stochastic
KL divergence minimization is not a convex problem for
a finite number of samples, while EL$_2$O is. In this
setting EL$_2$O is not only convex, but also linear,
so normal equations (or a single Newton update) give the complete solution.
While we use L$_2$ distance in this paper, L$_1$ distance differs
from KL divergence only in taking the absolute
value of the log posterior difference, and also
has no sampling noise. In terms of f-divergence
${\rm EL_2}$ corresponds to f-divergence $t(\ln t)^2$ and
${\rm EL_1}$ to $t|\ln t|$, in contrast to $t\ln t$ for KL divergence.
Second advantage of ${\rm EL_2O}$ is that its value can be used to quantify the quality of the
solution: when it is
small (less than 0.2 for our examples) $p$ is well described with $q$ and
we may exit optimization.
In contrast, in KL divergence based VI the value of the lower bound (ELBO) does not have an absolute
meaning since it is related to the normalization $\ln p(x)$ (free energy bound, \cite{JordanEtAl99}): while relative changes of ELBO are meaningful, the absolute value is not and other methods are needed to assess the quality of the answer \citep{YaoVSG18}. Even though we do not need it
explicitly, EL$_2$O can easily optimize on and output an
approximation to $\ln p(x)$. This can be useful for evidence or
Bayes factor evaluations, which we will pursue
elsewhere.
Sometimes we find low EL$_2$O already for a
full rank Gaussian and sometimes we need to go beyond it.
There is not much computational
benefit in using $q$ that is simpler than a full rank Gaussian,
as long as Cholesky decomposition and matrix inversion
are not a computational bottleneck: only the means are being
optimized and not the covariance matrix elements.
${\rm EL_2O}$ value provides a
diagnostic to asses the quality of the approximate posterior, so when it is large
one can extend the family of models $q$
to remedy the situation. The strategy we advocate is to improve $q$ until a low
value of ${\rm EL_2O}$ is reached.
However,
in contrast to recent trends in machine learning with
many layers of nonlinear transformations (e.g. normalizing
flows, \cite{RezendeMohamed15}), we advocate a single
transformation with few parameters only,
such that the number
of $\tilde{\mathcal{L}}_p$ evaluations is minimized,
and the analytic marginalization remains possible.
If this fails to reduce EL$_2$O one can improve the approximate posterior
by adding one
Gaussian mixture or non-bijective transformation at a
time.
Third advantage of ${\rm EL_2O}$ is its flexibility in choosing the sampling
proposal $\tilde{p}$, which distinguishes it from the KL divergence based VI,
which can only sample
from $q$, and requires reparametrization trick for optimization \citep{KingmaWelling13,RezendeMW14}. This trick is not needed for ${\rm EL_2O}$ optimization.
For $\tilde{p}$ one can use $q$, and iterate on it, which gives results
identical to VI in the large sampling density limit.
But we can also use true $p$ if we
have a way to evaluate its samples, which is not only possible
but easier than sampling from $q$ for forward model problems.
We can also sample from both $p$ and $q$,
which may avoid some of the pitfalls of the other sampling
approaches.
Because we can choose different sampling proposals we
can also address the quality of the results as a
function of this choice.
In the forward model
example where sampling from $p$ is easy we
have found that for simple $q$ (with ${\rm EL_2O} > 0.4$)
there was a difference between sampling from $q$
versus sampling from
$p$, the latter giving overall better results.
These differences were reduced as we increased the
expressivity of $q$, for example when going from FR Gaussian to NL+FR, consistent with the statement above that more
expressive $q$ improves the results and in the limit of very expressive
$q$ we approach exact inference. Samples from $q$ and
samples from $p$ gave nearly the same EL$_2$O value for the same $\mathcal{L}_q$,
suggesting that sampling from
$q$ may not be a fundamental limitation of EL$_2$O divergence based
variational methods,
but more a limitation of using
insufficiently expressive forms
of $q$. Recently, several divergences have been introduced
(e.g. \cite{RanganathTAB16,DiengTRPB17})
to counter these suggested problems of KL divergence, but here
we argue that with
sufficiently expressive $q$ this may not be an issue
for EL$_2$O method.
We also do not observe that in EL$_2$O
sampling from $q$ leads to
a significantly narrower approximation than sampling from $p$ once we
go to NL+FR for $q$, in contrast
to the arguments in the context of FRVI \citep{Bishop07}.
In this example
further improvement when
sampling from $p+q$ was negligible.
While this is based on a limited set of examples
and
deserves further study, we expect that with sufficiently
expressive $q$ EL$_2$O value can be driven to zero and we approach
exact inference, so that for most problems
we only need to
consider sampling from $\tilde{p}=q$.
Fourth advantage of ${\rm EL_2O}$ is its ability to use gradient and Hessian information (and even higher
order derivatives if available), while preserving its sampling noise free nature.
A significant trend in recent years, in machine learning, statistics and
scientific computing,
has been the development of analytic gradients and Hessians of $\tilde{\mathcal{L}}_p$ using
methods such as backpropagation.
${\rm EL_2O}$ can
take advantage of this and we present both the gradient based version and the gradient and Hessian based version. When ${\rm EL_2O}$ is using only the gradient information it can be
related to Fisher divergence \citep{Hammad78}.
The Hessian version converges especially rapidly, as every sample gives $M(M+3)/2+1$
constraints for $M$ dimensions, enough to fit a full rank
Gaussian component in a Gaussian mixture model. Moreover,
no optimization is needed to
determine the covariance matrix of the Gaussian $q$, as its inverse is simply given by averaging
the Hessian over the samples. When Hessian is not available
one can use the gradient information as in equation \ref{gradsig} to achieve the same.
For nonlinear least squares problems Hessian in the
Gauss-Newton approximation can be obtained at the same cost as the gradient, and we found this approximation to give reliable
posteriors in a realistic scientific application. For harder problems,
where MAP,
MFVI and FRVI with Gaussian $q$ all fail, we found the
Gaussian mixture model to work well.
In our applications
we combine full rank Gaussian mixtures model
with one dimensional nonlinear transforms to fit a
general posterior, while at the same time also being able to do analytic
marginalization over any parameters.
In most applications the number of iterations was
comparable to the number of likelihood evaluations: we start
with a single value for the mean
MAP strategy for the burn in, then slowly increase the
number of samples we average over by reusing samples from
past iterations, typically to about 10-15 samples, if
gradient and Hessian are available.
In a realistic scientific
application in the field of cosmology,
with 13 parameters and no analytic derivatives for 4 of them, we obtained good
posteriors with about 25 iterations, with 5 calls each to obtain the finite difference gradients, as compared to $10^5$ iterations for MCMC.
This was a particularly difficult problem for MCMC,
which did not even converge until we restarted it at
${\rm EL_2O}$ solution, and this problem was the original motivation for this work. Using the same parametrization on
ADVI, with $2\times 10^4$ calls, we obtained
worse posteriors despite 200 times higher computational cost, a consequence of noise in KL divergence it is minimizing.
For many of the parameters the posteriors are
very non-Gaussian, and we found remarkable agreement between
${\rm EL_2O}$ and MCMC using full rank Gaussian and a nonlinear
transformation with two or three parameters for each dimension.
In this
example evaluation of $10^5$ samples was feasible and we were able to
compare the results to MCMC, but in many realistic
situations MCMC would not even be feasible, and methods such as ${\rm EL_2O}$ may be
one of the few possible alternatives.
More generally, given the ubiquitousness of KL divergence in many
applications in statistics and machine learning,
${\rm EL_2O}$ may also be useful as an alternative to KL divergence beyond the
posterior inference applications described in this paper.
One important application where EL$_2$O has
an advantage over MCMC is calculation of normalization or
evidence $p(x)$. As discussed in this paper,
one of the optimization
outputs of EL$_2$O procedure is $\ln \tilde p$,
an approximation to
$\ln p(x)$, which can in turn be used for
model comparison by comparing the evidences
between the
different models.
\section{Acknowledgments}
We thank Andrew Charman, Yu Feng, Ryan Giordano, He Jia, Francois Lanusse, Patrick McDonald, Jeffrey Regier and
Matias Zaldarriaga for useful discussions. US is supported by grants NASA NNX15AL17G, 80NSSC18K1274, NSF 1814370, and NSF 1839217.
| {
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Медицинска нега транссексуалних особа и поред настојања законодавства многих земаља и даље није део конвенционалног медицинског образовања а никада кроз досадашњу историју није била део конвенционалне медицине.
Ако се узму у обзир бројна нерешена питања транссексуалаца у друштву, чини се да би био потребан још снажнији курикулум везан за транссексуалне особе, како би се поред недостатака знања избегао и негативни став сваког лекара о трансексуалној оријентацији.
Значај
Иамјући у виду да највећи број транссексуалних мушкараца и жена одлучују да остваре трајну социјалну улогу као припадници пола са којим се идентификују, и да се много транссексуалних људи такође одлучује за различите врсте медицинских модификација сопствених тела, све више са намеће потреба да се још на студијама лекари обуче из ове области. Тим више што се ове модификације сматрају терапијама промене пола и често укључују хормоне и пластичне операције. Како цео процес промене нечије физичке и социјалне појаве пола — транзиција — обично траје неколико година, то захтева обимну медицинску негу транссексуалних особа.
Став медицине
Стандардна, тестирана и прихваћена ефикасна пракса од стране медицинске заједнице у целини је лечење транссексуалаца признала као део стандарде здравствене заштите. Корисно је пратити логику овог стандарда, која се заснива на следеећем. Транссексуалца медицина обично дијагностикује као "Поремећај родног идентитета" која има своју шифру у списку болести (F64.0 ICD-10). На основу овог званичног става медицине, у једном делу света прихваћено је психолошко, психијатријско и медицинско лечење за ову дијагнозу, на основу одговарајућих стандарда за дијагнозу, лечење и негу. Ови стандарди указују на то да транссексуалци треба да добију адекватан и ефикасан третман за поремећај родног идентитета, укључујући саветодавне, медицинске и хируршке методе. Примјењујући ове стандарде, већина практичара је закључила да су третмани трансексуалаца заиста медицински потребни, али само ако се спроводе по признатим стандардима за медицински третман транссексуалаца.
Трошкови лечења транссексуалаца
Просечни трошкови мушко-женских операције транссексуалаца у Америци крећу се око 17.000 долара. Додавањем трошкова од око 1.000 долара за терапију, 1.500 долара за хормоне и 500 долара за посете лекару и лабораторијске тестове, укупан трошак износи око 20.000 долара током двогодишњег периода "транзиције".
Након завршетка операције, текући трошкови драматично падају и покривају само ниво одржавања хормона.
Ако ови поступци нису покривени осигурањем, транссексуалац мора лично да их платити. Већина хирурга захтијева плаћање унапред. Људи морају годинама уштеде новац како би завршили своју "транзицију". Они са нижим платним пословима често никад не могу приуштити операцију.
Епидемиологија
Иако је транссексуалност веома ретка, јер према ДСМ-4, њена учестаслост е на нивоу 1 : 30.000, медицинска струка мора имати адекватан одос према овим особама. Међутим епидемиолошка истраживања указују на бројне проблеме.
Здравствене и социјалне неједнакости које срећемо у свакодневном животу, код појединачних социјалних категорија, у које спадају и транссексуалне особе, укорењене су у везама које су дефинисане; расом, класом, старошћу, инвалидитетом, религијом, полом, родним идентитетом али и многим недовољно истраженим специфичним везама унутар једне хомогене групе. У медицинској, али и другој литератури, постоји велики број веродостојних доказа, о неоправданом консензусу међу професионалним медицинским организацијама, које праве разлике у здравственој заштити транссексуалаца као пацијента, али и у њиховим породицама, што захтева хитно решавање овог проблема у свим земљама света.
Стање на глобалном нивоу
Здравље је јавно добро и припада свима па и мањинским групама. Здравље је само по себи приоритетни циљ јер представља основни ресурс (инпут) економског развоја сваког друштва . Зато су дискриминаторски закони и амандмани (не тако ретки и у многим другим срединама широм света) несхватљиви јер су озбиљно угрозили доношења одлука о јавном здрављу и здравственој заштити, правима посета болницама, приступу здравственом осигурање и правној заштити из области здравствен и социјалне заштите транссексуалних особа у односу на хетеросексуалне..
Проблеми у здравственом образовању
Да медицинска нега трансексуалних особа није део конвенционалног медицинског образовања потврдила је, упркос својим ограничењима, и једна од најсвеобухватнија студија до сада, која се бавила проучавањем знања међу студентима медицине у Канади о здравственом систему и начину збрињавања транссексуалних особа, која је идентификовала празнину у образовању о транссексуалном здрављу у канадским медицинским школама.
Иако су студенти медицине, у начелу, заинтересовани за здравље транссексуалца, они нису адекватно припремљени у медицинским школама да се осећају пријатно, и да адекватно поуздано и без предрасуда брину о трансродним особама.
Овакво стање у здравственом образовању лекара у многим земаљама света утиче више фактора:
избор тема које су обухваћене у наставном програну,
начин на који је наставни материјал уведен,
начин на који се материјал испитује,
да студенти долазе у медицинску школу без осећаја да су спремни да брину о трансродној заједници,
да претходно медицинско образовање не чини будуће лекаре довољно спремним за лечење транссексуалаца.
Перспективе
С обзиром да трансродне заједнице представљају знатан дио становништва развијеног света, студенати медицине ће вероватно наићи на трансродне особе у њиховој будућој каријери, без обзира на специјалност којом ће се бавити. Стога, делотворна обука о трансродној медицини не би требало да буде ограничена на одређени посебни програм или опционалну радионицу. Уместо тога, темељни увод у здравствено стање транссексуалаца требало би да буде интегрисан у наставне програме медицинске школе, и све будуће лекаре треба упознати са овом материјом. То би помогло да се сви студенти укључе у активан рад како би медицинска заједница у будућности боље одговорила на потребе трансродних особа.
У том смислу будуће студије медицине треба да процене степен дефицита и на строжи начин иузуче ову област, а потом предложе и ефикасније мере интервенције намењене решавању дефицитарног знања у овој области.
Извори
Спољашње везе
Сексуална оријентација и медицина
Интерсексуална медицина | {
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Q: How To Quickly Loop WAV Files Without Delay In Python? I'm trying to loop a WAV file so it seamlessly plays in Python, but I didn't have any luck with:
os.system('aplay /home/pi/Desktop/F1/sample.wav')
There is a small delay between playback of this tiny (under 1000kb) WAV file when using a while loop. Is there another library that I could import to do this faster and natively in Python? Could my problem be solved with buffering, and if so, how do I accomplish this with said library? This is on a Raspberry Pi 3 B+, by the way.
| {
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Committing to sober living can be a daunting first step in recovery. Many individuals worry about how sober living affect them. The unknown can be scary. The more knowledge you have — including what to expect when entering a program – the easier the transition and journey can be.
Here, we have compiled answers to many of the questions commonly asked about Independence Lodge.
What are the requirements for living at Independence Lodge?
How do I know that the houses at Independence Lodge are safe and sober?
The easiest way to experience the comfort and safety of the houses at Independence Lodge is to schedule a tour, or to check out our houses with our online tour. All of our houses are regularly inspected by the Pennsylvania Alliance of Recovery Residences as well as the Bucks County Recovery House Association. Residents are randomly drug tested at a minimum of twice a month, if there is any suspicion of intoxication, or if management has identified relapse warning signs.
What if I relapse while staying at Independence Lodge?
The safety of the community at Independence Lodge is our highest priority, and any relapse will result in an immediate discharge from our program. However, we recognize that each resident's journey of recovery is different, and sometimes 'testing the waters' is part of that journey. We are located near several 24/7 drug and alcohol intake sites, and our staff will offer the resident assistance in getting back to a detox or inpatient program. Residents who are in good standing when they relapse, and successfully complete a treatment program may be offered readmission into our program.
Our houses are all within walking distance (less than a mile) of the two local bus lines and/or the Trenton and West Trenton SEPTA regional rail lines.
No, during the first 30 days at Independence Lodge new residents take part in a probationary period, commonly referred to as 'blackout'. This 30 day period is designed to foster new relationships within the house, and to ease the transition from 24 hour inpatient care to the outside world. During blackout the new resident may only leave the house with a senior member of Indepence Lodge, his sponsor, or an immediate family member. The new resident must also adhere to a curfew of 11pm, Sunday – Thursday, and midnight, Friday & Saturday.
If the new resident successfully completes all requirements of the 30 day probationary program (employment, sponsor, homegroup, meeting attendance, etc.), and is in good standing within the Independence Lodge program, then the blackout restrictions are lifted. The resident's curfew is extended to to midnight, Sunday – Thursday, and 2am, Friday & Saturday.
Residents of Independence Lodge may not go to casinos or strip clubs for ANY reason.
Yes, however all visitors must be approved by management. Immediate family members are welcome to tour the houses and meet the other residents. Women outside of immediate family members are not allowed in the houses under any circumstances, and we ask that all visitors remain in the common areas.
Many of our residents have cars, and those with cars are expected to help new residents with rides to appointments and meetings. However, we don't allow new residents in their first 30 days to have a vehicle unless it is an absolute necessity, and only with approval from the House Owner or Director of Client Services. After the resident successfully completes his blackout period, he is free to have a vehicle if it is properly inspected, insured, and if the resident has a valid driver's license.
Can I bring my entire wardrobe?
No, please keep in mind that you will be sharing your room and closet space with other people. Bring only the essential clothes and shoes you need for work, physical activity, and one or two outfits for going out. There's nothing wrong with taking care of how you look on the outside, but remember that the primary reason you're coming to Independence Lodge is to develop and grow on the inside.
Do I need to bring quarters for laundry?
No, each house has a washer and dryer provided at no cost. Detergent and dryer sheets are provided as well.
Blankets, a pillow, and sheets are all provided. However, you are welcome to bring your own bedding if you wish. All beds at Independence Lodge are standard twins.
Are phones or laptops allowed?
Yes, personal electronics are welcomed. We do reserve the right to restrict a resident's access to social media, dating apps, or other sites if we believe such access is putting the resident's recovery in danger.
Yes, each house at Independence Lodge is given a weekly budget for groceries, and house necessities (laundry detergent, toilet paper, cleaning supplies). The management team uses this weekly budget to shop for food for dinners Sunday through Thursday. Residents volunteer to cook the five weekly meals. Each house also has a basic pantry which usually includes staples like bread, peanut butter, eggs, coffee, and iced tea mix.
If there are ethical, religious, or health reasons that you cannot, or do not eat certain foods (vegetarianism, kosher, food allergies, etc.) our management staff will be happy to accommodate to the best of their ability, however it is not their responsibility to do this FOR you. It is important that you become the number one advocate for yourself and your dietary needs. Speak with our Director of Client Services to develop a plan of action that best accommodates your dietary restrictions.
Yes, each house has a small space for individual storage of dry food. There is limited storage for refrigerated or frozen food, so we recommend that you do not arrive with any food that must be kept cold. Food may not be stored in the bedrooms. | {
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{"url":"http:\/\/aimsciences.org\/article\/doi\/10.3934\/dcds.2011.31.913","text":"# American Institute of Mathematical Sciences\n\n2011,\u00a031(3):\u00a0913-940. doi:\u00a010.3934\/dcds.2011.31.913\n\n## On the index problem of $C^1$-generic wild homoclinic classes in dimension three\n\n 1 Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1- Komaba Meguro-ku Tokyo 153-8914, Japan\n\nReceived\u00a0 January 2010 Revised\u00a0 July 2011 Published\u00a0 August 2011\n\nWe study the dynamics of homoclinic classes on three dimensional manifolds under the robust absence of dominated splittings. We prove that, $C^1$-generically, if such a homoclinic class contains a volume-expanding periodic point, then it contains a hyperbolic periodic point whose index (dimension of the unstable manifold) is equal to two.\nCitation: Katsutoshi Shinohara. On the index problem of $C^1$-generic wild homoclinic classes in dimension three. Discrete & Continuous Dynamical Systems - A, 2011, 31 (3) : 913-940. doi: 10.3934\/dcds.2011.31.913\n##### References:\n [1] F. Abdenur, Ch. Bonatti, S. Crovisier, L. D\u00edaz and L. Wen, Periodic points and homoclinic classes,, Ergodic Theory Dynam. Systems, 27 (2007), 1. doi:\u00a010.1017\/S0143385706000538. [2] C. Bonatti and S. Crovisier, R\u00e9currence et g\u00e9n\u00e9ricit\u00e9,, Invent. Math., 158 (2004), 33. [3] C. Bonatti and L. D\u00edaz, On maximal transitive sets of generic diffeomorphisms,, Publ. Math. Inst. Hautes \u00c9tudes Sci., 96 (2002), 171. [4] C. Bonatti and L. D\u00edaz, Robust heterodimensional cycles and $C^1$-generic dynamics,, J. Inst. Math. Jussieu, 7 (2008), 469. doi:\u00a010.1017\/S1474748008000030. [5] C. Bonatti, L. D\u00edaz and S. Kiriki, Stabilization of heterodimensional cycles,, preprint, (). [6] C. Bonatti, L. D\u00edaz and E. Pujals, A $C^1$-generic dichotomy for diffeomorphisms: Weak forms of hyperbolicity or infinitely many sinks or sources,, Ann. of Math. (2), 158 (2003), 355. doi:\u00a010.4007\/annals.2003.158.355. [7] C. Bonatti, L. D\u00edaz and M. Viana, \"Dynamics Beyond Uniform Hyperbolicity. A Global Geometric and Probabilistic Perspective,\", Encyclopaedia of Mathematical Sciences, 102 (2005). [8] S. Gan, A necessary and sufficient condition for the existence of dominated splitting with a given index,, Trends in Mathematics, 7 (2004), 143. [9] N. Gourmelon, A Franks' lemma that preserves invariant manifolds,, Preprint, (). [10] N. Gourmelon, Generation of homoclinic tangencies by $C^1$-perturbations,, Discrete Contin. Dyn. Syst., 26 (2010), 1. doi:\u00a010.3934\/dcds.2010.26.1. [11] R. Ma\u00f1\u00e9, An ergodic closing lemma,, Ann. of Math. (2), 116 (1982), 503. [12] J. Palis, Open questions leading to a global perspective in dynamics,, Nonlinearity, 21 (2008). doi:\u00a010.1088\/0951-7715\/21\/4\/T01. [13] E. Pujals and M. Sambarino, Homoclinic tangencies and hyperbolicity for surface diffeomorphisms,, Ann. of Math. (2), 151 (2000), 961. doi:\u00a010.2307\/121127. [14] C. Robinson, \"Dynamical Systems. Stability, Symbolic Dynamics, and Chaos,\", 2nd edition, (1999). [15] S. Smale, Differentiable dynamical systems,, Bull. Amer. Math. Soc., 73 (1967), 747. doi:\u00a010.1090\/S0002-9904-1967-11798-1. [16] K. Shinohara, An example of C1-generically wild homoclinic classes with index deficiency,, Nonlinearity, 24 (2011), 1961. doi:\u00a010.1088\/0951-7715\/24\/7\/003. [17] L. Wen, Homoclinic tangencies and dominated splittings,, Nonlinearity, 15 (2002), 1445. doi:\u00a010.1088\/0951-7715\/15\/5\/306.\n\nshow all references\n\n##### References:\n [1] F. Abdenur, Ch. Bonatti, S. Crovisier, L. D\u00edaz and L. Wen, Periodic points and homoclinic classes,, Ergodic Theory Dynam. Systems, 27 (2007), 1. doi:\u00a010.1017\/S0143385706000538. [2] C. Bonatti and S. Crovisier, R\u00e9currence et g\u00e9n\u00e9ricit\u00e9,, Invent. Math., 158 (2004), 33. [3] C. Bonatti and L. D\u00edaz, On maximal transitive sets of generic diffeomorphisms,, Publ. Math. Inst. Hautes \u00c9tudes Sci., 96 (2002), 171. [4] C. Bonatti and L. D\u00edaz, Robust heterodimensional cycles and $C^1$-generic dynamics,, J. Inst. Math. Jussieu, 7 (2008), 469. doi:\u00a010.1017\/S1474748008000030. [5] C. Bonatti, L. D\u00edaz and S. Kiriki, Stabilization of heterodimensional cycles,, preprint, (). [6] C. Bonatti, L. D\u00edaz and E. Pujals, A $C^1$-generic dichotomy for diffeomorphisms: Weak forms of hyperbolicity or infinitely many sinks or sources,, Ann. of Math. (2), 158 (2003), 355. doi:\u00a010.4007\/annals.2003.158.355. [7] C. Bonatti, L. D\u00edaz and M. Viana, \"Dynamics Beyond Uniform Hyperbolicity. A Global Geometric and Probabilistic Perspective,\", Encyclopaedia of Mathematical Sciences, 102 (2005). [8] S. Gan, A necessary and sufficient condition for the existence of dominated splitting with a given index,, Trends in Mathematics, 7 (2004), 143. [9] N. Gourmelon, A Franks' lemma that preserves invariant manifolds,, Preprint, (). [10] N. Gourmelon, Generation of homoclinic tangencies by $C^1$-perturbations,, Discrete Contin. Dyn. Syst., 26 (2010), 1. doi:\u00a010.3934\/dcds.2010.26.1. [11] R. Ma\u00f1\u00e9, An ergodic closing lemma,, Ann. of Math. (2), 116 (1982), 503. [12] J. Palis, Open questions leading to a global perspective in dynamics,, Nonlinearity, 21 (2008). doi:\u00a010.1088\/0951-7715\/21\/4\/T01. [13] E. Pujals and M. Sambarino, Homoclinic tangencies and hyperbolicity for surface diffeomorphisms,, Ann. of Math. (2), 151 (2000), 961. doi:\u00a010.2307\/121127. [14] C. Robinson, \"Dynamical Systems. Stability, Symbolic Dynamics, and Chaos,\", 2nd edition, (1999). [15] S. Smale, Differentiable dynamical systems,, Bull. Amer. Math. Soc., 73 (1967), 747. doi:\u00a010.1090\/S0002-9904-1967-11798-1. [16] K. Shinohara, An example of C1-generically wild homoclinic classes with index deficiency,, Nonlinearity, 24 (2011), 1961. doi:\u00a010.1088\/0951-7715\/24\/7\/003. [17] L. Wen, Homoclinic tangencies and dominated splittings,, Nonlinearity, 15 (2002), 1445. doi:\u00a010.1088\/0951-7715\/15\/5\/306.\n [1] Zhiqin Qiao, Deming Zhu, Qiuying Lu. Bifurcation of a heterodimensional cycle with weak inclination flip. Discrete & Continuous Dynamical Systems - B, 2012, 17 (3) : 1009-1025. doi: 10.3934\/dcdsb.2012.17.1009 [2] Dante Carrasco-Olivera, Bernardo San Mart\u00edn. Robust attractors without dominated splitting on manifolds with boundary. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4555-4563. doi: 10.3934\/dcds.2014.34.4555 [3] Eleonora Catsigeras, Xueting Tian. Dominated splitting, partial hyperbolicity and positive entropy. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 4739-4759. doi: 10.3934\/dcds.2016006 [4] Wenxiang Sun, Xueting Tian. Dominated splitting and Pesin's entropy formula. Discrete & Continuous Dynamical Systems - A, 2012, 32 (4) : 1421-1434. doi: 10.3934\/dcds.2012.32.1421 [5] Lorenzo J. D\u00edaz, Jorge Rocha. How do hyperbolic homoclinic classes collide at heterodimensional cycles?. Discrete & Continuous Dynamical Systems - A, 2007, 17 (3) : 589-627. doi: 10.3934\/dcds.2007.17.589 [6] Xinsheng Wang, Lin Wang, Yujun Zhu. Formula of entropy along unstable foliations for $C^1$ diffeomorphisms with dominated splitting. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 2125-2140. doi: 10.3934\/dcds.2018087 [7] Shaobo Gan, Kazuhiro Sakai, Lan Wen. $C^1$ -stably weakly shadowing homoclinic classes admit dominated splittings. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 205-216. doi: 10.3934\/dcds.2010.27.205 [8] Lijun Wei, Xiang Zhang. Limit cycle bifurcations near generalized homoclinic loop in piecewise smooth differential systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2803-2825. doi: 10.3934\/dcds.2016.36.2803 [9] Karsten Matthies. Exponentially small splitting of homoclinic orbits of parabolic differential equations under periodic forcing. Discrete & Continuous Dynamical Systems - A, 2003, 9 (3) : 585-602. doi: 10.3934\/dcds.2003.9.585 [10] Amadeu Delshams, Pere Guti\u00e9rrez. Exponentially small splitting for whiskered tori in Hamiltonian systems: continuation of transverse homoclinic orbits. Discrete & Continuous Dynamical Systems - A, 2004, 11 (4) : 757-783. doi: 10.3934\/dcds.2004.11.757 [11] Eleonora Catsigeras, Heber Enrich. SRB measures of certain almost hyperbolic diffeomorphisms with a tangency. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 177-202. doi: 10.3934\/dcds.2001.7.177 [12] Shin Kiriki, Yusuke Nishizawa, Teruhiko Soma. Heterodimensional tangencies on cycles leading to strange attractors. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 285-300. doi: 10.3934\/dcds.2010.27.285 [13] Thomas Y. Hou, Dong Liang. Multiscale analysis for convection dominated transport equations. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 281-298. doi: 10.3934\/dcds.2009.23.281 [14] Dmitry N. Kozlov. Cobounding odd cycle colorings. Electronic Research Announcements, 2006, 12: 53-55. [15] Dan Liu, Shigui Ruan, Deming Zhu. Nongeneric bifurcations near heterodimensional cycles with inclination flip in $\\mathbb{R}^4$. Discrete & Continuous Dynamical Systems - S, 2011, 4 (6) : 1511-1532. doi: 10.3934\/dcdss.2011.4.1511 [16] V. Kumar Murty, Ying Zong. Splitting of abelian varieties. Advances in Mathematics of Communications, 2014, 8 (4) : 511-519. doi: 10.3934\/amc.2014.8.511 [17] Claudio Qureshi, Daniel Panario, Rodrigo Martins. Cycle structure of iterating Redei functions. Advances in Mathematics of Communications, 2017, 11 (2) : 397-407. doi: 10.3934\/amc.2017034 [18] Zheng-Ru Zhang, Tao Tang. An adaptive mesh redistribution algorithm for convection-dominated problems. Communications on Pure & Applied Analysis, 2002, 1 (3) : 341-357. doi: 10.3934\/cpaa.2002.1.341 [19] Jingyu Li, Chuangchuang Liang. Viscosity dominated limit of global solutions to a hyperbolic equation in MEMS. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 833-849. doi: 10.3934\/dcds.2016.36.833 [20] Eskil Hansen, Alexander Ostermann. Dimension splitting for time dependent operators. Conference Publications, 2009, 2009 (Special) : 322-332. doi: 10.3934\/proc.2009.2009.322\n\n2017\u00a0Impact Factor:\u00a01.179","date":"2018-07-20 20:02:01","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7460391521453857, \"perplexity\": 6159.832167542169}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-30\/segments\/1531676591831.57\/warc\/CC-MAIN-20180720193850-20180720213850-00244.warc.gz\"}"} | null | null |
Sideroxylon montanum is a species of plant in the family Sapotaceae. It is endemic to Jamaica. It is threatened by habitat loss.
References
Flora of Jamaica
montanum
Near threatened plants
Endemic flora of Jamaica
Taxonomy articles created by Polbot | {
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} | 4,326 |
Јагода Трухелка (Осијек, 5. фебруар 1864. — Загреб, 17. децембар 1957) је била хрватска учитељица и књижевница за децу и омладину чешког и немачког порекла.
После завршене учитељске школе у Загребу, била је наставница на девојачким школама у Осијеку и Загребу, па управница у Бањалуци и Сарајеву. Писала је романе и приповетке описујући живот жене, сељанке и малограђанке, патријархално Сарајево и амбијент старог Загреба. Највреднији део њеног књижевног стваралашва је дечја литература.
Значајнија дела
Тугомила, Загреб (1894)
Војача (1899)
У царству душе, Осијек (1910)
Златко, роман једног дечака, Загреб (1934)
Трилогија Златни данци, Загреб (1942)
Госпине трешње (1943)
Црни и бијели дани (1944)
Рођени 1864.
Умрли 1957.
Хрватски књижевници
Осјечани
Чеси у Хрватској | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,803 |
Q: Merge RDF .ttl files into one file database - filtering and keeping only the data/triples needed I need to merge 1000+ .ttl files into one file database. How can I merge them with filtering the data in the source files and keep only the data needed in the target file?
Thanks
A: There's a number of options, but the simplest way is probably to have use a Turtle parser to read all the files, and let that parser pass its output to a handler which does the filtering before in turn passing the data to a Turtle writer.
Something like this would probably work (using RDF4J):
RDFWriter writer = org.eclipse.rdf4j.rio.Rio.createWriter(RDFFormat.TURTLE, outFile);
writer.startRDF();
for (File file : // loop over your 100+ input files) {
Model data = Rio.parse(new FileInputStream(file), "", RDFFormat.TURTLE);
for (Statement st: data) {
if (// you want to keep this statement) {
writer.handleStatement(st);
}
}
}
writer.endRDF();
Alternatively, just load all the files into an RDF Repository, and use SPARQL queries to get the data out and save to an output file, or if you prefer: use SPARQL updates to remove the data you don't want before exporting the entire repository to a file.
Something along these lines (again using RDF4J):
Repository rep = ... // your RDF repository, e.g. an in-memory store or native RDF database
try (RepositoryConnection conn = rep.getConnection()) {
// load all files into the database
for (File file: // loop over input files) {
conn.add(file, "", RDFFormat.TURTLE);
}
// do a sparql update to remove all instances of ex:Foo
conn.prepareUpdate("DELETE WHERE { ?s a ex:Foo; ?p ?o }").execute();
// export to file
con.export(Rio.createWriter(RDFFormat.TURTLE, outFile));
} finally {
rep.shutDown();
}
Depending on the amount of data / the size of your files, you may need to extend this basic setup a bit (for example by using transactions instead of just letting the connection auto-commit). But you get the general idea, hopefully.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,523 |
\section{Introduction}
\label{sec:intro}
Active Galactic Nuclei (AGN) are characterized by large amplitude and rapid variability, especially in the X-ray band, which is probably originating in the inner regions of the accretion disk and the hot corona.
One of the most common tools for examining AGN variability
is the \emph{Power Spectral Density Function} (PSD).
Early attempts to measure the AGN X-ray PSDs showed that they have a power-law like shape with a slope of $\sim 1.5$ \citep{Green93, Lawrence93}. This result is indicative of
a scale-invariant \emph{red-noise} process, on timescales ranging from a few hours to
years, with no evidence of periodicities.
In recent years it has become increasingly clear that there exists at least one characteristic timescale in the AGN X-ray PSDs. This timescale reveals itself in the form of ``frequency breaks" ($\nu_{\rm br}$) in the PSD, where the slope changes from a value of $\sim -1$ below the ``break", to $\sim -2$ at frequencies higher than $\nu_{\rm br}$ (see e.g. \citealt{Uttley02, Markowitz03}). In at least one case, namely Ark 564, a second break, where the slope changes from $\sim -1$ to zero, is also detected \citep{Papadakis02,McHardy07}. These time scales may be linked to the characteristic disk time scales like the dynamical, thermal or viscuous timescale, and appear to correlate with the BH mass and accretion rate \citep{McHardy06, Koerding07}. Thus variability measurements represent a tool to investigate both the physics of the accretion process, as well as the fundamental parameters ($M_{BH}$, $\dot{m}$) of the active nucleus.
So far, our knowledge of the X-ray variability properties of AGNs is mainly based on the study of a few nearby, X-ray bright objects, which have been monitored extensively with {\it RXTE} over many years, and for which there also exist day-long, high signal-to-noise (S/N) {\it XMM-Newton} light curves. At the same time, deep multi-cycle surveys
(e.g. \citealt{Alexander2003, Brunner2008, Comastri2011, Xue2011};
also see \citealt{Brandt2005} and references therein), have been accumulating observations of
intermediate and high ($z>0.5$) redshift AGN, thus offering
the opportunity to explore AGN variability at high redshift as well. However, due to the sparse sampling, and the low flux of most AGN detected in these surveys, it is not possible to use PSD techniques to study the variability properties of these objects. For that reason, a different statistic, namely the \textit{excess variance} \citep{Nandra97,Turner1999,Edelson2002} has been used to parametrize the variability properties of the high redshift AGN \citep{Almaini, Paolillo04, Papadakis08}.
Strictly speaking, the excess variance is a \textit{maximum likelihood}
estimator of the intrinsic light curve variance only in the case of uniform sampling and identical and normally distributed measurement errors \citep{Almaini}. A detailed discussion of the statistical properties of the \textit{excess variance} and its performance in the case of red noise PSDs of various slopes and "break" frequencies, and of different S/N ratios, can be found in \citet{Vaughan03}. These authors however considered the case of continuously sampled data only, such as those provided by long XMM observations of nearby AGNs. Instead, in deep multi-cycle surveys, the effects
of sparse and uneven sampling must be taken taken into account when investigating the statistical properties of the excess variance.
The goal of this work is to investigate the performance of the excess variance as a measure of the intrinsic AGN variability. In particular, we consider sources similar to those observed in multi-epoch surveys and characterized by extreme sparsity, due to the observing strategy and orbital visibility of the targets. We measure the bias and the expected scatter of the excess variance measurements, and we investigate the dependence of the bias on the sampling pattern and gap length, as well as on the S/N ratio of the light curve. We believe that our results will be useful to researchers who wish to study the variability properties of high redshift AGN (an area which is still largely unexplored), as well as to understand the possible limitations of the existing data, and to correct (in a statistical sense) for some effects that the uneven sampling introduces in the estimation of the intrinsic variability. Finally, our results could be of use in the determination of the optimal observing strategy either for future surveys with the current X-ray satellites or for future X-ray missions.
The paper is organized as follows:
in \S\ref{section:EXV} we define the variability estimator;
in \S\ref{section:AGN_lc}
we describe the Monte Carlo simulations of AGN lightcurves both reproducing the
pattern of the XMM-Newton observations of the CDFS and
further testing more favorable observing strategies.
The applications to future X-ray surveys
are presented in \S\ref{section:Future Perspectives}
while our results are discussed in \S \ref{sec:conclusions}.
\section{Normalized Excess Variance}
\label{section:EXV}
The variability of accreting systems is usually investigated through the use
of the PSD, which gives the light curve variance per Hz at each temporal frequency.
AGN exhibit a power-law PSD,
as $S(f) \propto f^{- \beta}$, where $S(f)$ is the power at frequency $f$,
with slopes usually in the range $1 \lesssim \beta \lesssim 2$.
A proper derivation of the PSD (or of the lightcurve variance, see below) is intrinsically difficult:
extrapolations from any single realization can be misleading due to the stochastic
nature of any red-noise lightcurve \citep[see discussion in][]{Vaughan03}. For real data
this task is further affected by the signal-to-noise ratio
of the data, the finite length of the observation and by the sampling pattern.
The analysis of present day light curves of distant AGNs is difficult
since these sources are usually serendipitously detected in deep surveys.
As a result, their light curves are characterized by low signal-to-noise
ratio as well as sparse sampling. In such cases, instead of trying to
derive the PSD, it is easier (and often only possible) to estimate the total light curve variance using
the so called \textit{excess variance}, which is defined as \citep{Nandra97}:
\begin{equation}
\sigma^2_{NXS} = \frac{1}{N \overline{x}^2} \sum_{i=1}^{N} \,
[(x_i - \overline{x})^2 - \sigma_{err,i}^2],
\label{eq:exvar2}
\end{equation}
where $x_i$ and $\sigma_{err,i}$ are the count rate and its error in i-th bin, $\overline{x}$ is the mean count rate , and $N$ is the number of bins used to estimate $\sigma^2_{NXS}$. With this normalization we are able to compare excess variance estimates derived from different segments of a particular lightcurve or from lightcurves of different sources. The statistic $\sigma^2_{NXS}$ is an estimate of the (squared) fraction of the total flux per bin that is variable, corrected for the experimental noise. According to the Parseval theorem, the contribution to the intrinsic
variance due to variations between the shortest and longest time scales sampled, which $\sigma^2_{NXS}$ measures, should be roughly equal to the integral of the intrinsic PSD between the shortest and longest frequencies sampled.
The error\footnote{Note there was a typographical error in \citet{Nandra97}, in that the equation for the error on $\sigma^2_{NXS}$ should have had the quantity inside the rms summation squared, as clarified by \cite{Turner1999}. Also see \cite{Edelson2002} and \cite{Vaughan03} for alternative expressions and a discussions of the different formulae.} on $\sigma^2_{NXS}$, asymptotically for large $N$, is given by the variance of the quantity $(x_i - \overline{x})^2 - \sigma_{err,i}^2$, i.e.
\begin{eqnarray}
\Delta \sigma^2_{NXS} & = & S_D / [\overline{x}^2 (N)^{1/2}], \label{eq:exvar_err} \\
S_D & = & \frac{1}{N-1} \sum_{i=1}^{N} \,
\{[(x_i - \overline{x})^2 - \sigma_{err,i}^2]-
\sigma^2_{NXS} \overline{x}^2\}^2.
\nonumber
\end{eqnarray}
As mentioned earlier, the performance of the excess variance, under various intrinsic PSD models, in the case of evenly sampled light curves has already been investigated by Vaughan et al. (2003); here we intend to explore instead the performance in case of sparse sampling and low S/N.
We also point out that if each point in the lightcurve has equal weight,
then $\sigma^2_{NXS}$ is indeed a maximum-likelihood estimator of the lightcurve variance.
This is not true anymore in cases where the errors differ significantly from point to point,
and a numerical approach is needed in order to obtain the maximum-likelihood estimate of the intrinsic variance \citep[see details in][]{Almaini}. We will explore this case as well in the following sections.
\section{Monte Carlo Simulations of AGN lightcurves}\label{sec:3}
\label{section:AGN_lc}
\begin{figure*}
\begin{center}
\includegraphics[width=9cm]{figure1.ps}
\caption{Simulated AGN lightcurve according to the input parameters in Table \ref{tab:input_param},
reproducing a continuous sampling (black crosses) and the sampling pattern of
the XMM-Newton observations of the CDFS (red circles). Mean count rate and excess variance estimates refer
to the particular simulation extracted from a set of 5000 simulations.}
\label{fig:lc_sparsyXMM}
\end{center}
\end{figure*}
\subsection{The algorithm and the simulated CDFS light curves}
In order to quantify the bias and the uncertainty of the excess variance
as an estimator of the intrinsic source variance in the case of very
unevenly sampled light curves of faint sources, we performed Monte
Carlo simulations modifying the
original code of \citet{Timmer95}, that generates red-noise data
with a power law PSD, in order to reproduce the real data extraction
process including filtering and background subtraction.
We simulated, for each AGN, the actual lightcurve measurement:
we first create an intrinsic AGN lightcurve with the above algorithm,
following the appropriate PSD. Then we add to the AGN count rate,
in each time bin, the contribution from the expected background,
randomly adding Poisson fluctuations to both terms. A second
\emph{local} background estimate is also generated
(including again Poisson fluctuations), and then subtracted from
the AGN, as done in real data.
In order to account for the effect of red noise leak, which transfers
power from low to high frequencies, we generate lightcurves which
are 5 times longer than the largest timescale sampled by the data,
and extract a segment of the required length. We verified that
extending the simulated lightcurves further does not significantly
changes our results, while increasing considerably the processing time.
In order for our experiment to be as close as possible to reality,
we performed Monte Carlo simulation of AGN lightcurves assuming
the sampling pattern and uncertainties of the
XMM-Newton observations of the CDFS. In particular we take into
account only the first 1 Ms observations, taken between 2001 and
2002, to study a worst-case scenario before discussing, in the
next sections, more favourable ones. As a starting point, we simulated
red-noise lightcurves with intrinsic count rate and variance of one
of the brightest AGN observed by XMM-Newton
in the CDFS \citep[source id 68 from][]{Gia02} at $z\sim 0.54$,
using as input the set of parameters reported in Table \ref{tab:input_param}.
The source has a soft (and hard) flux of $\sim 5 \times 10^{-14}$ erg/s/cm$^2$, i.e. we expect 10-20 of these sources per square degree, according to, e.g. \cite{Hasinger93,Luo08}. Compared to other bright AGNs in the field, this source has the advantage of being fairly isolated and thus its flux and variability can be robustly estimated.
We explore PSD slopes ranging from 1 to 3;
in the following we are going to show
the results for simulations with $\beta=1.5$,
but we will discuss the results in all the other cases as well.
Fig. \ref{fig:lc_sparsyXMM} shows an example of a simulated lightcurve:
the red points highlight the sampling pattern of the XMM-Newton
observations, compared to the whole underlying lightcurve.
The first group of points corresponds to the two observations of July 2001 with an effective exposure, after filtering high background periods, of $\sim 80$ ks and the second (with more data points) to the six observations of January 2002 for an additional $900$ ks. The whole simulated lightcurve with continuous sampling (black crosses) thus spans $\sim 1.5\times 10^7 $sec, i.e.
about 6 months, out of which the actual XMM-CDFS observations (red circles) sample $\sim 9.8\times 10^5$ sec ($\sim$ 11 days). This type of observing pattern is driven primarily by the typical scheduling requirements of deep multi-cycle campaigns, and thus represents a recurring, although undesirable, observing scheme which has been the only available to astronomers until the 2009 extended XMM observing campaign of the CDFS \citep{Comastri2011}.
The figure also reports the mean count rate and excess variance measured
over the whole lightcurve and over the intervals sampled by the
XMM observations, for this specific realization. As discussed in more detail below, when sampling the whole lightcurve the measured values reproduce the input parameters, while in the case of sparse sampling we obtain biased results.
\begin{table}
\centering
\caption{Input parameters of the simulated AGN lightcurves}
\label{tab:input_param}
\begin{tabular}{l l}
\hline
\hline
Power-law PSD index ($\beta$) & 1,1.5,2,2.5,3 \\
Number of simulations ($N$) & 5000 \\
Mean count rate & 0.1 cnt/s \\
Time resolution ($\Delta t$) & 10 ks \\
Intrinsic lightcurve variance ($\sigma^2_{in}$) & $0.042$ \\
Background level & 0.06 cnt/s \\
\hline
\end{tabular}
\end{table}
\begin{figure}
\includegraphics[height=6cm]{figure2.ps}
\caption{Excess variance distribution based on a set of 5000 simulated AGN lightcurves,
such as the one shown in Fig.\ref{fig:lc_sparsyXMM}, reproducing the sampling pattern
of the XMM-Newton observation
of the CDFS (solid black line), compared to the expected input
value (vertical red line). The dotted red line represents the same distribution corrected for the true intrinsic mean
count rate (see discussion in the text), while the \textit{maximum likelihood} approach is shown by the dashed blue
line. The $\sigma^2_{NXS}$ distribution for a continuous sampled lightcurve is not shown here since it is an
extremely narrow distribution peaked on the intrinsic value of the variance.}
\label{fig:histoexv}
\end{figure}
\subsection{The distribution of $\sigma^2_{NXS}$ in the case of sparsely sampled light curves}\label{sec:3.2}
In Fig. \ref{fig:histoexv} we present the excess variance distribution
of a set of 5000 simulations of sparsely sampled lightcurves
such as the one shown in Fig.\ref{fig:lc_sparsyXMM}, for the case of an
intrinsic PSD with power-law slope $\beta=1.5$ (solid line).
The dashed line in the same
figure represents instead the distribution of the \textit{maximum likelihood} variance
estimator as proposed by \citet{Almaini}. The vertical dot dashed line
in Fig.\ref{fig:histoexv} marks the intrinsic variance ($\sigma^2_{intrinsic}=0.042$, i.e. 20.5\% r.m.s.).
Although the errors on each point of the lightcurve
are not identical, the sample distribution of the variance measured through
the numerical estimate of \citet{Almaini} does not differ much from the
distribution of the excess variance, at such count rate levels. Both distributions
in fact are highly peaked at values smaller than the intrinsic source variance.
The median value of the $\sigma^2_{NXS}$ distribution is listed
in Table \ref{tab:tablemaxlik} for simulations with (1) continuous sampling,
(2) sparse sampling, (3) sparse sampling using the maximum-likelihood estimator and
(4) correcting for the true mean count rate.
The lower and upper quartiles
of the distribution within 90\% are in brackets.
Both the \textit{maximum likelihood} and $\sigma^2_{NXS}$ are thus ''biased" estimators of the intrinsic source variance. In addition, both distributions are very broad,
and highly skewed towards large positive values. Clearly, an individual measurement of
neither $\sigma^2_{NXS}$ nor its \textit{maximum likelihood} equivalent, can be considered
as a reliable estimate of the intrinsic source variance. We also note that using Eq.\ref{eq:exvar_err} the median error on $\sigma^2_{NXS}$ is equal to
$\sim 0.006$ , i.e. the formal error tends to underestimate the true scatter and
does not account for the asymmetry of the distribution, as it does not include the effect of the sparse sampling.
As shown in Table \ref{tab:tablemaxlik}, in case of continuous sampled AGN lightcurve, as expected the distribution of $\sigma^2_{NXS}$ is quite narrow
and strongly peaked to the intrinsic source variance.
Very similar results are found also assuming different index $\beta$ of the power-law PSD.
We conclude that the sparse sampling does indeed results in a biased distribution of the excess variance, and increases the "uncertainty" on each individual value.
Using a sparse sampling pattern the variance of the lightcurve is underestimated,
mainly because each realization badly reproduces
the intrinsic mean count rate; in fact
the value derived from the sparsely sampled data will
always be closer to the sampled points than the true
mean, thus minimizing the variance\footnote{Note that while the mean count rate
that we measure for \textit{each individual realization} of the sparsely sampled lightcurves is
biased, and thus minimizes the variance, its distribution over the entire set of 5000 simulations
peaks at the expected input value.}.
To demonstrate this point, we fixed the average count rate $\overline{x}$ in Eq.
(\ref{eq:exvar2}) to its intrinsic value, finding that in such case the mean output variance
approaches \emph{on average} the input value (see Table \ref{tab:tablemaxlik}
and dotted red line in Fig. \ref{fig:histoexv}), while still retaining the large scatter.
\begin{table*}
\begin{center}
\caption{Median $\sigma^2_{NXV}$ and bias for continuous and sparse sampling}
\label{tab:tablemaxlik}
\begin{tabular}{c c c c c c}
\hline
$\beta$ & Continuous & Sparse & Max.likelihood & Mean Corr. & $b$ \\
& (1) & (2) & (3) & (4) & (5) \\
\hline
1 & $0.0418 (0.0415,0.0420)$ & $0.030 (0.004, 0.043)$ & $0.017 (0.005, 0.036)$ & $0.040(0.015, 0.050)$ & 1.4(1.,10.4)\\
1.5 & $0.0418 (0.0415,0.0420)$ & $0.023 (0.007, 0.035)$ & $0.015 (0.008, 0.035)$ & $0.041 (0.021, 0.070)$ & 1.8(1.2,6.0)\\
2 & $0.0418 (0.0415,0.0420)$ & $0.025 (0.005, 0.042)$ & $0.016 (0.007, 0.043)$ & $0.052 (0.022, 0.078)$ & 1.7(1.,8.36)\\
2.5 & $0.0418 (0.0415,0.0420)$ & $0.032 (0.008, 0.042)$ & $0.018 (0.010, 0.052)$ & $0.064 (0.030, 0.110)$ & 1.3(0.9,5.2)\\
3 & $0.0418 (0.0415,0.0420)$ & $0.037 (0.011, 0.061)$ & $0.020 (0.012, 0.054)$ & $0.070 (0.033, 0.150)$ & 1.1(0.7,3.8)\\
\end{tabular}
\end{center}
\end{table*}
\begin{figure}
\includegraphics[height=6cm]{figure3.ps}
\caption{Bias distribution based on a set of 5000 simulated AGN lightcurves such as shown in fig. \ref{fig:lc_sparsyXMM},
reproducing the sampling pattern of the XMM-Newton observation of the CDFS.}
\label{fig:histobias}
\end{figure}
\begin{table*}
\begin{center}
\caption{Median $\sigma_{NXV}^2$ and bias as a function of S/N ratio for $\beta=1.5$}
\begin{tabular}{c c c c c}
\hline
Mcr & $\frac{S}{N}$ & Source Flux & Median $\sigma^2_{NXS}$ & $b$\\
cnt/s (cnt/bin) & &
\begin{small}
(erg s$^{-1}$cm$^{-2}$)
\end{small} & & \\
\hline
0.1 (1000) & 25 & 6.25 $\times 10^{-13}$ & $0.022(0.007,0.034)$ & $1.9(1.2, 6)$\\
0.05 (500) & 22.6 & 3.12 $\times 10^{-13}$ & $0.022(0.007,0.034)$ & $1.9(1.2,6)$\\
0.01 (100) & 6.3 & 6.25 $\times 10^{-14}$ & $0.022(0.005,0.036)$ & $1.9(0.007,10.2)$\\
0.005 (50) & 3.4 & 3.12 $\times 10^{-14}$ & $0.021(0.002,0.038)$ & $2(1,21)$\\
0.002 (20) & 1.4 & 1.25 $\times 10^{-14}$ & $0.016(-0.54,0.66)$ & $2.52(0.06,\infty)$\\
0.001 (10) & 0.8 & 6.25 $\times 10^{-15}$ & $<0$ & ... \\
\hline
\end{tabular}
\end{center}
\label{tab:4}
\end{table*}
\subsection{The $\sigma^2_{NXS}$ bias}
The numerical experiment we discussed above can be used in principle to correct
the measured variances, in order to retrieve the true intrinsic value. To this end,
for each of the 5000
simulated lightcurves we computed the ratio between the intrinsic variance
$\sigma_{in}^2$ and the actual excess variance measured for the particular
simulated lightcurve $\sigma_{sim}^2$. The sample distribution of this ratios is plotted in
Fig. \ref{fig:histobias}. This distribution has a large scatter and it is highly asymmetric,
due to the large scatter and highly skewed nature of the
$\sigma_{sim}^2$ distribution itself. For about
$\lesssim 25\%$ of the simulated light curves this ratio is $\leq 1$,
but for the majority of them it is $> 1$.
We can then define the "median bias" of the estimated excess variance, which in essence
indicates the correction factor that is needed to retrieve the intrinsic variance, as follows:
\begin{equation}\label{eq:b}
b=\frac{\sigma_{in}^2} {med(\sigma_{sim}^2)}
\end{equation}
where $med(\sigma_{sim}^2)$ is the median of the $\sigma_{sim}^2$ distribution.
This definition of the bias is similar to the \citet{Almaini} definition,
although in the latter case, the authors
defined the bias using the standard deviation instead of the variance of the lightcurve.
They used an \textit{average} correction factor using lightcurves spread over periods
from 2 to 14 days, in the range
1-1.34 with the largest values for the faintest QSO with only two
widely spaced temporal bins.
In the case of the sampling pattern reproducing the XMM-Newton
observations of the bright source n.68, we derived $b=1.8$ for $\beta=1.5$.
We also calculated the bias values for different intrinsic power spectra,
finding that the bias changes for different slopes but in all the cases
the intrinsic variance is $\lesssim$ 2 times larger than the median excess
variance $\sigma_{sim}^2$ that we measure from the sparse lightcurves (see Table \ref{tab:tablemaxlik}).
If the bias was know a priori, it could be used to rescale the measured excess variance,
and correct for the effects of both the red noise leak and sampling pattern in the
measurement of this quantity. However, the bias of the individual realizations has a large scatter due to
the large and strongly asymmetric $\sigma_{sim}^2$ distribution, an effect which
has not been properly considered in previous works.
The bias factors shown in Table \ref{tab:tablemaxlik} are \textit{median} values over
5000 simulations while the individual excess variances can differ much more from the intrinsic variance.
Therefore, given the large and skewed
distribution shown in Fig. \ref{fig:histoexv},
the bias on individual lightcurve can be 2-3 times higher than the one estimated
using Eq. \ref{eq:b} and the extreme care must be employed
when inferring the variability parameters from single observations of AGN with
such extreme sampling patterns.
\begin{table}
\begin{center}
\caption{Median $\sigma_{NXV}^2$ and bias as a function of gap length}
\label{tab:gaps_exv_bias}
\begin{tabular}{c c c }
\hline
Temporal Gap & Median $\sigma^2_{NXS}$ & $b$\\
(days) & & \\
\hline
5.8 & 0.039(0.027,0.050) & 1.02(0.74,1.48)\\
11.6 & 0.036(0.022,0.050) & 1.12(0.80,1.82)\\
28.9 & 0.030(0.016,0.050) & 1.32(0.80,2.54) \\
57.9 & 0.027(0.13,0.52) & 1.48(0.76, 3.07) \\
115.7 & 0.024(0.10,0.50) & 1.62(0.80, 4.03) \\
231.5 & 0.021(0.07, 0.47) & 1.90(0.85, 5.71) \\
\hline
\end{tabular}
\end{center}
\end{table}
\section{What affects the observed variability bias?}\label{sec:bias dep}
\subsection{Bias Dependence on the source flux}\label{subsec:bias flux}
As discussed in the previous section, strongly unevenly sampled lightcurve
produces a biased estimate of the intrinsic lightcurve variance. Such bias
derives mainly from the inability of our data to constrain the average source
flux due to the red noise character of the AGN PSD, which implies
larger power at lower frequencies.
More importantly, a sparse sampling
produces a wide excess variance distribution, indicating that \textit{each individual measurement}
could differ significantly from the intrinsic variance, even if an average correction is applied to our measurement.
Obviously we expect a
dependence of the bias and its scatter on the source flux
as a result of the white noise introduced by Poisson fluctuations.
To estimate such effects, we simulated lightcurves
assuming different average count rates, corresponding to fluxes smaller
than the one of the source 68, as is the case for the bulk of the AGN population detected in the CDFS.
Table \ref{tab:4} shows the bias dependence on the source flux,
in case of the XMM-CDFS observation pattern, fixing the PSD slope
$\beta=1.5$.
The excess variance is the median of the distribution based on a set 5000 simulations
while the bias is estimated using Eq. \ref{eq:b} with the errors coming from the 90\% upper and lower quartiles of the
excess variance distribution. Conversion factors from counts to fluxes were calculated assuming a power law spectrum with $\alpha_{ph}=1.4$ and
$n_H=8 \times 10^{19}$ cm$^{-2}$.\\
The excess variance estimates and bias factor
do not change significantly with the source flux
down to count rates of $\sim 0.005$ cnt/s (which correspond
to a S/N ratio per bin of 3.4 given the assumed XMM background),
while the width of the excess variance distribution increases.
At lower S/N levels the bias increases up to a point where we are
not able to detect variability anymore,
since the excess variance distribution is wide and the median value becomes negative.
We verified that the results do not depend on the specific value of $\beta$ that we use.
This result suggests that a minimum S/N ratio per bin $\gtrsim 1.5-2$ is advisable
for estimating the intrinsic excess variance in case of
sparse sampled ligthcurves. Moreover we verified that
the same bias is observed when using the
maximum-likelihood approach proposed by \citet{Almaini},
for all the considered S/N ratios. Note that
in the low count regime the Almaini approach cannot predict a
negative intrinsic variance by construction and thus yields a ML value of 0.
\subsection{Bias Dependence on the Gap Length}\label{subsec:bias gap}
Apart from such dependence on sampling pattern and source flux,
we expect that the
bias will change as a function of the gap length.
To test this effect on the intrinsic variance estimator, we simulated AGN
lightcurves with the total exposure time of the XMM-CDFS observations
(440 ks) sampled by two blocks of observations of 220 ks each,
using the input parameters shown in Table \ref{tab:input_param}.
The temporal gap between the observations ranges from the extreme
case of $\sim$ 7 months, similar to the gap in the XMM-CDFS pattern,
to the more favourable case of $\sim 6$ days.
Table \ref{tab:gaps_exv_bias} summarizes our results, where the excess variance and bias
estimates are as before the median of the distribution based on a set of 5000
lightcurves simulations, fixing $\beta=1.5$, while the bias is estimated using Eq. \ref{eq:b}
and the errors are derived from the quartiles errors of the $\sigma^2_{NXS}$ distribution.\\
As expected both the median bias and the width of the excess variance
distribution increase with increasing
gap length; the same trend is observed assuming different power-law slope values.
Thus again, as discussed in \S\ref{sec:3} for the XMM sampling pattern,
because of the large uncertainties associated to the excess variance estimate, each individual
lightcurve measurement yields an extremely poor estimate of the intrinsic source variability,
and such uncertainties increases as a function of the gap length, as shown by the errors in Table \ref{tab:gaps_exv_bias}.
In such cases the only way to make a more robust estimate is to collect
repeated observations of the same source, in order to lower the statistical uncertainties (assuming that the process producing the variability is stationary). Alternatively large samples of sources may provide a less biased ensemble estimate, assuming that the underlying PSD is similar for all sources.
\subsection{Ensemble Excess Variance Estimate}
\begin{figure}
\includegraphics[height=6cm]{figure4a.ps}
\includegraphics[height=6cm]{figure4b.ps}
\caption{\textit{Upper panel}: Distribution of the mean-$\sigma_{NXV}^2$ estimated by binning 5000 simulated excess variance (adopting the XMM sampling pattern of \S\ref{subsec:bias flux}), in groups of 5, 10 20 and 50 points (according to the legend). The inset shows the median values of the binned distributions and their st.dev. The simulations are performed by assuming a count rate of 0.1 cnt/s and $\beta$ = 1.5.\textit{Lower panel}: Distribution of the errors on the mean-$\sigma_{NXV}^2$ estimated as the st.dev. of the points within each bin, divided by $\sqrt{n-1}$. The inset reports the mean values of the distribution for the different binning.}
\label{fig:binningSN1}
\end{figure}
\begin{figure}
\includegraphics[height=6cm]{figure5a.ps}
\includegraphics[height=6cm]{figure5b.ps}
\caption{As figure 4 but for 0.001 cnt/s}
\label{fig:binningSN}
\end{figure}
A collection of several observations of the same source or a large sample of AGN may produce a
less biased estimate of the AGN variability under some particular assumptions (stationary variability process or same PSD for all AGN).
In order to verify how reliably we can constrain the source variance through repeated/multiple
observations, we binned the 5000 simulated excess variances obtained by using the XMM pattern
as described in \S\ref{subsec:bias flux}, in groups of 5, 10, 20, 50-points.
For each bin we estimated the mean excess variance and its standard deviation.
The distributions of the 5, 10, 20 and 50-points binned mean-$\sigma_{NXV}^2$ and of its standard deviation
are shown in Fig. \ref{fig:binningSN1} for a count rate of 0.1 cnt/s and $\beta=1.5$.
The resulting mean-$\sigma_{NXV}^2$ distributions do not
peak on the intrinsic variance, as the individual realizations are anyway biased due to the sparse sampling.
However, these distributions are now more symmetric and roughly Gaussian.
A Kolmogorov-Smirnov test performed on the 5, 10, 20 and 50-points mean-$\sigma_{NXV}^2$
distributions indicates that only for the 5-points grouping we can reject
the hypothesis of Gaussian distribution at $>95\%$ level.
Furthermore if we compare the standard of the binned distributions in the upper panel
of Fig.4 (whose values are shown in the inset as the errors) to the scatter of the individual
realizations in each of the n=5,10,20 and 50 points bins, we find that such scatter (divided by $\sqrt{n-1}$)
is on average representative of the uncertainty on the binned mean-$\sigma_{NXV}^2$;
in fact the error in the upper panel is equal to the mean value of the distribution in the
lower panel (due to the central limit theorem). In practice this means that when binning
our data, we can estimate the uncertainty on each mean-$\sigma_{NXV}^2$ simply from the scatter of the individual points composing each bin.
According to the results described in \S\ref{subsec:bias flux}, we expect that the spread
of the distributions of mean excess variances increases with the decreasing source flux.
In fact, down to count rates of $\sim 0.005$ cnt/s (S/N=3.4), the 10, 20 and 50-points mean-$\sigma_{NXV}^2$
distributions are still Gaussian but the errors rise such as the discrepancy between the median values
of the mean-$\sigma_{NXV}^2$ distributions and the intrinsic variance.
We verified that this trend does not depend on $\beta$.
These results imply that if one bins together 10, 20 or 50 excess variances estimated for a moderately
bright AGN sample, then the corresponding binned mean $\sigma_{NXV}^2$ is roughly
a Gaussian variable, and the associated uncertainty is equal to the scatter of the individual binned $\sigma_{NXV}^2$,
divided by $\sqrt{n-1}$, irrespective of $\beta$.
However at low fluxes (count rates $< 0.002$ cnt/s, S/N$<$1.4), the errors become dominant and the
scatter on the mean excess variance is $>100\%$ (see Fig. \ref{fig:binningSN}).
Similarly we expect a dependence of the average excess variances on gap length.
To test such effect we applied the same binning method on 5000 simulated excess variances obtained by using the sampling pattern described in \ref{subsec:bias gap}.
For temporal gaps below $\sim 58$ days, the mean-$\sigma_{NXV}^2$ distributions obtained with the
5, 10, 20 and 50-points binning are Gaussian, while increasing the gap length up to $\sim$ 7 months,
the 5-points mean excess variance distribution becomes not Gaussian at $>95\%$. As before the means of these distributions do not peat at the input variances (e.g. they are biased) and the discrepancy respect to the intrinsic variance increases with the temporal gap, as does the uncertainty on the mean values of the distributions.
\subsection{Uniform and Progressive Sampling}
\label{subsec:Uniform_prog_Sampling}
In order to test more favourable scenarios, better suited to reduce
the bias in AGN variability estimates,
we generated two additional sets of lightcurves with the input parameters shown
in Table \ref{tab:input_param}, and adopting different sampling patterns, which span
the same maximum timescale as the XMM observations described in \S \ref{sec:3}:
\begin{enumerate}
\item Uniform sampling, consisting in 9 observations of 50 ks each
separated by constant temporal gaps of 1900 ks
($\sim 20$ days, fig. \ref{fig:lc_uni_prog}, \emph{upper panel});
\item Progressive sampling, where the observations are separated
by increasing lags according to the expression $gap = 2^n \times 10$ ks,
with $n=1,2,..,8$ (fig. \ref{fig:lc_uni_prog}, \emph{lower panel});
\end{enumerate}
\begin{figure}
\includegraphics[height=6cm]{figure6a.ps}
\includegraphics[height=6cm]{figure6b.ps}
\caption{Simulated AGN lightcurves (black crosses) with the uniform (\emph{upper panel}) and progressive (\emph{lower panel}) sampling schemes marked by red circles. The figure also reports the mean count rate and excess variance measured for the particular simulation over the whole lightcurve and over the intervals with uniform and progressive sampling.}
\label{fig:lc_uni_prog}
\end{figure}
\begin{figure}
\includegraphics[height=6cm]{figure7.ps}
\caption{Excess variance distribution for N=5000 lightcurves simulations of uniform (solid line) and progressive (dotted line) sampling: the
errors are the 90\% upper and lower quartiles of the $\sigma_{XNV}^2$ distribution. The uniform sampling removes the bias, in fact the mean of the distribution is in agreement with the expected value of the excess variance (red line) equal to the input parameter of the simulation ($\sigma \sim 20\%$). For a progressive sampling, the bias in the intrinsic variance slightly increases.}
\label{fig:histo_uni_prog}
\end{figure}
Fig. \ref{fig:histo_uni_prog} shows the normalized excess variance
distribution derived from 5000 simulations for the two sampling
schemes described above. We observe that the distributions are now more symmetric,
and closer to Gaussian, than was the case for the original XMM pattern (Fig.\ref{fig:lc_sparsyXMM}).
A regular sampling pattern also minimizes the median bias
($b=1.01(0.80, 1.32)$) in the intrinsic variance estimates
with a median $\sigma_{XNV}^2$ consistent with the expected value,
although the individual measurement still have
uncertainties of $\sim 25\%$.
For the progressive sampling the median bias is somewhat larger ($b=1.19(0.88,1.80)$).
as this sampling pattern favors short time scales, while the dominant contribution to the
total variance is due to longer ones.
Clearly, if we consider a sparsely sampled lightcurve, the preferable observing scheme
is thus a regular pattern with temporal gaps not much longer than the length of
each observation. In this situation the observations can be used to estimate
the intrinsic source variance {even from single observations, although with significant
uncertainties.
The progressive sampling may be preferred if we intend to trace the
whole PSD (as opposed to just the variance), but such measurements requires
higher S/N ratios and repeated measurements to average over the intrinsic
scatter of any stochastic process.
\begin{table*}
\begin{center}
\caption{Median $\sigma^2_{NXV}$ and bias as a function of S/N ratio for a future mission described in \S \ref{section:Future Perspectives}}
\label{tab:mcr_exv_bias}
\begin{tabular}{c c c c c}
\hline
Mcr & $\frac{S}{N}$ & Source Flux & Median $\sigma^2_{NXS}$ & $b$\\
cnt/s (cnt/bin) & &
\begin{small}
(erg s$^{-1}$cm$^{-2}$)
\end{small} & & \\
\hline
0.1 (1000) & 38 & $4 \times 10^{-14}$ & $0.037(0.027,0.048)$ & $1.1(0.9,1.5)$\\
0.01 (100) & 9.3 & $4 \times 10^{-15}$ & $0.037(0.026, 0.047)$ & $1.1(0.9,1.6)$\\
0.005 (50) & 7.2 & $2 \times 10^{-15}$ & $0.036(0.024,0.048)$ & $1.2(0.9,1.7)$\\
0.002 (20) & 3.9 & $8 \times 10^{-16}$ & $0.035(0.015,0.055)$ & $1.2(0.7,2.8)$\\
0.001 (10) & 2.7 & $4 \times 10^{-16}$ & $0.033(0.006,0.073)$ & $1.3(0.6,6.9)$\\
\hline
\end{tabular}
\end{center}
\end{table*}
\section{Constrains on the observing strategy of future X-ray surveys}
\label{section:Future Perspectives}
Several missions have been proposed over the past few years
to study high redshift AGNs; most of these are designed to have
larger effective area than current X-ray missions, wider Field-of-View
and, depending on the planned orbit, lower background.
For instance the \emph{International X-ray Observatory} \cite[IXO,][]{Barcons2011}
and its evolution \emph{Athena}\footnote{\emph{http://www.mpe.mpg.de/athena/workshop\_mpe\_2011/index.php}},
the \emph{Wide Field X-ray Telescope} \citep[WFXT,][]{Mur10},
all represent missions capable of performing AGN surveys with higher
speed than \textit{Chandra} or XMM.
The results discussed in \S \ref{subsec:Uniform_prog_Sampling}
allows to explore the capabilities of such future X-ray missions in the
time domain. In particular we examine the expectations for deep, wide-area surveys,
which will allow to probe the highest redshift and faintest AGN
populations at the expense of a continuous temporal coverage.
To investigate the capabilities of such missions in measuring AGN variability, we
present here the performance of a mission with 1 $m^2$ effective area,
1 sq.deg. FOV and the low background allowed by a low earth orbit,
very similar to the WFXT design \citep{Ros10}.
This results in a large number of moderate
and high redshift AGN \citep[see e.g.][]{Pao10}.
We used a total observing time of $\sim 400$ ks and we evaluated the performance that can
be expected assuming a uniform sampling scheme similar to the one
presented in \S \ref{subsec:Uniform_prog_Sampling}.
Figures \ref{fig:lcWFXT} and \ref{fig:histoWFXT} represent an example of a possible observing
scheme for the survey, where observations of 50 ks each are spread
evenly over $\sim 6$ months and the corresponding excesses variance and bias distributions,
respectively.
\begin{figure}
\includegraphics[height=6cm]{figure8.ps}
\caption{Simulated AGN lightcurve, sampled in 50 ks observations spread uniformly on $\sim 6$ months,
as expected from future large effective area mission such as those described in the text.}
\label{fig:lcWFXT}
\end{figure}
\begin{figure}
\includegraphics[height=6cm]{figure9a.ps}
\includegraphics[height=6cm]{figure9b.ps}
\caption{Excess variance (\emph{upper panel}) and bias (\emph{lower panel}) distribution based on a set of 5000 simulated lightcurves, such as the one shown in Figure \ref{fig:lcWFXT}; with this observing strategy we are able to retrieve the intrinsic variance with an uncertainty of $\sim 25\%$. }
\label{fig:histoWFXT}
\end{figure}
In order to verify the performance of such type of mission for faint AGN populations,
we explored the dependence of the measured excess variance on different values
of the source mean count rate.
The results are summarized in Table \ref{tab:mcr_exv_bias}.
The excess variance remains relatively small
($\lesssim 20\%$) even at the lower count rate levels. Compared to the case discussed in
\S \ref{section:AGN_lc} however, we are now able to detect variability
at flux levels\footnote{Conversion factors from counts to fluxes
were calculated assuming a power law spectrum with $\alpha_{ph}=1.4$ for
an unabsorbed AGN at z=0.} more than one order of magnitude lower than XMM,
using approximately the same observing time, thus allowing variability studies for hundreds
of AGNs per square degree.
Such good performances are due in part to the larger effective area, and
in part to the low background made possible by the considered low-earth orbital configuration.
\section{Discussion and Conclusions}
\label{sec:conclusions}
In this paper we discussed the performance of current and
future deep survey X-ray missions
in the time domain and their ability to measure AGN variability, using
realistic simulations that reproduce the real data properties.
We show that the \textit{excess variance} is a biased estimator of the intrinsic lightcurve
variance in sub-optimal observing conditions, such as those characterizing
the 2001-2002 XMM-Newton observation of the CDFS.
The same bias is observed when using alternative
estimators of the intrinsic lightcurve variance, as suggested by, e.g., \citet{Almaini}.
In fact we find that when the sampling pattern
is very sparse, the intrinsic variance of the lightcurve is underestimated,
mainly because each realization badly reproduces the intrinsic mean count rate.
Due to the red noise nature of the AGN PSD, this bias strongly depends on the temporal
gaps between observations on the longest timescales, while it is less sensitive to the detailed
distribution of the data points on short timescales.
Furthermore, for a fixed sampling pattern, the bias does not change with the source flux
as long as the S/N ratio per bin is $\gtrsim 1.5$; for lower values
we are hardly able to detect variability at all,
due to the increasing contribution of Poisson noise to the total variance.
We then suggest as rule of thumb, to use sources with a S/N ratio per bin
above 1.5-2, in estimating the intrinsic variance for sparse sampled lightcurves.
We further verified that the bias depends only mildly on the power-law PSD index,
with a peak for $\beta=1.5$, and anyway remains below 2 for all slopes tested here.
While in principle we can use simulations, such as those described here, to
correct the measured quantities and estimate the intrinsic variance,
we point out that the uncertainties on the bias factor can be very large
in the case of irregular sampling, and the bias distribution is very
asymmetrical, so that each individual lightcurve yields a very
poor estimate of the intrinsic AGN properties.
On the other hand we showed that binning together excess variances
in groups of 10, 20 and 50 points, produces mean values
that are approximately Gaussian distributed and its uncertainty can simply be estimated
from the scatter of the individual points composing each bin.
These results are irrespective of the power law slope $\beta$, the temporal gap, and of the S/N, even if the
the spread of the mean excess variance distributions increases with the gap length and with
decreasing S/N.
Unevenly observing patterns as the ones discussed in
\S \ref{sec:3} and \S \ref{sec:bias dep}, are often
due to the scheduling requirements of deep multi-cycle campaigns;
in order to show the benefits deriving from a proper observing strategy,
we tested two regular observing schemes,
which allow us to span the same maximum timescale as the XMM-Newton
observation of the CDFS; we find that such schemes significantly
reduce the bias in the \textit{excess variance} estimates and
produce more symmetrical distribution, with uncertainties
that range from $\sim 100\%$ down to $\sim 20\%$ for the brightest sources.
Uniform sampling patterns are those producing the best results, although
different schemes sampling a larger range of timescales may be desirable
to derive a full PSD.
Finally we showed that for future X-ray mission,
a properly designed observing strategy may allow to
measure variability for hundreds of sources per square degree.
Such dataset would largely overlap with the spectroscopic sample
\citep[e.g.][]{Gilli11}, thus resulting thousand of AGNs with both temporal and
spectroscopic informations.
Since the individual variance estimates will still be affected by significant uncertainties,
a large dataset will be essential in order to constrain the average timing properties of high
redshift AGNs (provided that the AGN population shares the same intrinsic properties).
Several dedicated timing missions have also been proposed in the X-ray regime
such as Lobster or LOFT \citep{Feroci10}. In such cases the continuous monitoring ensures a
sampling pattern very close to a continuous lightcurve yielding unbiased variability
estimates with small uncertainties, thanks to the possibility to average out the
scatter intrinsic to any stochastic process. This type of analysis however will be possible only
for the brightest (and mostly nearby) sources due to the small angular resolution of such missions.
We want to stress that the simulations presented here do not include
additional systematics, such as for instance vignetting
and PSF variation across the FOV. The readers are then encouraged to
explore their specific science cases using simulations
that closely reproduce their specific sampling pattern, S/N ratio, background contamination etc.
Furthermore, the observing strategy of future missions will likely be decided
based on additional scientific requirements, such as the need to discover and
trace transients with variable decay timescales, or to follow up observations made by
observatories at other wavelengths (e.g. LSST), which may require to adopt strategies
that are sub-optimal for AGN studies with respect to those discussed here.
\section*{Acknowledgements}
VA acknowledge support by the German Deutsche Forschungsgemeinschaft,
DFG Leibniz Prize (FKZ HA 1850/28-1). MP acknowledges support from the Italian PRIN 2009.
| {
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} | 5,221 |
Sobór Przemienienia Pańskiego (ukr. Спасо-Преображенський собор) – prawosławny sobór w mieście Dniepr, jedna z dwóch katedr eparchii dniepropetrowskiej Ukraińskiego Kościoła Prawosławnego Patriarchatu Moskiewskiego.
Budowę świątyni zaplanowano jako jedną z pierwszych budowli nowo powstającego miasta Jekaterynosław. Projekt powstał w 1786 i miała to być największa tego typu budowla na świecie. Kamień węgielny miała wmurować sama Katarzyna II w czasie oficjalnego otwarcia budowy nowego miasta – w dniu 9 maja 1787. Wraz ze śmiercią carycy budowa zamarła i pierwotnego projektu nigdy nie zrealizowano.
Po zmianie projektu i zmniejszeniu planowanej świątyni budowę zakończono w 1835. W latach 1930–1988 w Soborze nie odprawiano nabożeństw, z wyjątkiem 1941, kiedy podczas okupacji niemieckiej tymczasowo zezwolono na prowadzenie służby. W tym samym roku w czasie walk pomiędzy wojskami sowieckimi i hitlerowskimi na podwórze przed świątynią przyniesiono kilkadziesiąt ciał zabitych z pobliskich ulic. Znajdują się one w braterskiej mogile w pobliżu głównego wejścia do świątyni.
W latach 1975–1988 w świątyni działało muzeum religii i ateizmu. 21 stycznia 1992 świątynię oficjalnie przekazano Ukraińskiej Cerkwi Prawosławnej i od razu przystąpiono do jej restauracji.
Na terenie świątyni pochowanych jest szereg prawosławnych duchownych (biskupi Andrzej (Komarow), Kronid (Miszczenko), Warłaam (Iljuszczenko), księża Władimir Kapustianski, Konstantin Ohijenko, Fiodor Duplenko, Andriej Kirpicznikow).
Liturgie w soborze odprawiane są codziennie o godzinie 7 i 9 rano.
Przypisy
Galeria
Linki zewnętrzne
Widok na google maps
Szczegółowa historia świątyni – po rosyjsku
Cerkwie eparchii dniepropetrowskiej
Przemienienia, Sobór
Prawosławne katedry na Ukrainie
Świątynie pod wezwaniem Przemienienia Pańskiego
Zabytkowe cerkwie w obwodzie dniepropetrowskim | {
"redpajama_set_name": "RedPajamaWikipedia"
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The data relating to real estate for sale displayed on this Website comes in part from the Broker Reciprocity Program (BR Program) of M.L.S. of Naples, Inc., under License No. [insert LICENSEE's License Number]. Properties listed with brokerage firms other than John R Wood Properties are marked with the "BR House" logo. Detailed information about such properties includes the name of the brokerage firm with which the seller has listed the property. The properties displayed may not be all the properties listed with brokerage firms participating in the M.L.S. of Naples, Inc. BR Program, or contained in the database compilation of the M.L.S. of Naples, Inc.
IDX information is provided exclusively for consumers' personal, non-commercial use, it may not be used for any purpose other than to identify prospective properties consumers may be interested in purchasing, and the data is deemed reliable but is not guaranteed accurate by the MLS. Copyright 2019 Marco Island Association of REALTORS.
The data relating to real estate for sale on this limited electronic display comes in part from the Southwest Florida Multiple Listing Services. Properties listed with brokerage firms other than are marked with the BR Program Icon or the BR House Icon and detailed information about them includes the name of the Listing Brokers. The properties displayed may not be all the properties available through the BR Program. The source of this real property information is the copyrighted and proprietary database compilation of the participating Southwest Florida MLS organizations and is Copyright 2017 Southwest Florida MLS organizations. All rights reserved. The accuracy of this information is not warranted or guaranteed. This information should be independently verified if any person intends to engage in a transaction in reliance upon it. Some properties that appear for sale on this limited electronic display may no longer be available. For the most current information, contact Data last updated 4/25/2019 4:40 PM CST. | {
"redpajama_set_name": "RedPajamaC4"
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Prestwick Burgh Hall, also known as Prestwick Freeman's Hall and Prestwick Freemen's Hall, is a municipal building in Kirk Street, Prestwick, Scotland. The structure, which served as the meeting place of Prestwick Burgh Council, is a Category B listed building.
History
The first municipal building in Prestwick was an early 18th century tolbooth. The tolbooth was used as the offices and meeting place of the chancellor and the two bailies who administered the town: they were elected annually by the 36 freemen of the burgh who owned of land in and around the town. The current building was commissioned by the freemen of Prestwick for use as the local burgh school. It was designed in the Gothic Revival style, built in ashlar stone and was completed in 1837.
The design involved a symmetrical main frontage with three bays facing onto the corner of The Cross and Kirk Street; the central bay, which projected forward, featured a porch with an arched doorway and an octagonal tower above. The tower was fenestrated with a lancet window on the first floor and featured a clock face in the stage above which was surmounted by a spire. There were lancet windows in the outer bays. The ground floor was initially used as a prison and the first floor was used by the burgh school which accommodated some 60 children.
By the late 19th century the burgh council had assumed most of the functions of the freemen and the building had become the burgh hall. It continued in that use until the burgh council established the municipal buildings in Links Road in the late 1930s. The former burgh hall was then acquired by Ayrshire County Council and became their local district offices.
After the steeple was found to be structurally unsound, it was removed in 2011: a firm of consultants advised that it should be rebuilt using new masonry and that the original stone should be used as template. Although residents lobbied for it to be restored South Ayrshire Council failed to attract support from the Heritage Lottery Fund, or any other charitable source, to carry out the necessary works.
See also
List of listed buildings in Prestwick, South Ayrshire
References
Government buildings completed in 1837
City chambers and town halls in Scotland
Prestwick
Category B listed buildings in South Ayrshire
Clock towers in the United Kingdom | {
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Q: Scraping websites with Python3 (Scrapy, BS4) does yield incomplete data. Can not figure out why Some time ago I have set up a web scraper using BS4, logging the value of a whisky each day
import requests
from bs4 import BeautifulSoup
def getPrice() -> float:
try:
URL = "https://www.thewhiskyexchange.com/p/2940/suntory-yamazaki-12-year-old"
website = requests.get(URL)
except:
print("ERROR requesting Price")
try:
soup = BeautifulSoup(website.content, 'html.parser')
price = str(soup.find("p", class_="product-action__price").next)
price = float(price[1::])
return price
except:
print("ERROR parsing Price")
This worked as intended. The request contained the complete website and the correct value was extracted.
I was now trying to scrape other sites for data on other whiskys this time using SCRAPY.
I tried the following URLS:
https://www.thegrandwhiskyauction.com/past-auctions/q-macallan/180-per-page/relevance
https://www.ebay.de/sch/i.html?_sacat=0&LH_Complete=1&_udlo=&_udhi=&_samilow=&_samihi=&_sadis=10&_fpos=&LH_SALE_CURRENCY=0&_sop=12&_dmd=1&_fosrp=1&_nkw=macallan&rt=nc
import scrapy
class QuotesSpider(scrapy.Spider):
name = "whisky"
def start_requests(self):
user_agent = 'Mozilla/5.0 (Windows NT 6.3; WOW64) AppleWebKit/537.36 (KHTML, like Gecko) Chrome/44.0.2403.157 Safari/537.36'
urls = [
'https://www.thegrandwhiskyauction.com/past-auctions/q-macallan/180-per-page/relevance',
]
for url in urls:
yield scrapy.Request(url=url, callback=self.parse)
def parse(self, response):
page = response.url.split("/")[-2]
filename = f'whisky-{page}.html'
#data = response.css('.itemDetails').getall()
with open(filename, 'wb') as f:
f.write(response.body)
I just customized the basic example from the tutorial to create the fast prototype above.
However it did not return the complete website. The body of the response did miss several tags and especially the content I was looking for.
I tried to solve this with BS4 again like this:
import requests
from bs4 import BeautifulSoup
URL = "https://www.thegrandwhiskyauction.com/past-auctions/q-macallan/180-per-page/relevance"
website = requests.get(URL)
soup = BeautifulSoup(website.content, 'html.parser')
with open("whiskeySoup.html", 'w') as f:
f.write(str(soup.body))
To my surprise this produced the same result. The request and its body did not contain the complete website, missing all the data I was looking for.
I also included a user-agent header since I learned that some sites recognize requests from bots and spiders and do not deliver all their data. However, this did not solve the problem.
I am unable to figure out or debug why the data requested from those URLs is incomplete.
Is there a way to solve this using SCRAPY?
A: A lot of websites heavily relies on javascript to generate the final html page of website. When you send request to server it returns html code with some script web browsers like chrome, Firefox and others process that javascript code and the final html that you can see appears. But when you are using scrapy, request or some library they do not come with the functionality of executing the javascript code and hence the html code is different in html, and as the crawler sees the webpage.
If you want to see how crawler sees the website ( the html code of webpage as seen by crawler ) you can run command 'scrapy view {url}' this will open page in browser or if you want to get the html code of webpage as seen by crawler you can run command 'scrapy fetch {url}'. When you are working with scrapy it is good idea to open the url in shell ( the command is 'scrapy shell {url}' ) and then test your extracting desired content logic there with xpath or css method ( response.css('some_css').css('again_some_css'). ) and then finally add this logic to your final crawler. If you want to see what response you got in shell. you can just type view(response) and it will open the response received in browser. I hope that is clear. But if you want to process the javascript before finally processing the response ( when it is necessary ) you can use selenium which is headless browser or splash which is lightweight web browser. selenium is pretty easy to use.
Edit 1. For the first url : go to scrapy shell and check the css path div.bidPrice::text. Inside that you will see that content inside is generated dynamically and there is no html code and content is being generated dynamically.
| {
"redpajama_set_name": "RedPajamaStackExchange"
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Le prix du Maréchal-Foch, de la fondation de la Société des amis du maréchal Foch, est un prix de l'Académie française biennal d'histoire et de sociologie créé en 1955 et .
Ferdinand Foch est un général, maréchal de France et membre de l'Académie française, né le à Tarbes, dans les Hautes-Pyrénées, et mort le à Paris.
Lauréats
Notes et références
Liens externes
Prix de fondations décernés par l'Académie française sur le site de l'Académie française.
Prix littéraire de l'Académie française
Fondation en 1955 | {
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{"url":"https:\/\/cs-syd.eu\/posts\/2015-07-19-super-user-spark","text":"How to synchronise and back up your dotfiles with the Super User Spark\n\nDate 2015-07-19\n\nAlmost a year ago I wrote a program to synchronise and backup your dotfile called Super User Stone. I'm very pleased to finally deprecate that tool and announce the Super User Spark.\n\nI realise that the name is somewhat unimaginative but I had to find a way to not have to rename my SUS depot.\n\nAfter the installation of the Super User Spark, you will find yourself with a new binary in one of the directories of your PATH: spark. spark was essentially created for the exact same reason as the Super User Stone. spark allows you to synchronise and back up dotfiles. This way you only ever need one directory of dotfiles and deploy it on all your systems.\n\nA quick demo\n\nTo give you a good overview of the power of the spark language, we will require a more complex example than the one I used to demonstrate the Super User Spark. This demonstration will not be a comprehensive overview of the usage of spark. For more information, see the spark usage page.\n\nSay you use both Bash and Xmonad. You might then have these dotfiles on your systems:\n\n- \/home\/user\n|- .bashrc\n|- .bash_aliases\n|- lib\n|- Keys.hs\n\n\nIn this example, you're using a different kind of keyboard layout on your desktop and your laptop. Because of this, you use a different set of key bindings for Xmonad that you define in two different Keys.hs files. Moreover, on your laptop, you use different aliases, which you define in a different bash_aliases file. You would then build a SUS depot as follows that's shared among, let's say, two systems: desktop and laptop.\n\n- depot\n|- spark.sus\n|- shared\n| |- bashrc\n| |- bash_aliases\n| |- Keys.hs\n|- desktop\n| |- Keys.hs\n|- laptop\n| |- bash_aliases\n\n\nThe real difference between Stone and Spark is the file that you add to your depot. spark uses a domain specific language to allows you to specify how you want your files to be deployed. The full specification of the language can be found in the source code repository. The card will then internally be compiled to instructions for deployment. In this example the content of spark.sus, for this example depot, would look like this:\n\n# Call the card \"sus\".\n# This will only matter once you have more than one card.\ncard sus {\n\n# First look for the file in $HOST. # If the file is not found there, look in shared. alternatives$(HOST) shared\n\n# Any deployment will go into the home directory.\ninto ~\n\n# A block (between braces) allows you to keep the 'into' and 'outof' declarations\n# that are in effect but making the ones from the block local.\n{\noutof xmonad # From here on all deployments will come out of the 'xmonad' dir.\ninto .xmonad # This 'into' statement compounds with the previous to '~\/.xmonad'.\n\n{\ninto lib\n\nKeys.hs -> Keys.hs\n}\n}\n\n# After this block, everything is as it would have been before the block.\nbashrc -> .bashrc\nbash_aliases -> .bash_aliases\n}\n\n\nThis card will compile to the following deployments on the laptop system:\n\n\"\/home\/user\/sus-depot\/shared\/xmonad\/xmonad.hs\" l-> \"\/home\/user\/.xmonad\/xmonad.hs\"\n\"\/home\/user\/sus-depot\/shared\/bashrc\" l-> \"\/home\/user\/.bashrc\"\n\"\/home\/user\/sus-depot\/laptop\/bash_aliases\" l-> \"\/home\/user\/.bash_aliases\"\n\n\n... and these on the desktop system:\n\n\"\/home\/user\/sus-depot\/shared\/xmonad\/xmonad.hs\" l-> \"\/home\/user\/.xmonad\/xmonad.hs\"\n\nTo see a more elaborate example of how spark can be used, have a look at my personal SUS depot. One important feature that wasn't mentioned in this post are sparkoff's. These allow you to really make your spark configuration modular.","date":"2018-12-15 23:58:14","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.24915406107902527, \"perplexity\": 2723.5460091163472}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-51\/segments\/1544376827137.61\/warc\/CC-MAIN-20181215222234-20181216004234-00313.warc.gz\"}"} | null | null |
Q: using function to find max value in an array Can someone please explain what I'm doing wrong. I need to use an array to find the max value of the percentages array and display that amount and the year in the corresponding years array.
#include <stdio.h>
#include <stdlib.h>
#include <ctype.h>
int findHighNumber (double percentages[], int elements);
int main (void)
{
int index;
int elements;
int years[] = {2000, 2002, 2004, 2006, 2008, 2010, 2012};
double percentages[] = {6.7, 6.6, 8, 9, 11.3, 14.7, 14.6};
int sampleSizes[] = {187761, 444050, 172335, 308038, 337093, 1000, 346978};
int high = findHighNumber(percentages, elements);
printf ("%i had the highest percentage with autism at %.2lf%%.\n", years[high], percentages[high]);
return 0;
}
int findHighNumber (double percentages[], int elements)
{
int index, high;
high = percentages[0];
for (index = 1; index < elements; index++)
if (high < percentages[index])
{
high = percentages[index];
}
return high;
}
A: I'm not sure if this is a typo here, but it appears while calling
int high = findHighNumber(percentages, elements);
you don't have a variable elements defined there. That said, you have another syntax error (missing the closing }) near
if (high < percentages[index])
{
high = percentages[index]; ///after this.
Finally, inside findHighNumber (), elements is local to the function, so, the only usage as elements = index+1; is useless.
A: First, you have to declare the elements variable in main():
int elements = 7;
or even better:
int elements = sizeof(percentages) / sizeof(*percentages);
which computes the array size automatically.
Then, don't modify the elements variable in you loop.
If you do it, each time the current maximum is changed, you program your loop to stop after the next element, potentially missing the real maximum which can be located at the end of the array.
Then, like Bob said, you should return the index of the maximum, this way, you can retrieve the corresponding sampleSize, or year.
When you do years[high], you're using a percentage as an array index which can take any real value in [0,100], instead of an index which must stay in {0, 1, 2, 3, 4, 5, 6}.
What's more, you store high in an integer variable, resulting in a truncation of it's value. You may prefer to store it as a double variable.
So findHighNumber() can become:
int findHighNumber(double percentages[], int elements)
{
int index;
double high;
int high_index;
high_index = 0;
high = percentages[high_index];
for (index = 1; index < elements; index++)
if (high < percentages[index]) {
high = percentages[index];
high_index = index;
}
return high_index;
}
The main() function now can become something like:
int main (void)
{
int years[] = {2000, 2002, 2004, 2006, 2008, 2010, 2012};
double percentages[] = {6.7, 6.6, 8, 9, 11.3, 14.7, 14.6};
int sampleSizes[] = {187761, 444050, 172335, 308038, 337093, 1000,
46978};
int elements = sizeof(percentages) / sizeof(*percentages);
int high_index = findHighNumber(percentages, elements);
printf ("%i had the highest percentage with autism at %.2lf%% (sample"
"size was %d).\n", years[high_index], percentages[high_index],
sampleSizes[high_index]);
return 0;
}
Then, some small advices:
*
*putting a prototype of findHighNumber() is not needed, just define the function above the places where it's needed. This way, when you really have to put a prototype for a function, it a hint saying that you were forced to do so, (for example for mutually recursive functions), plus it shortens the code.
*you should define a, for example, struct autism_sample like that:
struct autism_sample {
int year;
int sample_size;
double percentage;
};
This way, you just have to define one array:
struct autism_sample autism_samples[] = {
{
.year = 2000,
.sample_size = 187761,
.percentage = 6.7,
},
{
.year = 2002,
.sample_size = 444050,
.percentage = 6.6,
},
...
};
This way, your data is organized in a more logical way, less error-prone to maintain and you gain the choice, in the implementation of findHighNumber(), to return either the index of the maximum or directly, a pointer to the element holding the maximum, the index becomes then useless.
What's more, it's easier to (un)serialize...
A: Given the way you are storing and using data, you should return the index where the higher element is, not its value.
Also you have to pass the right size of the array (and don't change it inside the function), while you are passing an uninitialized value.
A: You should send the index instead of the value.
int findIndexOfMax (double percentages[], int elements) {
int index = 0;
int highIndex = 0;
double high = 0.0;
high = percentages[highIndex];
for (index = 1; index < elements; index++)
{
if (high < percentages[index])
{
high = percentages[index];
highIndex = index;
}
}
return highIndex;
}
A: You are returning the wrong value .. you should not return what's the highest value but you should return the index of it.
or you you can simply print the result that you want in the function you're calling.
Also, I don't see that you have initialized the variable elements.
| {
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\section{Introduction}
Stars are born in dark clouds and giant molecular clouds (GMCs) which consist primarily of molecular hydrogen \citep[e.g.,][]{Cohen79,Tatematsu93, Myers95,Mardones97,Ohashi97, Dobashi05, Buckle12,Liu12}. The densest portions of the clouds are known as dense cores or molecular cloud cores where stars form through the gravitational collapse. To investigate such dense cores, the millimeter dust continuum emission is a good tracer \citep{Motte98, Johnstone06,Kauffmann08,Belloche11}. The observations in molecular lines are also required to investigate the dynamical states of the dense cores and to divide the cores overlapped in the same line of sight. Thus, a dense core survey both in the dust continuum and molecular line emission toward the overall cloud is essential to understand the formation and evolution of the dense cores.
In the last twenty years, many authors have investigated the dense cores in the dust continuum or molecular line emission toward the $\rho$ Ophiuchi, Taurus, L1333, Chameleon I, Lupus III, CrA, Pipe nebula, and Southern Coalsack regions \citep{Motte98, Tachihara02, Onishi02, Dzib13, Alvaro13}.
However, dense core surveys covering the entire extent of one continuous cloud have been limited.
There are mapping data both in the dust continuum and molecular line emission toward the Orion-A giant molecular cloud (Orion-A GMC) which is the nearest GMC \citep[$D$= 400 pc][]{Menten07, Sandstrom07, Hirota08} and the best-studied one. Thus, the Orion-A GMC is one of the best regions to investigate the distributions and physical properties of the dense cores.
In the Orion-A GMC, a large filamentary structure with a length of several $\times$ 10 pc is seen along the north-south direction \citep{Bally87,Nagahama98}; the filamentary structure is known as the integral-shaped filament. Recently, \citet{Polychroni13} identified filaments and dense cores in L 1641 N using the Herschel PACS 70/160 $\mu$m and SPIRE 250/350/500 $\mu$m data and found that most (71\%) of the prestellar cores are located along the filaments. Many authors have carried out observations in the dust continuum emission at 850 $\mu$m, 1.2 mm, and 1.3 mm to investigate the physical properties of dense cores in the Orion-A GMC \citep{Chini97, Johnstone99, Nutter07, Davis09}. Observations in dense-gas tracers such as the H$^{13}$CO$^{+}$, N$_{2}$H$^{+}$, C$^{18}$O, and CS emission lines have also been carried out \citep{Tatematsu93,Tatematsu98,Tatematsu08,Ikeda07}. These observations have, however, focused only on the integral-shaped filament. Hence, the distribution of the dense cores on the outside of the integral-shaped filament has not been revealed.
Previous authors \citep{Shimajiri11, Nakamura12, Shimajiri14} have presented wide, sensitive 1.1-mm dust-continuum and C$^{18}$O ($J$ = 1--0) line maps of the northern part of the Orion-A GMC with the AzTEC camera mounted on the Atacama Submillimeter Telescope Experiment (ASTE) 10-m telescope and with BEARS mounted on the Nobeyama Radio Observatory (NRO) 45 m telescope, respectively. In the 1.1 mm dust continuum map, we found the following new substructures in addition to the well-known integral-shaped filament. In the OMC-2/3 region, a filamentary structure is found to the east of the integral-shape filament. A shell-like structure around the HII region, M 43, and a filamentary structure associated with Dark Lane South Filament (DLSF), which is know as the photon-dominated region (PDR), \citep{Rodriguez01}, are detected. In the southmost region where an active cluster-forming region of L 1641 N is located, four filamentary structures with a length of $\sim$ 0.5 -- 2.0 pc are seen almost aligned with each other. In the C$^{18}$O ($J$=1--0) map, we found that the overall distribution of C$^{18}$O is similar to that of the 1.1 mm dust continuum emission.
In this paper, we present a catalog of the identified cores in the 1.1-mm dust continuum and C$^{18}$O ($J$=1--0) emission line using the Clumpfind method and their properties of the peak flux density, radius, mass, density, and aspect ratio. The 1.1 mm dust continuum and C$^{18}$O emission data are from \citet{Shimajiri11} and \citet{Shimajiri14}, respectively. We compare the physical properties of the identified C$^{18}$O cores in the OMC-1, OMC-2/3, OMC-4, Dark Lane South Filament (DLSF), and bending structure regions to investigate region-to-region variations. In addition, we compare the spatial distributions of the AzTEC/ASTE 1.1 mm dust, BEARS/NRO 45m C$^{18}$O, SCUBA/JCMT\footnote[2]{James Clerk Maxwell Telescope} 850 $\mu$m \citep{Nutter07}, BEARS/NRO 45m H$^{13}$CO$^+$ \citep{Ikeda07}, and BEARS/NRO 45m N$_2$H$^+$ \citep{Tatematsu08} cores to unveil their physical relationships.
\section{Catalogs}
\subsection{1.1-mm dust continuum emission}\label{aztec}
\subsubsection{Core identification in the AzTEC 1.1 mm dust continuum map}\label{1.1mm_ID}
The overall distribution of the AzTEC 1.1-mm dust continuum emission was revealed by \citet{Shimajiri11}.
Here, we identify cores using the two-dimensional Clumpfind method \citep{Williams94}. This algorithm has been widely used for the identification of cores and clumps \citep[e.g.,][]{Kirk06, Rathborne09, Ikeda11,Tanaka13}.
The algorithm works well with reasonable parameters to identify cores or clumps, although several authors have pointed out some shortcomings of the clumpfind algorithm \citep{Pineda09}. \citet{Pineda09} examined the behavior of the algorithm by changing the threshold level from 3$\sigma$ to 20$\sigma$, a wider range than \citet{Williams94} did, and found that the power-law index of the core mass function (CMF) sensitively depends on the threshold for the higher threshold range $>$ 5$\sigma$. \citet{Ikeda09}, however, demonstrated the weak dependence of core properties and CMF in Orion A on the threshold in a reasonable range from 2$\sigma$ to 5$\sigma$ levels, which was also shown by \citet{Pineda09}.
As described in Sections \ref{sect:properties_aztec} and \ref{sect:properties_c18o}, the physical properties (radius, mass, velocity width, etc.) of the condensations identified by the Clumpfind in this paper are similar to those of the star forming dense cores traced by the H$^{13}$CO$^+$ line \citep[ and references therein. See also Appendix \ref{sect:dependence} of this paper]{Ikeda07}. In this paper,
we will therefore define cores identified by the Clumpfind with a similar threshold value as they used.
The noise distribution of the AzTEC 1.1 mm continuum emission is not uniform over the image and is higher in the outer region. Before applying the Clumpfind \citep{Williams94}, we cut off the outer edge of the AzTEC image, where the coverage is less than 30\% and the noise level is higher by a factor of 1.3 than that in the central part. This is because the observations were made by the raster scan that boresights in azimuth and elevation and because the AzTEC 1.1 mm dust continuum image was obtained by mosaicing observations of two fields. The noise level is $\sim$ 9 mJy beam$^{-1}$ in the central region and $\sim$ 12 mJy beam$^{-1}$ in the outer region of the trimmed image. We applied the Clumpfind method to the AzTEC 1.1 mm dust continuum image with the criteria that the threshold should be the 2$\sigma$ level and the depth of the valley between adjacent peaks should be larger than the 2$\sigma$ interval. We adopted 9 mJy beam$^{-1}$ which is the noise level (= 1$\sigma$) in the central region as the noise level in the Clumpfind. Next, we removed cores whose FWHM sizes are less than the effective angular resolution ($\sim$ 36$\arcsec$). Furthermore, we only took cores having peak intensities above the 4$\sigma$ noise level. As described by \citet{Shimajiri11}, the emission around the central Orion-KL region could not be reconstructed as an accurate structure with the AzTEC data-reduction technique, because the continuum emission around Orion-KL is too bright. Thus, we removed cores in the central Orion-KL region. As a result, we identified 619 dust cores(see also Appendix \ref{sect:c18oID}), as shown in Figure \ref{mass}. Here, we note that the 257 cores are located in the C$^{18}$O observed region, excluding the central part of the Orion-KL region (see Figure \ref{fig:c18ocores}).
To investigate the performance of the FRUIT data reduction, \citet{Shimajiri11} performed a simulated source extraction in which Gaussian sources with various FWHM sizes and total flux densities were artificially embedded in the Orion data, and obtained the following result:
The larger FWHM size of the model source, the lower the recovered fraction of the input total flux density of the source. In the case that the FWHM size of the input source is under 150$\arcsec$ ($\sim$0.3pc), the output total flux density is underestimated by less than 20\%. The total flux density of the source with a peak flux density under 20 Jy is underestimated by less than 10\%. Moreover, the restored image of the 1.1 mm dust continuum emission is consistent with that of the SCUBA 850 $\mu$m dust continuum emission. Consequently, the total flux densities of the sources in our map should be recovered by more than 80\%, because the peak flux density FWHM size of the identified 1.1 mm dust cores are smaller than 20 Jy and 0.2 pc, respectively, as described in Section \ref{sect:properties_aztec} (see also Appendix \ref{sect:negative}).
\subsubsection{Physical properties of the 1.1 mm dust cores}\label{sect:properties_aztec}
The mass of the 1.1 mm dust core ($\equiv$ $M_{\rm H_2}$) was derived from the total flux density at 1.1 mm, $F_{\nu}$, on the assumption that all the 1.1 mm continuum emission arises from dust and that the emission is optically thin, using the formula,
\begin{equation}
M_{\rm H_2}=\frac{F_{\nu}D^2}{\kappa _{\nu}B_{\nu}(T_{\rm d})}. \label{dustmass}
\end{equation}
\noindent We adopted the dust mass opacity of
$\kappa _{\nu }=0.1 \left(\frac{\nu}{10^{12} {\rm Hz}} \right) ^\beta$ cm$^{2}$ g$^{-1}$ with $\beta $=2 \citep{Hildebrand83,Chini97} and $D$ = 400 pc. For the dust temperature, we adopted $T_{\rm d}$=20 K \citep{Cesaroni94}.
We determined the apparent core radius $R_{\rm obs}$ as
\begin{equation}
R_{\rm obs}=\left(\frac{A}{\pi}\right)^{\frac{1}{2}}, \label{Robs}
\end{equation}
\noindent assuming that the core is a sphere. Here, $A$ is the projected area of the core, derived by the Clumpfind. We further estimated the core radius $R_{\rm core}$ corrected for the beam size on the assumption that the core has a Gaussian intensity profile as,
\begin{equation} \label{eq:Rcore}
R_{\rm core}= \left\{\vbox to 24pt{} R_{\rm obs}^2 - \left[\vbox to 21pt{} \frac{\Delta \theta /2}{\sqrt{2\ln2}}(2\ln \frac{T_{\rm peak}}{\Delta I})^{1/2}\right]^2 \right\} ^{1/2}, \label{Rcore}
\end{equation}
\noindent where $\Delta \theta$ (=36$\arcsec$) is the effective beam size of the AzTEC 1.1 mm dust continuum map, $T_{\rm peak}$ is the peak intensity of the core, and $\Delta I$ is the threshold level in the core identification \citep[see][]{Williams94}. We note that the grid size of the map was set to the effective angular resolution in the Clumpfind analysis. The mean gas density ($\equiv$ $n$) of the core was derived as,
\begin{equation}\label{eq:n}
n=\frac{3M_{\rm H_2}}{4\pi \mu m_{\rm H} R_{\rm core}^3}, \label{density}
\end{equation}
\noindent where $\mu$ is the mean molecular weight per free particle taken to be 2.33 and $m_{\rm H}$ is the mass of a hydrogen atom. The range of the radius $R_{\rm core}$, mass $M_{\rm H_2}$, and density $n$ of the 1.1 mm dust cores are estimated to be 0.01 -- 0.2 pc, 0.6 -- 1.2 $\times$10$^2$ $M_{\odot}$, and 0.3 $\times$ 10$^4$ -- 9.2 $\times$ 10$^6$ cm$^{-3}$, respectively.
Figure \ref{dust_core_hist} shows histograms of the radius, mass, density of the 1.1 mm dust cores. In the distributions of $R_{\rm core}$, $M_{\rm H_2}$, and $n$ of the 1.1 mm dust cores, peaks are seen at 0.09 pc, 1.3 $M_{\odot}$, and 1.0 $\times$10$^4$ cm$^{-3}$, respectively.
The uncertainty of $R_{\rm core}$ is 0.07 pc, which is derived from the uncertainty in the estimation of the core projected area.
As estimated in Section \ref{1.1mm_ID}, the uncertainty of the total flux density of the identified 1.1mm core should be less than 20\%, since the size of the core is less than 0.2 pc. Since the uncertainty of the total flux density of the identified 1.1mm core should be less than 20\% (see Section \ref{1.1mm_ID}), the uncertainty of $M_{\rm H_2}$ is 20\%.
We summarized the mean, minimum, and maximum values of each physical parameter in Table \ref{Physical_properties_1mm}.
Table \ref{table_cores} shows the physical properties of all the identified cores.
\subsection{The C$^{18}$O ($J$=1--0) emission line} \label{c18o}
\subsubsection{Core identification in the C$^{18}$O map} \label{clumpfind}
The overall distribution of the C$^{18}$O emission and the velocity structure are described by \citet{Shimajiri14}. To identify cores from the C$^{18}$O ($J$=1--0) data, we applied the Clumpfind algorithm \citep{Williams94} to the C$^{18}$O ($J$=1--0) cube data with an angular resolution of 26$\arcsec$.4 and a velocity channel width of 0.104 km s$^{-1}$. Here, note that we applied the Gaussian gridding convolution function (GCF) with 22$\arcsec$.5 FWHM size to the original C$^{18}$O data, resulting in the effective angular resolution of 26$\arcsec$.4, and did no smoothing in the velocity space.
We adopted the criteria that the threshold should be the 2$\sigma$ level (1$\sigma$= 0.19 K in $T_{\rm MB}$) and the depth of the valley between adjacent peaks should be larger than the 2$\sigma$ interval as suggested by \citet{Williams94}.
We also followed the additional criteria introduced by \citet{Ikeda07} and rejected ambiguous or fake core candidates whose sizes and velocity widths are smaller than the spatial and velocity resolutions, respectively: a core must contain two or more continuous velocity channels,
and they must have at least 3 pixels whose intensities are above the 3$\sigma$ level, and
in addition the pixels must be connected to one another in both the space and velocity domains.
As a result, we identified 235 cores in total (see Figure \ref{fig:c18ocores}).
\subsubsection{Physical properties of the C$^{18}$O cores} \label{sect:properties_c18o}
We estimated the radius $R_{\rm core}$, velocity width in FWHM $dV_{\rm core}$, LTE (Local Thermodynamic Equilibrium) mass $M_{\rm LTE}$, virial mass $M_{\rm VIR}$, and mean density $n$ of the C$^{18}$O cores. The definitions of these parameters are the same as those used by \citet{Ikeda07} and \citet{Ikeda09}. {We estimated the core radius $R_{\rm core}$ using equation (\ref{eq:Rcore}).
To obtain the observed velocity width $dV_{\rm obs}$, we calculated a velocity dispersion within the C$^{18}$O cores, and then we multiplied the factor $\sqrt{8 {\rm ln} 2}$ to convert the core velocity dispersion to the FWHM line width on the assumption of a Gaussian profile. Thus, the observed velocity width $dV_{\rm obs}$ is given by
\begin{equation}
dV_{\rm obs} = \sqrt{8 {\rm ln} 2} \Biggr[\frac{\Sigma_i v_i^2 I_i}{\Sigma_i I_i} - \Biggl(\frac{\Sigma_i v_i I_i}{\Sigma_i I_i}\Biggr)^2\Biggr]^{1/2},
\end{equation}
\noindent where $v_i$ and $I_i$ are the radial velocity and intensity of the $i$-th pixel in each core, respectively.
The velocity width $dV_{\rm core}$ should be corrected for the velocity resolution, $dV_{\rm spec}$=0.104 km s$^{-1}$ as
\begin{equation}
dV_{\rm core} = \sqrt{dV_{\rm obs}^2 - dV_{\rm spec}^2}.
\end{equation}
In the mass estimation, we adopted $T_{\rm ex}$ = 20 K \citep{Cesaroni94}. For the fractional abundance of C$^{18}$O relative to H$_2$, $X_{\rm C^{18}O}$, we adopted 1.7 $\times$ 10$^{-7}$ \citep{Frerking82}. Assuming that the C$^{18}$O($J$ =1--0) emission is optically thin, we have
\begin{equation}\label{eq:LTE}
M_{\rm LTE} = 3.47 \times 10^{-2}
\Biggl(\frac{X_{\rm C^{18}O}}{1.7 \times 10^{-7}} \Biggr)^{-1}
T_{\rm ex}e^{5.27/T_{\rm ex}}\\
\Biggl(\frac{D}{400\ {\rm pc}}\Biggr)^2
\Biggl(\frac{\Delta \theta}{26\arcsec.4}\Biggr)^2
\Biggl(\frac{\eta_{\rm MB}}{0.4}\Biggr)^{-1}
\Biggl(\frac{\Sigma_{i}T_{\rm A}^{\ast}\Delta V_{i}}{\rm K\ km s^{-1}}\Biggr) M_{\odot},
\end{equation}
\noindent where $\Sigma$$_i$$T_{\rm A}$$^{\ast}$$\Delta$$V_{i}$ is the total integrated intensity of the core. We adopted a main beam efficiency $\eta_{\rm MB}$ of 38\% for the 2010 season data and 36\% for the 2013 season data \citep[see][]{Shimajiri14}. Note that we adopted the grid spacing of the data cube, $\Delta \theta$, set to the effective angular resolution of 26$\arcsec$.4. The mean gas density of the core was derived by
\begin{equation}\label{eq:n}
n=\frac{3M_{\rm LTE}}{4\pi \mu m_{\rm H} R_{\rm core}^3}. \label{density}
\end{equation}
The virial mass assuming that the cores have spherical shapes and the virial ratio are estimated as
\begin{equation}
M_{\rm VIR} = 209 \Biggl(\frac{R_{\rm core}}{\rm pc}\Biggr) \Biggl(\frac{dV_{\rm core}}{\rm km\ s^{-1}}\Biggr)^2 M_{\odot}
\end{equation}
\noindent and
\begin{equation}
\mathcal{R}_{\rm vir} = \frac{M_{\rm VIR}}{M_{\rm LTE}},
\end{equation}
\noindent respectively. The range of $R_{\rm core}$, $dV_{\rm core}$, $M_{\rm LTE}$, and $n$ are 0.13 -- 0.34 pc, 0.31 -- 1.31 km s$^{-1}$, 1.0 -- 61.8 $M_{\odot}$, and (0.8 -- 17.5) $\times$ 10$^{3}$ cm$^{-3}$, respectively (see Table \ref{Physical_properties_c18o}). Figure \ref{dust_core_hist} shows histograms of $R_{\rm core}$, $M_{\rm LTE}$, and $n$ of the C$^{18}$O cores: peaks are seen at 0.22 pc, 15.1 $M_{\odot}$, and 2.9 $\times$10$^3$ cm$^{-3}$, respectively.
The uncertainty of $R_{\rm core}$ is 0.05 pc derived from the uncertainty in the estimation of the core projected area.
The uncertainty of $dV_{\rm core}$ is 0.104 km s$^{-1}$, corresponding to the velocity resolution. The uncertainty of $M_{\rm LTE}$ is a factor of 6, which is derived from the uncertainty in $X_{\rm C^{18}O}$ \citep{Shimajiri14}.
The physical properties of the individual C$^{18}$O cores are listed in Table \ref{table_c18o_cores}.
Figures \ref{fig5} (a) and (b) show virial ratio - LTE mass and virial ratio - density relations of the identified C$^{18}$O cores, respectively. We consider that the cores with
a $\mathcal{R}_{\rm vir}$ value
less than three are under virial equilibrium according to \citet{Ikeda09}
With the increasing LTE mass and density, the virial ratio decreases; a similar trend between the LTE mass and the virial ratio was also found by \citet{Dobashi96}, \citet{Yonekura97}, and \citet{Ikeda07}. Most of the bound C$^{18}$O cores are distributed in the filamentary structure. On the other hand, most of the unbound C$^{18}$O cores are distributed outside the filamentary structure (see Section \ref{compare}). Figure \ref{fig5} (c) shows a LTE mass - $R_{\rm core}$ relation of the C$^{18}$O cores. The best-fit power-law functions for the unbound and bound cores are log$_{10}$($M_{\rm LTE}$/$M_{\odot}$) = (3.8 $\pm$ 13.6)log$_{10}$($R_{\rm core}$/pc)+(3.2 $\pm$ 9.3) and log$_{10}$($M_{\rm LTE}$/$M_{\odot}$) = (6.6 $\pm$ 16.5)log$_{10}$($R_{\rm core}$/pc)+(5.3 $\pm$ 10.5), respectively.\footnote[3]{The $\chi$-square fitting method with uncertainties in both the x and y coordinates was applied by using the IDL MPFITEXY tool \citep{Williams10}.} No significant difference can be seen between the data distributions for the unbound and bound core due to the larger uncertainty in the estimation of the LTE mass, although the trend that the LTE mass of the bound core is larger than that of the unbound core can be recognized in Figure \ref{fig5} (c).
Figure \ref{fig5} (d) shows a $dV_{\rm core}$ - $R_{\rm core}$ relation of the identified C$^{18}$O cores.
The best-fit power-law functions for the unbound and bound cores are
log$_{10}$($dV_{\rm core}$/km s$^{-1}$) = (2.7 $\pm$ 14.6)log$_{10}$($R_{\rm core}$/pc)+(1.6 $\pm$ 10.0) and log$_{10}$($dV_{\rm core}$/km s$^{-1}$) = (2.4 $\pm$ 8.3)log$_{10}$($R_{\rm core}$/pc)+(1.3 $\pm$ 5.3), respectively. No significant difference can be seen between the data distributions for the unbound and bound cores. }
Figure \ref{fig:dec_vsys} shows the change of the systemic velocity of the cores (i.e., the peak C$^{18}$O velocity) along the declination, which is best fitted by the relation ($V_{\rm sys}$/km s$^{-1}$)= (5.1 $\pm$ 0.3) (Dec/deg) + (36.5 $\pm$ 1.8). This result suggests a presence of a large-scale velocity gradient along the south-north direction ($\sim$ 0.7 km s$^{-1}$ pc$^{-1}$) (cf., the velocity gradient along the integral-shaped filament is estimated to be 1.0 km s$^{-1}$ pc$^{-1}$ in the $^{12}$CO, $^{13}$CO, H$^{13}$CO$^+$, CS lines \citep{Bally87, Tatematsu93, Ikeda07, Shimajiri11, Buckle12, Shimajiri14}).
\section{Discussion}
\subsection{Mass distribution of the 1.1 mm dust cores}
Region-to-region variation of the dust core mass distribution can be recognized in Figure \ref{mass}. The high-mass cores ($M_{\rm H_2}$ $\ge$ 10.0 $M_{\odot}$) are mainly located in the integral-shaped filament. On the other hand, the intermediate-mass (1.0 $M_{\odot}$ $\le$ $M_{\rm H_2}$ $<$ 10.0 $M_{\odot}$) and low-mass ($M_{\rm H_2}$ $<$ 1.0 $M_{\odot}$) cores tend to appear on the outside of the integral-shaped filament. One of the reasons why the high-mass cores are concentrated in the integral-shaped filament is that the 1.1 mm dust cores in the integral-shaped filament could not be resolved in the AzTEC observations with an angular resolution of $\sim$ 36$\arcsec$ (corresponding to $\sim$ 0.07 pc at 400 pc). In fact, previous SCUBA 850 $\mu$m observations with an angular resolution of $\sim$ 14$\arcsec$ \citep{Nutter07} resolved each 13 1.1 mm dust core into two or three smaller cores (also see Table \ref{table_cores}).
Furthermore, interferometer observations with a high angular resolution of $\sim$ 1 -- 3$\arcsec$ revealed that cores in OMC-2/FIR 4, OMC-2/FIR 6, and L1641 N consist of several smaller cores \citep{Shimajiri08, Shimajiri09, Stanke07}. Observations with interferometers such as the Atacama Large Millimeter/Submilimeter Array (ALMA) are crucial to unveil the internal structures of the cores and confirm whether the trend of the region-to-region variation of the dust core mass distribution is significant.
\subsection{Cross identification between the 1.1 mm dust cores and the cataloged YSOs}\label{spitzer}
Recently, \citet{Megeath12} identified young stellar objects (YSOs) in the Orion A and B molecular clouds from the infrared array camera (IRAC)/Spitzer and multi-band imaging photometer for Spitzer (MIPS)/Spitzer data. They classified 2991 YSOs as pre-main sequence stars with disks and 488 YSOs as protostars using the criterion that the spectral index $\alpha$ (=$\lambda F_{\lambda}$/$d\lambda$) $\ge$ -0.3 for protostars and $\alpha$ $<$ -0.3 for pre-main sequence (PMS) stars with disks. The active galactic nucleus (AGN), galaxies with polycyclic aromatic hydrocarbons (PAH) emission, outflow shock knots and stars contaminated by PAH emission are excluded from these YSO catalog \citep[see][]{Megeath12}. The AzTEC map includes 1801 of 2991 pre-main sequence stars with disks and 202 of 488 protostars. The central part of the Orion-KL region, which could not be well reconstructed by the AzTEC, includes 122 pre-main sequence stars with disks and 22 protostars. We removed these pre-main sequence stars with disks and protostars from the following comparison with our cores.
Figure \ref{fig:cores_YSOs} shows a histogram of
the separations between the YSOs cataloged by \citet{Megeath12} and the peak positions of the 1.1 mm dust cores nearest to them. In the figure, only the protostars and PMSs with separations less than 36$\arcsec$, which is the angular resolution of the AzTEC 1.1 mm dust continuum data, are counted. The distribution of the protostars has a peak at a separation of 7.5$\arcsec$ and decreases to 15$\arcsec$. Thus, we adopted the 15$\arcsec$ separation as a criterion to identify the YSOs associated with the 1.1 mm dust cores.
As a result, 50/1679 (3.0\%) of the pre-main sequence stars with disks are associated with the 1.1 mm dust cores and 49/180 (36.1\%) of the protostars associated with the cores. Figures \ref{postageA}--\ref{postageC} show close-up images of the 1.1 mm dust cores associated with the YSOs. We note that the spatial resolution of the AzTEC 1.1 mm dust continuum map is $\sim$ 36$\arcsec$ (corresponding to $\sim$ 0.07 pc at 400 pc), which cannot resolve each dense core in a cluster-forming region where many dense cores are concentrated in small areas. This effect should lower the detection rate. For example, previous dust continuum observations in the 1.3-mm dust continuum emission with an angular resolution of 11$\arcsec$ found eleven dust cores in the OMC-2 region \citep{Chini97}, but only three cores are identified in our AzTEC 1.1 mm dust continuum map. In addition, we cannot exclude the possibility that the dust cores and the YSOs overlap by chance on the same line of sight. However, this possibility is thought to be small, since the YSOs are likely to be in the Orion-A GMC.
To investigate the region-to-region variation of the environments and evolutionary phases of star formation, we compared the number density of the 1.1 mm dust cores, protostars, and pre-main sequence stars in OMC-1, OMC-2/3, OMC-4, DLSF, the bending structure, and the southern part in the 1.1 mm dust continuum map. The OMC-1 region is known to be a high-mass star-forming region \citep{Furuya09,Bally11,Lee13}. The OMC-2/3 region is known to be an intermediate-mass star-forming region \citep{Takahashi06,Takahashi08,Takahashi09,Takahashi12,Takahashi13, Shimajiri08, Shimajiri09}. The DLSF is influenced by the far ultraviolet (FUV) radiation from the trapezium cluster \citep{Rodriguez01, Shimajiri11,Shimajiri13, Shimajiri14}. In the southern area of the 1.1 mm dust continuum map, cloud-cloud collision is suggested to be occurring by \citet{Nakamura12}. We summarize numbers and number densities of the 1.1 mm dust cores, protostars, and pre-main sequence stars in each region in Table \ref{Spitzer_dist}. Although the six areas/regions are not uniquely and rigorously defined and this is likely to introduce some uncertainties in statistical analyses and discussions, the number density of the 1.1 mm dust cores in the southern part of the 1.1 mm dust continuum map (4.2 cores pc$^{-2}$) is found to be the lowest.
In the integral-shaped filament, the number densities of the protostars in OMC-1, OMC-2/3, and OMC-4 (6.3 -- 8.0 protostars pc$^{-2}$) are similar. On the other hand, the number densities of protostars in the other regions of DLSF, the bending structure, and the southern part are 3 - 27 times lower than those in the integral-shaped filament. This is consistent with the fact that the OMC-1, OMC-2/3, and OMC-4 regions are more active star-forming regions than the DLSF, bending structure, and southern regions. The number density ratios of the 1.1 mm dust cores without YSOs to YSOs including protostars and pre-main sequence stars with disk in OMC-2/3 (6.8 cores pc$^{-2}$/41.8 YSOs pc$^{-2}$ = 0.16) and OMC-4 (7.4 cores pc$^{-2}$/81.3 YSOs pc$^{-2}$ = 0.09) are 2.6--6.0 times lower than those in the other regions of DLSF (11.1 cores pc$^{-2}$/20.3 YSOs pc$^{-2}$= 0.54), the bending structure (8.4 cores pc$^{-2}$/18.1 YSOs pc$^{-2}$= 0.46), and the southern part (4.2 cores pc$^{-2}$/8.6 YSOs pc$^{-2}$ = 0.49). Thus, we speculate that the DLSF, bending structure, and southern regions are in younger evolutionary stages than that of the integral-shaped filament.
This interpretation is supported by the spatial variation of the mean density of the 1.1mm dust core.
The mean densities of the 1.1mm dust cores in OMC-2/3 and OMC-4 are 6.2 $\times$ 10$^4$ and 18.9 $\times$ 10$^4$ cm$^{-3}$, respectively, while those in the bending, DLSF, and south regions are 1.7 $\times$ 10$^4$, 2.1 $\times$ 10$^4$, and 3.0 $\times$ 10$^4$ cm$^{-3}$, respectively. Consequently, the mean densities of the 1.1 mm dust cores in the integral-shaped filament is 2.1 -- 11.1 times higher than those in the DLSF, bending structure, and southern regions.
Since the gas density is generally thought to increase as star formation progresses, the difference in mean density suggests that the integral-shaped filament is most evolved.
We must consider another possibility that the OMC-2/3 and OMC-4 regions are forming higher mass stars. In fact, the OMC-2/3 region is known as an intermediate star forming region \citep{Takahashi06}. However, as shown in Fig. \ref{fig6} (c), the difference in the $M_{\rm LTE}$ distribution between OMC-2/3 and Bending as well as between DLSF and OMC-4 is not significant, suggesting that the stars with similar masses will form on the assumption of the same star formation rate among the regions.
\subsection{Comparison of the core properties in the OMC-1, OMC-2/3, OMC-4, DLSF, and bending structure regions} \label{}
In the Orion-A GMC, the environments and evolutional phases of star formation have considerable region-to-region variations as discussed in Section \ref{spitzer}. To investigate the influence of the different environment and evolutional phase on physical properties of the dense cores, we compare the physical properties of the C$^{18}$O cores in the OMC-1, OMC-2/3, OMC-4, DLSF, and bending structure regions.
Figure \ref{fig6} shows histograms of the radius, velocity width, LTE mass, and virial ratio of the C$^{18}$O cores, respectively, in the five regions.
To quantitatively examine the similarities of the physical properties among the five regions, we applied the Kolmogorov-Smirnov (KS) test, which considers the maximum deviation between the distributions of two samples \citep[e.g.,][]{Wall12}, to the five histograms in each panel (see Tables \ref{KStest_Rcore_dV} and \ref{KStest_Ratio_MLTE}).
The results of the KS test show that there is no significant difference among the five regions for most of the physical properties.
Although the $R_{\rm core}$ values in DLSF are relatively small as shown in Fig. \ref{fig6} (a), the KS test shows no significant difference for the $R_{\rm core}$ values due to the small sample for DLSF.
The $dV_{\rm core}$ distribution in DLSF is significantly different from that in the OMC-2/3 region with a significant level of five percent ($p$-value = 3.1\%). The $dV_{\rm core}$ value in DLSF is relatively small as shown in Fig. \ref{fig6} (b).
The LTE-mass distribution in DLSF is significantly different from those in OMC-1 and OMC-2/3 with a significant level of five percent ($p$-value = 0.7\%).
As described in the above, the LTE-mass distributions in DLSF and OMC-4 have two distinct peaks at $M_{\rm LTE}$ = 2.9 $M_{\odot}$ and 24.2 $M_{\odot}$.
\subsection{Effects of external pressure and internal magnetic field on the dynamical states of the C$^{18}$O cores}
The external pressure and internal magnetic ($B$) field in the cores are important factors to determine their dynamical states. Recently, \citet{Li13} performed high angular (5$\arcsec$) resolution observations with a velocity resolution of 0.6 km s$^{-1}$ in NH$_3$ using the Very Large Array (VLA) and Green Bank Telescope (GBT) toward the OMC-2/3 region (c.f., the mean velocity width of the C$^{18}$O cores is 0.61 km s$^{-1}$). They found that most of the massive cores are supercritical from the comparison between mass and critical mass, suggesting that cores will collapse or fragment. The critical mass $M_{\rm critical}$ is defined as $M_{\rm critical}$=$M_{\rm J}$ + $M_{\Phi}$, where $M_{\rm J}$ and $M_{\Phi}$ are the Jeans mass and the maximum mass that can be supported by a steady $B$ field \citep[e.g.,][]{McKee92}. Here, we investigate the dynamical states of the C$^{18}$O cores with the external pressure and the internal magnetic field according to \citet{Li13}.
The Jeans mass is estimated as
\begin{equation}
M_{\rm J} = 1.182 \frac{\sigma^4}{G^{3/2}P_{\rm ic}^{1/2}},
\end{equation}
\noindent
where $G$ is the gravitational constant, $\sigma$ is the one-dimensional velocity dispersion within the core, and $P_{\rm ic}$ is the external pressure. The pressure $P_{\rm ic}$ can be expressed as $P_{\rm ic} =n_{\rm ic} \mu m_{\rm H} \sigma_{\rm ic}^2$.
We considered the tenuous gas traced by the $^{13}$CO (1--0) emission line as the external gas of the C$^{18}$O cores. Thus, we adopted the density $n_{\rm ic}$ of 2.0 $\times$ 10$^3$ cm$^{-3}$ \citep{Nagahama98} and the velocity dispersion $\sigma_{\rm ic}$ of 0.67 km s$^{-1}$ \citep{Shimajiri14}. The velocity dispersion in the core, $\sigma$, can be estimated from the core velocity width $dV_{\rm core}$ as $\sigma$ = $dV_{\rm core}$ / $\sqrt{8 {\rm ln}2}$, assuming a Gaussian velocity profile. As a result, the Jeans mass $M_{\rm J}$ of the C$^{18}$O cores is estimated to be 0.2 -- 55.6 $M_{\odot}$ (see Table \ref{Physical_properties_c18o}). On the other hand, the maximum mass can be estimated using the formula,
\begin{equation}
M_{\Phi} = c_{\Phi}\frac{\pi B R_{\rm core}^2}{G^{1/2}},
\end{equation}
\noindent where $c_{\Phi}$ is a non-dimensional scaling factor and is 0.12 for an axisymmetric isothermal cloud \citep{Tomisaka88} and $B$ is the magnetic field strength. \citet{Crutcher99} derived the field strength from the observations of the Zeeman effect in CN toward OMC-1 and found $B$ $\sim$ 0.19 -- 0.36 mG.
Here, we adopted 0.1 mG as the field strength according to \citet{Li13}.
Thus, the estimated maximum mass could be the lower limit. As a result, the maximum mass $M_{\Phi}$ of the C$^{18}$O cores is estimated to be 0.1 -- 0.8 $M_{\odot}$ (see Table \ref{Physical_properties_c18o}).
\noindent Here we define the critical mass ratio $\mathcal{R}_{\rm c}$ as,
\begin{equation}
\mathcal{R}_{\rm c} = \frac{M_{\rm LTE}}{M_{\rm J} + M_{\Phi}}.
\end{equation}
We list $M_{\rm J}$, $M_{\Phi}$, and $\mathcal{R}_{\rm c}$ of each C$^{18}$O core in Table \ref{table_c18o_cores}.
Figure \ref{fig:virial_critical} shows a relation between the virial and critical mass ratios of the C$^{18}$O cores.
The $\mathcal{R}_{\rm c}$ value decreases with the increasing the $\mathcal{R}_{\rm vir}$ value. The difference between $\mathcal{R}_{\rm vir}$ -- $\mathcal{R}_{\rm c}$ relations for the bound and unbound C$^{18}$O cores are recognized in Fig. \ref{fig:virial_critical}. The slopes for the bound C$^{18}$O cores are larger than those for the unbound C$^{18}$O cores.
The best-fit power-law functions are $\mathcal{R}_{\rm c}$ = 6.6 $\pm$ 0.12 $\mathcal{R}_{\rm vir}$$^{-1.16 \pm 0.04 }$ for the unbound C$^{18}$O cores and $\mathcal{R}_{\rm c}$ = 12.3 $\pm$ 2.9 $\mathcal{R}_{\rm vir}$$^{-1.56 \pm 0.18}$ for the bound C$^{18}$O cores.
We found that all the bound C$^{18}$O cores are supercritical ($\mathcal{R}_{\rm c}$ $\ge$ 1) and 25 of the 61 unbound C$^{18}$O cores are subcritical ($\mathcal{R}_{\rm c}$ $<$ 1) as seen in Fig. \ref{fig:virial_critical}.
Here, we note that there are large uncertainties in the estimates of the Jeans mass owing to the relation of $M_{\rm J}$ $\propto$ $\sigma^4$ and in the estimation of the maximum mass owing to the large uncertainty in $B$. For further investigation of the dynamical states of the dense cores, observations with higher spectral resolution and measurements of the magnetic field strength would be crucial.
\subsection{Comparison between the 1.1 mm dust and C$^{18}$O cores} \label{compare}
In Sections \ref{aztec} and \ref{c18o}, we have identified the 257 dust and 213 C$^{18}$O cores using the Clumpfind method and estimated their physical properties in the C$^{18}$O observing area. The mean $R_{\rm core}$ value of the C$^{18}$O cores (0.22 $\pm$ 0.04 pc) is 2.4 times larger than that of the 1.1 mm dust cores (0.09 $\pm$ 0.03 pc). The 1.1 mm dust continuum emission probably traces the inner part of the dense cores, while the C$^{18}$O emission line probably traces the outer part of the dense cores. The $n$ value range of the 1.1 mm dust cores ((0.3 -- 915.0) $\times$ 10$^{4}$ cm$^{-3}$) is 13 times larger than that of the C$^{18}$O cores ((0.8 -- 17.5) $\times$ 10$^{3}$ cm$^{-3}$).
We compared the spatial distribution of the 1.1 mm dust cores with that of the C$^{18}$O cores. Figures \ref{omc23_id} -- \ref{orion_w_id} show the positions of the 213 C$^{18}$O cores on the AzTEC 1.1 mm dust continuum map in the OMC-2/3, OMC-4, DLSF, and bending structure regions.
Figure \ref{orion_omc1_id} shows the positions of the C$^{18}$O cores in the OMC-1 region where the 1.1 mm image could not be well reconstructed there. Figure \ref{orion_s_id} shows the positions of the 1.1 mm dust cores in the southern part of the 1.1 mm dust continuum map where there is no C$^{18}$O data. We found that the spatial relation between the 1.1 mm dust and C$^{18}$O cores can be categorized into the following four types
as shown in Figure \ref{category_sample}:
\begin{enumerate}
\item[]{[Category A] The peak positions of the 1.1 mm dust and C$^{18}$O cores agree with each other within the 1.1 mm map resolution of 36$\arcsec$.}
\item[]{[Category B] Several C$^{18}$O cores are distributed around the peak positions of the 1.1 mm dust cores within the 1.1 mm map resolution of 36$\arcsec$.}
\item[]{[Category C] The C$^{18}$O cores not associated with any 1.1 mm dust cores.}
\item[]{[Category D] The 1.1 mm dust cores not associated with any C$^{18}$O cores.}
\end{enumerate}
Table \ref{category_table} summarizes the numbers of the 1.1 mm dust and C$^{18}$O cores in each category. We found 69 pairs of the dust and C$^{18}$O cores in Category A. In this category, one C$^{18}$O core seems to be associated with one 1.1 mm dust core. In Category B, there are 23 C$^{18}$O cores (10.8\%) and 10 dust cores (3.9\%). There are three possible explanations for the B type cores.
One is the several cores are overlapped in the same line of sight. The $V_{\rm LSR}$ is different among the C$^{18}$O cores which are associated with the same 1.1 mm dust core (see Tables \ref{table_cores} and \ref{table_c18o_cores}).
Furthermore, the peak positions of 6/10 dust cores (AzTEC-Ori 110, 232, 234, 242, 272, and 482) coincide with those on the C$^{18}$O integrated intensity map which includes all velocity components as well as the dust continuum map. These results suggest that several cores are distributed on the same line of sight.
Second is due to the poor angular resolution of the AzTEC data. The resolution of the AzTEC data (= 36$\arcsec$) is larger than in the C$^{18}$O data (= 26$\arcsec$). To confirm this possibility, we identified the C$^{18}$O cores using the C$^{18}$O data smoothed to the same angular resolution as the AzTEC 1.1mm map (=36$\arcsec$). As a result, 5 of 10 dust cores (AzTEC-Ori 162, 232, 242,273, and 482) are associated with one C$^{18}$O core identified on the 36$\arcsec$ map, although these dust cores are associated with two C$^{18}$O cores identified on the 26$\arcsec$.4 map. This result suggests that these 1.1 mm dust cores are not resolved due to the poor angular resolution.
Hence, distinct condensations resolved by the C$^{18}$O observations may remain unresolved by the AzTEC observations.
The other is the depletion of the C$^{18}$O molecules in the central parts of dust cores. Such depletion in the central part of the dust condensation has been reported in the B 68 and L 1498 regions \citep{Bergin02,Tafalla02}. Although the inter-core diffuse gas in the Orion-A GMC is warmer than those in low-mass star forming regions, the dense cores are well-shielded from nearby radiation and become cool enough for CO molecules to freeze onto dust grains. In fact, the CO depletion in the dense cores in the Orion-A GMC has been reported by several authors \citep{Ripple13,Tatematsu14, Ren14}.
In Category C, we identified 121 C$^{18}$O isolated cores. There are two possible origins of these cores.
First is due to the poor angular resolution of the AzTEC data.
Some 1.1 mm dust cores are associated with two cores in the 850 $\mu$m data with an angular resolution of 14$\arcsec$, suggesting that the 850 $\mu$m data with the higher angular resolution resolved the 1.1 mm dust cores.
Some 850 $\mu$m cores associated with the C$^{18}$O cores are not associated with any 1.1 mm dust cores. These cores are located between two 1.1 mm dust cores or on elongated structures in the 1.1 mm dust map as shown in Figs. \ref{omc23_id_all} - \ref{omc4_id_all}.
The reason why the 1.1 mm dust cores are not associated with any 850 $\mu$m cores is probably that the 850 $\mu$m counterparts in the 1.1 mm map are not identified due to the poor angular resolution. We note that
the smoothed 850 $\mu$m map with the same angular resolution as in the 1.1 mm map is quite consistent with the 1.1 mm map \citep[see Fig. 20 in][]{Shimajiri11}.
Second is that these C$^{18}$O cores do not have high enough column density to be detected in the dust continuum. Figure \ref{hist_column} shows the histogram of each column densities of the 1.1 mm dust, all C$^{18}$O cores, and C$^{18}$O cores in Category C. The column densities of each cores is estimated from the equation, $N_{\rm H_2}$ = $n$ $\times$ 2 $R_{\rm core}$.
The minimum column density of the 1.1 mm cores (2.4 $\times$ 10$^{21}$ cm$^{-2}$) is twice larger than that of the C$^{18}$O cores (1.0 $\times$ 10$^{21}$ cm$^{-2}$). The column density sensitivity of the C$^{18}$O data is higher than that of the 1.1 mm data. On the contrary, the mass sensitivity of the C$^{18}$O data is worse than that of the 1.1 mm data as shown in Fig. \ref{dust_core_hist} (b), since the mean radius $R_{\rm core}$ of the C$^{18}$O cores is twice larger than that of the 1.1 mm dust cores.
Although the column density estimation has uncertainties, there is a possibility that the C$^{18}$O cores lacking 1.1 mm cores are due to the lack of the sensitivity of the column density of the 1.1 mm data.
For the 81 bound C$^{18}$O cores (51.9\%), we speculate that
the central part of the cores have not yet evolved to reach the density $>$ $\sim$ 10$^4$ cm$^{-3}$ and are not detected in the dust continuum, as described in the beginning of this section. For the 40 unbound C$^{18}$O cores, we speculate that the unbound C$^{18}$O cores are transient structures created by turbulent compression and do not have high enough column density. Most of the unbound C$^{18}$O cores ($\sim$ 70.2\%) are in Category C and are not located on the integral-shaped filament. Here, we define the integral-shaped filament as the area having signal-to-noise ratios above 15 for the 1.1 mm flux density in the OMC 2, 3, and 4 regions for this study.
Recent three-dimensional Magnetohydrodynamic (MHD) simulations have suggested that the turbulent compression creates a local dense part that is gravitationally unbound and cannot produce stars \citep{Nakamura11}.
In Category D, there are 178 dust cores that are not associated with any C$^{18}$O cores
out of the 257 dust cores that were in the C$^{18}$O-observed region (excluding Orion-KL).
We discuss three possible origins of these cores as follows.
The first possible origin is that the C$^{18}$O molecule is selectively dissociated by the FUV radiation from the massive stars in the trapezium cluster and NU Ori.
The FUV intensity at the wavelengths of the dissociation lines for abundant CO decays rapidly on the surface of molecular clouds owing to very large optical depths of the FUV emission at these wavelengths \citep{Glassgold85,Yurimoto04,Liszt07, Rollig13}.
\citet{Bethell07} suggested that the photoionization may play a significant role at $A_{\rm v}$ $\sim$ 10 in the case that the cores are sufficiently clumpy using a reverse Monte Carlo radiative transfer code and spectral modeling. They also mentioned that the cosmic-ray ionization is dominant in such high $A_{\rm v}$ regions.
For less abundant C$^{18}$O, which has shifted absorption lines owing to the difference in the vibrational-rotational energy levels, the decay of FUV is much lower.
The FUV radiation can penetrate the dense region owing to the clumpiness of the cloud.
\citet{Shimajiri13} have found that the distributions of the [CI] emission coincide with those of the $^{12}$CO emission in the PDRs of Orion bar, M43, and DLSF as well as the entire of the cloud, suggesting that these PDRs and the entire of the Orion A cloud have the clumpy structures \citep{Spaans96,Kramer08}.
For the three PDRs, the ionizing sources are the neighboring OB stars, because the 8 $\mu$m (PAH), 1.1 mm, and $^{12}$CO emission are located sequentially as a function of the distance from the OB stars, i.e., the edge-on view for the OB star/PDR system \citep{Shimajiri11}. This result suggests that these PDRs are located on the plane of the sky against OB stars.
As a result, C$^{18}$O molecules are expected to be selectively dissociated by FUV photons even in the inner part of the cloud.
It is possible that the structure of the cloud is clumpy and full of holes such that the mean extinction through the cloud from the perspective of the exciting stars is generally $A_{\rm V}$ $<<$ 5, even though the apparent extinction (based on the observed dust emission) is much greater.
Figure \ref{flux_comp} shows the correlation between the 1.1 mm dust and C$^{18}$O intensities at the position of the 1.1 mm dust cores. The 1.1 mm dust cores associated with the C$^{18}$O cores, which are categorized into A or B, have stronger C$^{18}$O intensity. The number of the 1.1 mm dust cores not associated with any C$^{18}$O cores (Category D) increases with decreasing C$^{18}$O peak intensity.
Most of the 1.1 mm dust cores (56/70 cores) in the PDRs are not associated with any C$^{18}$O cores.
Here, we defined the area of PDRs, DLSF, M43, and Regions A-D, as listed in Table \ref{area_definition} \citep[also see Regions A, B, C, and D shown by][]{Shimajiri11}.
The C$^{18}$O intensity at the position of the 1.1 mm dust cores in the PDRs is lower than the intensity expected from the best-fit line for the 1.1 mm dust cores associated with the C$^{18}$O cores as shown in Fig. \ref{flux_comp}.
The C$^{18}$O intensity at the position of several 1.1 mm dust cores not in PDRs is below the 3$\sigma$ level.
These dust cores are located around NGC1977 and DLSF. Thus, these cores seem to be also influenced by the FUV radiation.
These facts suggest that the C$^{18}$O molecule is selectively dissociated by the FUV radiation in the low 1.1 mm flux density range. \citet{Shimajiri14} found that the abundance ratio of $^{13}$CO to C$^{18}$O, $X_{\rm ^{13}CO}$/$X_{\rm C^{18}O}$, in the Orion A GMC decreases with the increasing C$^{18}$O column density,
implying that the effect of the selective FUV dissociation of C$^{18}$O becomes smaller in the higher 1.1 mm flux density range where it becomes hard for the FUV radiation to penetrate owing to the dust shielding.
Note that several 1.1 mm dust cores in the large 1.1 mm flux density range are categorized into D.
This might be because the FUV radiation can penetrate the large 1.1 mm flux density regions owing to the clumpiness of the cloud.
In fact, the $X_{\rm ^{13}CO}$/$X_{\rm C^{18}O}$ value is larger than the solar system value of 5.5 even in the inner part of the cloud \citep{Shimajiri14}.
Meanwhile, \citet{Shimajiri11} found that the $^{12}$CO peak intensity range in the DLSF region is $\sim$ 20 -- 50 K. Especially, at the outer layers of the cloud surface in DLSF, it increases up to 50 K. On the assumption that the $^{12}$CO ($J$=1--0) line is optically thick, the result shows that the the temperature is 20 -- 50 K in DLSF, suggesting that the regions with bright dust but faint C$^{18}$O emission have dust temperatures higher than the assumed 20 K. Thus, there is a possibility that the outer layers of the cloud surface is significantly heated, driving up dust emission and (self-shielded) $^{12}$CO emission, but the FUV radiation could still penetrate far enough into the unshielded C$^{18}$O layer to photodissociate the C$^{18}$O molecule without heating it.
The second possibility is the depletion of the C$^{18}$O molecule in the central part of the dust cores.
In this case, the C$^{18}$O peak positions are distributed around the center positions of the dust cores.
The peak positions of the C$^{18}$O cores disagree with those of the 1.1 mm dust cores \citep{Ripple13,Tatematsu14, Ren14}.
The third possibility is contaminations from the components by ambient gas and surrounding cores.
As shown in Fig. \ref{flux_comp}, many 1.1 mm dust cores are not associated with any C$^{18}$O cores, in spite of the fact that these C$^{18}$O intensities are more than 3$\sigma$ and are not located in PDRs.
In most cases, the C$^{18}$O cores and/or extended emission are distributed around the 1.1 mm dust cores categorized into D \citep[also see Fig. 3 (b) in][]{Shimajiri14}. There is a possibility that the C$^{18}$O cores associated with the 1.1 mm dust cores categorized into D are embedded in the components of other C$^{18}$O cores and/or extended emission and and can not be extracted as cores.
\subsection{Comparison among the 1.1 mm dust, 850 $\mu$m dust, H$^{13}$CO$^+$, and N$_2$H$^+$ cores}
We also compared the spatial distribution of the 1.1 mm dust cores with those of the SCUBA 850 $\mu$m dust \citep{Nutter07}, H$^{13}$CO$^+$ (1--0)\citep{Ikeda07}, and N$_2$H$^+$ (1--0) cores \citep{Tatematsu08}. The H$^{13}$CO$^+$ and N$_2$H$^+$ emission are known as the dense gas tracers \citep[e.g.,][]{Saito01,Takakuwa03,Maruta10, Friesen10,Johnstone10,Tanaka13}.
The angular resolution of the SCUBA 850 $\mu$m data is 14$\arcsec$ (0.03 pc) and \citet{Nutter07} identified the condensations having a peak flux density more than 5$\sigma$ relative to the local background as cores.
The H$^{13}$CO$^+$ data has an angular resolution of 21$\arcsec$ (0.04 pc) and \citet{Ikeda07} identified the H$^{13}$CO$^+$ dense cores by the Clumpfind.
The N$_2$H$^+$ observations with a telescope beam size of 17$\arcsec$.8 (0.03 pc) were performed with a grid spacing of 20$\arcsec$.55 (0.04 pc) and \citet{Tatematsu08} identified the N$_2$H$^+$ dense cores by eyes from the $F_1,F$ =0, 1--1, 2 component, which is an isolated component of the seven hyperfine components, and the most intense hyperfine component $F_1, F$=2, 3--1, 2.
In Table \ref{table_cores}, we summarize the SCUBA 850 $\mu$m dust, H$^{13}$CO$^+$, and N$_2$H$^+$ cores distributed around the peaks of the 1.1 mm cores within the spatial resolution of 36$\arcsec$.
Figures \ref{omc23_id_all} -- \ref{orion_s_id_all} show the comparison of the spatial distribution among the 1.1 mm dust, C$^{18}$O, 850 $\mu$m dust, H$^{13}$CO$^+$, and N$_2$H$^+$ cores. Table \ref{association_rate} summarizes the comparison.
The AzTEC map excluding the Orion KL region contains 198 850 $\mu$m dust cores and 202 H$^{13}$CO$^+$ cores. The overall spatial distribution of the 1.1 mm dust cores has good agreement with that of the 850 $\mu$m dust cores. We found that 133 of the 215 850 $\mu$m dust cores (61.9\%) are associated with the 1.1 mm dust cores. However, 82 850 $\mu$m dust cores (38.1\%) are not detected in the 1.1 mm continuum emission, probably because of the insufficient sensitivity and beam dilution in the 1.1 mm map.
The lowest mass of the dense cores detected in the SCUBA 850 $\mu$m observations is 0.13 -- 0.15 $M_{\odot}$, which is four times smaller than those detected in the 1.1 mm dust continuum observations. In addition, the angular resolution of the SCUBA 850 $\mu$m observations is 2.6 times higher than that of the 1.1 mm dust continuum observations.
These facts indicate that the sensitivity of the SCUBA data is ten times higher than that of the AzTEC 1.1 mm dust continuum data.
We found that 56.7\% (119/210) of the H$^{13}$CO$^+$ cores are associated with the 1.1 mm dust cores. The fraction is 1.3 times higher than that for the C$^{18}$O cores, in spite of the fact that the mass detection limit of the H$^{13}$CO$^+$ observations is twice higher than that of the C$^{18}$O observations.
Furthermore, the mean density of the 1.1mm dust cores (5.5 $\times$ 10$^4$ cm$^{-3}$) is closer to
that of the H$^{13}$CO$^+$ cores (1.6 $\times$ 10$^4$ cm$^{-3}$) and smaller than that of the C$^{18}$O cores (0.4 $\times$ 10$^4$ cm$^{-3}$). Thus, the 1.1 mm dust continuum and the H$^{13}$CO$^+$ emission line are thought to trace the similar density area. These facts suggest that the H$^{13}$CO$^+$ emission is a better tracer of the dense cores than the C$^{18}$O emission.
In the comparison of the spatial distributions between the 1.1 mm dust and N$_2$H$^+$ cores, the high fraction of 77.8\% (21/27) of the N$_2$H$^+$ cores are found to be associated with the 1.1 mm dust cores. Such a high fraction implies that both the N$_2$H$^+$ and H$^{13}$CO$^+$ emission can trace well the dense cores compared to the C$^{18}$O emission. The remaining six N$_2$H$^+$ cores are not associated with any 1.1 mm dust cores probably owing to the insufficient spatial resolution of the 1.1 mm map. In fact, the six cores are located in the extended feature in the 1.1 mm map, and are associated with the SCUBA 850 $\mu$m dust cores. We note that the number of the N$_2$H$^+$ cores is the smallest and the lowest mass of the detected N$_2$H$^+$ cores is 7.3 $M_{\odot}$, which is much larger than those of the 1.1 mm, 850 $\mu$m, and H$^{13}$CO$^+$ cores.
\section{Summary}
The main results of this study are summarized as follows:
\begin{enumerate}
\item{We have cataloged 619 dust cores in the AzTEC 1.1 mm dust continuum using the Clumpfind method. The ranges of the radius $R_{\rm core}$, mass $M_{\rm H_2}$, and density $n$ of these cores are estimated to be 0.01--0.2 pc, 0.6 -- 1.2 $\times$ 10$^2$ $M_{\odot}$, and 0.3 $\times$ 10$^4$ -- 9.2 $\times$ 10$^6$ cm$^{-3}$, respectively. The high-mass cores ($M_{\rm H_2}$ $\ge$ 10.0 $M_{\odot}$) are located in the integral-shaped filament. On the other hand, the intermediate-mass (1.0 $M_{\odot}$ $\le$ $M_{\rm H_2}$ $<$ 10.0 $M_{\odot}$) and low-mass ($M_{\rm H_2}$ $<$ 1.0 $M_{\odot}$) cores are mainly located on the outside of the filament.}
\item{The distribution of the C$^{18}$O ($J$=1--0) emission is similar to that of the 1.1 mm dust continuum emission. We have identified 235 C$^{18}$O cores from the C$^{18}$O data by the Clumpfind algorithm. The ranges of $R_{\rm core}$, $dV_{\rm core}$, $M_{\rm LTE}$, and $n$ are 0.13 -- 0.34 pc, 0.31 -- 1.31 km s$^{-1}$, 1.0 -- 61.8 $M_{\odot}$, and (0.8 -- 17.5) $\times$ 10$^3$ cm$^{-3}$, respectively. In the 235 C$^{18}$O cores, 61 cores are gravitationally unbound, while 174 cores are gravitationally bound.}
\item{
We performed the core identification with various step sizes and threshold levels in a reasonable range of 2$\sigma$ to 5$\sigma$ in order to investigate the influence of the Clumpfind parameters on the core properties. The number of the identified C$^{18}$O cores significantly decreases with increasing step size, while the core number weakly depends on the threshold level.
The $R_{\rm core}$, $dV_{\rm core}$, and $M_{\rm LTE}$ values gradually increase with increasing step size, but do not depend on the threshold level.
}
\item{The LTE mass vs. virial ratio and density vs. virial ratio relations of the identified C$^{18}$O cores show that the virial ratio tends to decrease with the increasing LTE mass and density. The best-fit power-law functions for the unbound and bound cores are ($M_{\rm LTE}$/$M_{\odot}$) = (58.2 $\pm$ 27.3)($R_{\rm core}$/pc)$^{1.7 \pm 0.3}$ and ($M_{\rm LTE}$/$M_{\odot}$) = (325.9 $\pm$ 130.4)($R_{\rm core}$/pc)$^{2.1 \pm 0.3}$, respectively. The coefficient for the bound cores is significantly larger than for the unbound cores. The difference between the data distributions for the unbound and bound cores cannot be seen in the $dV$ vs. $R_{\rm core}$ relation. }
\item{We compared the physical properties of the C$^{18}$O cores among the OMC-1, OMC-2/3, OMC-4, DSLF, and bending structure regions. The Kolmogorov-Smirnov (KS) test showed that there is no significant difference among the five regions for each property, although the physical environments in the regions are very different from each other.}
\item{We investigated the dynamical states of the C$^{18}$O cores with the external pressure and internal magnetic field. We found that all the bound C$^{18}$O cores are supercritical ($\mathcal{R}_{\rm c}$ $\ge$ 1) and 25 of the 61 unbound C$^{18}$O cores are subcritical ($\mathcal{R}_{\rm c}$ $<$ 1), although there are large uncertainties in the estimation of the Jeans mass owing to the relation of $M_{\rm J}$ $\propto$ $\sigma^4$.}
\item{We examined the spatial relations between the 1.1 mm dust and C$^{18}$O cores. We found that the relations can be categorized into the following four groups. First, one C$^{18}$O core is associated with one 1.1 mm dust core. Second, two or more C$^{18}$O cores are associated with one dust core. Third, there are isolated C$^{18}$O cores which are not associated with any dust core. Fourth, 1.1 mm dust cores not associated with any C$^{18}$O cores also exist.}
\item{We compared the spatial distributions of the 1.1 mm dust, 850 $\mu$m dust, C$^{18}$O, H$^{13}$CO$^+$, and N$_2$H$^+$ cores. The overall distribution of the 1.1 mm dust cores is found to have good agreement with that of the 850 $\mu$m dust cores. In addition, the N$_2$H$^+$ and H$^{13}$CO$^+$ emission are found to trace well the dense dust cores compared to the C$^{18}$O emission.}
\end{enumerate}
\acknowledgments
We acknowledge the anonymous referee for providing helpful suggestions to improve this paper.
Y. Shimajiri was financially supported by a Research Fellowship from the JSPS for Young Scientists. This work was supported by JSPS KAKENHI Grant Number 90610551.
Part of this work was supported by the French National Research Agency (Grant no. ANR–11–BS56–0010–STARFICH).
K. S., T. S., and F. N., were supported by JSPS KAKENHI Grant Numbers, 26287030, 26610045, 26350186, and 24244017.
M. M., and T. T., were supported by MEXT KAKENHI No. 23103004.
{\it Facilities:} \facility{ASTE (AzTEC)}, \facility{Nobeyama 45m (BEARS)}.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,355 |
Anarchy and Art
# Anarchy and Art
From the Paris Commune to the Fall of the Berlin Wall
Allan Antliff
ANARCHY AND ART
Copyright © 2007 by Allan Antliff
All rights reserved. No part of this book may be reproduced or used in any form by any means—graphic, electronic or mechanical—without the prior written permission of the publisher, except by a reviewer, who may use brief excerpts in a review, or in the case of photocopying in Canada, a license from Access Copyright.
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The publisher gratefully acknowledges the support of the Canada Council for the Arts and the British Columbia Arts Council for its publishing program, and the Government of Canada through the Book Publishing Industry Development Program and the Government of British Columbia through the Book Publishing Tax Credit Program for its publishing activities.
Efforts have been made to locate copyright holders of source material wherever possible. The publisher welcomes hearing from any copyright holders of material used in this book who have not been contacted.
Text and cover design by Shyla Seller
Cover art: _Strike Zone_ by Richard Mock, 1991
Printed and bound in Canada
**Library and Archives Canada Cataloguing in Publication:**
Antliff, Allan, 1957-
Anarchy and art: from the Paris Commune to the fall of the Berlin Wall / Allan Antliff.
Includes bibliographical references and index.
ISBN 978-1-55152-218-0 (pbk.)
EISBN 978-1-55152-300-2
1. Art—Political aspects. 2. Anarchism in art. 3. Art and society. 4. Artists—Political activity. 5. Anarchism. I. Title.
N72.P6A67 2007 701'.03 C2007-902269-3
_Fight until the unborn lives, and the dying, die!_
—Bayard Boyesen
## CONTENTS
Acknowledgments
Introduction
CHAPTER 1 **A Beautiful Dream:** _Courbet's Realism and the Paris Commune of 1871_
CHAPTER 2 **Wandering:** _Neo-Impressionists and Depictions of the Dispossessed_
CHAPTER 3 **Obscenity:** _The Advent of Dada in New York_
CHAPTER 4 **True Creators:** _Russian Artists of the Anarchist Revolution_
CHAPTER 5 **Death to Art!:** _The Post-Anarchist Aftermath_
CHAPTER 6 **Gay Anarchy:** _Sexual Politics in the Crucible of McCarthyism_
CHAPTER 7 **Breakout from the Prison House of Modernism:** _An Interview with Susan Simensky Bietila_
CHAPTER 8 **With Open Eyes:** _Anarchism and the Fall of the Berlin Wall_
Bibliography
Index
## ACKNOWLEDGMENTS
Research for this book was aided by grants from the Social Sciences and Humanities Research Council, the University of Victoria, and the support of the Canada Research Chair program. I am particularly indebted to Wolf Edwards, Patricia Leighten, Mark Antliff, Richard Day, and Robyn Roslak for many invaluable insights. In addition, weekly discussions at the Victoria Anarchist Reading Circle were a constant stimulation. Finally, my thanks goes to Brian Lam, Shyla Seller, Robert Ballantyne, and Bethanne Grabham of Arsenal Pulp Press, who saw this book to completion with grace and élan.
## INTRODUCTION
As its title indicates, this book was written in part to call attention to and encourage the development of an emerging field in art history: the study of anarchism in art. Though there are many monographs on artists who have identified as anarchists, to date broader surveys of the relationship between anarchism and art are few and far between. In part, this is because anarchist art has been perceived generally as one facet in a larger project—"leftist" art—with the result that differences between it and other traditions have often been glossed over or ignored altogether. This book, therefore, is a step toward the foregrounding of art production as it relates to historical, philosophical, social, and political issues from an anarchist perspective.
From European anarchism's beginnings in the nineteenth century, the arts have been an integral part of the movement, as evidenced by Pierre-Joseph Proudhon's willingness in the 1860s to write an entire book in defense of the anarchist artist Gustave Courbet. In similar fashion, Peter Kropotkin's pamphlet "Appeal to the Young" (1880) counted artists as key players in the social revolution, and addressed them with this stirring call:
... if your heart really beats in unison with that of humanity, if like a true poet you have an ear for Life, then, gazing out upon this sea of sorrow whose tide sweeps up around you, face to face with these people dying of hunger, in the presence of these many corpses piled up in these mines, and these mutilated bodies lying in heaps on the barricades, in full view of the desperate battle which is being fought, amid the cries of pain from the conquered and the orgies of the victors, of heroism in conflict with cowardice, of noble determination face to face with contemptible cunning—you cannot remain neutral. You will come and take the side of the oppressed because you know that the beautiful, the sublime, the spirit of life itself are on the side of those who fight for light, for humanity, for justice!1
These positive views regarding the importance of art carry forward into the early twentieth century, when American anarchist Emma Goldman asserted: "Any mode of creative work which with true perception portrays social wrongs earnestly and boldly is a greater menace ... and a more powerful inspiration than the wildest harangue of the soapbox orator."2 And we find this attitude echoed by anarchist theorists and activists up to the present day.
Why, then, has the anarchist movement attributed such importance to art? To answer this question, we need to examine the role of the individual in anarchist theory. In 1900, Goldman closed her essay, "Anarchism: What It Really Stands For," with the following reflections:
Anarchism, then, really stands for the liberation of the human mind from the domination of religion; the liberation of the human body from the domination of property; liberation from the shackles and restraint of government. Anarchism stands for a social order based on the free grouping of individuals for the purpose of producing real social wealth, an order that will guarantee to every human being free access to the earth and full enjoyment of the necessities of life, according to individual desires, tastes, and inclinations.3
Goldman's statement points to how anarchism widens the field of political action far beyond the economic and class-based focus of Marxism and the socialist currents influenced by it.4 She critiques religion for oppressing us psychologically, capitalist economics for en-dangering our corporal well-being, and forms of government that shut down our freedoms. She also asserts that the purpose of anarchism is to liberate humanity from these tyrannies. And most importantly for our purposes, she predicts that in an anarchist social order, individuals will differentiate endlessly, according to their "desires, tastes and inclinations."
Goldman counted Kropotkin amongst her most important influences, so it is appropriate we turn to him for further insight. For Kropotkin, anarchism is synonymous with "variety, conflict."5 In an anarchist society, "anti-social" behavior would inevitably arise, as it does at present; the difference being this behavior, if judged as reprehensible, would be dealt with according to anarchist principles.6 More positively, the refusal to "model individuals according to an abstract idea" or "mutilate them by religion, law or government" allowed for a specifically anarchist type of ethics to flourish.7 This entailed the unceasing interrogation of existing social norms in recognition that morals are social constructs, and that there are no absolutes guiding ethical behavior. Kropotkin characterized anarchist ethics as "a superabundance of life, which demands to be exercised, to give itself ... the consciousness of power."8 He continued: "Be strong. Overflow with emotional and intellectual energy, and you will spread your intelligence, your love, your energy of action broadcast among others!"9 In sum, the anarchist subject's power, situated socially, is not reactive; it is generative. Kropotkin wants power to "overflow"; it has to if a free social order is to be realized. We find the same perspective articulated in the early 1870s by Michael Bakunin—who most famously declared "the destructive urge is also a creative urge"—in his reflections on freedom and equality:
I am free only when all human beings surrounding me—men and women alike—are equally free. The freedom of others, far from limiting or negating my liberty, is on the contrary its necessary condition and confirmation. I become free in the true sense only by virtue of the liberty of others, so much so that the greater the number of free people surrounding me the deeper and greater and more extensive their liberty, the deeper and larger becomes my liberty.10
Anarchist social theory develops out of this perspective. Bakunin goes on to theorize the necessity of socializing property in the name of individual liberty. Rejecting both state-adjudicated socialism and capitalism, he declares, "we are convinced that freedom without socialism is privilege and injustice, and that socialism without freedom is slavery and brutality."11 Kropotkin similarly argued for the necessity of socializing property, while Proudhon supported the institution of private ownership on a small scale on the condition that it never become an instrument of domination.12
Configuring art within this tradition, it follows that, aesthetically speaking, diversity is inevitable: after all, the artist's creative freedom goes hand in hand with a politics that refuses power over others or hierarchical relations that would dictate what is and is not acceptable. The artist is also radically reflexive, because anarchists create art in tandem with the transformation of society anarchically, and they interrogate it with this aspiration in mind, giving rise to creative activity that enriches the field of art production and the libertarian social project.
This, then, is the terrain we will be exploring. Adopting an episodic approach, I discuss European and American art from the era of the Paris Commune through World Wars I and II to the fall of the Berlin wall. Each chapter examines the engagement of anarchist artists with a range of issues, including aesthetics, war and violence, sexual liberation, ecological crisis, militarism, state authoritarianism, and feminism. Throughout, the interface of art production and anarchism as a catalyst for social liberation has been my main preoccupation. In the spirit that gave rise to the art under examination, I have tried to ensure my reflections are accessible to the general reader as well as specialists.
### NOTES TO THE INTRODUCTION
1 Peter Kropotkin, "Appeal to the Young," in _Kropotkin's Revolutionary Pamphlets,_ Roger N. Baldwin, ed. (New York: Dover Press, 1970): 273.
2 Emma Goldman, _The Social Significance of Modern Drama_ (New York: Applause Theater Book Publishers, 1987): 1-2.
3 Emma Goldman, "Anarchism: What it Really Stands For," in _Anarchism and Other Essays,_ with a new introduction by Richard Drinnon (New York: Dover Press, 1969): 62.
4 This aspect of anarchism has been noted by a number of contemporary theorists. See, for example, Richard Day, _Gramsci is Dead: Anarchist Currents in the Newest Social Movements_ (London: Pluto Press, 2005): 15-16 and Todd May, _The Political Philosophy of Poststructuralist Anarchism_ (University Park, PA: Pennsylvania State University Press, 1994): 50.
5 Peter Kropotkin, "Anarchism: Its Philosophy and Ideal" (1896), in _Kropotkin's Revolutionary Pamphlets,_ 143.
6 Peter Kropotkin, "Anarchist Morality" (1891), in _Kropotkin's Revolutionary Pamphlets,_ 106.
7 Ibid., 113.
8 Ibid., 108.
9 Ibid., 109.
10 Michael Bakunin, _The Political Philosophy of Bakunin,_ G.P. Maximoff, ed. (New York: The Free Press, 1953): 267.
11 Ibid., 269.
12 On Kropotkin and Proudhon, see Allan Antliff, _Anarchist Modernism: Art, Politics and the First American Avant-Garde_ (Chicago: University of Chicago Press, 2001): 3-5.
## CHAPTER 1
A BEAUTIFUL DREAM _Courbet's Realism and the Paris Commune of 1871_
The 1871 Paris Commune, which pitted the armed populace of Paris against the armies of an arch-conservative government ensconced in Versailles, is a much studied episode in the history of European radicalism. That anarchist debates concerning art played a role in the event is rarely acknowledged. But art was an issue. In the preceding years, government attacks against Gustave Courbet—painter, anarchist, and future communard—had inspired philosopher and economist Pierre-Joseph Proudhon to pen his last major statement, _Du principe de Part et de sa destination social_ (1865). In _Du principe,_ Proudhon praised Courbet for extending the dialectical interplay between anarchist social criticism and society's transformation into the artistic realm.1 The same year the book was published, Proudhon's position was countered by the young journalist (and future novelist) Emile Zola, who argued that anarchism and art was an aesthetic issue, not a social one.2 How this debate found its resolution is the subject of this chapter.
The story begins in the 1840s, when Paris became a haven for a small collection of political refugees known as the "radical Hegelians." This group of activists were then transforming the German academic Wilhelm Hegel's philosophy of history into a radical theory of social change which challenged the sanctity of the church, the system of monarchical rule, and capitalist property relations. Principal among them were the Russian Michael Bakunin, who arrived in France to avoid forcible extradition to Russia, and the Germans Karl Marx and Karl Grün, who had been forced out of Germany for their journalistic activities. In Paris, they all sought out and befriended Proudhon, then a French working-class economist in his mid-thirties who had only recently settled in the capital himself, in 1838. Proudhon had come to Paris under the auspices of a grant from the Academie de Besançon to study languages and political economy; however, academic sponsorship came to an abrupt halt after he published his stinging critique of capitalism and the state entitled _What is Property? An Inquiry into the Principle of Right and of Government_ in 1840. Answering his leading query in no uncertain terms, Proudhon declared that "property is theft" and denounced "the government of man by man" in favor of a society based on "equality, law, independence, and proportionality"—principles which he argued found their highest perfection in the social union of "order and anarchy."3 In one simple and compelling statement the anarchist movement was born, delivered in a message that rang as a clarion call throughout leftist Europe.4
_Pierre-Joseph Proudhon. Lithograph after a photograph by Felix Nadar_.
Proudhon and his new friends met in the humble apartments, ale houses, and coffeehouses of working-class Paris, where they engaged in passionate discussions that turned on two issues: the critique of idealism mounted by the radical Hegelian philosopher Ludwig Feuerbach, and the related concept of dialectics, which was central to the Hegelian theory of historical change.5 Briefly, Hegel had posited that world history was driven by an unfolding process of alienation in which a divine "world spirit" manifested itself in partial and incomplete forms of self-knowledge, objectified in human consciousness as Reason and Freedom. This spirit was gradually emerging to complete self-consciousness and self-definition through a dialectical process in which incomplete forms of self-consciousness manifest in human history were formulated, negated, and then reconciled in successively higher and more inclusive syntheses—syntheses that in turn were destined to themselves be negated and subsumed. History progressed along this dialectical path until the world spirit achieved total self-knowledge, at which time its own objectification and self-alienation would cease. In this scheme of dialectical progress, humanity played a key but sublimated role in the world spirit's development. "Man," wrote Hegel, "is an object of existence in himself only in virtue of the divine in him—that which is at the outset reason and which, in view of its activity and power of self-determination, [is] called freedom."6 Society created institutions such as the state, and practices such as art, religion, and philosophy, in order to objectify these principles of spirit in the world, thus preparing the way for the reconciliation of the world spirit with itself through historical progress.7
Hegel argued that the dialectical manifestation of the world spirit's self-consciousness could only be recognized in retrospect, and that the future forms of reason and freedom could not be predicted. In other words, this was a philosophy of the status quo in which the current social state of affairs was justified as the latest manifestation of the world spirit's unfolding self-consciousness. Thus, in Hegel's treatment, the dialectic of human history was driven by a force external to it—the world spirit—which paradoxically ensured that history's development was out of humanity's hands.8
The radical Hegelians questioned this notion by utilizing the principles of reason and freedom to critically distinguish "the actual and rational features of the universe from the illusionary, irrational ones."9 In Germany, for example, they rejected the prevailing monarchist political order and argued for the adoption of the bourgeois-democratic and republican principles of the French Revolution.10 The radical Hegelians also introduced human agency into the dialectical process, equating their social critique with the dialectic of negation. In Lesek Kolakowski's words, they believed "the dialectic of negation ... must address itself to the future, being not merely a clue to understanding the world but an instrument of active criticism; it must project itself into unfulfilled historical possibilities, and be transformed from thought into action."11
Feuerbach's critique completed the radicals' revision of Hegel's grand scheme. Feuerbach argued that the divine world spirit was a fiction, and that the real dialectic driving history hitherto had been a process of human estrangement from our essence in which ideals born of human experience were continuously objectified in the form of metaphysical concepts attributed to otherworldly deities, such as goodness, justice, and love.12 Humanity's self-negation through objectification could only be overcome by recognizing that such metaphysical principles were an illusion, since no ideals existed apart from humanity. "The species," wrote Feuerbach, "is the last measure of truth... what is true is what is in agreement with the essence of the species, what is false is what disagrees with it."13 Freedom, therefore, resided in our ability to realize these humanized ideals in the world. Once humanity recognized that the principles that constituted its essence were inseparable from its sensuous, historical experience, humanity could unite existence with essence, and life with truth.14 Feuerbach characterized his philosophy as "anthropological" to signal that, finally, our metaphysical ideals had been brought down to earth and subsumed into humanity's sensuous, historical essence.
Proudhon was introduced to Feuerbach's critique by Karl Griin in the fall of 1844.15 In _The Socialist Movement in France_ (1845), Griin described his meetings with Proudhon and the French anarchist's eagerness to discuss German philosophy. Proudhon had already gained a cursory grasp of Hegel and was greatly relieved when Griin told him how Feuerbach's criticism dissolved the Hegelian "bombast."16 Griin outlined Feuerbach's revision of Hegel for Proudhon and ended the conversation declaring his "anthropology" was "metaphysics in action," to which Proudhon excitedly replied, "I am going to show that _political economy_ is metaphysics in action."17
Feuerbach provided Proudhon with the philosophical basis for sweeping the metaphysical ethics of religion and philosophy aside in favor of moral principles logically "synthesized" from experience. In his 1858 publication, _Justice in the Revolution and in the Church,_ Proudhon wrote, "metaphysics of the ideal taught by Fiche, Schelling, and Hegel is nothing: when these men, whose philosophy has rightly gained distinction, fancied they were deducing an _a priori_ they were only, unknown to themselves, synthesizing experience."18 Proudhon described his method of arriving at moral judgments as human-centered and anti-metaphysical:
As far as I am concerned morality exists for itself; it is not derived from any dogma, or from any theory. With man consciousness/conscience is the dominant faculty, the sovereign power, and the others are useful to it as instruments or servants ... it is not at all from any metaphysics, from any poetry or from any theodicy that I deduce the rules of my life or my sociability; on the contrary, it is from the dictates of my consciousness/conscience that I should rather deduce the laws of my understanding.19
Feuerbach's dialectical and anthropological neo-Hegelianism, which underpinned Proudhon's concept of critical synthesis, led the French anarchist to justify revolutions as the supreme attempt to realize moral goals through social change. In his 1851 publication, _The General Idea of the Revolution in the Nineteenth Century,_ Proudhon called revolution "an act of sovereign justice, in the order of moral facts, springing out of the necessity of things, and in consequence carrying with it its own justification."20 "Springing out of the necessity of things," moral imperatives changed as society changed: in this critical method, "justice" took on a radically contingent, historical, and social character.
Apart from Feuerbach, Proudhon was also indebted to the theory of dialectics espoused by the German philosopher Immanuel Kant.21 In _The Critique of Pure Reason_ (1781), Kant claimed he had proved the inability of human reason to know the world as it is, meaning the world conceived apart from the perspective of the knower.22 Reason, he argued, could not transcend the boundaries of the sensible, and the dialectical nature of human reason was proof of this fact. Kant held that from any premise we could derive both a proposition and its negation. This dialectical opposition exposed the falsehood of the premise which gave birth to it, leading him to conclude that we could never attain the transcendental knowledge necessary for knowing the world in its totality. Kant called these dialectical constructs "antinomies."23
In Proudhon's anti-metaphysical reformulation of the Kantian dialectic, the social critic, guided by the imperatives of reason, deduced moral syntheses from dialectical contradictions found in society. The means by which a synthesis was transformed from a moral-based deduction of social contradictions to a resolution of those contradictions was through social transformation. Proudhon argued social contradictions, and the moral solutions the social critic deduced from these contradictions, were historically contingent and ever-changing.24 In Proudhon's system, the free exercise of human reason in every social sphere came to the fore as the progressive force in history, a position which led him to argue that freedom from all coercion was the necessary prerequisite for realizing a just society. In James Rubin's words, "Proudhon held that anarchy (that is an-archy, the absence of authority) was the only possible condition for social progress."25 Proudhon's anarchist philosophy of art followed from this critique of metaphysical idealism. He codified his position in _Du principe de I'art,_ which was published the year of his death in 1865.
In the opening chapter, Proudhon informed his readers that the book was inspired by the French government's refusal to exhibit Gustave Courbet's painting, _Return from the Conference_ , at the official state art exhibition of 1863.26 Courbet was an old friend of Proudhon and a long-standing participant in the radical culture of Paris. His artistic notoriety stemmed from the years 1848-1851, when the French monarchy was overthrown and a Republican government was briefly instituted. In 1851, Courbet created a scandal at the state's annual art exhibition, where he exhibited two immense paintings depicting banal scenes from the life of the French peasantry, painted in a style akin to popular woodblock prints.27 The upper-class public were accustomed to works such as Jean-Léon Gérôme's _Greek Interior_ of 1850, which offered slickly painted "classical" titillations far removed from the social realities of the day. Courbet's _Stonebreakers_ (see color plate 1) and _Burial at Ornans_ (both painted in 1849-50 and exhibited in 1851), therefore, came as a shock. Displayed in the state salon where elite culture was traditionally celebrated, these paintings shattered the artistic boundaries between rich and poor, cultured and uncultured, and as a result they were roundly condemned for their rude subject matter, rough and "unfinished" brushwork, shallow perspectives, and overall lack of painterly decorum.28
_Gustave Courbet,_ Return from the Conference, _1862-1863, destroyed 1945. Oil on canvas. From Rait,_ Gustave Courbet, peintre, 1906.
But artistic crudity was not the sole reason for the heated objections to Courbet's work. During the short-lived Republic, the workers of Paris and Lyon engaged in violent agitation for the state to adopt Proudhon's call for "national workshops" that would guarantee them employment, and the impoverished French peasantry were in a perpetual state of unrest against landlords in the countryside. Beset by growing working-class radicalism, the Parisian upper classes saw Courbet's paintings as an affront to establishment values in art and a political provocation against their power. Eventually they solved the problem of social unrest by throwing their lot in with the dictatorship of Louis Napoleon III, nephew of Napoleon Bonaparte, who pro-claimed himself emperor after a coup d'état in 1851.29
A Barricade at Neuilly, _Paris, 1871. Lithograph. From Ernest Alfred Vizetelly_ , My Adventures in the Paris Commune, _1914_.
The Republic may have been defeated; however, throughout Napoleon Ill's reign, from 1851 to 1870, Courbet continued to paint in the same uncompromising manner. He called his new style "realism," and paid tribute to himself and his accomplishment in a huge retrospective painting in 1855 entitled _The Painter's Studio: A Real Allegory._ In the center, Courbet depicted himself painting a landscape, observed by an admiring nude model. The model is "real," but also an allegorical figure of the painter's muse (nature). Behind the artist are the patrons, comrades, writers, and philosophers who inspired him—notably Charles Baudelaire on the far left (reading) and Proudhon, who surveys the scene from the back of the room. Facing the painter are the products of the corrupt and degenerate society he critiqued, including destitute workers, a businessman, and Napoleon III himself, with his hunting dog and gun.30
Courbet's _Return from the Conference,_ which depicted drunken clerics on their way home from a religious gathering, was another realist tour de force, in this instance directed against the degenerate institution of the church. Refused a showing in the 1863 state exhibition and maligned by establishment critics, the painting provoked a tremendous storm of indignation, leading Courbet, who regarded the work as the artistic equivalent to Proudhon's own critical "synthesis" of society's wrongs, to ask the anarchist philosopher to defend it.31
In _Du principe de I'art,_ which is based in part on a lively correspondence between painter and philosopher, Proudhon recounted Courbet's rebuke of the establishment critics who vilified _Return from the Conference._ The artist had condemned them "for misrepresenting ... the high mission of art, for moral depravity, and for prostituting [art] with their _idealism."_ "Who is wrong," Proudhon asked, "the so-called _realist_ Courbet, or his detractors, the champions of the ideal?"32 His purpose was to critically resolve this question.
First, he turned his attention to the issue of idealism. Proudhon, following Feuerbach, viewed metaphysical knowledge as an impossibility, and this informed his critique of artistic idealism, in which he attacked the idea that metaphysical ideas could spring, fully-formed, from the imagination of the artist. Art, Proudhon argued, was made up of specific forms, subjects, and images. The idealized subject in art, therefore, was inseparable from the real objects it represented.33 Thus, there was no metaphysical "separation of the real and the ideal" as Courbet's "idealist" critics maintained.34
Proudhon then took up the question of realism. By the early 1860s, other artists were also painting in a realist style; however, they tended to temper the aesthetic crudeness associated with Courbet and chose subject matter from everyday life that, though "real," would not of-fend. Proudhon criticized the artists of this "realist" camp, accusing them of maintaining that art should slavishly imitate reality. This, he argued, was a falsification of what art was.35 A photograph, for example, could capture an image, but it could not replicate the power of the artist to magnify the qualities of character residing in a subject or imbue an inanimate object with meaning. Any "realist" aesthetic that imitated the photograph was "the death of art," Proudhon concluded.36
_Emile Zola, n.d. Photograph._
In his earlier writings, Proudhon posited an anti-metaphysical, moral critique as the basis for social advancement. In _Du prindpe de l'art,_ he argued that art could, potentially, become a vehicle for such a critique. Art was a product of idealism, albeit idealism in a Proudhonian sense, because the creative imagination of the artist, like art's subject matter, was inseparable from the real world. Courbet not only recognized this fact; _his_ realism turned art to critical ends in the interest of social advancement, bringing realism into line with Proudhon's prognosis for social reconstruction through critiques deduced from the material conditions of contemporary society. As such, Courbet's painting stood in stark contrast to both "photographic realism" and the "metaphysical" art of Gérôme and his ilk, wherein ir-rational and self-indulgent pursuit of otherworldly "chimeras" such as "beauty" elevated artistic contemplation to an ideal in-and-of-itself, rendering the critical power of human abstraction and reason "useless."37 "Our idealism," wrote Proudhon, "consists of improving humanity ... not according to types deduced a priori ... but according to the givens supplied continuously from experience."38
Recognition of art's relationship to society, therefore, was the prerequisite for the free exercise of the artist's critical reason. In Feuerbachian terms, the artist gained freedom from the condition of self-alienation engendered by a metaphysical world-view by taking up the cause of improving society through art. "Art," wrote Proudhon, is "an idealist representation of nature and of ourselves, whose goal is the physical and moral perfection of our species."39 It would "progress as reason and humanity progress,"40 revealing, "at last,"
... man, the citizen and scientist, the producer, in his true dignity, which has too long been ignored; from now on art will work for the physical and moral improvement of the species, and it will do this, not by means of obscure hieroglyphics, erotic figures, or useless images of spirituality, but by means of vivid, intelligent representations of ourselves. The task of art, I say, is to warn us, to praise us, to teach us, to make us blush by confronting us with a mirror of our own conscience. Infinite in its data, infinite in its development, such an art will be safe from all spontaneous corruption. Such an art cannot possibly degenerate or perish.41
Proudhon's public defense of Courbet in a lengthy book set the stage for the ambitious Parisian journalist Emile Zola to make his entrance. Zola was then establishing his reputation as a fiercely independent champion of radical politics and artistic independence.42 And his critical rejoinder to Proudhon's thesis—a book review entitled "Proudhon and Courbet"—was nothing if not audacious. Zola opened his review declaring that he too supported "the free manifestation of individual thoughts—what Proudhon calls anarchy."43 But here, the similarities ended.44 Proudhon, Zola argued, was trapped by his method, which proceeded from a desire for the reign of equality and liberty in society to a logical deduction of the type of art that would bring about such a society.45 The rigors of "logic" determined that Proudhon could only imagine one kind of artist: an artist who contributed to the anarchist struggle through the exercise of critical reason in the service of the social good.46 The result was an impoverished definition of art. The author of _Du principle de l'art_ defined art as "an idealization of nature and ourselves, whose goal is the physical and moral perfection of our species."47 But this was an oppressive tautology which could broach no unruly deviation on the part of the artist from art's stated goal. Proudhon, concluded Zola:
Burning the Guillotine on the Place Voltaire, _Paris_ , _1871. Lithograph. From Ernest Alfred Vizetelly,_ My Adventures in the Paris Commune, _1914_.
Poses this as his general thesis. I public, I humanity, I have the right to guide the artist and to require of him what pleases me; he is not to be himself, he must be me, he must think only as I do and work only for me. The artist himself is nothing, he is everything through humanity and for humanity. In a word, individual feeling, the free expression of a personality, are forbidden.48
Here, Zola's support for "the free expression of the personality" came head-to-head with the Feuerbach-derived underpinnings of Proudhon's notion of artistic anarchism. In _Du principe de I'art,_ Proudhon had reasoned, step by step, from a repudiation of photo-graphic realism and metaphysical idealism in art to a reformulation which tied art inextricably to the improvement of society. Individual freedom only entered the realm of art to the degree that the artist mounted a moral critique. Zola quite rightly pointed out that Proudhon's concept of artistic liberty was tied to a historical mission, and thus found its sole libertarian legitimation in relation to it.
For Zola, on the other hand, the locus of freedom was the autonomous individual. In his words, "My art is a negation of society, an affirmation of the individual, independent of all rules and all social obligations."49 Proudhon argued that moral imperatives derived from the study of society should shape art, and that Zola parried by marshaling a radical subjectivism in which the imagination of the artist stood in for the metaphysical realm of the ideal. "I will have Proudhon note," Zola wrote, "that our ideas are absolute.... We achieve perfection in a single bound; in our imagination, we arrive at the ideal state. Consequently it can be understood that we have little care for the world. We are fully in heaven and we are not coming down."50
"A work of art," he continued, "exists only through its originality."51 Content in a work of art was of secondary importance because it always derived from something else—either the external world or traditional subject matter. The true measure of artistic freedom was style, and in this regard the artist's manipulation of formal elements such as color, texture, light, etc. was the only aspect of a painting that was unique—original—in a word, individual.
These are the terms in which Zola appreciated Courbet.52 "My Courbet is an individual," he wrote, while praising the artist's youthful decision to cease imitating "Flemish and Renaissance masters" as the mark of his "rebellious nature."53 Even realism was transformed into an extension of the artist's individualism. Courbet had become a realist because he "felt drawn through his physical being ... toward the material world surrounding him."54 Zola drove the point home in a vivid description of a studio visit:
I was confronted with a tightly constructed manner of painting, broad, extremely polished and honest. The figures were true without being vulgar; the fleshy parts, firm and supple, were powerfully alive; the backgrounds were airy and endowed the figures with astounding vigour. The slightly muted coloration has an almost sweet harmony, while the exactness of tones, the breath of technique, establish the planes and help set off each detail in a surprising way. I see again these energy-filled canvases, unified, solidly constructed, true to life and as beautiful as truth.55
Having established the libertarian primacy of style, Zola caustically ridiculed Proudhon for emphasizing the exact opposite, namely the subject matter. Proudhon saw Courbet "from the point of view of pure thought, outside of all painterly qualities. For him a canvas is a subject; paint it red or green, he could not care less.... He [always] obliges the painting to mean something; about the form, not a word."56 The problem, Zola concluded, was that Proudhon failed to understand that "Courbet exists through himself, and not through the subjects he has chosen." "As for me," he continued, "it is not the tree, the face, the scene I am shown that moves me: it is the man revealed through the work, it is the forceful, unique individual who has discovered how to create, alongside God's world, a personal world."57
Zola defined a work of art as _"a fragment of creation seen through a temperament"_ [Zola's emphasis].58 For him, the "fragment" was secondary to "temperament" and the index of temperament was style. Equating the exercise of temperament with freedom, Zola turned stylistic originality into an anarchist act. Here, the politics of art imploded into the art object as the artist strove to assert personal freedom through stylistic innovation. The contrast with Proudhon's artist, who could not approach a condition of freedom except through social critique, seemed unequivocal. It took the Paris Commune (March 18 to May 28, 1871) to resolve the debate.
The Commune was an outgrowth of the defeat of Emperor Napoleon III following an ill-advised declaration of war with Prussia in July 1870. Prussia formed an alliance with other German principalities and invaded France. Defeat followed defeat and on September 2, 1870, Napoleon III surrendered with most of his army just out-side Paris. Insurrectionary demonstrations in the capital led to his capitulation and the proclamation of a republican Government of National Defense on September 4. This government signed an un-popular armistice with the Germans on January 28, 1871. In early February, a newly elected and overwhelmingly conservative National Assembly transferred the seat of government from Paris to Versailles. While the conservatives consolidated at Versailles, radicals in Paris rejected the armistice and called for the socialization of the economy under a communal form of government. The Paris Commune was elected by 275,000 Parisians in March at the same time as parallel communes were declared in Lyons, Marseilles, Toulouse, Narbonne, Le Creusot, and St. Etienne. Later that month, National Assembly troops moved against the regional communes and by mid-April, Paris was left isolated.59
Paris in Flames, _1871. Lithograph. From Ernest Alfred Vizetelly_ , My Adventures in the Paris Commune, _1914_.
In Paris, the Communards implemented a host of popular measures such as the expropriation of workshops and their transfer to worker-owned cooperatives, the requisition of empty buildings for public housing, and paying top government officials the equivalent of a skilled worker's wage. Grassroots clubs in the city districts _(arrondissements)_ infused Paris with the spirit of direct democracy. The founding dedication of one such club to "uphold the rights of the people, to accomplish their political education, so that they might govern themselves," communicates the anarchic tenor of city.60 In April, the Commune attempted to rally the rest of the country to its cause in an appeal "to the people of France" calling for:
the total autonomy of the commune extended to every township in France, and the ensuring to each the fullness of his rights and to every Frenchman the free expression of his faculties and aptitudes as man, as citizen and as worker; the commune's autonomy is to be restricted only by the right to an equal autonomy for all the other communes that adhere to the contract that will ensure the unity of France.61
Courbet, who regarded the Paris Commune as a first step towards Proudhon's anarchist program, threw himself into the effort.62 On April 30, at the Commune's height, he wrote,
I am, thanks to the people of Paris, up to my neck in politics: president of the Federation of Artists, member of the Commune, delegate to the Office of the Mayor, delegate to [the Ministry of] Public Education, four of the most important offices in Paris.... Paris is a true paradise! No police, no non-sense, no exaction of any kind, no arguments! Everything in Paris rolls along like clock-work. If only it could stay like this forever. In short, it is a beautiful dream. All government bodies are organized federally and run themselves.63
_Bodies of Communards photographed at a Cemetery,_ Paris, 1871. _Photograph from Ernest Alfred Vizetelly,_ My Adventures in the Paris Commune, _1914_.
The Federation of Artists had been formed on April 13 at Courbet's instigation. Its first act was to issue a manifesto declaring complete freedom of expression, an end to government interference in the arts, and equality amongst the membership.64 Complete freedom of expression: for Courbet, there was no conflict between Zola's advocacy of freedom through style and Proudhon's advocacy of freedom through critique—an anarchist future could accommodate both.
On May 21, the French army finally breached the Commune's defenses and, backed by relentless bombardments, poured into the city centre. Barricades slowed the army down as the Communards fought back street by street and vast swaths of Paris were engulfed in flames. Once the Paris defenses had been breached, the military went on a killing rampage: fifteen- to thirty-thousand Parisians were slaughtered before the fighting finally ended on May 28; 38,000 more were arrested in its aftermath, including Courbet.65 He spent six months in jail and his property was seized. Penniless upon his release and threatened with re-arrest, Courbet fled to Switzerland, where he died in exile on December 31, 1877.
### NOTES TO CHAPTER I
1 Pierre-Joseph Proudhon, _Du principe de I'art et de sa destination social_ (Paris, 1865).
2 Emile Zola, "Proudhon and Courbet," _My Hatreds,_ trans. with an introduction by Paloma Paves-Yashinsky and Jack Yashinsky (Lewiston: The Edwin Mellen Press, 1992), 9-21. The review was originally published in two instalments—on July 26, 1865 and August 31, 1865—in _Le Salut Public._
3 Pierre-Joseph Proudhon, _What is Property? An Inquiry into the Principle of Right and of Government_ (New York: Dover Press, 1970): 277, 280, 286.
4 The state was also paying attention: in 1842, Proudhon was charged with "attacking property, inciting contempt for government, and offending against religion and morals." Thanks to a skillful self-defence, he was acquitted. See Henri de Lebac, _The Un-Marxian Socialist: A Study of Proudhon,_ R.E. Scantlebury, trans. (New York: Sheed & Ward, 1948): 7.
5 Bakunin was in Paris from March 1844 until November 1847 and he and Proudhon engaged in lengthy discussions about Hegel's dialectics, sometimes through the night. See Brian Morris, _Bakunin: the Philosophy of Freedom_ (Montreal: Black Rose Books, 1993): 12-13; Marx lived in Paris from 1844 until his expulsion from France in February, 1845. Marx first met Proudhon in September 1844 and they also discussed Hegel's philosophy; James H. Billington, _Fire in the Minds of Men: Origins of the Revolutionary Faith_ (New York: Basic Books, 1980): 289-290; Karl Griin was Proudhon's third important source of information on Hegel and Feuerbach. Griin got to know Proudhon shortly after arriving in Paris in September 1844; Billington, 291.
6 Hegel, quoted in Lesek Kolakowski, _Main Currents in Marxism: The Founders_ (Oxford: Oxford University Press, 1978): 72.
7 Ibid, 73.
8 Ibid, 82.
9 Ibid.
10 Ibid, 83.
11 Ibid, 85.
12 David McLellan, _The Young Hegelians and Karl Marx_ (New York: Frederick A. Praeger, 1969): 89.
13 Feuerbach quoted in ibid, 92.
14 Ludwig Feuerbach, "Provisional Theses for the Reformation of Philosophy," (1843), _The Young Hegelians: An Anthology,_ Lawrence S. Stepelevich, ed. (Cambridge: Cambridge University Press, 1983): 164.
15 George Woodcock, _Pierre-Joseph Proudhon: A Biography_ (Montreal: Black Rose Books, 1987): 87-88.
16 Grün quoted in de Lebac, 134, note 33. According to Steven Vincent, Griin also provided Proudhon with translations of his article on Feuerbach, "Louis Feuerbach and the Socialists," (1845); Louis Feuerbach's preface to the second edition of his _Essence of Christianity_ (1841); and a commentary on Feuerbach's philosophy by his brother Frederic Feuerbach, entitled _The Religion of the Future_ (1843-45). See Stephen Vincent, _Pierre-Joseph Proudhon and the Rise of French Republican Socialism_ (New York: Oxford University Press, 1984): 94-95.
17 Grül quoted in Woodcock, 88.
18 Pierre-Joseph Proudhon, _De la justice dans la revolution et dans l'eglise_ (Paris, 1858): 198.
19 Ibid, 492-93.
20 Pierre-Joseph Proudhon, _General Idea of the Revolution in the Nineteenth Century_ (London: Pluto Press, 1989): 40.
21 Proudhon read the French translation of the _Critique of Pure Reason_ in the winter of 1840 and again in the winter of 1841. See Vincent, 62, 72.
22 Roger Scruton, _Kant_ (Oxford: Oxford University Press, 1982): 46.
23 Ibid., 48-49.
24 Bakunin also rejected the higher, subsuming synthesis of the Hegelian dialectical triad. See Robert M. Cutler, "Introduction," _The Basic Bakunin: Writings 1869-1871,_ Robert M. Cutler, trans. and ed. (New York: Prometheus Books, 1992). Cutler's is the best interpretation of Bakunin's dialectics I have encountered. Proudhon's reconciliation of Kant's critical dialectic with Feuerbach's radical Hegelian dialectic is a point of contention. Some state that the Hegelian model of the dialectic was rejected by Proudhon in favour of Kant's. Others read Proudhon's Feuerbach-influenced dialectic as Hegelian. Either way, Proudhon ends up being branded as a "confused" theorist and this has hampered discussion of his aesthetics. See, for example, T. J. Clark, _The Absolute Bourgeois: Artists and Politics in France_ — _1848-1851_ (Princeton: Princeton University Press, 1982): 166-167.
25 James Henry Rubin, _Realism and Social Vision in Courbet and Proudhon_ (Princeton: Princeton University Press, 1980): 34.
26 Proudhon, _Du principle de l'art,_ 1.
27 Courbet first gained recognition in 1848 when he won a state prize for the genre painting _After Dinner at Ornans_ (1847).
28 See T. J. Clark, _Image of the People: Gustave Courbet and the 1848 Revolution_ (London: Thames and Hudson, 1999).
29 Ibid.
30 Rubin, 38-47.
31 Ibid., 164.
32 Proudhon, _Du principe de I'art,_ 3.
33 Ibid., 31.
34 Ibid.
35 Ibid., 38.
36 Ibid., 39, 40-42.
37 Ibid., 199. Proudhon also condemned idealist "art-for-art's-sake" trends amongst anti-academic artists. See André Reszler, _L'esthétique anarchiste_ (Paris: Presses Universitaires de France, 1973): 20.
38 Proudhon, _Du principe de I'art,_ 199.
39 Ibid., 43.
40 Ibid., 84.
41 Ibid.
42 Nina Athanassoglou-Kallmyer, "An Artistic and Political Manifesto for Cezanne," _The Art Bulletin_ (September, 1990): 484.
43 Ibid., 485. Zola's association of artistic individualism with anarchism continued into the 1870s and '80s. See Ann Lecomte-Himy, "L'Artist de Tempérment chez Zola et Devant la Public: Essai D'Analyse Lexicologique et Sémiologique," _Émile Zola and the Arts,_ Jean-Max Guieu and Alison Hilton, eds. (Washington: Georgetown University Press, 1988): 85-98.
44 "I am diametrically opposed to Proudhon: he wants art to be the nation's product, I require that it be the product of an individual"; Zola, "Proudhon and Courbet," _My Hatreds,_ 14.
45. Ibid., 9.
45 Ibid.
46 Ibid., 11.
47 Ibid.
48 Ibid.
49 Ibid., 20.
50 Ibid., 21.
51 Ibid., 12.
52 Ibid., 17.
53 Ibid.
54 Ibid.
55 Ibid., 18.
56 Ibid., 19.
57 Ibid.
58 Ibid., 12. Arguably, the Proudhon-Zola debate marks an important juncture in the history of nineteenth-century aesthetic theory. The formulation of an anti-metaphysical subjectivist aesthetic is commonly reserved for Friedrich Nietzsche. See Andrew Bowie, _Aesthetics and Subjectivity: from Kant to Nietzsche_ (Manchester: Manchester University Press, 1990).
59 _The Paris Commune of 1871,_ Eugene Schulkind, ed. (London: Jonathan Cape, 1972): 19-23.
60 "Organe des Clubes," _Bulletin Communal_ no. 1 (May 6, 1871) in Schulkind, 127.
61 "An Official Proclamation of the Commune (April 19, _187 i)" Journal officeil de la Commune,_ April 20, 1871) in Schulkind, 150.
62 In an open letter entitled "To the Artists of Paris" published on April 6, Courbet wrote that the "apostles" of the revolution were the workers and its "Christ" was Proudhon. This letter was accompanied by a call for a meeting the next day to form the Federation of Artists of Paris. Gustave Courbet, "To the Artists of Paris," _Journal officiel de la Commune_ (April 6, 1871) in _Letters of Gustave Courbet,_ P. Ten-Doesschate Chu, ed. (Chicago: University of Chicago Press, 1992): 409.
63 Gustave Courbet to his family, Charenton, April 30, 1871 in _Letters of Gustave Courbet,_ 416.
64 The manifesto is cited in Jean Péidier, _La Commune et les Artistes_ (Paris: Nouvelles Editions Latines, 1980): 64.
65 Ernest Alfred Vizetelly, _My Adventures in the Commune: Paris 1871_ (London: Chatto and Windus, 1914): 353-354.
## CHAPTER 2
WANDERING _Neo-Impressionists and Depictions of the Dispossessed_
_At the bottom of the stairs lay the mariners of the street cur-rent, the tramps who had fallen out of the crowd life, who refused to obey—they had abandoned time, possessions, labor, slavery. They walked and slept in counter-rhythm to the world._
—Anaïs Nin, "Houseboat," 19411
Following the fall of the Paris Commune in 1871, successive Republican governments presided over an explosive expansion of French indus-trial capitalism which quickly eroded older, more rural forms of production and community life. The economic juggernaut was made possible thanks to a new infrastructure of rail lines and roads which spread through the countryside, bringing economic transformation to hitherto relatively untouched areas.2 It came with a price: in villages, towns, and hamlets throughout France, the products of local craftsmen were displaced by cheap goods mass-produced in factories, and small-scale farms geared to the material needs and ecological capaci-ties of the local community were undermined by imported produce from abroad and the reconfiguration of agricultural production on a large-scale, export-oriented basis. This process was augmented by a great economic depression that lasted from 1873 to 1896, a crisis which forced artisans and peasants into debt, and from there to the mines, factories, mills, and urban centers that fed the industrial capitalist monolith.3
_Gustave Marissiaux_ , The Slag Heap, _ca 1904. Photograph_.
Roger Magraw writes that as the old skills and rural communities died, "uprooted, alienated, deskilled workers took refuge in consumerism, or, more often, in drink, crime, and domestic violence."4 But many of the displaced refused to be victims; instead, they entered into a state of revolt against encroaching capitalist servitude, articulated in the form of an anarchist critique of marginalization and the cruel existence of the dispossessed.
Nowhere was this critique more clearly encapsulated than in the art of the neo-impressionists. The term "neo-impressionism" was coined in 1886 by the anarchist art critic Felix Feneon to characterize the stylistic evolution of a group of Paris-based painters whose ranks included Paul Signac, Camille Pissarro, Lucien Pissarro, Georges Seurat, Anna Bloch, Charles Angrand, Maximilien Luce, Albert Dubois-Pillet, and Henri Edmond Cross; shortly thereafter, the group expanded to include Théo Van Rysselberghe and a circle of artists based in Belgium. The difference between impressionism and neo-impressionism, Signac would later explain, was the neo-impressionists' "scientific" application of color, as opposed to "instinctual"; a second difference was that, politically speaking, almost all of the neoimpressionists were avowed anarchists whose paintings and graphic contributions to journals such as _Le Père Pinard, L'en dehors, La Plume, L'Assiette Au Beurre,_ and _Les Temps Nouveaux_ played a key agitational role in the movement.5
Take, for example, _Les Errants (The Wanderers)_ (1897) (see color plate 2), a lithograph produced by Théo Van Rysselberghe for an album of prints issued by _Les Temps Nouveaux_ (see color plate 2). Van Rysselberghe's title came from a poem of the same name by the anarchist playwright Emile Verhaeren. In the corner of the print is a passage from Verhaeren's poem which reads: "Thus the poor people cart their misery for great distances over the plains of the earth ..." In the late 1880s and early 1890s, the workers of Belgium had repeatedly risen up in a series of mass strikes, riots, and violent clashes with the police and army. The first such incident erupted in the industrial city of Liege, where a commemoration of the Paris Commune led to full-scale rioting that spread throughout the country's industrial mining region.6 We can better grasp the desperation of the region's workers through photographs of their living hell—the prosperous (from a bourgeois perspective) towns where workers were reduced to combing slag heaps for bits of coal "after hours." Men, women, and children worked ten- to thirteen-hour days, six days a week, in the mines and mills of Belgium; they were paid at or below subsistence level, and if there was no work, they starved.7
Van Rysselberghe's _Wanderers_ are refugees displaced by poverty, the police, and the army. In the 1890s, thousands of such families were forced to tramp the roads of Belgium by grinding unemployment, lock-outs, or brutal acts of government suppression; "They cart their misery for great distances," Verhaeren wrote. Enraged at the injustice, Van Rysselberghe depicted these outcasts in their most abject moment of defeat, condemned to wandering without end in a world ruled by an economic system that "capitalizes everything, assimilates everything, and makes it its own."8
But to where might they have wandered? Perhaps to the city, to join the multitudes of unemployed and underemployed. Henri Lebasque's lithograph, _Provocation_ (1900), distributed by _Les Temps Nouveaux,_ bears testimony to the kind of marginalization awaiting them in the great marketplaces of capital. A stark critique of starvation in the face of capitalism's bountiful plenitude, the provocation is the commodification of bread, humanity's most basic sustenance. A child stands weak and listless, staring at loaves displayed in a brightly-lit shop window; business prospers while the child is hungry. Similar testimony to the inhumane nature of capitalism is captured in a drawing for _Les Temps Nouveaux's_ July 1907 issue by George Bradberry, depicting an emaciated tramp who pauses to stare at fat cows chewing their cud. "The starving man," reads the caption, "envies the satiated beasts!" And so the rural outcast stands mute by the field—valueless, penniless, and "worthless."
Whilst some anarchist artists focused on the dispossessed's plight, others chose to portray the oppression of work under capitalism. In 1889, Camille Pissarro created a small booklet entitled _Social Turpitudes,_ which depicted the drudgery of emergent forms of urban wage labor. Among them is an image of seamstresses subject to the watchful eye of a supervisor. They hunch over piecework in a debtor's prison where they have been condemned by their poverty to endless, repetitious tasks such as this. Pissarro also depicted the brutalization of day-laborers; an illustration for the May 1893 issue of _La Plume,_ for example, shows the back-breaking drudgery of stevedores who spent their lives—when they could obtain work—shoveling and hauling coal.
_Image Not Available_
_Henri Lebasque,_ Provocation, _1900_. © _Estate of Henri Lebasque / SODRAC (2006) Lithograph from_ Album Les Temps Nouveau. _Private collection_.
_Image Not Available_
_Maximilien Luce_ , The Factory Chimneys: Couillet Near Charleroi, _1898-1899. Oil on canvas. © Estate of Maximilien Luce / SODRAC (2006)_
Thus far I have discussed the neo-impressionists' damning criticism of industrial capitalist labor and the injustice of working-class destitution. However, this was not the sum total of their viewpoint; they also pointed to different possibilities lying dormant in Europe's besieged pre-capitalist ways of life. Here, critique was wed to utopia, and the condition of wandering took on new meaning.
The latter theme emerges in a painting by Maximilien Luce en-titled _The Factory Chimneys: Couillet Near Charleroi_ (1898-1899). Luce was an uncompromising working-class militant who was briefly imprisoned for his anarchist activities in 1894. Toward the end of the 1890s, he traveled through northern France and Belgium, recording his impressions of the mining towns and factories.9 An exhibition of his paintings held in 1891 led one anarchist art critic to note that he found in Luce's work "the bleeding soul of the people, the life of the multitudes anguished and inflamed by suffering and bitterness."10
_Factory Chimneys_ is dominated by the grim industrial capitalist inferno of Couillet, where treeless streets of rooming houses disgorged workers daily into the mills. But in the corner of the painting, a man and boy walk away from the entrapment of this inferno. Their destination is unnamed; their pur-pose, undetermined. They might be setting out on a journey, or perhaps they seek momentary respite from the grey, polluted environment they leave behind. In any event, they are passing from one world to another—the rhythm of capital gives way to the rhythm of nature.
_Peter Kropokin, 1899. Photograph._
Luce and the neo-impressionists were fully aware of the violence that emergent capitalism did to nature's rhythms, and the crippling contortions its industries imposed on humanity. They read the critiques of Elisée Reclus and Peter Kropotkin, both of whom condemned the disequilibrium of industrial capitalism as a violation of harmonious social relations and, ultimately, of humanity's relationship to the earth.11 Writing in 1864, Reclus observed:
The barbarian pillages the earth; he exploits it violently and fails to restore its riches, in the end rendering it uninhabitable. The truly civilized man understands that his interest is bound up with the interest of everyone and with that of nature.12
Nineteenth-century anarchists sought to end this barbarism in the name of a social order in which property would be held in common and social and ecological devastation would come to an end. Harmony entailed a freedom that respected and nurtured differences while sustaining the good of the whole. Just as mutual aid undergirded the diverse interrelatedness of plants, insects, and animals, so humanity could realize a greater diversity through cooperation.13 However, for many, this farsighted and demanding vision seemed to run against the grain of history.14
Where, then, could anarchism find a sure footing in society? In the first instance, among other anarchists. Reclus wrote of anarchists' obligation "to free ourselves personally from all preconceived or im-posed ideas, and gradually group around ourselves comrades who live and act in the same fashion." Such "small and intelligent societies," he argued, could form the basis of a greater harmonious social order.15
However, communities of anarchists were not the sole social force working against the industrial capitalist leviathan. Reclus and others looked to the surviving patterns of communal existence among the peasantry, where the traces of a different social rhythm still prevailed. Camille Pissarro's great neo-impressionist paintings, such as _Apple Picking at Eragmy-sur-Epte_ (1888) (see color plate 3), capture the cadence of this life, where work was relatively untouched by the regula-tory regime of capitalized production. These workers take their time; they pause to chat amongst themselves, and their activity is voluntary and cooperative. Here, humanity transforms the world through cultivation rather than destruction.
Thus, everyday life approaches a condition of harmony akin to anarchism—or so the anarchist writer and critic Octave Mirbeau thought. For him, Pissarro's canvases depict a world animated by "the ideal," where the cities of capital, "booming as they may be, are no more perceptible, having no more planetary importance, behind the fold of terrain that hides them, than the lark's nest in the bottom of a furrow."16 Without a doubt, these paintings verge on utopian. We know that Pissarro and other anarchist artists also depicted the brutalization of landless peasant laborers on the large capitalized farms of rural France. However, the neo-impressionists were equally enthralled by the lifecycle they encountered in Europe's small hamlets and land-holdings, where self-sufficiency and pre-capitalist ways still persisted.
In fact, the technique of the neo-impressionists was suffused with anarchist politics. Their application of unique and discrete colors on the canvas—the small dots of paint that give their paintings their soft glow and shimmering radiance—accorded to scientific principles of vision, so as to produce an overall harmonious effect. This painterly technique was their analogue for the harmony in freedom that could unite humanity and, in turn, reconcile us with nature. In her masterful study of the movement, Robyn Roslak writes that the visual syn-thesis of the neo-impressionist canvas represented:
... the progressive process through which harmony and variety in unity (terms which defined the ideal anarchist social structure) were achieved. These, of course, were the very terms which the neo-impressionists and their critics used to describe neo-impressionist painting. There, individual spots of paint, akin to the human individuals in anarcho-communist social theory, are amassed to form unified, harmonious, synthetic compositions, which appear as such because of the way in which the discrete colors are scientifically applied to compliment one another while preserving their own, unique character.17
Thus, the neo-impressionists fused politics with reality, giving their ideals a material presence in the form of social critiques on canvas that pointed toward an anarchic future.
Of course, this future could not be achieved without revolution. And the anarchists knew that among the masses of displaced and dispossessed workers, the memory of revolts and the hope of revolution remained intact. In fact, many anarchist militants came from the ranks of these working-class itinerants, who played a key role in the movement as they traveled from place to place spreading revolutionary ideas through pamphlets, songs, and conversations.18 In 1896, HenriEdmond Cross paid homage to one such anarchist in an illustration, _The Wanderer,_ issued by _Les Temps Nouveaux._ Copies of this print may well have circulated the length and breadth of France and beyond.
In it, the "Wanderer" sits alone, caught up by a visionary revelry, which is depicted behind him. The revolution has been won, and workers are throwing the insignia of capitalist oppression—flags and other symbols of authority—into a raging bonfire. These workers, and the wanderer himself, are surrounded by a beautiful neo-impressionist landscape: harmony in freedom has banished tyranny.
_Henri-Edmond Cross,_ The Wanderer, _1896. Lithograph from_ Les Temps Nouveaux. _Private Collection._
Anarchists such as those in Cross's _The Wanderer_ were outcasts, but they also were free. Their freedom resided in a day-to-day life apart from capital, as well as the revolutionary vision they propagated to those encountered along the way. Like Nin's tramps, they too abandoned time, possessions, labor, and slavery in a refusal to obey. And, like them, they existed in counter-rhythm to a society in which their ideals were deemed valueless. But they also struggled for a better world.
### NOTES TO CHAPTER 2
1 Anaïs Nin, _Under a Glass Bell_ (Chicago: Swallow Press, 1946): 111.
2 Roger Magraw, _A History of the French Working Class: Workers and the Bourgeois Republic,_ 1871-1939 (Oxford: Blackwell Publishers, 1992): 5-7.
3 Magraw, ibid., 5.
4 Ibid., 11.
5 On the neo-impressionists' politics, see Robyn Roslak, _Neo-Impressionism and Anarchism in Fin-de-siècle France: Painting, Politics and Landscape_ (London: Ashgate, 2007) and John Hutton, _Neo-Impressionism and the Search for Solid Ground: Art, Science and Anarchism in Fin-de Siècle France_ (Baton Rouge, LA: Louisiana State Press, 1994). Signac outlines the difference between impressionism and neoimpressionism in Paul Signac, _D'Eugène Delacroix au néoimpressionisme_ (Paris: 1899): 100, 102. _Le Père_ Pinard was founded in 1889 by Emile Pouget as a weekly targeting Parisian workers. In the early 1890s its readership approached 100,000, according to police estimates. _L'en debors_ began in 1891 and was edited by the anarchist-individualist Zo d'Ax. _La Plume_ , edited by Léon Deschamps, was launched in 1891 and combined essays on art and poetry with anarchist theory. _L'Assiette Au Beurre_ (founded 1901) was an illustrated publication edited by Samuel Schwartz and André de Joncières that featured contributions from anarchist artists of various orientations. _Les Temps Nouveaux_ , edited by Jean Grave, was the flagship journal of French anarchist communism from its founding in 1895. On _Le Père Pinard, L'en debors_ , and La Plume, see Richard D. Sonn, _Anarchism and Cultural Politics in Fin de Siècle France_ (Lincoln, NE: University of Nebraska Press, 1989): 17-36. On _L'Assiette au Beurre and_ Les _Temps Nouveaux_ , see Patricia Leighten, "Réveil anarchiste: Salon Painting, Political Satire, Modernist Art," _Modernism/Modernity_ 2 no. 2 (1995): 26-27.
6 Stephen H. Goddard, _Les XX and the Belgium Avant-Garde_ (Lawrence, KS: Spencer Museum of Art, 1992): 24.
7 Ibid., 56, 69-70, notes 6,7.
8 Jacques Camatte, _The World We Must Leave_ (New York: Autonomedia, 1995): 39.
9 Sonn, 145.
10 Georges Darien, "Maximilien Luce," _La Plume_ LVII (1891): 300.
11 John Clark and Camille Martin outline the ecological foundations of anarchist communism in _Anarchy, Geography, Modernity: The Radical Social Thought of Elisée Reclus_ , john Clark and Camille Martin, eds. and trans. (Lanham, MA: Lexington Books, 2004): 3-113.
12 Elisée Reclus, "Du Sentiment de la nature dans les societes modernes," _La Revue des deux mondes_ 1 (December, 1864) quoted in Marie Fleming, _The Geography of Freedom_ (Montreal: Black Rose Books, 1988): 114.
13 Peter Kropotkin outlines this thesis in Peter Kropotkin, _Mutual Aid_ (Montreal: Black Rose Books, 1988).
14 As the nineteenth century drew to a close, Marxism, which argued the spread of industrial capitalism was the necessary precursor to socialism, drew millions into its fold. On Marxism and industrial capitalism, see Jean Baudrillard, _The Mirror of Production_ (St. Louis: Telos Press, 1975).
15 Reclus to Clara Koettlitz, April 12, 1895 quoted in Fleming, 175.
16 Octave Mirbeau, "Camille Pissarro," _L'Art dans les deux mondes_ 8 (January 10, 1891), quoted in Martha Ward _Pissarro, Neo-impressionism, and the Spaces of the Avant-Garde_ (Chicago: University of Chicago Press, 1996): 181.
17 Robyn Sue Roslak, Scientific Aesthetics and the Aestheticized Earth: the Parallel Vision of the Neo-Impressionist Landscape and Anarcho-Communist Social Theory (PhD diss., University of California at Los Angeles, 1987): 204.
18 For a period discussion of itinerant anarchists and their role in the movement, see Felix Dubois, _The Anarchist Peril_ (London: T. Fisher Unwin, 1894).
## CHAPTER 3
OBSCENITY _The Advent of Dada in New York_
Many are familiar with the early twentieth-century Dada movement, when anti-war artists from a range of countries attacked the social and cultural order that had given rise to World War I. In the current literature, it is commonplace to date Dada's New York beginnings to French artist Francis Picabia's arrival in June of 1915 and his five "object portraits" published in the July-August issue of the avant-garde art journal _291._1 These depictions are rightly singled out because they embody so many of the definitive features of Dadaist production in New York, in that their evocation of industrialism and commercialism violated the conventions defining art while simultaneously setting off a chain of associative readings that transgressed the subject at hand.
Where I part with the prevailing view, however, is in regard to Picabia's attitude towards the United States and, by extension, the role ascribed to him in the development of American modernism. Art historian Wanda Corn and others have argued that the object portraits were a celebratory incorporation of American popular culture into high art, a broadening, if you will, of the modernist landscape to include the American point of view.2 However, this reading downplays the complexity of Picabia's portraits as well as the dissident politics that inspired them. By way of reply, this chapter focuses on one of these works, _Portrait of a Young American Girl in a State of Nudity_ (1915), as a case study in how the advent of Dada in New York was bound up with an anarchist critique of contemporary American culture and the distinctive type of modernity it embodied.
Picabia was a Paris-based painter who had first visited the United States in the company of his wife, Gabrielle Buffet-Picabia, in 1913 to attend the opening of a large-scale exhibition of European and American modernism known as the Armory Show. The exhibition began in New York (February 17-March 15) and then traveled to Chicago (March 24-April 16) before closing in Boston (April 28-May 19). The Picabias arrived in New York on January 20 and stayed on through February and March before departing home for France on April 10. In New York, the couple met with the photographer Alfred Steiglitz, who ran a small non-commercial gallery (known as "291," after its Fifth Avenue address) and journal— _Camera Work_ —to show-case the latest experiments in European and American modernism. During their stay, the Picabias grew close to Steiglitz and many others in his circle, notably the Mexican caricaturist Marius de Zayas, the anarchist journalist Hutchins Hapgood, and Paul Haviland, a wealthy art collector and photographer.3
Prior to his New York visit, Picabia had been exhibiting for just under two years with a group of Parisian cubists (the "salon cubists") led by painter and theorist Albert Gleizes, who co-authored the seminal statement, _Cubism_ (1912), with fellow painter Jean Metzinger. Gleizes was the group's organizational dynamo who arranged exhibitions, promoted the movement in the press, and discouraged any aesthetic deviations.4 The group's successes in Paris ensured that when the American organizers of the Armory Show went to France in a quest for modern art, they would return with a substantial list of cubist paintings and sculpture. At the Armory Show, Picabia exhibited four paintings, including _Dances at the Spring_ (1912) (see color plate 4), alongside work by Gleizes, Marcel Duchamp, Raymond DuchampVillon, Roger de La Fresnaye, Fernand Leger, and Jacques Villon.5
Picabia's paintings were textbook examples of the cubist aesthetic circa 1912, which followed the metaphysical tenets of the French philosopher Henri Bergson. Briefly, Bergson argued that the conventional scientific view of the world—which filtered perception through clock time, Newtonian physics, and Euclidian geometry—was a false-hood. Developing a metaphysics to counter it, he posited that the true state of matter could only be grasped through a suspension of the intellect so as to open us to in-tuition. According to Bergson, artists were more capable than others of entering into a sympathetic, intuitive relationship with the world, a claim that was also trumpeted by Gleizes and the cubists, who based their style on these principles.6
Image Not Available
_Francis Picabia_ , Portrait d'une Jeune Fille Américaine dans 1'état de Nudité. 291, _nos 5-6, July-August 1915. © Estate of Francis Picabia / SODRAC (2006)._
_Francis Picabia, ca 1913. Photograph_.
In _Time and Free Will_ (1889), _Creative Evolution_ (1907), and other works, Bergson argued that matter was actually energy in a condition of flux and interpenetration and that each moment in time was qualitatively different from the last, like the condition of matter itself. This was the reality that cubism depicts. In Picabia's _Dances at the Spring,_ for example, the dancers' bodies appear to break up and merge with their surroundings because the painter is trying to represent the dynamism of matter in space and time as filtered through his artistic intuition.7
So things stood in 1912. However, upon arriving in New York, Picabia made a dramatic break with the cubist movement. As recorded by Hutchins Hapgood in an interview for the _Globe_ newspaper, Picabia argued that the artist's role was not to "mirror the external world" but rather "to make real, by plastic means, internal mental states."8 Picabia explained that he could no longer follow cubism because the cubists were "slaves to the strange desire to reproduce" the external world, just like the old masters whose works hung in the dusty halls of the Louvre.9
While his cubist paintings were on exhibit in the Armory Show, he began working in a new "post-cubist" style.10 This was an "unfettered, spontaneous, ever-varying means of expression in form and color waves," painted "according to the commands, the needs, the inspi-ration of the impression, the mood received."11 The results—sixteen watercolors, including _New York Perceived Through the Body_ (1913)—were exhibited at a one-man show (with catalogue) which opened at Steiglitz's 291 gallery on March 17, two days after the New York leg of the Armory Show closed.12 In a special catalogue statement summing up his new aesthetic, Picabia claimed to be unleashing "the mysterious feelings of his ego." An article on "The Latest Evolution in Art and Picabia" published in Stieglitz's in-house journal, _Camera Work,_ went further. Here, his style was characterized as "the real Anarchy, needed and foreseen."13
What prompted Picabia to reject cubism in favor of abstraction? The impetus can be traced to a second ex-cubist, Marcel Duchamp. In the summer of 1912, Duchamp left Paris for Munich, where he studied Max Stirner's anarchist-individualist manifesto, _The Ego and Its Own,_ a materialist critique of metaphysics and an assertion of libertarian individualism.14 Stirner argued that the metaphysical thinking underpinning religion and notions of truth laid the foundation for the hierarchical division of society into those with knowledge and those without. From here, an entire train of economic, social, and political inequalities ensued, all of which were antithetical to anarchism.15 Combatting metaphysics, Stirner countered that ideas are indelibly grounded in our corporal being. The egoist, therefore, recognized no metaphysical realms or absolute truths separate from experience. Indeed, Stirner deemed the very notion of an "I" to be a form of meta-physical alienation from the self. Libertarian "egoism," Stirner wrote, "is not that the ego is all, but the ego _destroys_ all. Only the self-dissolving ego . . . the _finite_ ego, is really I. [The philosopher] Fichte speaks of the 'absolute' ego, but I speak of me, the transitory ego."16 Once conscious of its freedom, this self-determining, value-creating ego inevitably came to a "self-consciousness _against_ the state" and its oppresive laws and regulations.17 As Stirner put it, "there exists not even one truth, not right, not freedom, humanity, etc., that has stability before me, and to which I subject myself."18 He concluded:
I am the _owner_ of my might, and I am so when I know myself as _unique._ In the _unique one_ the owner himself returns into his creative nothing, out of which he is born. Every higher essence above me, be it God, be it human, weakens the feeling of my uniqueness, and pales before the sun of this consciousness. If I concern myself for myself, the unique one, then my concern rests on its transitory, mortal creator, who consumes himself, and I may say: I have set my affair on nothing.19
Years later, Duchamp related that reading Stirner in Munich brought about his "complete liberation."20 He and Picabia were very close, and upon returning to Paris that fall, they likely discussed Stirner's ideas at length.21 In any event, scarcely three months later, Picabia was introducing New Yorkers to "the mysterious feelings of his ego" in free-flowing expressions "cut loose" from cubist "convention" and its "established body of laws and accepted values."22
If Stirnerist egoism pushed Picabia to adopt a new expressive style, it also, evidently, reinforced his predilection for challenging the statist and religious mores of his day: Picabia was an archhedonist who engaged in numerous extra-marital affairs and excessive drug taking. "He went to smoke opium almost every night," Duchamp later re-called, "[and] I knew that he drank enormously too."23 This hedonism would take a decidedly political turn after Picabia returned from New York.
For example, in 1913, he lent his name to anarchist-led protests in Paris against the censorship of a newly unveiled monument at the famous Père Lachaise cemetery honoring the era's most notorious homosexual, Oscar Wilde (1854-1900). Objecting to the monument's prominent genitalia and the very idea that a disgraced homosexual merited any memorial, officials had covered the statue with a tarpaulin and fixed a metal plate over the offending organ. In response, a group of Parisian-based anarchist-individualist artists who called themselves the Artistocrats mounted a campaign in defence of the monument. Writing in their journal _Action d'art,_ they celebrated Wilde's sexuality as a healthy expression of egoist anarchism and condemned the censors as sex-negative perverts whose prudery went against the laws of nature. They also published a full-page anti-censorship statement with Picabia's name listed among its signatories.24
Picabia was shortly to marshal his own protest against the same censorious forces besieging the Wilde monument. His first full-scale exercise in artistic "egoism" upon returning from New York was an attack on the repression of sexual impulses under the moral regime of Catholicism. The imposing canvas, the enigmatically titled _Edtaonisl (Ecclesiastic)_ (see color plate 5), was exhibited at the Autumn Salon of November 1913. The painting's subject was a Dominican priest whom Picabia had witnessed during the sea voyage to New York furtively watching the rehearsals of a Parisian exotic dancer.25
Asked to explain his painting, Picabia related that he had fused impressions of the "rhythm of the dancer, the beating heart of the clergyman, the bridge [of the ocean-liner] ... and the immensity of the ocean."26 Here, Picabia echoed a central tenet of Stirner's philosophy: that the idealist notion of a "soul" separate from the body fostered self-alienation and the suppression of our natural desires and pleasures.27 Rooted in sensation, his painterly critique of alienation played havoc with artistic conventions, metaphysical idealism, and the priest's vow of chastity. Thus, he brought the moral cornerstone of Catholicism into disrepute while at the same time leaving critics dumbstruck be-fore one of modernism's earliest examples of full-blown abstraction.28
When World War I began in August 1914, Picabia was conscripted, but initially avoided the trenches by arranging enlistment as a chauffeur for a French cavalry general behind the lines, first at Bordeaux and then Paris. Increasingly endangered by the war's progression (he was deemed fit for the infantry), he next secured an assignment as an army purchasing agent and was sent overseas in the summer of 1915 to procure supplies in the Caribbean.29 Promptly abandoning his mission upon reaching New York in June 1915, he reconnected with de Zayas, who had just launched the satiric avant-garde monthly _291_ that March. While still in Paris, Picabia had contributed _Girl Born Without a Mother_ (ca 1915), a loosely sketched depiction of rods and springs erupting in ill-defined activity, to _291'_ June 1915 issue. This was followed in the July-August edition by the five meticulously executed "object portraits," including the drawing of a spark plug, _Portrait of a Young American Girl in a State of Nudity_ (1915), mentioned at the beginning of this chapter.30
Read sequentially, the portraits are witty and sometimes caustic commentaries on the personalities associated with de Zayas' journal. The portrait gracing the cover of _291,_ for example, is a broken camera with lens extended, whose bellows has become detached from the armature and is collapsing. Attached to the side of the camera is an automobile brake set in park, and a gearshift resting in neutral. The lens strains toward the word "Ideal" printed above it in Gothic script; beside the apparatus is stenciled, _Here, This is Steiglitz / Faith and Love_ (1915). Steiglitz, whom Picabia had befriended during his first New York excursion in 1913, had a long history of opposing modern art's commercialization, which he feared would compromise the artist's creative integrity. For over a decade he had run his gallery as a non-commercial venue where New Yorkers could gain exposure to modern photography, sculpture, and painting and, if Steiglitz deemed them sincere in their admiration, purchase a work at prices that varied widely according to the means of the admirer and other considerations.31 Picabia had exhibited at this gallery and was intimately familiar with its workings, as was de Zayas. Indeed, de Zayas had named his journal after Steiglitz's gallery to signal his allegiance to the latter's ideals; however, by the summer of 1915, he was rethinking his position.
De Zayas saw a need for a more conventional approach, believing modernists of quality could maintain their independence regardless of commercial pressures if their art was effectively promoted by sympathetic professionals who respected their freedom and paid them well for sales. When Picabia arrived in New York, he joined the debate on the side of de Zayas and as a result, a rift developed. By July, de Zayas had decided to usurp the preeminence of Steiglitz's project and embark on a new venture, to be located in midtown Manhattan and christened the "Modern Gallery" (the gallery eventually opened in October 1915).32 Exasperated by Steiglitz's continuing hostility, Picabia and de Zayas evidently called him to account. The July-August cover of _291_ suggested Steiglitz's efforts to popularize modernism on his terms were as exhausted as a broken camera. It was also a potshot at propriety: the sagging bellows resemble a slackened and impotent penis, incapable of achieving an erection.33
Inside the journal were four more drawings. The first, a self-portrait, was a car horn entitled _The Saint of Saints_ — _This is a Portrait About Me_ (1915). The horn is positioned against what appears to be an automobile cylinder and spark plug depicted in cross-section; Picabia loved fast cars and here he is, blowing his own horn as the avant-garde artist who is more "advanced" than anyone else. The horn was followed by the spark plug rendering _Portrait of a Young Girl in a State of Nudity,_ which referred to artist Agnes Ernst Meyer's role as the "spark" that had started the journal by agreeing to bankroll it behind the scenes. The next portrait, _De Zayas, De Zayas!_ (1915), plays on the editor's vision in founding a journal devoted to satire. It consists of electrical wiring linking an improbable ensemble of objects—a corset, automobile headlights, an electrical post, and a gyrating mechanical device—all of which "work" to create illumination. The final portrait, _Voila Haviland the Poet as He Sees Himself (1915),_ depicts his friend Paul Haviland as an electric lamp with no plug; earlier, in June of that year, the Franco-American Haviland had been forced to "unplug" himself from participating in _291_ following a summons from his father to at-tend to the family business in France. These insular references would have eluded most of _291'_ readers; they have only come to light thanks to painstaking art historical research.34
In an interview for the _New York Tribune_ in October 1915, Picabia described his new portrait style as revelatory, relating that upon disembarking in New York, he had been struck by America's "vast mechanical development." "I have enlisted the machinery of the modern world, and introduced it into my studio," he provocatively argued, be-cause "the machine" is "more than a mere adjunct to life. It is really a part of human life—perhaps its very soul."35
Ascribing such significance to machines underlines the multifaceted complexity of Picabia's new style, in which he furthered his rejection of cubism and his hostility toward censorious social institutions in a critique with America as its cipher. This was a remarkable exercise in artistic iconoclasm, but to grasp its ramifications, we need to examine _Young American Girl_ more closely from a cubist perspective.
As we have seen, the cubists trumpeted their style as the product of an intuitive, anti-intellectual, and qualitative experience of reality. They even went so far as to equate a cubist artwork, born of qualitative experience, to a living organism.36 The antithesis of qualitative perception was the utilitarian state of mind, which quantified, ordered, and standardized nature. According to Bergson, this type of thinking was unartistic, but could nonetheless provide occasion for a distinctive kind of artistry, namely the comic; this is the theme of his book _Laughter_ (1900), which analyzed humor's relation to our lesser utilitarian minds.
Bergson's thesis focused on moments of disjuncture when manifestations of living, organic, qualitative being get mixed up with the in-organic, quantitative, and lifeless. He characterized the type of being that is the antithesis of a living entity as "readymade" and "mechanical." "The attitudes, gestures, and movements of the human body," he wrote, become laughable in "exact proportion as the body reminds us of a mere machine."37 Contrasts of automatism with natural movement, "the rigid, ready-made, and mechanical" with "the supple, ever-changing, and living," were "the defects that laughter singles out."38 Here, we find one of the satirical themes in _Young American Girl_ and the other object portraits; Picabia drew on Bergson's thesis concerning humor to mount what was, in effect, a parody of cubism.
A cubist portrait was understood to be an artistic exercise of profound sympathy, in which the artist captured the sitter's unique personality through a process of intuition attuned to the life force of the subject, right down to her material dynamism. To quote Gleizes and Metzinger in _Cubism,_ by circumnavigating the intellect, the artist created a "sensitive passage between two subjective spaces" in which "the personality of the sitter" was "reflected back upon the understanding of the spectator."39 As such, the painting was a unique expression of a unique moment in the creative evolution of both the artist and the subject.
Picabia's _Young American Girl_ is the antithesis of this. It is, in cubist terms, completely art-less, lacking in emotion, empathy, or originality: or rather, it is a parodic inversion of Bergsonian cubist values, an exercise in mimicry that apes the painterly idealism it critiques in its guise as a humorous joke. To cite Bergson:
Whether we find reciprocal interference of series, inversion, or repetition, we see that the objective in comedy is always the same—to obtain what we have called a mechanization of life. You take a set of actions and relations and repeat it as it is, or turn it upside down, or transfer it bodily to another set with which it partially coincides—all these being processes that consist in looking upon life as a repeating mechanism, with reversible action and interchangeable parts.40
Cubism is made fun of, but this is not the only target. What, for instance, was Picabia saying about the United States when he represented Americans, including a "young girl," as machines? I would argue he was passing judgment, and that his assessment is less than flattering. Taking his comments to the _New York Tribune_ reporter as our starting point, Picabia suggests that Americans are distinguished as a nation by an advanced state of industrialism, which dominates them to such a degree that machine qualities have invaded their very souls, so to speak. Thus, Picabia inverts cubism's metaphysical reading of what it is to be human in order to clear the ground for addressing the culture of the United States critically. And through this elliptic process of mirroring he comments not only on its industrial prowess, but also on the mass marketing that drives it. Picabia's young American girl is a diagrammatically drawn spark plug—the sort of thing one could find in any auto-parts catalogue, newspaper, or magazine advertisement.41 However, if this is advertising, what of its content? Stripped of art's aura, the patina of beauty encapsulated by the girl's "state of nudity" implied something else: a marketing ploy that pointed to the portrait's encoded satiric function as sexual provocation.
This sexual stamp had Parisian roots. In all likelihood it was inspired in part by _Supermale_ (1902), a satirical (and certainly by American standards, obscene) novel by the French satirist Alfred Jarry.42 Jarry's book tells the story of an American scientist who creates a "perpetual motion food" which allows for, among other things, non-stop sex. In a challenge to her father, the scientist's young daughter—"a little slip of a girl"—demonstrates that she can achieve the same results through sheer force of will. The object of her affections is a machine-like "super-male" who is abnormally lacking in emotion. After a lengthy performance with the young girl, he dies while hooked up to another of her father's inventions, a love-inspiring machine.43
The Jarryesque subtext of Picabia's _Young American Girl,_ then, is its tongue-in-cheek presentation of feminine sexual allure Americanized, industrialized, and commercialized. Think of this portrait as a satire of American advertising in which Picabia shamelessly parades a young girl for sale in a "state of nudity" with a standard manufacturer's guarantee—FOR-EVER—of flawless performance in perpetuity. Certainly, this latter feature is what caught the attention of the _291_ circle. In an accompanying article on the object portraits, de Zayas wrote that modern art could only succeed in the United States if it adopted the features of commercialism—and then praised Picabia for inventing such an art.44
More to the point, Picabia had created an artistic means of attacking this commercialism on its own turf. _Young American Girl_ was a slap in the face of the puritanical artistic values underpinning the mass marketing of modern American femininity. In the early twentieth century, popular magazines and advertisements were filled with unsullied but curvaceous full-bodied beauties such as J.C. Leyendecker's vacationing golfers or popular magazine illustrator Howard Chandler Christy's virginal _American Girl_ (1912), from his "Liberty Bells" series. In the minds of both publishers and the general public, such idealizations made the marketing of femininity respectable, even aesthetically and culturally uplifting.45
_Alfred Jarry, 1896. Photograph by Felix Nadar._
Picabia's version of commercialized womanhood mirrored and mocked American marketing by stripping its prototype _Young American Girl_ down to her "essence" as a sexual commodity for sale. Here, the politics of censorship make their entrance, because Picabia's portrait can also be read as a challenge to the repressive anti-obscenity laws which were at the time regulating American artistic production, both high and low.
The spearhead of censorship was Anthony Comstock, Special Agent for the United States Postal Office and chief investigator for the New York Society for the Repression of Vice, an organization em-powered to arrest and charge anyone in possession of literature, photographs, or artwork it judged to be obscene.46 From 1873, when the Federal obscenity law ("The Comstock Act") was passed, Comstock and his agents had full power to search premises and seize materials at will. In the first two years alone, over 194,000 pictures and photo-graphs were confiscated and destroyed under their watch. At the same time, anti-vice organizations and state laws against vice mushroomed around the country.47
Art was not immune from the onslaught. One of Comstock's earliest raids targeted a fashionable New York art gallery for distributing reproductions of "objectionable, lewd, and obscene" work by "the modern French School" (the works in question were nudes by well-known French academics such as William-Adolphe Bouguereau).48 The repressiveness was such that by 1895, no less a figure than Kenyon Cox of the conservative New York Academy of Design was complaining bitterly about it.49 The illustration beauties of American commercial art, therefore, reflected more than native prudishness: they were carefully calibrated to sell a product while remaining firmly within the boundaries of what the censors deemed respectable.
_Howard Chandler Christy,_ The American Girl, Liberty Belles, _1912. Oil on canvas._
When did censorship in the United States become a concern of Picabia's? As we have seen, upon returning to France in the summer of 1913, he had challenged censorious moralizing in his own country by signing the petition in support of the Wilde monument and exhibiting the anti-clerical _Edtaonisl_ that November. Earlier still, however, he was involved in another obscenity controversy. In March 1913, while Picabia was in New York, the Armory Show traveled to the Chicago Art Institute, where conservatives reacted by filling the press with letters and articles condemning the exhibition, in particular calling the room given over to the French cubists "obscene," "lewd," "immoral," and "indecent."50 Chicago's anti-vice Law and Order League called for the exhibition's closure, and civic figures such as clergymen and high school instructors concurred.51 The mayor of Chicago even joined the bandwagon by visiting the exhibition, where he singled out Picabia's _Dances at the Spring_ and made fun of it in the company of reporters.52
Finally, in early April, the Illinois Senate sent a vice investigator to examine the artwork. In a front-page news story for the _Chicago American,_ the investigator declared cubism to be "lewd" and feared for its "immoral effect on other artists."53 Based on the investigator's report, the State Lieutenant Governor ordered the Illinois Senate's anti-vice "White Slave Commission" (white slavery was a popular term for prostitution) to determine whether or not the art was "harmful to public mores" (in the end the exhibit was allowed to continue).54 The level of hostility was so intense that the alarmed New York organizers published a hastily compiled pamphlet entitled _For and Against,_ which reprinted, among other items, a statement by Picabia that had accompanied his "post-cubist" exhibition at Steiglitz's 291 gallery.55
Recall that when this scandal reached its boiling point in early April, Picabia was spending considerable time with the journalist Hutchins Hapgood. Hapgood was a member of the Free Speech League, an organization founded in 1902 with the express purpose of challenging Comstock and the anti-obscenity laws.56 Picabia might well have discussed the Armory Show's reception in Chicago with Hapgood or, for that matter, with any of the artists and critics in the Steiglitz circle. The depths of their pro-vice contempt for censorship are ably summed up in a caricature published in _The Revolutionary Almanac_ (1914) by Hippolyte Havel, a friend of Steiglitz and Hapgood who also knew Picabia.57 Entitled _Saint Anthony, Guardian of Morals_ (ca 1914), the illustration accompanies a story in which Comstock muses on the trials of life in a world of "nudity and shamelessness," where "good and chastity" go unrewarded while those with "neither conscience nor care" feast on "sinful life's joys." Frustrated by this state of affairs, he dreams of overthrowing God—who has evidently acquired a taste for vice—in order to decree that the entire universe be surveyed, from dawn to sunset, by a censoring army of "emissaries, spies, and detectives."58 One can well imagine Picabia chortling at the joke.
In 1915, the Chicago events of two years earlier might have seemed like a distant memory were it not for the fact that in March, just a few months before Picabia's arrival in June, none other than Comstock himself again raised the hackles of New Yorkers by raiding an art exhibition at a popular Greenwich Village restaurant run by Havel and his lover Polly Holiday.59 The art in question was a series of nudes by Clara Tice, a young artist who had studied under the anarchist painter Robert Henri.60 During the raid, outraged patrons blocked Comstock's officers until one of them, Alan Norton, editor of an ir-reverent monthly magazine entitled _Rogue,_ announced that he was purchasing the entire collection on the spot.61 This threw a wrench into Comstock's rights of seizure and brought the raid to an end, though charges were laid.
**THE GUARDIAN OF MORALS**
Saint Anthony, The Guardian of Morals, _ca 1914. From Hippolyte Havel, ed._ , The Revolutionary Almanac, 1914.
The incident got front-page coverage the next day in the _New York Tribune_ and rekindled the New York modernists' fight against Comstock's censorious regime.62 In May and July of 1915, the art promoter and publisher Guido Bruno defied the law by mounting two exhibitions in his Greenwich Village gallery featuring drawings of nudes by Tice.63 In the wake of her subsequent trial (and acquittal) on obscenity charges in September, he staged a mock court event where Tice defended herself before a cross-section of New York's avant-garde.64 By the time Picabia arrived in New York, therefore, the Tice affair would very likely have caught his attention, given his past encounters with the American drive to suppress "vice" during the Armory Show.
Let us return, then, to Picabia's _Young American Girl_ and consider it more closely from this perspective. In her 1997 study _Suspended License,_ Elizabeth Childs has observed that "the history of censorship is not just a matter of institutional solutions to embarrassing or threatening art; it is also a history of individual artistic decisions made in the face of such policies."65 Childs enumerates a range of artistic responses to such repression, from self-censorship to avoid persecution to courting it by breaking the rules anyway. And then there are the more subterranean tactics of "working inside or around a censorious art system." "These tactics," she writes, "include changing the venue for the exhibition of the work; changing the image itself; changing the context for viewing the work; appearing to follow the rules while en-coding prohibited sentiment in art; or perhaps the most effective ploy, turning the tables and attacking the censoring institution through the art itself."66
_Young American Girl_ encodes _and_ attacks. Read as a brazen "advertisement" of a young girl's naked sexuality, the portrait is nothing less than a pornographic outrage worthy of the severest prosecution. Yet it isn't, literally—one has to interpret it as such. Here, Picabia echoed one of the most telling accusations American radicals leveled against Comstock—pornography is in the mind of the beholder. Furthermore, in the course of doing so, he exposed the hypocrisy undergirding American censorship. The nakedness of this portrait was all about "marketing the product"; thus, in one fell swoop, Picabia put the lie to the puritanical grease facilitating the capitalization of sex for profit. Here, Bergson gets the last word. In _Laughter,_ he speculated:
Might not certain vices have the same relation to character that the rigidity of a fixed idea has to intellect? Whether as a moral kink or a crooked twist given to the will, vice has often the appearance of a curvature of the soul. Doubtless there are vices into which the soul plunges deeply with all its pregnant potency, which it rejuvenates and drags along with it into a moving circle of reincarnations. These are tragic vices. But the vice capable of making us comic is, on the contrary, that which is brought from without, like a ready-made frame into which we are to step. It lends us its own rigidity instead of borrowing from us our flexibility. We do not render it more complicated; on the contrary, it simplifies us.67
Whereas America was vice-ridden in the worst sense, Picabia was vice-ridden in the best sense. His subtle and mercurial ego, untouched by vice's "tragic" aspect, deployed the humorous side of the equation by way of parodying the "moral kink" of Comstockery as a mechanical simplification in a "readymade frame": _Portrait of a Young American Girl in a State of Nudity._
If we read Picabia's object portraits as a straightforward embrace of things American, we are missing the point. Certainly, he introduced a new field of expression to art in the United States, but he did so in solidarity with anarchist currents of dissidence decidedly at odds with the establishment values that claimed purchase on America. In other words, the advent of Dada in New York was as much a political event as an artistic one.
### NOTES TO CHAPTER 3
1 Rudolf E. Kuenzli, "Introduction," _New York Dada,_ Rudolf E. Kuenzli, ed. (New York: Willis Locker and Owens, 1986): 2-3.
2 Wanda Corn, _The Great American Thing: Modern Art and National Identity, 1915-1935_ (Berkeley: University of California Press, 1999): 64-66.
3 On Picabia and Gabrielle Buffet-Picabia's visit, see the leading authority on New York Dada, Francis M. Naumann, _New York Dada_ (New York: Harry N. Abrams, 1994): 19-20.
4 The leading role of Gleizes is discussed in Mark Antliff, _Inventing Bergson: Cultural Politics and the Parisian Avant-Garde_ (Princeton: Princeton University Press, 1993): passim.
5 See the catalogue listings of works exhibited in _The Armory Show 50th Anniversary Exhibition_ (New York: Clarke and Way, 1963): 190, 200.
6 On cubist aesthetics as codified by Gleizes and Metzinger, see Antliff, _Inventing Bergson,_ 39—66.
7 Ibid., 46-48.
8 Hutchins Hapgood, "A Paris Painter," _New York Globe,_ February 20, 1913, 8.
9 "Picabia, Art Rebel, Here to Teach New Movement," _New York Times_ (February 16, 1913): sect. 5, 9.
10 The style was destined to stir considerable controversy. See Ineana B. Leavens, _From "291" to Zurich: The Birth of Dada_ (Ann Arbor, MI: UMI Research Press, 1983): 74-75.
11 "A Post-Cubist's Impressions of New York," _New York Tribune_ (March 9, 1913): part 11, 1.
12 The exhibition ran from March 17 to April 5.
13 The preface was reprinted as "Cubism by a Cubist: Francis Picabia in the Preface to the Catalogue of his New York Exhibition," _For and Against: Views on the International Exhibition held in New York and Chicago,_ Frederick James Gregg, ed. (New York: Association of American Painters and Sculptors, Inc., 1913): 45. See also Maurice Aisen, "The Latest Evolution in Art and Picabia," _Camera Work,_ Special Number (June 1913): 21.
14 Stirner's influence is discussed in Francis M. Naumann, "Marcel Duchamp: A Reconciliation of Opposites," _Marcel Duchamp: Artist of the Century,_ Rudolf E. Kuenzli and Francis M. Naumann, eds. (Cambridge, MA: MIT Press, 1989): 29-32.
15 Max Stirner, _The Ego and Its Own_ (London: A.C. Fifield, 1915): 180-190; 473.
16 Ibid., 237.
17 Ibid., 361.
18 Ibid., 463
19 Ibid., 490.
20 Naumann, "A Reconciliation of Opposites," 29.
21 For example, Duchamp and Picabia might have conversed about Stirner during a trip to Jura, France in October of that year, just prior to Picabia's trip to the United States. The Jura trip is discussed in William A. Camfield, _Francis Picabia: His Art, Life and Times_ (Princeton, Princeton University Press, 1979): 35.
22 "A Post- Cubist's Impressions of New York," 1.
23 Pierre Cabanne, _Dialogues with Marcel Duchamp_ (New York: Da Capo, 1987): 32.
24 Mark Antliff, "Cubism, Futurism, Anarchism: The 'Aestheticism' of the Action d'art Group, 1906-1920," _Oxford Art Journal no._ 21 (1998): 109.
25 Virginia Spate, _Orphism: The Evolution of Non-Figurative Painting in Paris, 1910-1914_ (Oxford: Oxford University Press, 1978): 328.
26 Camfield, 61.
27 The mind-body fusion is the basis for Stirner's materialist critique of the "soul"—a self-alienating concept used by Christianity to suppress our sensual inclinations. See Allan Antliff, _Anarchist Modernism: Art, Politics, and the First American AvantGarde_ (Chicago: University of Chicago Press, 2001): 77; Stirner, 451-453.
28 The critical reception is discussed in Camfield, 59.
29 Ibid., 71.
30 Naumann, _New York Dada,_ 59-60. According to Michel Sanouillet, the first three issues of _291_ (March, April, May) were mailed to Picabia in Paris, and he created his drawing at the request of de Zayas for inclusion in the June issue. Michel Sanouillet, "Picabia's First Trip to New York," _Dada New York: New World for Old,_ Martin Ignatius Gaughan, ed. (New York: G. K. Hall, 2003): 118.
31 I discuss the exploitive commercial pressures on American modernists and various attempts to overcome them in Antliff, _Anarchist Modernism,_ 24; 32-33; 54-55.
32 Marius de Zayas, _How, When, and Why Modern Art Came to New York,_ Francis M. Naumann, ed. (Cambridge, MA: MIT Press, 1996): 90-96.
33 Richard Whelan, _Alfred Steiglitz: a Biography_ (New York: Little, Brown, 1995): 348-349.
34 Naumann, _New York Dada,_ 60-61.
35 "French Artists Spur on an American Art," _New York Tribune_ (October 24, 1915): 2.
36 Antliff, _Inventing Bergson,_ 35.
37 Henri Bergson, _Laughter: An Essay on the Meaning of the Comic_ (1900) in _Comedy,_ Wylie Sypher, ed. (Baltimore: Johns Hopkins University Press): 79.
38 Ibid., 145.
39 Gleizes and Metzinger, _Cubism_ quoted in Antliff, _Inventing Bergson,_ 48.
40 Bergson, 126.
41 William Innes Homer has traced Picabia's advertisement sources. See William Innes Homer, "Picabia's 'Jeune fille américaine dans I'etat de nudite' and 'Her Friends'," _Art Bulletin_ 58 (March 1975): 110-115.
42 Linda Henderson, _Duchamp in Context: Science and Technology in the Large Glass and Related Works_ (Princeton: Princeton University Press, 1998): 47-51.
43 Alfred Jarry, _The Supermale_ (New York: New Directions, 1977).
44 Maurice de Zayas, "New York At First Did Not See," _291_ 5-6 (July-August 1915): n.p.
45 In her exhaustive study of such representations, Martha Banta singles out Christy's series as the prototype ideal. Martha Banta, _Imagining American Women:_ _Idea and Ideals in Cultural History_ (New York: Columbia University Press, 1987): 206-211.
46 Nicola Beisel, _Imperiled Innocents: Anthony Comstock and Family Reproduction in Victorian America_ (Princeton: Princeton University Press, 1997): 3.
47 Jane Clapp, _Art Censorship: A Chronology of Proscribed and Prescribed Art_ (Metuchen, NJ: Scarecrow Press, 1972): entry 1874 (b), 151.
48 Ibid., entry 1887, 160-161. Some of the artists are listed in Nicola Beisel, _Imperiled Innocents: Anthony Comstock and Family Reproduction in Victorian America_ (Princeton: Princeton University Press, 1997): 168.
49 Kenyon Cox to Mr. Fraser, April 2, 1896, Century Collection, Manuscripts Division, New York Public Library.
50 See, for example, "May Bar Youngsters From Cubists Show," _Chicago Record Herald_ (March 27, 1913): 22; "Art Institute Censured by Pastor for Display of 'Vulgar' Pictures," _Chicago Record Herald_ (April 8, 1913): 9.
51 Milton W. Brown, _The Story of the Armory Show_ (New York: Abbeville Press, 1988): 208-209.
52 "'I See It'" Says Mayor at Cubist Art Show," _New York Herald_ (March 28, 1913): 9.
53 "Cubist Art Called Lewd: Investigator for Senate Vice Commission Fears Immoral Effect on Artists," _Chicago American_ (April 3, 1913): 1.
54 "Slave's Commission to Probe Cubist Art," _The Inter-Ocean_ (April 2, 1913): 10. Reporters who were invited along recorded that Picabia's _Dancers at the Spring_ was again singled out for derisive comment during the commission's visit; "Art of Cubists Staggers Critics in State Senate," _The Inter-Ocean_ (April 3, 1913): 1.
55 Picabia, "Cubism by a Cubist," 45-48.
56 Hutchins Hapgood, _A Victorian in the Modern World_ (New York: Harcourt, Brace and Co., 1939): 279. The League's founding and activities leading up to World War I are discussed in David M. Rabban, _Free Speech in Its Forgotten Years, 1870-1920_ (Cambridge: Cambridge University Press, 1997): 44-64.
57 Antliff, _Anarchist Modernism,_ 97-99.
58 "The Confiscated Picture," _The Revolutionary Almanac,_ Hippolyte Havel, ed. (New York: Rabelais Press, 1914): 70-71.
59 Marie T. Keller, "Clara Tice, 'Queen of Greenwich Village,'" _Women in Dada: Essays on Sex, Gender, and Identity,_ Naomi Sawelson-Gorse. ed. (Cambridge, MA: MIT Press, 1998): 414. On Havel and Holiday see Antliff, _Anarchist Modernism,_ 80.
60 Keller, 415.
61 Ibid., 414.
62 Ibid., 417-418.
63 Ibid., 414.
64 Ibid., 418.
65 Elizabeth C. Childs, "Introduction," _Suspended License: Censorship and the Visual Arts,_ Elizabeth C. Childs, ed. (Seattle: University of Washington Press, 1997): 15.
66 Ibid., 16.
67 Bergson, 69-70.
## CHAPTER 4
TRUE CREATORS _Russian Artists of the Anarchist Revolution_
"Three artists spent the night in the mansion, since outside the museum a studio was set aside for making art. As the artists told it, that memorial morning, 'We were awakened by shouts of, "We'll shoot! Hands up!"' Armed soldiers ordered them to get dressed, took them out to the courtyard and together with anarchists sent them off to the Kremlin." This is Aleksandr Rodchenko's description of a government raid on the anarchist-held Morozov mansion in Moscow in the early morning of April 12, 1918, published in _Anarkhiia (Anarchy)._ The report survives as an undated fragment in the New York Public Library, where North America's only copy of the short-lived revolutionary newspaper was allowed to disintegrate, neglected and forgotten, until the remains were microfilmed some years ago.1
The obscurity of _Anarkhiia_ mirrors the fate of Rodchenko's anarchism. Consult any history of the Russian avant-garde and you will read that the artistic left pledged allegiance to the "October Revolution," i.e. the Communist Party coup of 1917 and subsequent dictatorship.2 What this narrative buries, however, is a messy history of artistic rebellion and political repression which engulfed Rodchenko and others during the years 1917-1919, when anarchism, not Marxism, was the raison d'être of their art.
The Russian Revolution began in February 1917, after soldiers sent to quell anti-war strikes and rioting in the capital of St. Petersburg (renamed Petrograd during the war and later, Leningrad) joined the protestors instead. It was hardly surprising that revolt was in the air. When World War I broke out, the absolute monarchy of Nicholas II, Tsar of the Russian Empire, was allied with France and Britain. Russia's ill-equipped and under-supplied armies were sent to do battle against the formidable forces of Germany and Austria-Hungary. By February 1917—the beginning of the revolution—millions of Russian lives had been lost, the front was deep inside Russian territory, the economy was collapsing, and the government was in disarray.
Events in St. Petersburg precipitated a crisis during which Nicholas II was persuaded to abdicate in favor of an ad hoc provisional government formed by political leaders from a hitherto powerless parliament—the Duma—that the Tsar had created in the early 1900s. The provisional government was pro-capitalist and determined to continue Russia's failing war effort despite growing rebellion amongst the troops. But during the spring and summer of 1917, its power was increasingly usurped in the cities, towns, and rural regions by local councils ("soviets") made up of elected delegates representing the majority population—workers and peasants. The "workers' and peas-ants' soviets" amalgamated into regional federations, which in turn allied with urban-based factory committees formed by workers who were bent on asserting public control of their factories and workshops. Soldiers began forming soviets as well, and discipline in the army broke down, setting the stage for the events of October 1917. That fall, the Russian Communist Party, led by Lenin, secured majority representation in the soviets of Moscow and St. Petersburg. Rallying other radicals to its side under the slogan "all power to the soviets," the Communist Party spearheaded a successful coup under the auspices of a Soviet Military-Revolutionary Committee. On October 25, soldiers and armed workers stormed the headquarters of the unpopular provisional government centers in St. Petersburg and Moscow. The leaders of the provisional government fled, and the next day the Military-Revolutionary Committee, which was under the control of the Communist Party, announced that it was forming a new governing "Soviet of People's Commissars" made up exclusively of Communist Party members, with Lenin at the helm. This marked the beginning of Communist Party dictatorship in Soviet Russia, an era that would only come to an end with the fall of the Berlin wall. But before this dictatorship could consolidate, the Communists had to fight for four years (1917-1921) against the invading armies of Germany and Austria-Hungary, anarchist insurgency in Russia and the Ukraine, and a host of reactionary generals (known as the "whites") bent on restoring the Tsarist monarchy.
Let us return, then, to the night of April 11-12, 1918. The month before, on March 3, a delegation of Communists acting on behalf of the Soviet government concluded a separate peace with Germany and its allies at Brest-Litovsk by ceding a quarter of Russia's arable land, a quarter of its population, and three-quarters of its industry to the German and Austria-Hungarian empires.3 Prior to the conclusion of negotiations, the Communist Party had split into a Leninist "right" wing, which favored a separate peace, and a more popular "left" wing, which opposed the action. The position of the left echoed the sentiments of the majority of workers' and peasants' soviets, where negotiations with Germany were condemned and resolution after resolution called for a revolutionary war to defeat world capitalism.4
In the early months of 1918, anarchist opposition to the negotiations was adamant and unequivocal. In his book _The Russian Anarchists,_ Paul Avrich cites Aleksandr Ge, a prominent anarchist-communist, who delivered a speech at the Central Executive Committee of soviets on February 23 in which he threatened: "The anarchist-communists proclaim terror and partisan warfare on two fronts. It is better to die for the worldwide social revolution than to live as a result of an agreement with German Imperialism."5 Russian anarchist-syndicalists took the same position, calling for the organization of "relentless partisan warfare" by guerrilla detachments throughout the length and breadth of Russia.6 And they were serious: during February and early March, the local clubs of the Moscow Federation of Anarchists organized detachments of "Black Guards," armed with rifles, pistols, and grenades.7
In Moscow, there were at least twenty-five anarchist clubs where the detachments gathered. These clubs were more than meeting places; they were radical cultural institutions. For example, the "Dom' Anarkhiia" (House of Anarchy), where the federation's official paper _Anarkhiia_ was published, also featured a library and reading room, "proletarian art printing" facilities, a poetry circle, and a large theater hall in which plays were performed and lectures held.8 Many of the structures occupied by the anarchists had formerly housed the Moscow elite; when the anarchists moved in, they turned them into communes and invited workers to share their new quarters.9 The Morozov mansion had been the residence of a textile mill owner who was one of the richest men in Russia; under anarchist occupation, it served as commune, artists' studio, and a people's museum. Valuable porcelain, rare engravings, and other museum artworks were destroyed during the raid of April 12—an act that Rodchenko vigorously protested in his article "O Muzei Morozova."10
The ostensible reason for the April 12 government attacks on the Morozov mansion and other anarchist centers in Moscow was a series of expropriations conducted by the Black Guards in March and early April, but the real motivation was to shut down the movement in Russia.11 Russian anarchist Gregorii Maximov's study of the movement's repression contains a number of articles and documents which lay bare the Communist strategy.12 The government's political police force, known as the Cheka, issued an official release in the wake of the raids declaring that their purpose was to disarm "bands styling them-selves as Anarchists." "The All-Russian Committee Against Counter-Revolution (Cheka)," states the release, "invites all citizens who have suffered from the attacks of robber bands to appear at the militia headquarters for the purpose of identifying the hold-up men detained during the disarming of the Anarchist groups." Thus, the anarchists were criminalized.13 Simultaneously, the Moscow Council of People's Commissars, acting on behalf of the Moscow Soviet, branded them with an additional smear—"counter-revolutionaries." The Council's statement read:
Notwithstanding the most challenging trenchant ideological criticism of the soviets and the Soviet Power on the part of the anarchist papers _(Anarchy, Voice of Labor,_ etc.) the Moscow Soviet refrained from taking any repressive measures against the anarchists.... At the same time the Moscow Soviet had definite information to the effect that entire counter-revolutionary groups are joining the anarchist armed detachments, having for their aim the utilization of the latter for some kind of covert action against Soviet Power. And already the anarchist press and speakers called upon their followers to start upon this course of action directed against the Soviet.... The Council of People's Commissioners, the Soviet and Moscow Province and the Presidium of the city soviet of Moscow found themselves facing the necessity of liquidating the criminal adventure, of disarming the anarchist groups.14
"Liquidation" has an appropriate ring in light of subsequent events. During the Cheka raids, forty anarchists were killed or wounded, and over 500 were taken prisoner.15 In prison, they were stripped of their clothing and lined up for examination by "the well-to-do of the city"—invited, as we have seen, by the Cheka to identify "thugs and bandits."16 That morning, _Anarkhiia_ failed to appear, and the next day, the anarchist-syndicalist paper _Golos Truda (Voice of Labor)_ was shut down. By the end of the week, writes Maximov, "not a single anarchist publication was left in the city."17 Shortly afterwards, the Communists moved against anarchists in every region under their control. Maximov documents the progress of repression in late April and early May of 1918 as anarchists were rounded up, disarmed, and jailed, their publications suspended, and their clubs and communes destroyed.18 "The blow," concludes Maximov, "was well-aimed and well-timed" to cripple the movement when it was "still in the stage of becoming—of self-determination."19
In late April, the Moscow Federation of Anarchists regrouped and relaunched _Anarkhiia_ for a brief period—one of its early issues commemorated the raids with a poem ("That Day," reprinted at the end of this chapter) and a rough-hewn woodcut of a defiant anarchist raising the black standard—but in the new reality, anarchist organizations operated under threat of repression, with increasingly grave consequences.20 This persecution succeeded in breaking up the anarchist ranks.21 Some went underground to launch an anti-Communist bombing campaign that brought waves of arrests in 1919.22 Others joined the Communist Red Army and fought against "Whites"; a number even served in the government as loyal "Soviet-Anarchists," only to be jailed in the early 1920s.23 For a time, a Ukrainian anarchist insurrectionary army led by Nestor Makhno escaped the repression and provided refuge for those fleeing the Communist clamp-down, but when the civil war ended, it too was crushed.24
Who were the artists of the anarchist movement during these turbulent years? To Rodchenko's name, we can add a host of other avant-garde artists and theoreticians: Alexei Gan, who organized the House of Anarchy's "proletarian theater" group and championed the stage paintings of the sixteen-year-old anarchist baker, A. Lukashnin; Kazimir Malevich, leader of the suprematist school of painters; the non-objective painter Nadezhda Udalt'sova and Olga Rozanova (a suprematist); the poets Vladimir Mayaskovskii and Vasilii Kamenskii who, along with futurist painter David Burliuk, founded the anarchist "House of Free Art" club in Moscow; and Vladimir Tatlin, a trail-blazing sculptor.25
The journal in which these artists debated the events of their time and art's relation to the revolution was _Anarkhiia._ Why they chose _Anarkhiia,_ I would argue, is because the individualist, working-class orientation of both the journal and the Moscow Federation, which it represented, echoed the sentiments of the artists themselves. Take, for example, the "Open Letter to the Workers" and "Decree No. 1" which Burliuk, Mayaskovskii, and Kamenskii issued in their newspaper, _Gazeta Futuristov,_ which published its first—and only—issue on March 15, 1918.26 The Moscow Federation of Anarchists counted this newspaper among its publications and welcomed the group's House of Free Art—founded in late March in a requisitioned restaurant—as its newest club (predictably, it was forced to close after the Cheka raids).27 The "Open Letter" published in the _Gazeta_ proclaimed that futurism was the artistic wing of "socialism/anarchism," and that a "revolution of the psyche" would overthrow the calcified artistic practises of bourgeois culture. "Decree No. 1 on the Democratization of Art" condemned art's confinement in upper-class "palaces, galleries, salons, libraries and theaters" and announced that spontaneous artistic expression—art in the streets—was the way forward.28
The libertarian initiatives of these artists were welcomed in the Moscow Federation; its secretary, Lev Chernyi, supported an "associational" anarchism based on the philosophy of Max Stirner.29 Chernyi's position—that only the free association of people in federated groups could provide the foundation for an anarchist society—was shared by _Anarkhiia's_ editor, German Askarov.30 In the previous chapter, I out-lined the salient features of Max Stirner's anarchism, notably its materialistic rejection of metaphysics. I would add that among the classes of his day Stirner singled out the workers—the "unstable, restless, changeable" individuals who owed nothing to the state or capitalism—as the one segment of society capable of solidarity with those "intellectual vagabonds" who approached the condition of anarchistic egoism which he propagated.31 Liberation for the workers did not lie in their consciousness of themselves as a class, as Marx claimed; it would only come if they embraced the egoistic attitude of the "vagabond" and shook off the social and moral conventions that yoked them to an exploitative order.32 In other words, the true revolution lay in each worker's egoistic psyche: this would set the revolt against the state in motion. Once the struggle for a new, stateless order was underway, the vastness of the working class would ensure the bourgeoisie's defeat. "If _labor_ becomes _free"_ , Stirner concluded, "the state is lost."33 Hostility to abstract reasoning and bourgeois culture, militant individualism, and a belief in a new libertarian and working-class era: these positions de-fined the anarchism of the Moscow Federation. And in 1918, they set the terms for debating the relation of art to the anarchist revolution in the pages of _Anarkhiia._
On March 25, _1918, Anarkhiia_ published a "Letter to Our Comrades, the Futurists" that resonated with the Federation's antipathy for the culture of the bourgeoisie and the role of art under its patronage. The author, Baian Plamen (a pseudonym), criticized "socially passive" Futurists in the anarchist ranks who proclaimed their radicalism while serving "the bourgeois way of life" by decorating the cafes of the wealthy and designing useless "artifacts."34 This was a swipe at artists Rodchenko, Udalt'sova, and Tatlin, who, from July 1917 to January 1918, had designed and furnished a Moscow cafe-theater ("The Cafe of the Revolutionary City") for Nikolai Filippov, a wealthy capitalist who owned most of Moscow's bakeries. Under the direction of the Futurist Georgii Yakulov the artists renovated Filippov's haunt in the latest avant-garde style.35 Rodchenko contributed hand-crafted lamps and other decorative elements; stylish tables and benches were made; and Tatlin and Udalt'sova organized the construction of relief elements projecting from the cafe's ceiling and walls.36
_Vladimir Tatlin,_ Selection of Materials: Iron, Stucco, Glass, Asphalt, _1914. Whereabouts unknown. From Punin,_ Tatlin (protiv kubizma), _1921._
The establishment opened on January 30, 1918, and quickly became notorious as Moscow's most radical artistic experiment.37 However, where the artists saw "revolution," Plamen saw a sellout. The criticism stung, and Tatlin quickly rushed to the defence with a rejoinder—"My Answer to 'Letter to the Futurists'"—in _Anarkhiia's_ March 29 issue. Tatlin's reply is important and worth quoting in full:
I agree with you that the futurists are too busy with cafes and embroidery of various quality for emperors and ladies. I explain this by a 3/5 loss of focus in their artistic vision. Since 1912 I have been appealing to members of my profession to improve their eyesight. Having reconstructed corner and center reliefs of a superior type, I cast aside as unnecessary a number of 'isms'—the chronic sickness of contemporary art. I am waiting for well-equipped artistic 'depots' where an artist's psychic machine might be repaired as necessary. I appeal to all those in my guild to pass through the suggested gateway and throw off the old to admit a breath of anarchy.38
Tatlin concurred with Plemen that futurist art for the ruling class—"emperors and ladies"—was undesirable. He also condemned contemporary art's "isms" as a "sickness." Finally, he claimed to have discovered, in his art, a "gateway" for "throwing] off the old to admit a breath of anarchy." Tatlin was certainly familiar with the "isms" of the avant-garde. Prior to World War I, he had painted in a variety of modernist styles ranging from fauvism to cubism, but in the winter of 1913-1914, he developed a new form of relief sculpture that trans-formed the terms of avant-garde experimentation:
Tatlin was very interested in analyzing the construction and architectonics of the world. He arrived at a fundamental artistic discovery [with the reliefs]; non-figurative forms of various colors and textures _(fakturas)_ were removed from the surface of the picture into the space in front of the picture, at first without divorcing them from the plain background. The represented relation in space of each of these components of the picture was thus turned into the real context of each component, showing how they really relate in real space.... Tatlin called these kinds of compositions 'selections of materials' because the abstract picture that was turned into painterly relief was no longer painted with a brush but composed out of materials of various structural and painterly characteristics.
The next step was to break away from the surface of the picture. Now the composition was involved with real space (in front of the surface that served as a backdrop, or in between two sur-faces perpendicular to each other) and was supported only by a wire or a stiffly bent pivot. This was the first "sculpture without a pedestal," which at the same time inevitably showed architectural characteristics because of the real structural relations that developed between the various components of the construction. Tatlin called these creations counter-reliefs.39
_Vladimir Tatlin,_ Corner Counter-Relief, 15) 14-15. _Whereabouts unknown. From_ Vladimir Evgrafovich Tatlin, _1915._
The materialism of each element (surface, texture, color, etc.) in a work: this is the "gateway" through which Tatlin urged his comrades to pass. It remained for Rodchenko to give this passage an explicitly Stirnerist valiance.
In 1918, Rodchenko was well versed in Tatlin's ideas, having met the artist in 1915 and collaborated with him on numerous projects, including Filippov's cafe. He had also conducted his own experimentations with the properties of paint on canvas throughout 1915-17, and by 1918 this was the element he made his own. Here is Rodchenko's description of his paintings, published on April 28, 1918 in _Anarkhita:_
Designing vertical plane surfaces, painted a suitable color, and intersecting them with lines of depth, I discover that color serves merely as a useful convention for separating one plane from another, and for bringing out those elements which indicate depth and its intersections....
Taking into consideration only the projections of principal and central lines very different from the parallel peripheral lines or those that enter in depth, I completely neglect both the quality and the combination of colors....
Constructing projections on ovals, circles, and ellipses, I often distinguish only the extremities of the projections with color, which gives me the possibility of emphasizing the value of the projections and the color, used as an auxiliary means and not an end.
By thoroughly studying the projection in depth, height, and breadth, I discover an infinite number of possibilities for construction outside the limits of time.40
During this same period, Kazimir Malevich, leader of the "suprematist" group of artists whose ranks included Rodchenko, was experimenting with the same painterly values in his own abstractions (see color plate 6). Malevich's non-objective style, first manifested in the form of a stark black square painted on a white background, was rooted in the metaphysical mysticism of theosophy and notions of a "fourth" dimension beyond the sensate third. Malevich argued that humanity was evolving toward a higher state of being that would unite us with all living things, and ultimately, the universe itself. Evoking the third dimensions of space and depth on two-dimensional surfaces using non-objective forms such as circles, triangles, squares, and lines, suprematist paintings functioned as an analogue for the perception of this "higher" dimension—a dimension apprehended by a consciousness that was irrational rather than rational, "felt" rather grasped analytically. The hallmark of this consciousness was "simultaneity"; freed of third-dimensional moorings, things once separate and distinct merged, defying all logic and common sense.41 This illogic formed the basis for the poetics of suprematism's most important literary allies, Alexei Kruchenykh and Velimir Khlebnikov, who utilized transrational language _(zaum)_ to create "unresolved dissonances" that tapped our inner psyche and opened us to "simultaneity."42
At the turn of the century, many radicals, anarchists included, mixed spirituality and politics. For Malevich and his cohorts, the Russian revolution signaled a breakthrough into suprematist consciousness, an idea he promoted in _Anarkhiia,_ where he declaimed suprematist "egoism" as the visionary individualism of the anarchist revolution. For example, _Anarkhiia's_ March 27, 1918 issue featured a proclamation by Malevich entitled "To the New Limit":
We are revealing new pages of art in anarchy's new dawns.... The ensign of anarchy is the ensign of our "ego," and our spirit, like a free wind, will make our creative work flutter in the broad spaces of the soul. You who are bold and young.... Wash off the touch of dominating authorities. And, clean, meet, and build the world in awareness of your day.43
Asserting the revolutionary hegemony of suprematism, Malevich was more than ready to take on his non-objectivist rivals. In the same issue of _Anarkhiia_ (March 28) in which Tatlin's reply to "Plamen" appeared, he published his own "Reply" in which he blasted the futurists' "counter-revolutionary" activities and dismissed their anarchism as a "revolt" against existing conditions that paled in comparison with the suprematists' spiritual-artistic revolution, which had pushed humanity to "the limit of an absolutely new world."44
The year 1918 also saw Malevich embark on an unprecedented series of paintings. This cycle—his _White on White_ paintings—was unveiled on April 27, 1919 at the "10th State Exhibition of Non-objective Creation and Suprematism" (see color plate 7). Malevich's accompanying statement on "Suprematism" elucidated the aim of his latest work.45 Hitherto the suprematists had painted color forms floating against a white ground. Non-objective form and pure color had overcome the old artistic practice of representation and its methods of color-mixing that simulated "things and objects." However, the persistence of color frustrated Malevich, because aesthetic deliberations over the arrangement of color were far removed from the higher suprematist state of mind.46 Even if an artist's work was "constructed abstractly but based on color interrelations," Malevich wrote, his will would remain "locked up" by "the walls of aesthetic planes, instead of being able to penetrate philosophically."47 The move to _White on White_ broke through this limitation, liberating the artist to approach a revolutionary, suprematist consciousness in a medium from which the old world was finally, completely purged. Devoid of color, the _White on White_ forms dissolved into a void and Malevich's egoist "will" was free to soar, uninhibited, beyond the known world:
I am free only when—by means of critical and philosophical substantiation—my will can extract a substantiation of new phenomena from what already exists. I have breached the blue lampshades of color limitations, and have passed into the white beyond: follow me, comrade aviators!... The white free depths, eternity, is before you.48
As we have seen, Malevich was just as committed to the anarchist revolution as Rodchenko, however at the April 1919 exhibition it be-came clear to all concerned that Rodchenko's _Black on Black_ paintings and his manifesto, "Rodchenko's System" (reprinted at the end of this chapter) were being pitted against Malevich's "Suprematism" statement and his _White on White_ series. During the days leading up to the exhibition Rodchenko's wife and fellow non-objectivist, Varvara Stepanova, kept a diary where she discussed the critical purpose of Rodchenko's work. Throughout, Stepanova called Rodchenko "Anti," a pseudonym he used in _Anarkhiia._49
The exhibition, wrote Stepanova, was "a contest between Anti and Malevich, the rest are rubbish. Malevich has hung five white canvases, Anti black ones."50 Stepanova praised "Anti" for his powerful distillation of "pure painterly effects, without being obscured by incidental elements, not even by color." She also recorded her (and presumably Rodchenko's) view of the implications the _Black on Black_ series held for Malevich. "Anti's works" were "a new step in painting after suprematism.... The destruction of the square and a new form, the intensification of painting for its own sake, as a professional feature, a new interesting facture and just painting, not a smooth coating in a single color, the most unrewarding—black."51 We can probe the anarchist foundations of the "destruction of the square"—clearly a reference to Malevich—through a reading of "Rodchenko's System" as a step-by-step process of egoistic affirmation and negation.52
"Rodchenko's System" opened with Stirner's most fundamental materialist axiom, "At the basis of my cause I have placed nothing,"53 and its fifth aphorism was another passage from Stirner: "I devour it the moment I advance the thesis, and I am the 'l' only when I devour it.... The fact that I devour myself shows merely that I exist." These aphorisms are important for Rodchenko's manifesto, but to grasp their import we have to return, once more, to _The Ego and Its Own._
In the previous chapter, we saw how positing the notion of an "I," as Stirner argued, assumed there was an absolute condition of "being" that transcended our uniqueness. Such "Absolute thinking," wrote Stirner, "is that thinking which forgets that it is _my_ thinking, that _I_ think, and that it exists only through _me._ But I, as I, swallow up again what is mine, am its master; it is only my _opinion,_ which I can at any moment _change,_ i.e. annihilate, take back into myself and consume."54 For Stirner, the sensuous, devouring ego was the irreducible core of uniqueness and the cornerstone of the mastering "I" that had no essence, that was, in effect, the "nothing" at the foundation of his philosophy. "I am not an ego along with other egos," wrote Stirner:
I am unique. Hence my wants too are unique, and my deeds; in short, everything about me is unique. And it is only as this unique that I take everything for my own, as I set myself to work, and develop myself, only as this. I do not develop man, nor as man, but as I, I develop—myself.
This is the meaning of the— _unique one_.55
_Aleksandr Rodchenko,_ Black on Black #81, 1918. _Oil on canvas. State Russian Museum, St. Petersburg._
Suprematism celebrated the evolution of a mystifying abstraction, humanity. Malevich's anarchist "ego" was a manifestation of a dawning collective consciousness that penetrated a realm which was unabashedly metaphysical. Far from asserting uniqueness, the transrationalism of Malevich and his poetic allies sought to break down the 'false' barriers separating the self from a hidden "fourth dimension" outside of time and the material world. In Stirnerist terms, this was just one more instance of groveling subservience to a mysterious "higher" condition apart from the self.
Quoting Stirner, Rodchenko set himself against all this. For his second aphorism ("colors disappear—everything merges into black"), he borrowed a passage from Kruchenykh's transrational play, _Gly-Gly_ , in which Malevich and Kruchenykh both figured as dramatis personae.56 Putting this "transrational" poet into service to trumpet his thesis was an egoistic put-down that would not have been lost on Malevich.
An aphorism from the German psychologist Otto Weininger's book _Uber die letzten Dinge_ (1907) and two quotations from Walt Whitman's _Leaves of Grass_ (1855) served the same end. Here, Rodchenko transformed Weininger's psychological insight into an elliptic commentary on himself. By "murdering" suprematism, he was achieving "self-justification" of a consummate egoistic sort, since, following Stirner, the "self" that justified the act was itself devoid of an "essence": it was the "nothing" that the "murderer" aspired to "prove." Finally, the Whitman passages, which praised the invigorating role death plays in the process of life, indicated that Rodchenko's "voyage of the soul" necessitated both creation (his paintings) and negation (again, suprematism) and introduced the affirmative section of "Rodchenko's System."
In the closing section, alluding to his debt to Tatlin, Rodchenko attributed his own "assent" to the downfall of all "isms" whose "funeral bells" were rung by the _Black on Black_ series. From this point on, the motive of his work would be "invention (analysis)" utilizing the material constituents of the object ("painting is the body") to "create something new from painting." Once through Tatlin's "gateway," Rodchenko stripped the canvas of metaphysics and distilled its base elements, the painterly "body" and the creative "spirit." Having mastered the "isms" of the avant-garde, he would now master painting itself, moment by moment, in a process of free invention. These were the qualities Stepanova celebrated in her diary, where she wrote that "Anti," the "analyst," and "inventor," created work that presented nothing but "painting." The _Black on Blacks_ "[left] no room for color," and their _facture_ gained an extraordinary presence as a result. In her diary, Stepanova related that the "lustrous, matt, flaky, uneven, smooth" surfaces of the _Black on Blacks_ so impressed fellow anarchist Udal'tsova that she asked for one to be taken down so that she could feel it.57 The exhibition, Stepanova concluded, was a tremendous success for "Anti" and "his mastery, his facture."58
In early 1919, Rodchenko celebrated his creative egoism, but could painterly anarchism combat terror, repression, and ideological assaults? Rodchenko's plight recalls the plaintive objections he once raised in _Anarkhiia_ during the revolution's hopeful early days. Attending a meeting of the Communist-dominated "Proletarian Culture" organization, he heard a vitriolic speech on "proletarian art" from one "comrade Zalevskii" that condemned cubism and futurism as the "last word in bourgeois art" and the antithesis of working-class culture. The pre-revolutionary avant-garde, countered Rodchenko, were "daring inventors" who, though "hungry and starving" under the old order, had produced "revolutionary creations." The bourgeoisie "hated" the cubists and futurists because it "want[ed] to see only itself and its taste in the mirror of art." Now Zalevskii demanded that the workers emulate their oppressors. "But the worker," wrote Rodchenko, does not want to "strangle his brother, the rebellious artist." "I am sure," he concluded, "that working people want true creators, not submissive bureaucrats."59 Rodchenko voiced his objections freely because he ad-dressed a large working-class readership from the platform of a still-viable anarchist movement. Though beset by adversaries, he could still appeal to the readers of _Anarkhiia_ for support and rally other artists to the cause. But as Communist power progressed, these freedoms were shut down.
Rodchenko's capitulation came in March of 1921, when he, Gan, and Stepanova joined with Konstantin Medunetskii, Karl Ioganson, Gregorii Stenberg, and Vladimir Stenberg to form "The First Working Group of Constructivists." The group drew up a manifesto wherein they dedicated themselves to "Soviet construction" guided by "scientific communism, based on the theory of historical materialism." Repudiating artistic anarchism circa 1918, they declared that art had no role to play in the "social production of the future culture" because "it arose from the mainstreams of individualism."60
Arguably, the date of this declaration—March 18—was not coincidental. The night before, the Communist Party had crushed the last flicker of resistance to its rule at the island fortress of Kronstandt, where an anarchist-led soviet had called for a revolution against the Communist dictatorship and then held out for sixteen days until its rebellious inhabitants were subdued by Red Army detachments.61 Alarmed by anarchist involvement in the rebellion, the Cheka swept the streets of Russia, throwing hundreds of anarchists, including Askarov and Chernyi, into prison.62 Plainly, the time was ripe for a retreat into Marxist orthodoxy.
_The following accompanied the exhibition of Rodchenko's_ Black on Black _paintings in 1919:_
Rodchenko's System
_At the basis of my cause I have placed nothing._
—M. Stirner, _The Ego and Its Own_
_Colors disappear_ — _everything merges into black._
—A. Kruchenykh, _Gly-Gly_
_Muscle and pluck forever!_
_What invigorates life invigorates death,_
_And the dead advance as much as the living advance._
—Walt Whitman, _Leaves of Grass_
_Murder serves as a self-justification for the murderer; he thereby aspires to prove that nothing exists._
—Otto Weininger, _Aphorisms_
_... I devour it the moment I advance the thesis, and I am the_ "I" _only when I devour it.... The fact that I devour myself shows merely that I exist._
—M. Stirner
_Gliding o'er all, through all,_
_Through Nature, Time, and Space,_
_As a ship on the waters advancing,_
_The voyage of the soul_ — _not life alone,_
_Death, many deaths I'll sing._
—Walt Whitman, _Leaves of Grass_
The downfall of all the "isms" of painting marked the beginning of my ascent.
To the sound of the funeral bells of color painting, the last "ism" is accompanied on its way to eternal peace, the last love and hope collapse, and I leave the house of dead truths. The motive power is not synthesis but invention (analysis). Painting is the body, creativity, the spirit. My business is to create something new from painting, to examine what I practice practically. Literature and philosophy are for the specialists in these areas, but I am the inventor of new discoveries in painting.
Christopher Columbus was neither a writer nor a philosopher, he was merely the discoverer of new countries.
—Aleksandr Rodchenko, "Rodchenko's System," Tenth State Exhibition: Non-objective Creation and Suprematism, Moscow, 1919
"That Day"
Shots.
Shots.
A crackling machine gun.
Again.
Guns!
God! What is it? Why?
October; its the same as then.
5 a.m.
Morning.
Jump out of bed.
Devils. Don't know. What they crushed.
The Clubs.
People, dull and rude.
Don't know
who
they
killed.
They're bandits—they say.—Criminal dirt, gathered at midnight.
People.
Can't [see] their faces.
—published in _Anarkhiia_ (April 23, 1918)
### NOTES TO CHAPTER 4
All citations from _Anarkhiia_ are from the New York Public Library holding unless otherwise noted.
1 Aleksandr Rodchenko, "O Muzei Morozova, II," _Anarkhiia_ , n.d. This was the second section of a two-part article. April 12, 1918 saw the first wave of raids on anarchist centers throughout Moscow. The raid on the Morozov mansion fits that pattern: the artists were awakened at night by armed detachments, rounded up with the anarchists occupying the building, and shipped off for interrogation.
2 With the notable exception of the German art historian Hubertus Gassner. See Hubertus Gassner, "The Constructivists: Modernism on the Way to Modernization," _The Great Utopia: The Russian and Soviet Avant-Garde, 1915-1932,_ exh. cat. (New York: Solomon R. Guggenheim Museum, 1992): 298-319.
3 Paul Avrich, _The Russian Anarchists_ (Princeton: Princeton University Press, 1967), 182.
4 Ronald I. Kowalski, _The Bolshevik Party in Conflict: The Left Opposition of 1918_ (Pittsburgh: University of Pittsburgh Press, 1991): 146-154.
5 Avrich, _The Russian Anarchists,_ 182.
6 Ibid.
7 Ibid., 183.
8 Gregorii Maximov, _The Guillotine at Work_ (Chicago: Alexander Berkman Memorial Fund, 1940): 406.
9 Ibid., 408.
10 Maximov, _The Guillotine at Work,_ 40; Aleksandr Rodchenko, "O Muzei Morozova," _Anarkhiia,_ n.d.
11 Avrich, _The Russian Anarchists,_ 184.
12 "The April Pogrom in Moscow" in Maximov, _The Guillotine at Work,_ 383-393.
13 "Release of the Extraordinary Committee to Struggle Against the Counter-Revolution (Cheka)," _Znamia Truda_ (April 13, 1918) in Maximov, _The Guillotine at Work,_ 383.
14 "An Official Communication," _Znamia Truda_ (April 13, 1918) in Maximov, _The Guillotine at Work,_ 384-385.
15 Avrich, _The Russian Anarchists,_ 184.
16 Maximov, _The Guillotine at Work,_ 357.
17 Ibid., 356.
18 "Pogroms Follow in Petrograd and in the Provinces," in Maximov, _The Guillotine at Work,_ 396-404.
19 Maximov, _The Guillotine at Work,_ 410.
20 "Tot' Den'," _Anarkhiia,_ (April 23, 1918). The artist and poet are not identified. Avrich writes that the cycle of arrests, executions, and imprisonments intensified in 1919, and that by 1920 "the Cheka dragnet had swept the entire country," effectively snuffing out the anarchist movement. _The Anarchists in the Russian Revolution,_ Paul Avrich, ed. (Ithaca: Cornell University Paperbacks, 1973): 138.
21 Avrich, _The Russian Anarchists,_ 189-195.
22 Ibid., 188-189.
23 Avrich, _The Russian Anarchists,_ 200-203. In November 1921, the Moscow headquarters of the Soviet-Anarchists was raided and its leaders jailed. "The Persecution of the Anarchist Universalists" in Maximov, _The Guillotine at Work,_ 503-505.
24 Anarchist collaborations with the Bolsheviks are discussed in Avrich, _The Russian Anarchists,_ 196-203. For a history of the Ukrainian anarchist insurrection and its subsequent repression, see Peter Arshinov, _History of the Makbnovist Movement (1918-1921),_ Lorraine and Freddy Perlman, trans. (Detroit: Black and Red, 1974): passim.
25 Gassner, "The Constructivists," _The Great Utopia,_ 303. I have gleaned Can's activities from the pages of _Anarkhiia._ One issue advertised a lecture series to take place at the "House of Anarchy" beginning on March 25, 1918 in which Gan would lecture on "Art and Proletarian Theatre." Among the other speakers were Lev Chernyi, whose topics were the "History of Culture. Social History. Sociology. Associational Anarchism," and one of the Gordin brothers, who discussed the "Ethical Basis of Anarchism;" "Bezplathye kursy po anarkhizmy," _Anarkhiia,_ n.d. Can's article on "comrade Lukashin" includes a self-portrait by the young artist and an example of one of his paintings for the proletarian theater production of Leonid Andreyva's play _Savvy_ (1906). The play's anarchist hero sought to annihilate everything because the world was so corrupt no rehabilitation of the social order was possible—an interesting commentary on the attitude of the Federation of Anarchists at this juncture. Lukashin also belonged to the anarchist "Initiatory Group" whose members included Rodchenko and Tatlin; A[lexei].G[an]., "Tovarishch" Lukashin'," _Anarkhiia,_ n.d. The membership of Rodchenko and Tatlin is documented in an undated letter to the "Federal Council of Anarchist Groups" signed by V. Tatlin, A. Rodchenko, and A. Morgunov in German Karginov, _Rodchenko_ (London: Thames and Hudson, 1979): 60.
26 Gassner, "Constructivists," _The Great Utopia,_ 303.
27 Ibid.
28 "Decree No. 1 on the Democratization of Art," _Gazeta futuristov_ (March 15, 1918) in Gassner, "Constructivists," 303.
29 Avrich, _The Russian Anarchists, 177._ Chernyi's book on "Associational Anarchism" includes two chapters dealing with anarchist egoism and collectivism; Lev Chernyi, _Novoe Napravlenie v Anarkhizme: Asosiatsionnii Anarkhism_ (Moscow: 1907; 2nd ed., New York, 1923): passim.
30 Avrich, _The Russian Anarchists, 177._
31 Stirner, _The Ego and Its Own,_ 148-149.
32 Stirner, 152.
33 Stirner, _The Ego and His Own,_ 361.
34 A.A. Strigalev and L.A. Zhadova, "Notes on Baian Plamen [pseudonym], 'Open Letter to the Futurists'," _Anarkhiia_ (March 25, 1918) in _Tatlin,_ Larissa Alekseevna Zhadova, ed." (New York: Rizzoli, 1988): 185.
35 Karginov, _Rodchenko,_ 91.
36 John Miller, _Vladimir Tatlin and the Russian Avant-Garde_ (New Haven: Yale University Press, 1983): 127-28.
37 Karginov, _Rodchenko,_ 92.
38 Vladimir Tatlin, "My Answer to 'Letter to the Futurists,'" _Anarkhiia_ (March 29, 1918) in _Tatlin,_ Zhadova, ed., 185.
39 A. A. Strigalev, "From Painting to the Construction of Matter," in _Tatlin,_ Zhadova, ed., 19.
40 Aleksandr Rodchenko, "The Dynamism of Planes," _Anarkhiia_ (April 28, 1918) in Selim O. Khan-Magomedov, _Rodchenko; The Complete Work,_ Vieri Quillici, ed. (Cambridge, MA: MIT Press, 1987): 32, note 10.
41 In suprematist theory, the passage of time as experienced in the third dimension was actually movement in the fourth, hence time was a fiction. Evoking the idea of two-dimensional forms moving into three dimensions alluded to our own movement in fourth-dimensional "hyperspace." Linda Dalrymple Henderson, _The Fourth Dimension and Non-Euclidian Geometry in Modern Art_ (Princeton: Princeton University Press, 1983): 274-299.
42 Charlotte Douglas, "Views from the New World: or A. Kruchenykh and K. Malevich: Theory and Painting," _The Ardis Anthology of Russian Futurism,_ Ellendea Proffer and Carl R. Proffer, eds. (Ann Arbor: Ardis Press, 1980): 361-362.
43 Kazimir Malevich, "To the New Limit," _Anarkhiia_ 28 (March 27, 1918) in K.S. Malevich, _Essays on Art: 1915-1933: Vol. 1,_ Troels Andersen, ed. (London: Rapp and Whitling, 1968): 56.
44 Kazimir Malevich, "Reply," _Anarkhiia_ 29 (March 29, 1918) in Malevich, _Essays on Art: Vol. 1,_ 52-54.
45 Kazimir Malevich, "Suprematism," _Statements from the Catalogue of the " Tenth State Exhibition: Nonobjective Creation and Suprematism"_ in _Russian Art of the AvantGarde: Theory and Criticism, 1902-1934,_ John E. Bowlt, ed. (New York: Viking Press, 1976): 143-145.
46 Ibid., 144.
47 Ibid.
48 Ibid.
49 _Aleksandr Rodchenko: Experiments for the Future: Diaries, Essays, Letters, and Other Writings,_ Alexander N. Larentiev, ed. (New York: Museum of Modern Art, 2002): 134, note 27.
50 Vavara Stepanova, "Notes from the Diary on the Preparation and Management of the Tenth and Nineteenth State Exhibitions," entry April 10, 1919 in _The Future is Our Only Goal,_ Peter Noever, ed. (Munich: Prestel-Verlag, 1991): 124.
51 Stepanova, "Notes," entry April 10, 1919, 124.
52 Aleksandr Rodchenko, "Rodchenko's System," _Tenth State Exhibition_ in Bowlt, ed., _Russian Art of the Avant- Garde,_ 149-151.
53 Translated as "All things are nothing to me" in Stirner, _The Ego and Its Own,_ 3.
54 Ibid., 453.
55 Ibid., 482-483.
56 John Bowlt ed., _Russian Art of the Avant-Garde,_ 305, note 2.
57 Stepanova, "Notes," entry April 10, 1919 in Noever, 125.
58 Stepanova, "Notes," entry April 29, 1919 in Noever, 126.
59 Aleksandr Rodchenko, "O doklad' T. Zalevskago v Proletkult'," _Anarkhiia,_ n.d.
60 "First Program of the Working Group of Constructivists" manifesto cited in Christina Lodder, _Russian Constructivism_ (New Haven; Yale University Press, 1983): 94.
61 Avrich, _The Anarchists in the Russian Revolution,_ 156.
62 Avrich, _The Russian Anarchists,_ 230-231.
_Image Not Available_
COLOR PLATE 1:
_Gustave Courbet,_ The Stonebreakers, 1849, _destroyed 1945. Oil on canvas. Dresden: Gemaldegalerie Neue Meister, 2522. Photo: Deutsche Fotothek Dresden_.
_Image Not Available_
COLOR PLATE 2:
_Théodore van Rysselberghe,_ **Les** Errants (The Wanderers), 1897, _Lithograph on paper, 46.8 cm × 57 cm. Jane Voorhees Zimmerli Art Museum, Rutgers, The State University of New Jersey. Herbert J. Littman Purchase Fund. Photograph by Jack Abraham; 1986.0888_.
_Image Not Available_
COLOR PLATE 3:
_Camille Pissarro,_ Apple Picking at Eragny-sur-Epte, _1888. Oil on canvas, 24"_ × 29". _Dallas Museum of Art, Munger Fund._
_Image Not Available_
COLOR PLATE 4:
_Francis Picabia,_ Dances at the Spring, 1912. _Oil on canvas, 47 ½" × 47½". © Estate of Francis Picabia / SODRAC (2006). Philadelphia Museum of An: The Louise and Walter Arensburg Collection, 1950-134-155_.
_Image Not Available_
COLOR PLATE 5:
_Francis Picabia_ , Edtaonisl (ecclesiastique), 1913. Oil on canvas, 118¾ _x_ x 118 ¾" ©Estate of Francis Picabia / SODRAC (2006), Art Institute of Chicago.
_Image Not Available_
COLOR PLATE 6:
_Kazimer Malevich,_ Suprematist Painting, 1915. _Oil on canvas, 101.5 cm × 62 cm. Stedelijk Museum, Amsterdam_.
_Image Not Available_
COLOR PLATE 7:
_Kazimer Malevich,_ Suprematist Painting (White on White), _1917-18. Oil on canvas, 97 cm_ × 70 _cm. Stedelijk Museum, Amsterdam._
_Image Not Available_
COLOR PLATE 8:
_Jess,_ If All the World Were Paper and All the Water Sink, 1962. _Oil on canvas, 38" × 56". Courtesy Odyssia Gallery, New York_.
_Image Not Available_
COLOR PLATE 9:
_Jess,_ The Mouse's Tale, 1951. Collage, 47" × 52 ". _Collection of San Francisco Museum_ of Modern An, Gift of _Frederic P. Snowden_.
COLOR PLATE 10:
_Susan Simensky Bietila, Bread and Puppet Theater performance at the anti-Vietnam War March to UN Headquarters, New York, April 15, 1967. Gelatin/silver print, 5¼_ × 7¾".
COLOR PLATE 11:
_Susan Simensky Bietila,_ Arrest at the Demonstration Against Military Recuiters, Brooklyn College, 1967, _2006. Pen and ink wash on paper, 8"_ × 7".
COLOR PLATE 12: _(facing page)_
_Susan Simensky Bietila,_ Drawing Resistance _Poster, with woodcut by Iris Pasic, 2001. II" × 17_ ".
COLOR PLATE 13: _(above)_
_Gee Vaucher,_ Nagasaki Nightmare, poster, _1980. Black gouache painting, 30 cm_ × _21 cm._
COLOR PLATE 14:
_Book cover image by Freddie Baer, for_ Against the Megamachine: Essays on Empire and its Enemies, _by David Watson, 1998. 6"_ × _9"._
COLOR PLATE 15:
_Richard Mock, untitled, 1995. Oil on linen, 24"_ × 22".
COLOR PLATE 16:
_Richard Mock,_ Strike Zone, _1991. Linocut print, 20"_ × 15¾"
## CHAPTER 5
DEATH TO ART! _The Post-Anarchist Aftermath_
_The old bureaucratism has been smashed,
but bureaucrats still remain._
—Joseph Stalin (April 1919)1
What do revolutionary artists do after they renounce anarchism? In the Soviet Union, the newly minted constructivists numbered amongst the most militant pro-Communist groups in the spectrum of post-revolution culture. The Higher State Artistic and Technical Workshops (Vkhutemas), founded in 1921, served as their base of operations through the 1920s (the workshops were closed in 1930).2 The Communist Party was firmly in control of the state apparatus; and from this point forward, artistic affairs were monitored through the cultural institutions of the government, which channelled money into art schools and served as the major patron for art commissions.
What distinguished the constructivists from others in the cultural scene was their outspoken rejection of traditional art-making.3 Vavara Stepanova's lecture on "The General Theory of Constructivism" (December 22, 1921) pinpoints the salient features, and the ways in which the artistic politics of anarchism were reworked to accord with the group's Marxism. Constructivism was codified as an anti-art movement which rejected the creation of art objects such as paintings and sculpture, as well as any role for aesthetics in "intellectual production."4 Traditionally, art was a product of "the illusions of individual consciousness" and served no social role apart from establishing "an ideal of beauty for a given epoch."5 In the pre-revolutionary era, how-ever, not all artists were caught up in the snare of aesthetics and idealism. Stepanova singled out the rise of an "analytical method" among certain avant-garde artists which revealed, for the first time, the art medium's materialistic foundations. "Art stopped being representational" thanks to the artists' "revolutionary-destructive activity, which stripped art down to its basic elements," thus causing "changes in the consciousness of those who work in art by confronting them with the problem of construction as a practical necessity."6 In sum, Tatlin's relief sculptures and Rodchenko's _Black on Black_ canvases were "cleansed" of their anarchist content by way of theoretical omission. "Once art has been purged of its aesthetical, philosophical, and religious tumours," wrote Stepanova, "we are left with its material bases, which will henceforth be organized by intellectual production."7
In his capacity as chief theoretician of the nascent movement, Alexei Gan took the same tack in his major statement, _Constructivism_ (1922).8 Dating constructivism's rise to the Communist coup—"October 25, 1917"—Gan roundly condemned art-making, past and present. "All so-called art," he wrote, "is permeated with the most reactionary idealism, is the product of extreme individualism" and "the hypocrisy of bourgeois culture." Declaiming "DEATH TO ART!" in capital letters, he banished painting and sculpture, along with capitalism and individualism, to the pre-October era.9 Of course, the history of art in the post-October anarchist movement problematized this maneuver, but no matter. In the era of the revolution, Gan argued, there had been "tendencies" amongst the avant-garde toward "the pure mastery of artistic labor of intellectual-material production." Those who instigated them, however, had never managed "to sever the umbilical cord that still held and joined them to the traditional art of the Old Believers [a reactionary sect of Russian Orthodoxy]. Constructivism has played the role of midwife."10 Midwife to what? "The practical reality of the Soviet system" that was the constructivist's "school, in which they carry out endless experiments tirelessly and unflinchingly."11
In the early years, constructivism's baby steps consisted of solving hypothetical problems. Rodchenko's _Oval Hanging Spatial Construction no. 12_ (ca 1920), for example, was an exercise in design which exploited the object's materials to realize a given problem with the maximum of economy. Rodchenko created this work using a single sheet of ply-wood, out of which he cut a series of concentric circles. Economy of construction was matched by the ease with which the three-dimensional object could be collapsed down into two-dimensions and stored.12 The constructivists later referred to this period of experimentation as their "laboratory" phase.13 The next step was to merge constructivist activity with industry so that technology and the real demands of the factory floor would dictate the design and purpose of the product. This phase got underway in the mid-1920s at the Higher State Artistic and Technical Workshops. For example, Rodchenko and his students worked on designs for mass-produced, multipur-pose furniture that maximized space usage in ultra-efficient workers' apartments and clubs.14 For her part, Stepanova taught in the textile faculty of the school, designed sports and factory uniforms, and took on a commission to design fabrics in a local factory in order to master the technologies of clothes production and dyeing. Throughout the 1920s, constructivist activity branched out into graphic design, architecture, and many other endeavors.15
In Communist theory, industrialism was the materialist base from which Russia's working class had emerged and upon which Soviet socialism would be built under the Party's disciplining guidance.16 It followed that during the 1920s, the drive to reorganize society in the mirror image of industry, "socialized" and regimented, was enthusiastically embraced by the constructivists. In this regard, no better ex-ample exists than constructivist theater director Vsevolod Meyerhold's staging of _The Magnanimous Cuckold_ in 1922.
Meyerhold was a well-known radical who had joined the Communist Party in 1918 and produced plays for the Red Army during the revolution. Recalled to Moscow in 1920 to help administer state theater programs, he was appointed director, in autumn 1921, of the newly formed State Higher Theater Workshops, sister organization to the Higher Artistic and Technical Workshops.17 Hundreds of students enrolled in the workshops, in which Meyerhold introduced constructivist principles into staging, and a new acting methodology called "Biomechanics" to his training program. Many of Meyerhold's stu-dents worked in factories during the day and trained and performed for his theater in the evening.18 The goal was to suffuse his productions with working-class content, giving them the requisite cultural stamp that would set them apart from pre-revolutionary theater.
_Alexei Gastev,_ Scientific Management Motion Study Photograph, _ca_ 1925. _From Rene Fulop-Miller,_ The Mind and Face of Bolshevism, 1927.
"Biomechanics" was based on techniques of movement then being promoted under the direction of the Communist bureaucrat and "proletarian poet," Alexei Gastev.19 Gastev spearheaded a state-financed program to introduce the latest form of labor organization, known as scientific management, to the Soviet workforce. Developed in America, scientific management, also known as "Taylorism" after its founder, Frederick Winslow Taylor, was a system of labor coordination which trained workers in efficiency of movement, breaking down work into easily executed tasks which enabled managers to speed up the pace of production exponentially.20 The movement generated a whole new layer of white collar management while at the same time facilitating the super-exploitation of workers through piece-work pay scales, impossible-to-achieve production targets, and on-the-job deskilling which destroyed trade unions. The authoritarian cultural values of scientific management are reflected in the intensified supervision of the worker, whose entire workday was under the thumb of one or more managers.21 Work was restructured around efficiency standards gained through the scientific study of exemplary laborers; for example, the movements of prize-winning speed typists were studied to determine the most efficient hand positions and related tasks such as the placement and insertion of typing paper.22
Armed with such standards, ambitious scientific management teams proposed to transform the workplace and keep it running smoothly ever after. Filmed motion studies were an additional aid: timers deter-mined the quickest movements, which were then recreated in three-dimensional models to assist training; light devices were attached to the body to facilitate the recording of movement. Another means of registering efficiency was to photograph movements with a timer attached to the light device; faster, more efficient movements left shorter dots of light. All the above methodologies were employed in the Soviet Union, where Gastev transformed scientific management into a mass movement.
_Alexei Gastev, n.d. Photograph_.
He set to work with the express blessings of Lenin and Leon Trotsky, both of whom were scientific management boosters. Trotsky, while head of the Red Army, was notorious for imposing scientific management and military-style organization in factories to maintain production during the revolution. To do so, he drew from a pool of well-paid managers imported from America and Europe, as well as home-grown experts, including Gastev.23 Praising scientific management for its disciplinary qualities in January 1920, Trotsky argued:
A whole number of features of militarism blend with what we call Taylorism. Compare the movements of a crowd and a military unit, one marching in ranks, the other in a disorderly way, and you'll see the advantage of an organized military formation. And so the positive, creative forces of Taylorism should be used and applied.24
Later that March, at the 9th Congress of the Russian Communist Party, Trotsky proposed augmenting the imposition of scientific management in the workplace by disciplining workers through blacklisting, penal battalions, and concentration camps.25
Lenin, who had studied scientific management before the war, shared Trotsky's enthusiasm.26 In "The Immediate Tasks of the Soviet Government" (April 28, 1918)—issued, we should note, in the immediate wake of the anti-anarchist Cheka raids—he wrote: "The possibility of building socialism will be determined precisely by our success in combining the Soviet government with the Soviet organization of administration with the modern achievements of capitalism. We must organize in Russia the study and teaching of the Taylor system and systematically try it out and adapt it to our purposes."27 To this end, he supported the creation of a Central Institute of Labor in 1920, with Gastev at the helm. During the 1920s, the Institute played an integral role in the introduction of scientific management throughout the industrial infrastructure.28
We can gauge Gastev's extremism from articles published during the civil war years. Gastev hailed scientific management as the organizational counterpart to machine production, and predicted a new Communist man would emerge from rationally organized production. Under a regime of scientific management, he wrote, "machines would be transformed from the managed into the managers" and norms established scientifically would permeate the life of the proletariat, right down to "aesthetic, mental, and sexual needs."29 He imagined the coming Communist society operating as a single industrial unit. Mechanized workers would be directed by an equally mechanized "special staff of engineers, designers, instructors, and head draftsmen who would work with the same regularity as the rest of the giant factory."30 "We must fearlessly state," he wrote, "that it is absolutely necessary for the present-day worker to mechanize his manual labor; that is, he must make his gestures resemble those of a machine.... Only the creation of a collective rhythm will provide the conditions for objective leadership."31 The goal?
... _mechanized collectivism._ The manifestations of this mechanized collectivism are so foreign to personality, so anonymous, that the movement of these collective complexes is similar to the movement of things, in which there is no longer any individual face but only regular uniform steps and faces devoid of expression, of a soul, of lyricism of emotion, measured not by the shout or a smile but by a pressure gauge or a speed gauge.32
This Marxist vision, in which individualism (anarchist or other-wise) was totally effaced by industrialized collectivism, struck a chord with the constructivists, including Meyerhold.33 Gastev's connections with Meyerhold are clear enough. During the 1920s, both men sat on the board of a scientific management propaganda organization called the League of Time, and Meyerhold produced time-management propaganda plays that toured the Soviet Union as part of the "Living Newspaper" theater program.34 Indeed, the "biomechanical" training methods practiced in Meyerhold's theater were indebted to Gastev's studies, as Meyerhold himself acknowledged.35 On June 11, 1922, Gastev published an article in which he called for the study of "that magnificent machine" the human organism through "a special science, biomechanics," in "laboratory" conditions.36 Meyerhold's theater was just such a laboratory, where the new regimes of movement for the workplace were collectivized into highly mechanized performances. In an interview, Meyerhold described biomechanics as the application of "the Taylor system" to acting, which was analogous to the labor of a skilled worker.37 This made theater useful in the building of socialism: biomechanics was disciplined training for the factory floor.38 His statement on "Biomechanics," published on June 12, 1922, goes further, evoking constructivism as a methodological foundation:
The work of the actor in an industrial society will be regarded as a means of production vital to the proper organization of labor of every citizen of that society.... In art our constant con-cern is the organization of raw material. Constructivism has
_Scientific Management Poster,_ Victory is Still Ahead of Us (Portrait of Alexei Gastev), _ca 1925. From Rene Fulop-Miller,_ The Mind and Face of Bolshevism, 1927.
forced the artist to become both artist and engineer. Art should be based on scientific principles; the entire creative act should be a conscious process. The art of the actor consists in organizing his material: that is, in his capacity to utilize correctly his body's means of expression. The actor embodies in himself both organizer and that which is organized.... The actor must train his material (the body) so that it is capable of executing instantaneously those tasks that are dictated externally (by the actor, the director).39
The constructivist imperative to do away with pre-revolutionary theatrics ("the 'inspirational' method and the method of 'authentic emotions'") and merge with industrialism determined not only the scientific method of acting, but also the features of the _Magnanimous Cuckold_ theater set.40 The set was designed at Meyerhold's request by Luibov Popova, a suprematist turned constructivist who taught "color construction" at Vhukutemas and "material formation" at Meyerhold's school.41 Popova did away with the conventional illusionism of staging in favor of the needs of the performance. Her utilitarian set was stripped of embellishment, to create an efficient workspace for the unfolding biomechanical action. In this sense, it was just like a factory interior—devoid of decorative features and designed for maximum efficiency of production.42
Even props played a productive role. Popova's stage helped to regulate the movements of the actors during a performance. The set featured giant wheels that periodically sped up and slowed down, and the motions of the worker-actors echoed the tempo set by Popova's revolving wheels, which mirrored the pace of the performance. In effect, the stage worked like a machine in a factory assembly line: in a factory, the machine regulates the pace of work, and the workers have to keep up with it. What we have, then, is a merger of constructivist principles of design with the creation of an actor-worker whose bodily efficiency emulated that of the factory.43 _The Magnanimous Cuckold_ was a spectacular success and led to further collaborations with Popova, Stepanova (who designed stage sets and costumes), and the constructivist architect Alexandr Vesnin.44
_Liubov Popova, 1919. Photograph_.
Theatrical achievements aside, with Gastev at the helm, scientific management swept through the Soviet economy, bringing piece-work quotas that could be adjusted by the management as it saw fit and thousands of "efficiency experts" to rationalize, control, and manage. A report in early 1925 observed "at present there exists no branch of state activity" where the principles of scientific management had not "penetrated."45 As the regime thrust forward, workers resisted—one newspaper reported in 1928 a "serious anti-rationalization mood," as evidenced by instances of workplace sabotage, plant occupations, and the forcible eviction of managers, who also had bricks thrown at them.46 However "labor discipline" only intensified under the command economy of five year plans (1928-1933; 1933-1937) when "piece-work rates were dropped below the level necessary for the minimal decencies of life" and "sixteen to seventeen hour" work days, including "voluntary work on holidays" was the rule.47
Scientific management as celebrated by the constructivists went hand-in-hand with brutal exploitation right down the social pyramid, but that didn't save the movement. In the early 1930s, socialist realism—representational art infused with Communist-dictated "socialist content"—came to the fore, and the Soviet art community was thinned by waves of purges targeting prominent figures for past deviations from the new artistic line.48 Amongst the constructivists, Meyerhold was arrested in 1939 and tortured into confessing involvement in an "anti-Soviet" conspiracy during the production of _The Magnanimous Cuckold_. Tried in secret, he was shot on February 2, 1940. Popova escaped this fate, having died of natural causes in 1924. Alexei Gan was shot in 1942. Tatlin reverted to producing innocuous portraits and flower paintings (he died in 1953). Gastev was arrested in 1938 and died in prison in 1941. As for Stepanova (d. 1958) and Rodchenko (d. 1956), they survived by retreating into designing propaganda books and magazines—this despite the fact that in the early 1930s, Rodchenko had made a great show of denouncing his former artistic "errors" and praising forced labor camps (which he photographed) for their "rehabilitating" role.49 How far "Anti" had fallen from his anarchist days can be gauged by his actions during a purge which swept through the higher echelons of Soviet Uzbekistan. In 1934, Rodchenko was commissioned to design an illustrated album, _Ten Years of Uzbekistan,_ commemorating a decade of Communist Party rule. This book, published in Russian and Uzbek, included glowing profiles of careerists like Yakov Peters, a sadistic former Chekaist who had overseen the repression of "counter-revolutionaries" in St. Petersburg during 1919.50 The purge of the Uzbek leadership began in 1937 and lasted through 1938. (Yakov, for example, was arrested and shot in 1938.) Upon learning of each arrest, Rodchenko dutifully ruined his personal copy of _Ten Years of Uzbekistan_ by "disappearing" the victim in thick black ink.51 "Death to Art!" had reached its apogee.
_Liubov Popova_ , Stage Set for The Magnanimous Cuckold, _Meyerhold Theater, Moscow, 1922._
_Liubov Popova_ , Stage Set for The Magnanimous Cuckold, _Meyerhold Theater, Moscow, 1922._
### NOTES TO CHAPTER 5
1 Stalin quoted in Mark R. Beissinger, _Scientific Management, Socialist Discipline and Soviet Power_ (Cambridge, MA: Harvard University Press, 1988): 19.
2 See the definative study on constructivism; Christina Lodder, _Russian Constructivism_ (New Haven: Yale University Press, 1983), 112.
3 The largest artists' organization of the period, the Association of Artists of Revolutionary Russia (1922-1932), practiced painting and sculpture in a realist style, as did lesser groups such as the Society of Easel Painters and the New Society of Painters. See Brandon Taylor, "On AkhRR," _Art of the soviets: Painting, Sculpture and Architecture in a One Party State, 1917-1992,_ Matthew Cullerne Bown and Brandon Taylor, eds. (Manchester: Manchester University Press, 1993): 51-72.
4 Varvara Stepanova, "The General Theory of Constructivism," (December 22, 1921) in Noever, 174.
5 Ibid.
6 Ibid.
7 Ibid., 177.
8 Gan's role is discussed in Maria Gough, _The Artist as Producer: Russian Constructivism in Revolution_ (Berkeley: University of California Press, 2005): 68.
9 Alexei Gan, _Constructivism_ (1922) in _The Tradition of Constructivism,_ Stephen Bann, ed. (New York: Da Capo Press, 1974): 36-37.
10 Ibid., 41.
11 Ibid., 40.
12 See Lodder, 24-29.
13 Ibid., 7.
14 Ibid., 133-139.
15 Ibid., 147-152, 263.
16 See V.I. Lenin, "The Immediate Tasks of the Soviet Government" (April 28, 1918), quoted in Maurice Brinton, _The Bolsheviks and Worker's Control_ (Montreal: Black Rose Press, 1975): 40-41.
17 Edward Braun, _Meyerhold: A Revolution in Theatre_ (London: Methuen, 1998): 160, 171.
18 Lancelot Lawton, _The Russian Revolution_ (London: Macmillan and Co., 1927): 376.
19 Braun, _Meyerhold: A Revolution in Theatre,_ 172.
20 Judith A. Merkle, _Management and Ideology: The Legacy of the International Scientific Management Movement_ (Berkeley: University of California Press, 1980): 11-15.
21 Ibid., 75-80.
22 Frank B. Gilbreth and L. M. Gilbreth, _Applied Motion Study: A Collection of Papers on the Efficient Method to Industrial Preparedness_ (New York: Sturgis and Walton, 1917): 37.
23 Beissinger, 32.
24 Leon Trotsky quoted in ibid.
25 Ibid.
26 Merkle, 105-109.
27 Lenin quoted in Merkle, 113.
28 Beissinger, 37-44.
29 Gastev quoted in Beissinger, 33.
30 Gastev quoted in Kurt Johannson, _Aleksej Gastev: Proletarian Bard of the Machine Age_ (Stockholm: Almqvist and Wiksell, 1983): 62.
31 Gastev quoted in ibid., 61-62.
32 Gastev quoted in Kendell E. Bailes, "Alexei Gastev and the Soviet Controversy over Taylorism, 1918-24," _Soviet Studies 29_ July 1977): 378.
33 Catherine Cook, _Russian Avant-Garde: Theories of Art, Architecture, and the City_ (London: Academy Group, 1995): 112.
34 Beissinger, 54-55.
35 Braun, _Meyerhold: A Revolution in Theatre,_ 172.
36 Johannson, 113.
37 Lawton, 375.
38 Ibid.
39 Vsevolod Meyerhold, "Biomechanics" (June 12, 1922), in _Meyerhold on Theatre,_ Edward Braun, ed. (New York: Hill and Wang, 1969): 199.
40 Ibid.
41 Dmitri V. Sarabianov and Natalia L. Adaskina, _Popova_ (New York: Harry N. Abrams, 1990): 198-200.
42 Luibov Popova, "Discussion of the Magnanimous Cuckold" (April 27, 1922), in _Popova,_ 378.
43 Ibid., 378-379.
44 Braun, _Meyerhold: A Revolution in Theatre,_ 184-185, 195.
45 Beissinger, 84.
46 Ibid., 89.
47 Merkle, 132.
48 Matthew Cullerne Bown, _Socialist Realist Painting_ (New Haven: Yale University Press, 1998): passim.
49 Alexandr Rodchenko, "Reconstructing the Artist" (1936), in Larentiev, 237-304.
50 David King, _The Commissar Vanishes: The Falsification of Photographs and Art in Stalin's Russia_ (New York: Metropolitan Books, 1997), 133.
51 Ibid., 126-133.
## CHAPTER 6
GAY ANARCHY _Sexual Politics in the Crucible of McCarthyism_
In the United States, as opposed to Europe, World War II is still remembered as the "good war" for a reason. Compared to the "relatively gentle breezes that reached America's shores," writes Ralph Levering, "the winds of war that pounded Europe and Asia from 1939 to 1945 were like a six-year-long hurricane":
A sizable portion of the cities on the great Eurasian landmass and its adjoining island states—Britain and Japan—were dam-aged severely. London was hit repeatedly early in the war by German bombers, and later by German rockets. Rotterdam and other cities in the Netherlands were bombed mercilessly in 1940. Berlin was pummelled by Allied bombers until, as American diplomat Robert Murphy observed, "the odor of death was everywhere"; other German cities, like Dresden and Stuttgart, were fire-bombed until tens of thousands of the residents were charred beyond recognition. To the east, fierce fighting virtually levelled thousands of cities and towns in Eastern Europe, Russia, and China, and a concerted American bombing campaign against Japan in the last years of the war turned large areas of compact Japanese population centres into rubble. Estimates of war related deaths in all the countries involved run as high as 55 million, of whom roughly 20 million were Russians.1
_The Sheldon-Claire Company_ , This is America..., _1942. Poster._
Protected by two oceans, the United States' economy boomed while civilian life (Japanese-Americans exempted) went on as usual. Peace and prosperity on the home front proved a boon to government propaganda, which mobilized the population around the myth that they lived in an equi-table, bountiful, democratic paradise bursting with freedoms. As one 1942 poster boasted, "This is America ...where every boy can dream of being President. Where free schools, free opportunity, free enterprise, have built the most de-cent nation on earth. A nation built upon the rights of all men."2
Seemingly, in the minds of the American public, the government could do no wrong. Indeed, on August 8, 1945, two days after the atomic bomb was dropped on Hiroshima, 85 percent of those surveyed favored the action, and 96.5 percent of those polled following the bombing of Nagasaki approved of future bombings, if required.3 Dwight Macdonald, the anarchist-pacifist editor of the New York journal _Politics_ (1944-1949), was one of the few to offer another pointof-view: dropping the bomb proved conclusively that the much vaunted democracies of the United States and Britain were stage-managed affairs run by bureaucrats completely devoid of any humanitarian sentiment. He wrote:
It seems fitting that the bomb was not developed by any of the totalitarian powers ... but by two democracies (Britain and the United States), the last major powers to continue to pay at least ideological respect to the humanitarian-democratic tradition. It also seems fitting that the heads of these governments are both colorless mediocrities. Average men elevated to their positions by the mechanics of the system. All this emphasizes that perfect automatism, that absolute lack of human consciousness or aims which our society is rapidly achieving.... The more commonplace the personalities and senseless the institutions the more grandiose the destruction.4
Protests from lone figures like Macdonald, however, could not stem post-war reaction as the United States government rallied its popu-lace for a new "Cold War" against its former ally, the Soviet Union. "Beginning in the late 1940s and continuing until the mid-1960s" the bellicose pro-America, pro-capitalist, anti-radical consensus encompassed "political parties, labor unions, and business groups, mass circulation magazines and daily newspapers, ethical and religious groups, veterans and professional organizations, and liberal and conservative interest groups."5 These were tough times to be an anarchist.
The American public's hostility towards radicalism was com-pounded by the sorry state of anarchism in the United States. Before World War II, the movement was already much diminished, thanks to government persecution during World War I and the mass appeal of the American Communist Party in the 1920s and '30s. The cataclysmic defeat of Spain's anarchist-syndicalist movement during the Spanish Civil War (1936-1939) was another bitter blow, and World War II brought further calamity: the Jewish-American wing of the movement supported the allied war effort in the name of fighting fascism while Italian- and Spanish-American anarchists denounced both sides as imperialists. By 1945, "the divisions caused by the war," writes Paul Avrich, "left the anarchists in a shambles, and what had once been a flourishing movement shrank to the proportions of a sect."6
Responding to these circumstances in the winter of 1945, Holly Cantine, co-editor with Dachine Rainer of the journal _The Retort_ (1942-1951), called for a return to anarchism's communal roots.7 _The Retort_ was published from Cantine's house near Woodstock, New York, an arts-and-crafts socialist-anarchist community of artists and intellectuals founded in the 1900s.8 Cantine had grown up in Woodstock and, after a brief stint of graduate studies at Columbia University, returned to build a house not far from the village. Rainer met Cantine at the offices of _Politics_ around 1945 and joined him at his home on Mount Tobias, not far from Woodstock, after a brief courtship.9 There, they lived self-sufficiently and worked to establish a community of the like-minded.10
_Holley Cantine and Dachine Rainer, Woodstock, NY, ca 1946-1950. Photograph._
In his editorial on strategies for building an American anarchist movement in the post-war era, Cantine proposed that activists abandon party politics and union-based organizing in favor of decentralized, non-hierarchical cooperative initiatives along the lines of his own efforts.11 In this way, the radical could serve as "the precursor of a new society, an individual who has broken with the values of the status quo, and has created for himself a new way of life based on a more equitable set of values."12 Cantine underlined that he was not proposing a retreat from the world. On the contrary, the spread of anarchism by example, he predicted, would inspire workers to walk away from "existing institutions, and cause them to collapse."13
_Robert Duncan, ca 1944. Photograph._
At least, that was his hope. In any event, immediately after the war his program found an echo in San Francisco, where former Woodstock resident and anarchist poet Robert Duncan helped found a weekly discussion group called the Libertarian Circle. The Circle began in early 1946, when Duncan and Philip Lamantia proposed founding an "open and above-board" anarchist discussion group to fellow anarchist Kenneth Rexroth. Rexroth had been living in San Francisco since 1927 and was well-known in the arts community for his poetry and literary criticism; with his backing, the project prospered.14 From humble beginnings in Rexroth's living room, the Libertarian Circle moved to the top floor of a building occupied by the Jewish anarchist group Arbeiter Ring, where they met weekly to discuss topics such as anarchism and literary mysticism, Emma Goldman, the Kronstadt revolt, and sexual anarchy (the latter necessitating two simultaneous discussions, "one upstairs, the overflow in the downstairs meeting hall").15 The group also rented a hall for monthly dances and augmented their activities with a weekly Poetry Forum. Every other Wednesday, one writer's work was read, followed by a discussion led by the poet himself; alternative Wednesdays were reserved for seminars in poetry and criticism led by Rexroth.16 In addition, the Libertarian Circle published a one-issue journal, _Ark,_ in 1947, which featured reproductions of paintings, prose essays and poetry, statements on anarchism by George Woodcock and Amon Hennacy, and an article by Duncan on the sexual politics of art entitled "Reviewing _View,_ An Attack" that took aim at America's premier surrealist journal, _View,_ launched in September 1940 by Charles Henry Ford.
Ford was a poet and surrealist enthusiast of independent means who, like Duncan, was "out" among his friends as homosexual.17 He ran the quarterly as an unabashedly commercial enterprise from a New York penthouse office located above a chic nightclub not far from the Museum of Modern Art. Each issue was linked to exhibitions in the city's high-end galleries, which helped to finance the magazine through advertising (alongside pitches for perfumes and lotions).18 Articles on surrealist theorists such as George Bataille jostled with advertisements for publications by Andre Breton; reproductions of artwork by Leonor Fini, Rene Magritte, and the sado-surrealist Hans Bellmer ("creator of a moveable woman whose body, taken apart and re-composed against the laws of nature ... revealed an acute sense of the marvellous allied to a profound nostalgia for childhood and periods dominated by femininity mixed with spurts of violent eroticism"); _View_ art postcards for sale; (illustrated) essays on topics such as "The American Macabre," "Shrinking the Heads," and "Geatano Zumbo and Death"; and, last but not least, a repeat full-page advertisement for Helena Rubinstein Galleries (New York-Paris-London), founded by the cosmetics industrialist, announcing _"art_ knows no frontiers, _beauty_ knows no limitations."19
The magazine was incorporated as a business _(View Inc.)_ and its office doubled as a bookstore and commercial gallery; it also sponsored jazz concerts and occasional select exhibitions. (Ford wrote about the "brilliant and chic" crowd attending these events.) But _View's_ reign as the _"world's leading journal of avant-garde art and literature"_ came to an unexpected halt in the summer of 1947, when its editor abruptly closed shop and departed for Europe.20
In the 1940s, Duncan knew Ford, having encountered him after moving to the New York area in 1939 to pursue a love affair begun during his undergraduate studies at Berkeley.21 The affair ended in 1940 and Duncan, at loose ends, relocated to Woodstock, where for a time he shared a house with James and Blanche Cooney, editors of _The Phoenix_ (1938-1940). Ideologically, the Cooneys stood for:
View _cover, Series V, No. 6, January 1946; cover image by John Tunnard._
... the unequivocal condemnation of Industrial forms of society, whether they be of Capitalist (with all its varying shades of Democratic, Liberal, Conservative, Technocratic, etc.), Marxian Communist, Fascist or Nazi variety [and] the un-swerving determination to serve under none of these degrading, deathly states, but to break away in small communities, in small precursors of a resurrection and renascence of mankind through a return to the dignity and purity and religiousness of a mode of life rooted in agriculture and the handicrafts.22
The Cooneys likened these communities of refuge to "Arks," and this is probably what Duncan and his collaborators had in mind when they named their post-war publication: a contrarian resistance to Cold War society.23 _Ark's_ opening editorial stated:
In direct opposition to the debasement of human values made flauntingly evident by the war, there is rising among writers in America, as elsewhere, a social consciousness which recognizes the integrity of the personality as the most substantial and considerable of values.... Present-day society, which is be-coming more and more subject to the state with its many forms of corrupt power and oppression, has become the real enemy of individual liberty. Because mutual aid and trust have been coldly, scientifically destroyed; because love, the well of being, has been methodically parched; because fear and greed have become the prime ethical movers, states and state-controlled societies continue to exist. Only the individual can cut himself free from this public evil. He can sever the forced relations between himself and the state, refuse to vote or go to war, refuse to accept the moral irresponsibility yoked into him. Today, at this catastrophic point in time, the validity if not the future of the anarchist position is more than ever established. It has become a polished mirror in which the falsehoods of political modes stand naked. No honest person, if he has looked into this mirror, can morally support a government of any description, whether it be a state-capitalist Soviet Union, a capitalist America, a fascist Spain, or any considered society wherein an idea is woven into a blanket of law and cast over a living people from above. Any inorganic thing made authority over the organic is morally weakening and makes annihilating warfare inevitable.... We believe that social transformation must be the aim of any revolutionary viewpoint, but we recognize the organic, spontaneous revolt of individuals as presupposing such a transformation.24
While living with the Cooneys, Duncan commuted down to New York to take part in Manhattan's social circles and meet with other literary radicals, including Dwight Macdonald. He also took new lovers, avoided serving in the war by declaring his homosexuality, spent a summer working as a dishwasher in Provincetown, traipsed about Florida, and gained a reputation as the poet of note who, in 1944, had signed his name to an article in _Politics_ on "The Homosexual in Society."25 This was an anarchist assertion of sexual libertariamsm and a critique of the pervasive "homosexual cult" amongst gays.26 Notions of difference that encouraged "snobbery and removal from the 'common sort,'" argued Duncan, merely encouraged gays to add their voices to a range of oppressions—including heterosexism—across the existing social landscape.27 The truly liberating "starting point" for "creative life and expression," countered Duncan, was:
... a devotion to human freedom, toward the liberation of human love, human conflicts, human aspirations. To do this one must disown all the special groups (nations, religions, sexes, races) that would claim allegiance. To hold this devotion every written word, every spoken word, every action, every purpose must be examined and considered. It must always be remembered that others, those who have surrendered their humanity, are not less than oneself. It must be always remembered that one's own honesty, one's battle against the inhumanity of one's own group (be it against patriotism, against bigotry, against, in this specific case, the homosexual cult) is a battle that cannot be won in the immediate scene. The forces of inhumanity are overwhelming, but only one's continued opposition can make any other order possible, will give an added strength for all those who desire freedom and equality to break at last those fetters that seem now so unbreakable.28
Duncan's attack on _View_ in 1947 was premised on this stance. He began by accusing Ford of marketing art as an "experience" spiced with "Freudian menace" for "wealthy dilettantes" who reveled in "the sheer expensively bought spectacle of it,"29 an approach to art that Duncan characterized as "aesthetic."30 By way of example, he pointed to _View's_ publication of an anti-war story, "The Buzzard," which described the hungering desires of a bird hovering over a battlefield; the story was sandwiched between Bataille-style "documents"—illustrated by photos of physically deformed people—reporting on human cruelties and abnormalities in Central America. Thus, "The Buzzard's" politics were aestheticized and subsumed within a nightmarish mix of facts, imagery, and fiction. "In the world of _View,"_ Duncan observed, "horror becomes an end in itself—not a rejection, but an acceptance, more than that, a tremulous embrace of what was horrible, a sensation which may be tasted by the reader for the vicarious thrill of it."31 Sadly, however, surrealism for consumers was doing very well on the magazine rack: "like The Buzzard, it _[View]_ draws its profit and substance from the battlefields, the misery and deformity of modern society; and in contemporary America, where the populace at large relishes the charnel havoc wrought upon the cities of Europe and Japan, _View's_ circulation booms."32
In sum, the magazine's success was symptomatic of a larger social sickness, but it was also symptomatic of Ford's internalization of the lies American society imposed on him. Accepting the belief in a singular "normal human being" propagated by an establishment so "hostile to individuation" that it "describes and debases any individual in terms of his deviation," Ford had embraced the identity of the deviant homosexual "freak" and was now "suing for little more than that the world allow him his 'freakishness.'"33 Tragically, this led him to prop up the very status quo he should oppose: "He turned to write for, and to live in the mi-lieu that might accept him and that at the same time had the power to provide a protection of a kind; he moved from the outcast legions of low Bohemia to high Bohemia on the margins of that ever curious and hungry section of society, the money-aristocracy."34 In this regard, Ford was following the example of Breton and the surrealists in exile who, upon arriving in New York in the early 1940s, had been "taken up and taken in by the culture collectors." "All the drama of the real political world" was then "played in charade to give excitement to the boredom of the rentiers" as Breton and company "capitalized upon their revolutionary personalities."35 Ford helped sell surrealism to this clientele, and Breton valued him for it.36
_"Two Documents," page spread from_ View, _May 1945._
To its credit, _View_ during these years was also "unremittingly hos-tile toward the State and its war." To this extent, it was "anarchistic—against the State."37 However, Ford's perpetual chewing on "the cud of fin-de-siecle diabolism" suggested he was "hostile not only to the State, but to the individual" as well.38 "The real menace in the shadow of which we all live—the twentieth-century State or what is so aptly called the Permanent War Economy," reasoned Duncan, was out to crush authentic individuality.39 _View_ raised no resistance to this, because "if to be an individual means individual responsibility, _View_ with its allegiance to the rentier-aristocracy is hostile to individualism."40 Rather than caving in to this social strata's "lust for a thrill" in light of its supposed "superiority to the mores of the modern State," Duncan called on Ford to cultivate the "potential awakening to productive and creative life" that anarchism sought.41 "Experience" should be "something the artist struggles to transform into a field for achieving his or her desires."42 Ford had lost sight of this, and with it, art's insurrectionary dimension.
_"A Paris..." page from_ View, _Vol. VI, Nos. 2-3, March-April 194.6._
Contrast Ford's actions with Duncan's. The Libertarian Circle and _Ark_ were communal endeavors free of capitalization whose purpose was to enliven the creative lives of the participants at the same time as they radicalized them. As such, they represented a step towards the exemplary "new way of life based on a more equitable set of values" that Cantine had called for in _Retort._ Both projects were destined to wind down in 1948, but this did not deter Duncan. In 1949, he helped start a third initiative, the Poetry Conference, where writers and artists gathered once a week for readings and discussions. Here, Duncan met and fell in love with a young painter, Jess Collins; the two formed an enduring relationship that lasted until Duncan's death in 1988.43
Born in California, Collins had been drafted into the atomic bomb "Manhattan Project" while studying chemistry at the California Institute of Technology. He was sent to the project's Oakridge, Tennessee facilities to work on enriching uranium, and on his twenty-second birthday—August 6, 1945—the United States dropped the fruits of his labor on Hiroshima. Upon release from the military, Collins returned to the California Institute and completed a Bachelor of Science degree in radiochemistry. He rejoined the nuclear war industry in 1948 as a technician, producing plutonium at the Hanford, Washington Atomic Energy Project, but in 1949 he abruptly quit and moved to San Francisco to study art. In a 1992 interview, Collins said that the decision was inspired by "a very strong and convincing dream that the world was going to completely destruct by the year 1975." "I'm sure," he continued, "the kind of work I was doing had some effect on my state of mind at the time."44
He later commemorated this vision during the Cuban missile standoff of October 1962, when the United States and the Soviet Union came within hours of starting a nuclear war. His painting _If All the World Were Paper and All the Water Sink_ (1962) (see color plate 8), depicts him (or perhaps Duncan) in profile, looking upon a sunlit glen where children are dancing in a circle. An owl bearing a key swoops down over what at first looks like a bucolic scene until we gaze into the distance, where a mushroom cloud is erupting skyward. For Collins, the circle dance is an act of sun worship symbolic of children's life-energy and their "wondrous ability to infinitely connect images and stories without having to segregate everything."45 The destructive power of the bomb, on the other hand, is the antithesis of nature's spontaneous, ever conjoining life-force: a weapon of mass destruction deployed for divisive political ends by government bureaucrats.
During his time at Hanford, Collins had begun painting as an "antidote to the science method."46 "I wanted to do something that was truly meaningful to me," he recalled, and art "was far more meaningful than making plutonium."47 We get a sense of what the term "meaningful" signifies when he goes on to describe the impact of his training under the anarchist abstract painter Clyfford Still at the California School of Arts, where Jess enrolled as a student in 1949-51. Still's lectures, Collins recalled, were "very moving in terms of my understanding of the passion of the immediate image and the difficulty an artist has in arousing a sense of spirit in a societal structure that tends to suppress it."48
Still was an accomplished figural painter who had turned to nonobjectivity in the 1940s. He explained why in 1963 when he dismissed the war and its aftermath as an era of "mechanism, power, and death," concluding "I see no point in adding to its mammoth arrogance the compliment of graphic homage."49 Still's pessimistic outlook had led him to retreat from social struggle; painting provided him with the one "limited arena" where his "negative dialectic of creative freedom" found resolution beyond the restrictions of the Cold War.50
But freedom was tenuous once an artwork left the studio, and in this regard, Still was particularly hostile toward establishment art critics and the cultural institutions they served. He saw them as operatives in the "authoritarian devices for social control" that dampened the spontaneous reaction of the public to art and the artist's capacity to be innovative. In the post-war years, when his work began attracting attention, he strenuously objected to mainstream art critics "shouting about individualism" in American art while they buried the dissident politics of his work under the weight of formalist aesthetics ("superficial value of material"). "Behind these reactions," he wrote, was "a body of history matured into dogma, authority, tradition. The totalitarian hegemony of this tradition I despise, its presumptions I reject."51 In an attempt to deal with the situation, Still insisted that his work was not to be shown to or discussed by anybody who lacked insight into the aesthetic and moral values his paintings embodied. In particular, he refused access to James Thrall Sobey (art critic), Clement Greenberg (art critic), and Alfred Barr (director of the Museum of Modern Art): all important players in the cultural marketing of non-objective painting as a symbol of American freedom during the Cold War.52
At the California School of the Arts, Still urged his students to adopt the same strategy, but apart from that they were free to follow their own path.53 According to Collins, though personally committed to abstraction, Still "never dictated an aesthetic" and encouraged his students to reject any preconceptions regarding what was "good, right or proper" in painting.54 Consequently, when Collins began noticing imaginative "scenes or fantasies" in his own non-objective paintings, he welcomed them as features worth exploring.55 Around 1951, just after moving in with Duncan, this led him to adopt collage "as a way ... to construct imaginary scenarios in a more realistic rather than a non-objective way."56 His first large-scale collage "paste up," _The Mouse's Tale_ (1951) (see color plate 9), was also a personal statement. Collins conceived of this work as a self-conscious reclaiming of the male body—the homosexual object of desire—from the domination of heterosexual macho presentations. As he put it, he was "showing innocent beauty," albeit with a decidedly gay sensibility.57
The gesture might not seem radical until put in context. Collins created his collage at a time when being gay was something associated with the criminal underworld, leftism, scandal, and mental illness. People lost their jobs and went to jail for it.58 Responding to this state of affairs, _The Mouse's Tale_ doesn't agitate, polemicise, protest, or pro-claim a position; rather, by celebrating male beauty, Collins followed Duncan's dictum in "The Homosexual in Society," to configure his own creativity as a starting point for "devotion to human freedom, toward the liberation of human love, human conflicts, human aspi-rations."59 This is the sense in which _The Mouse's Tale_ was radical—Collins' personal breakthrough into sexually-charged imagery and his refusal to participate in the American war machine unfolded along the same political continuum.
And there were further developments. In 1952, Collins and Duncan teamed up with the artist Harry Jacobus to found an independent artist-run gallery which they named after French satirist Alfred Jarry's (see Chapter Three) theatrical parody of a stupid, bumbling European bourgeois imperialist, "King Ubu."60 The agenda of King Ubu was resolutely non-commercial and experimental—to avoid any monetary schemes arising from the venture, all three founders agreed to run it for one year (December 1952-December 1953), and then close it down.61 One could say that Collins, Duncan, and Jacobus were realizing the anti-capitalist ethos propagated by Still by creating a non-commercial exhibition space where artists were subject only to the judgement of their peers or those who expressed enough interest to search out the gallery, which was located in a run-down section of the city. Over the course of its existence, the King Ubu hosted fif-teen exhibitions, two plays staged by Duncan, regular Sunday literary meetings involving readings by former Libertarian Circle participants (Rexroth, Lamantia, and others), two experimental film screenings, and musical performances.62
Apart from the King Ubu experiment and a host of similar projects that followed, the domestic sphere was also important. Here, through the 1950s and 1960s, Collins and Duncan hosted artistic events and circulated privately-produced publications free from the gaze of censors and critics.63 In effect, Collins and Duncan transformed their home into an community-building sphere for self-expression and ex-change between friends and acquaintances, which deepened mutual understanding while enriching lives on a profound level. Collins and Duncan enacted anarchism on an intimate scale, and their post-war art and activism were of a piece with this ethos: creating authentically, they inspired others to do the same.
### NOTES TO CHAPTER 6
1 Ralph B. Levering, _The Cold War: 1945-1987_ (Arlington Heights, IL: Harlan Davidson, 1988): 14.
2 I am quoting the text of a World War II poster, "This is America ... Keep it Free!" (1942) in William L. Bird, Jr. and Harry R. Rubenstein, _Designing for Victory: World War II Posters on the American Home Front_ (Princeton: Princeton Architectural Press, 1998): 85.
3 James Tracy, _Direct Action: Radical Pacifism from the Union Eight to the Chicago Seven_ (Chicago: University of Chicago Press, 1996): 51; Lawrence S. Wittner, _Rebels Against War: The American Peace Movement, 1941-1960_ (New York: Columbia University Press, 1969), 106.
4 Dwight Macdonald, "The Bomb," (September 1945) in _Memoirs of a Revolutionist_ (New York: Meridian Books, 1958): 177.
5 Levering, 56.
6 Paul Avrich, _Anarchist Voices_ (Princeton: Princeton University Press, 1995): 417.
7 Holley Cantine, Jr., "Editorials," _Retort 2_ (Winter 1945): 2-9
8 Dachine Rainer, "Holley Cantine: February 14, 1916-January 2, 1977," _Drunken Boat: Art, Rebellion, Anarchy,_ Max Bleckman, ed. (New York and Seattle: Autonomedia and Left Bank Books, 1994): 178.
9 Cantine's first companion, Dorothy Paul, served as co-editor of _The Retort_ from its founding in 1942 through to 1945. After Cantine began his relationship with Rainer, Paul departed and Rainer took on co-editorship. See ibid., 182.
10 Ibid., 184.
11 Cantine, Jr., "Editorials," 7.
12 Ibid., 3.
13 Ibid., 4.
14 On Rexroth's life, see Ken Kabb, _The Relevance of Rexroth_ (Berkeley; Bureau of Public Secrets, 1990): 49-55, and Kenneth Rexroth, "Interviews," _The San Francisco Beat Poets,_ David Meltzer, ed. (New York: Ballantine, 1971): 13. On the founding of the Libertarian Circle, see Ekbert Faas, _Young Robert Duncan: Portrait of the Poet as Homosexual in Society_ (Santa Barbara: Black Sparrow Press, 1983): 192.
15 Rexroth quoted in ibid.
16 Ibid., 193
17 Dikran Tashjian, _A Boat Load of Madmen: Surrealism and the American AvantGarde, 1920-1950_ (New York: Thames and Hudson, 1995): 157-160.
18 Ibid., 199-200. See, for example, the back cover of _View_ 5 (May 1945).
19 See _View,_ vols. 6-7 (February 1946-March 1947).
20 Ford quoted in Tashjian, 200-201.
21 Faas, 72.
22 James Peter Cooney, "Editorial," _The Phoenix 2_ (Spring 1939): 120.
23 Ibid., 123.
24 "Editorial," _Ark_ (1947) quoted in Lawrence Ferlinghetti and Nancy J. Peters, _Literary San Francisco: A Pictorial History from Its Beginnings to the Present Day_ (San Francisco: City Lights Books, 1980): 155-156.
25 Faas, 123.
26 Ibid., 61-145.
27 Robert Duncan, "The Homosexual in Society," _Politics_ 1 (August 1944): 209-111.
28 Ibid., 111.
29 Robert Duncan, "Reviewing View, An Attack," _The Ark_ 1 (1947): 63.
30 Ibid., 63-64.
31 Ibid., 64. French surrealist George Bataille's journal, _Documents_ (1929-1932), published many photo features of slaughterhouse scenes, killings, human deformities, etc., alongside celebratory articles on topics such as sadism, torture, rape, cannibalism, murder, and human sacrifice, usually visited upon helpless women or children. Such preoccupations are indicative of the movement. See Briony Fer, David Batchelor, and Paul Wood, _Realism, Rationalism, Surrealism: Art Between the Wars_ (New Haven: Yale University Press, 1993): 200-209 and Rudolf Kuenzli, "Surrealism and Misogyny," _Surrealism and Women,_ Mary Ann Caws, Rudolf Kuenzli, and Gwen Raaberg, eds. (Cambridge, MA: MIT Press, 1991): 17-26.
32 Duncan, "Reviewing _View"_ 64.
33 Ibid.
34 Ibid., 65.
35 Ibid., 66.
36 See Tashjian, 188-198.
37 Duncan, "Reviewing _View"_ 67.
38 Ibid.
39 Ibid., 65.
40 Ibid., 67.
41 Ibid.
42 Ibid., 63.
43 Patrick Frank, "San Francisco, 1952: Painters, Poets, Anarchism," _Drunken Boat: Art, Rebellion, Anarchy,_ 146.
44 Michael Auping, "An Interview with Jess," _Jew: A Grand Collage, 1951-1993_ (Buffalo: Albert-Knox Art Gallery, 1993): 19.
45 Collins quoted in Michael Auping, "Jess: A Grand Collage," _Jess: A Grand Collage, 1951-1993,_ 46.
46 See "Chronology," _Jess: A Grand Collage, 1951-1993,_ 234. Collins is cited in Frank, 146.
47 Auping, "An Interview with Jess," 19.
48 Ibid., 20.
49 Clyfford Still, "Statement (1952)" and "Statement (1963)," quoted in David Craven, _Abstract Expressionism as Cultural Critique: Dissent During the McCarthy Period_ (Cambridge: Cambridge University Press, 1999): 166.
50 David Anfam, "Clyfford Still's Art: Between the Quick and the Dead," _Clyfford Still: Paintings 1944-1960,_ James T. Demetrion, ed. (Washington: Hirshhorn Museum and Sculpture Garden, Smithsonian Institution, 2001): 36.
51 Clyfford Still, "Statement (1959)," _Theories of Modern Art,_ ed. Herschel. B. Chipp (Berkeley: University of California Press, 1969): 574-575.
52 Clyfford Still to Betty Parsons, March 20, 1948, quoted in Serge Guilbaut, _How_ _New York Stole the Idea of Modern Art: Abstract Expressionism, Freedom, and the Cold War,_ Arthur Goldhammer, trans. (Chicago: University of Chicago Press, 1983): 201. On the critical appropriation of abstract painting by Still and others to the cause of the Cold War, see ibid., 166-194.
53 Richard Candida Smith, _Utopia and Dissent: Art, Poetry, and Politics in California_ (Berkeley: University of California Press, 1995): 102.
54 Auping, "An Interview with Jess," 20.
55 Ibid.
56 Ibid.
57 Ibid., 25.
58 Jennifer Terry, _An American Obsession: Science, Medicine, and Homosexuality in Modern Society_ (Chicago: University of Chicago Press, 1999): 329-352.
59 Ibid., 111.
60 Seymour Howard, "King Ubu: Gallery in a Theater," _The Beat Generation Galleries and Beyond_ , John Hatsoulas, ed. (Davis, CA: John Natsoulas Press, 1996): 23. On Jarry's "Ubu" character, see Patricia Leighten, "The White Peril and L'Art negre: Picasso, Primitivism, and Anticolonialism," _Art Bulletin 72_ (December, 1990): 621-622.
61 Howard, 23.
62 Ibid., 26-27.
63 According to Robert J. Bertholf, domestic households became a locus of activity in San Francisco during the 1950s, thanks in part to the example of Collins and Duncan. Among other projects, Collins and Duncan held poetry readings and performances in their home. See Robert J. Bertholf, "The Concert: Robert Duncan Writing Out of Painting," _Jess: A Grand Collage, 1951-1993,_ 75-76. They also participated in a co-operative publication, _The Artist's View_ (1951-1954), which circulated privately amongst the participants. Each issue was produced by an individual in the group. See Smith, 236.
## CHAPTER 7
BREAKOUT FROM THE PRISON HOUSE OF MODERNISM _An Interview with Susan Simensky Bietila_
The 1960s are rightly viewed as a time of renewal in the history of twentieth-century anarchism, when mass uprisings in places as far flung as the United States, France, Czechoslovakia, and Mexico challenged the status quo in explicitly anti-authoritarian terms. In America, the civil rights movement merged with the anti-Vietnam War movement, giving rise to a richly diverse counterculture with strong anarchist currents that carried over into the 1970s. Marxist turned anarchist Murray Bookchin nicely encapsulated the gulf between the old left and the emergent counterculture. Recalling Marxist-dominated politics prior to the 1960s, he wrote:
'Life-style?'—the word was simply unknown. If we were asked by some crazy anarchists how we could hope to change society without changing ourselves, our relations with each other, and our organizational structure, we had one ritualistic answer: '... after the revolution.' 'After the revolution....'—this was our magic talisman. It expressed our incredibly naive belief that merely by 'abolishing' the economic relationships and institutions of capitalism we would thereby abolish the bourgeois family, the bourgeois state, and bourgeois attitudes towards sexuality, women, children, indeed toward people and life as a whole. (The gross deception here—a deception which lies at the very core of Marxism—is that changes in the pre-conditions of society and life are equivalent to changes in the conditions of society and life, a fallacy which blatantly mistakes the sufficient reason for the necessary reason.) And this 'beautiful revolution' would be realized by using bourgeois methods of organization and involved bourgeois relations between people. We totally failed to recognize that our methods and relations were subverting our goals, indeed our very personalities as revolutionaries.1
Contemporary "Youth Culture," on the other hand, was rife with potential: "In its demands for tribalism, free sexuality, community, mutual aid, ecstatic experience, and a balanced ecology," wrote Bookchin, it prefigured, "however inchoately, a joyous communist and classless society, freed from the trammels of hierarchy and domination, a society that would transcend the historic splits between town and country, individual and society, and mind and body."2
The 1960s did indeed mark a sea change, at least in terms of who was articulating what radicalism was. For example, this was the era when Noam Chomsky began speaking out against the foreign policies of the United States government from an anarchist viewpoint. While Chomsky critiqued American politics, Bookchin popularized anarchism's ecological dimension. At the time, anarchist-feminism was renewed in part thanks to the tireless efforts of Alex Kates Shulman and Richard Drinnon to promote the life and writings of Emma Goldman. In the previous chapters, I discussed gay poet Robert Duncan and his role in the American poetry scene; a second voice championing libertarian sexuality was social theorist Paul Goodman, author of the best-seller _Growing Up Absurd_ (1960). In addition, poets Diane di Prima, Gary Snyder, and others promoted connections between anarchism, poetics, and spirituality while John Cage explored its musical ramifications. Finally, the Living Theatre collective developed and popularized their distinctive variation of anarchist-pacifism in the United States and Europe.
I have long been interested in what role anarchist visual artists played during these years. I was fortunate, then, to make the acquaintance of one of anarchism's better-known contemporary artists, Susan Simensky Bietila. During the 1960s, she worked as an illustrator for the activist press in New York while completing an undergraduate degree in art under the tutelage of prominent abstractionist painters. In the following interview, conducted by email, she sheds considerable light on the ways in which the mainstream art world of the 1960s maintained a separation of art and politics at the same time as the American counterculture was failing to realize its anarchic potential.
What was it like growing up in New York?
I was born in 1947 and grew up working-class in Brooklyn. The community was largely Eastern European Jewish, and my family lived in a Federal Housing Project apartment. When I was a kid, I was recruited out of kindergarten on an art scholarship to the Brooklyn Museum School; by the time I was six, I was traveling on the subway by myself to Saturday art classes. I went to the High School of Music & Art and had studio classes as well as art history. There I met bohemian teens from Greenwich Village and heard about the existence of the anti-nuclear bomb group, Student Peace Union.3
My political activism began in the summer of 1964 when I worked at Camp Twin Link, run by covert Communist Party USA members. This same camp also ran a neighborhood after-school program I had attended as a child. It was linked to the Atlanta School of Social Work, which was a hotbed of civil rights activism and connected with the Student Non-violent Coordinating Committee.4 Other counselors were civil rights activists and college students who were in the Students for a Democratic Society [SDS].5 Many of them had been sent to this or other political summer camps as children by their leftist parents. At the camp, I was caring for five-year-old boys and less than delighted with this sort of work, being too poor to have been sent to summer camp myself, and feeling as if there was some hypocrisy at essentially being a nanny for radicals. I found out about the war in Vietnam and the United States' role in it from Paul Millman, another counselor at the camp who was in SDS at Antioch College. He scolded me for not reading the news and following world affairs, and I took this advice to heart. There was also an adjoining teen work camp—all black teens from projects near the one where I lived. They were there on scholarship, to be reformed out of their "gang-loving ways." They got to be kids, but I had to work. While there, I gained class awareness and developed a suspicion of traditional "left" politics.
In the fall of 1964, I went to Brooklyn College, City University of New York [CUNY], as part of the Scholars Program for students gifted in mathematics and the hard sciences, but soon became an art major.
New York was an intense place to be, politically speaking, during this time. What organizations were you involved in at CUNY?
The first group I was involved in was Brooklyn College's equivalent of the Berkeley Free Speech Movement—the Ad Hoc Committee for Academic Freedom, which included faculty as well as students.6 As at Berkeley, students who had braved confrontations during the civil rights voter registration drives in the south returned to school to find their own political expression severely restricted. The hypocrisy of this being the "norm" at a prestigious institution of higher learning fueled the creation of a powerful movement on campus.
Then SDS. I can't recall exactly when in 1965 the SDS chapter at Brooklyn College was formed. There were older students at Brooklyn College who had been politically active for several years and were in contact with the students who started SDS. There were also "travelers" who visited college campuses and helped organize. I was active in SDS from the time that there were a few hundred members nationally until tens of thousands were involved, and my political understanding grew exponentially. My thinking developed with the organization to the point where we named the United States government "imperialist" and called for defeat in Vietnam.
I was elected chapter "president"—or was volunteered, as the position was meaningless within the consensus dynamics of the group, but helpful for functioning on campus. All student groups had to be registered and approved by the student council and the administration required a "President—Vice President—Secretary—Treasurer" structure. Only officers could reserve rooms, and submit posters to be approved for display, etc. I was probably elected because I was safe in my standing as a student, since I was in the Scholar's Program and getting high grades. It was good theater to have the official spokesperson of the most radical group on campus be a fairly inarticulate seventeen-year-old girl who looked even younger than her age. It poked fun at the Administration's "Red Menace" fear-mongering stereotypes. Whatever the reasons for my selection, the trust the other student activists had in me bolstered my self-confidence.
The college president, Harry Gideonse, was at the time the head of Freedom House, a "liberal" anti-Communist think tank/academic wing of the government's drive to stamp out domestic radicalism following World War II.7 He had instituted bureaucracies to stifle freedom of political expression after conducting a more blatant reign of terror in the 1950s, when faculty were required to sign anti-Communist "loyalty oaths" and were subjected to political inquisitions. Many were fired for having unacceptable political ideas. Lots of students had been expelled as well. In 1950, Gideonse dissolved the student government and closed the college newspaper, _The Vanguard,_ using bogus excuses, but really because they were bases of opposition to his agenda. During the early 1960s, students were expelled for participating in an anti-nukes protest. I heard about it later in the 1960s from Jerry Badanes, who was active in Movement for a Democratic Society [MDS], the non-student wing of SDS. If my recall is correct, he was one of the students expelled. During an air raid drill, when students were supposed to go to basement areas marked as nuclear shelters, a number of students reclined on the steps of Boylan Hall, the main building, each holding a sun reflector, the kind used at the beach to get a speedy tan, as if from the flash of light that preceded the mushroom cloud. This was an act of civil disobedience, but more impressively, it was my first exposure to _détournement_8
When I started college, every leaflet and poster had to be approved by the Dean of Student Activities, Archie McGregor, or it would be torn down by employees from the Office of Student Activities. All invited "outside" speakers had to be approved—all films or presentations as well. Prohibitions against walking on the grass and a dress code for female students were rigorously enforced. There was a security guard in front of the library whose job was to turn away women wearing pants. It was quite the model police state. Inspired by the Free Speech Movement, some older students who had been involved in the civil rights movement and some faculty, particularly from the departments of sociology and philosophy, Professor Richard Mendes, and Dr. Sitton, came together to organize for academic freedom. I was on the steering committee of this group and a meeting was held at my house. Mother was awed; she served coffee and pastries. The student council got involved, as did the Young Democrats [youth recruitment wing of the Democratic Party]—both the student government and faculty council, and they overturned the censorship powers of the administration. Harry Gideonse decided that it was the opportune time to retire. This cleared the way for more freedom to agitate against the Vietnam War and the military draft, issues that had immediate impact on every male student.
The SDS chapter grew steadily and included students with various political leanings. It remained somewhat counter-cultural, overlapping with the bohemian, folk-singing, pot-smoking sector of the student community. Trotskyists and the Communist Party USA had their own student groups, which attracted few if any new students.9 The Progressive Labor Party [PLP], however, was active within our SDS chapter and caused distrust against us within the national organization, although anyone who took the trouble to get to know us realized quickly that, despite being very visible, the PLP didn't dominate the chapter.10 The head of student organizing for the PLP, Jeff Gordon, was in the Brooklyn College Chapter of SDS and was one of the people subpoenaed to appear before the House Un-American Activities Committee [HUAC], along with future Yippie Jerry Rubin, in 1966.11 Rubin was there because he had been one of the organizers of the Vietnam Day Committee which protested against the war in Berkeley. I went to Washington to demonstrate against HUAC shortly after the Brooklyn College SDS chapter was formed.
I was the representative of the Brooklyn College chapter to the New York Regional Council and to the National Council of SDS, which was as important for my political development as the grassroots organizing at Brooklyn College. It got me out of New York for the first time in my life; I met people from all over America, many from different backgrounds and cultures.
Although I was nowhere close to being a regional or national leader, I met other people like me, made interesting friends, and had opportunities I never would have otherwise had. Terry Davis, the chapter representative from the Borough of Manhattan Community College, who also grew up in a housing project, was my age and more worldly. She taught me to dance, and took me and Bobby Quidone, who was gay, to the Newport Folk Festival to talk to artists about doing a benefit for SDS. We got invited to all-night parties and road trips with affluent Argentine-Jewish Brecht Theatre aficionados, one of whom dated a playwright friend of SDS organizer Sarah Murphy. Sarah later married one of the student leaders of the 1968 uprising in Mexico City.12 I first heard of the situationists from them, I believe.13
So you were getting a real political education from the time you entered university. What impact did this have on your art and did politics seep into your art classes?
Though I was an activist, I didn't do college work with political con-tent. A number of artists from the New York abstractionist school taught at Brooklyn College. The art faculty whom I remember were Ad Reinhardt, Carl Holty, Philip Pearlstein, and Jimmy Ernst, son of the surrealist Max Ernst. David Sawin taught Art History along with Morris Dorsky. The department chairman was the well-known art historian, Milton W. Brown. Walter Rosenblum taught photography and was fine politically.
You photographed some early anti-Vietnam street theater by the Bread and Puppet Theater in 1966 for an assignment in Rosenblum's class (see color plate 10).14
Bread and Puppet made beautiful masks and decorated anti-war demonstrations with elegant pageantry. Their tone was mournful, grieving. I loved the technique, but was feeling anger at having been taken in by the myth of American democracy and was searching for a means of expression with more satirical bite.
I am intrigued by the fact that you studied with Carl Holty, Ad Reinhardt, and other prominent artists. Reinhardt was among the most vocal regarding the importance of abstraction in his generation, and participated in the founding of the New York-based Writers and Artists Protest organization that formed in 1965. In April and June of that year, they published two anti-Vietnam War advertisements in the _New York Times_ —"End Your Silence"—with hundreds of writers' and artists' signatures.15 Can you tell me more about instruction at Brooklyn College and how the artists approached the issue of art and politics in their capacity as teachers? Did they ever discuss political issues in relation to art? Or did they maintain a strict separation between the two?
There was so much of a separation that there was complete silence—not only political content, but narrative had no place in the critical discussion. I later became aware that many of the art faculty very actively opposed the war in Vietnam and marched in organized artists' contingents at the antiwar demonstrations, but this was never, ever discussed with me. Even though my political activity on campus was obvious and the art faculty knew when I left town for conferences and demonstrations, I was never invited to join the anti-war art groups. I assumed at the time that art students were not welcome.
None of the Art Department faculty joined the student-faculty antiwar group on campus or, for that matter, participated in the campus free speech movement. There was no discussion of politics in studio classes or in critiques [discussions of student work involving professors and students]. Studio critiques were completely formalist—composition and technique were the issues. Art history classes were barely better; heavy on rote slide identification, with art sealed off from the history of the world, the assumption being that art existed only within an "art world" where it had meaning only in relation to previous works of art. Artists were influenced only by other artists, with each school rebelling against the previous generation—an orderly evolution of styles with the present being the glorious and logical culmination of all high art that came before.
I did not accept these premises and felt that I was being fed McCarthyist dogma.16 It was not until the 1980s that I began to understand where all this was coming from, but at the time there was no forum for discussing or questioning the dogma. I just looked else-where for theoretical constructs that were enlightening and to art from previous historical periods for inspiration.
During the World War II era, the Brooklyn College Art Department had been greatly influenced by the German Bauhaus and was, according to the college's own official history, "blurring the lines between fine and applied art."17 This had ended before I arrived. Major changes in the curriculum were made in 1956 separating study into classes by discrete mediums—drawing, painting, printmaking, sculpture, etc.
When I told the Scholars Program that I was going to major in Studio Art, Milton W Brown, the Art Department Chair, called me in for a meeting and laid out a plan for my education. I was to take a few introductory classes and then tutorials with senior professors selected for me. Ad Reinhardt was later assigned to be my "mentor"—in charge of overseeing my progress.
At Brooklyn College, abstraction was the rage. It was considered somewhat scandalous by the media and got lots of attention, the controversy being about whether it was art or not. It was incomprehensible to the uninitiated—elitist and quite the commodity circus, in my opinion.
It was probably Reinhardt who encouraged me to go to galleries to see what was being shown at the time. I knew that there were parties at show openings. He evidently thought that it would be okay for me to show up at openings, but it didn't seem like something that I would ever do. I imagined feeling awkward, unwelcome, and out of place. It seemed like a career as an abstract painter was basically a sophisticated hustle, playing up to rich patrons, marketing oneself. What I was being encouraged to do as a painter was look at what was selling and then create my own "look"—one extremely similar to what everyone else was doing, but just different enough to not be out and out imitation.... Then, once it "sold," I was to stick with it as an identity—my own franchise. Well, I was busy creating an identity, but it was about being true to myself and resisting pressures to conform to social mores which were phony.
Besides, even sophisticated women seemed to have an impossible time being taken seriously as painters in the abstractionist boys' club. I sensed that it would be a fruitless effort as well as self-destructive. I imagined that getting involved in this world would include sleeping with old guys and putting up with the current expectation of feminine behavior. Reinhardt was probably sincere about encouraging me to start going to upscale 57th Street galleries and meeting people in the art world, but I had a visceral reaction against the whole art-as-commodity marketing thing and thought that such circles would be where these forces would operate with the greatest intensity.
Reinhardt never talked to me about politics. I never knew the extent of his political involvement until long after his death [Ad Reinhardt died in 1967]. I was being taught to develop color-and-form presentations on canvas and when I got a harmonious push-pull balance of form, that was the content. I think that Reinhardt himself believed in following a particular abstractionist trajectory, which led to a form of spiritual purism in painting. I could see the logic of his ideas, but disagreed with his premises about the history of art and with his ideas about art's social role. I never argued with him about these ideas. I did not have the words or the confidence to articulate my objections until after I had done political artwork for publication.
I felt that my art classes were anachronistic and that my political activism was where the real learning was happening. I wanted the same intellectual excitement in my art-making, but I had no role models. None of the other art students were doing anything with provocative content. I was given a studio all to myself and was isolated from other students in the department.
As time went on, I spent less and less time painting and was quite unproductive. I recall listening to Reinhardt's critiques of my paintings in the spring of 1967 and becoming more aware of composition and color, but being totally frustrated with my work.
I asked questions in art history classes about political content in art and was referred repeatedly to Marxist Arnold Hauser's _Social History of Art;_ I found it less than forthcoming regarding real questions about the impact of art and the role of the artist in society.18
I talked with Jimmy Ernst [son of German surrealist painter Max Ernst] about doing political art, expecting for some reason that he might be sympathetic, having escaped from the Nazis ... but he was cold as ice. Every professor said the same thing: "Art never changed society." "All political art is propaganda and not good art." "Why would you want to create propaganda?" Some also implied that "propaganda" meant "pro-Communist Party dictatorship."
In an interview with Jeanne Seigel for a series called "Great Artists in America Today," that aired June 13, 1967 on New York's WBAI radio station, Reinhardt said:
I think an artist should participate in any protests against war as a human being. There's no way they can participate as an artist without being almost fraudulent or self-mocking about what they're doing. There are no good images or good ideas that one can make. There are no effective paintings or objects that one can make against the war. There's been a complete exhaustion of images. A broken doll with red paint poured over it or a piece of barbed wire may seem to be a symbol or something like that, but that's not the realm of the fine artist anyway."19
I gather that the art education you were getting at Brooklyn College convinced you that "modernism" as codified by Reinhardt and others had no political relevance?
Actually, I was convinced that "modernism" as presented by Reinhardt, Ernst, etc., was very political, reactionary art promoting a McCarthyite attack on the ability of art to be an accessible form of social discourse and making it, in fact, an elitist commodity. Looking at documents from the period, I found an interview with Ad Reinhardt in which his opinions are even more blatant—a radio panel discussion between Reinhardt and artists Leon Golub, Allan D'Arcangelo, and Marc Morrel following the Angry Arts events of early 1967.20 The discussion was broadcast on August 10, just twenty days before Reinhardt died. In the debate, Reinhardt says:
I'm not so sure just from a political and social point of view what protest images do and I would raise a question. I suppose this is an advertising or communications problem. In no case in recent decades has the statement of protest art had anything to do with the statement in the fine arts.21
As an artist, you can only reach those people who are willing to meet you more than half way. At least that's the fine artist's problem now. Another kind of artist who has techniques of communication or who wants to affect people like an advertising artist or a poster artist or somebody who wants to get a strong reaction, that's another matter. You don't know exactly how effective that is.22
Imagination is the word used for an idea man in an advertising agency. You don't have imagination in the fine arts.23
In response to a question by Marc Morrel—"How do you look at a painting?"—he replies:
Only as a painting, of course. I don't see how a painter can look at painting except as a painting. Then you know the artist is involved in certain tricks in colors and forms. But one artist doesn't look at another artist ever as somebody who's had some kind of experience. That's for laymen, the idea that an artist expresses some life experience he's had.24 If you are saying that an artist's impulse comes from some life experience first it wouldn't be true. An artist comes from some other artist or some art experience first.25
Goya is only important because of his relation to Manet.26
Reinhardt called Picasso's anti-war mural _Guernica_ [1937] "just a cubist, surrealist painting of some kind. It doesn't tell you anything about the Spanish war [the Spanish Civil War, 1936-39].... Actually, I'm against interpretation anyway, but the most interesting or at least the most relevant interpretation seems to be the psychoanalytic one in which Picasso reveals himself to be an open book."27
When Leon Golub stated that the figures in _Guernica_ "have a tremendous effectiveness on me even today," Reinhardt responded, "The Spanish war was lost." Golub replied, "Paintings don't change wars. They show feelings about wars." Reinhart responded again, "It didn't explain anything about Spain to anyone."28
So, in summary, his position was that: (1) An artist's life experience does not impact on the artist's work; (2) Imagination has no role in the fine arts; (3) Art is not a means of communication; (4) Art which at-tempts to communicate is not "fine art," but advertising or poster art; (5) Looking for meaning in a painting is only for ignorant "laymen"; and (6) Art intended as socially critical satire is inevitably co-opted and successfully exploited by those it was made to criticize.
What an angry, thoroughly negative man. Every single thing that he is against, I advocate. Quite extraordinary.
Tell me more about the divorce between art-making and activism at Brooklyn College.
My studio was a cell at the top of a tower, where, working in complete isolation, it felt more and more like a prison. It occurred to me one day that doing abstract paintings was incredibly isolating as well as boring. To me it just seemed absurd, considering all that was going on in the mid-1960s and what was being questioned. Politics, gender, all the power relationships in society were up for grabs at that point and it was a very exciting time. I was sure that there was a way to do artwork which wasn't isolating, where the art went out into the city and you got to actually see people's reactions. I was driven to find a way to do this.
As far as college thwarting my political art goes, the academic structure at Brooklyn—being immersed in the separation of art and politics—was probably more powerful than merely preaching anti-politics dogma. I'm thinking of the way the Art Department was structured: subjects based upon techniques—painting, drawing, printmaking—and then art history divided by periods, with art analyzed mainly within the confines of the art that came before it. There was no structural place for the analysis of art socially: no discussion, no actual interaction between the art student's work and the public.
In 1966, Brooklyn College SDS was active against the draft and against the war.29 We produced new leaflets every few days and handed them out during class changes. Early on, people would turn away in disgust. Some crumpled up the leaflets and threw them at us. Some called us Communists with great animosity and others took the leaflet and threw it on the ground or in the trash. But as time went on and more soldiers died, students began to pay more attention. They stayed and argued and were more knowledgeable about the war. They argued the domino theory [the American government asserted that South East Asian countries would "fall like dominos" if the war in Vietnam was lost], but were not really sold on it.
Anticipation of being drafted loomed. Within SDS, there were running discussions about the class nature of the draft, and the perceived "immorality" of the protection being in university gave. As the war required more troops, the government Draft Board instituted an exam in early 1966 to cull the college students with low grades and take away their student exemptions. We picketed out front during the exam and students joined the demo as they exited the test. The exam was so crass. An arbitrary grade would decide who would live and who would die.
Military recruiters were scheduled to set up on campus in the lobby of Boylan Hall. SDS planned a sit-in. I helped plan it, but decided not to go, afraid of the reaction at home if I were arrested. My father had gone ballistic when I was arrested the year before at a party of activist kids invaded by 200 police. Eighty-eight of us spent the night in the "tombs" [notorious basement-level holding cells in downtown Manhattan's police headquarters]. William Kunstler was our lawyer, and all the charges were dropped and records expunged. So I went to English class in the same building as the sit-in. I could hear the chants echoing through the halls and felt so torn.... When class ended, I walked out the door and was grabbed by three burly cops who had been escorted to my class by the Dean of Student Activities, Archie McGregor himself. They lifted me under the arms and shoved me downstairs to the sit-in, where they promptly declared that I was under arrest. I was lifted up by four cops, one holding each limb, and thrown into a waiting paddy wagon. I landed on a pile of my friends. After that wagon was full, they pulled up another and packed the activists in. We sat waiting for the vans to drive off and nothing happened. We couldn't see what was happening. All the people at the sit-in seemed to have been arrested, but the paddy wagons sat. We heard a buzz of people and felt surges of tumultuous activity outside. Chants of "Cops off campus" rose intermittently. Five hours later, we were finally taken to the police station to be booked (See color plate 11).
Meanwhile, previously uninvolved students and faculty were up in arms. "Cops on campus" was denounced as a disgrace and the faculty demanded that the administration drop the charges against us immediately. My art professors organized a bail fund to get me out and called my parents to tell them that I had done nothing wrong. We spent the night locked up. In the morning, when we were released, we returned to campus to find 5,000 students waiting for us. Students who had never been to a demonstration before organized a student strike and shut down the school. We held a spontaneous rally and heard about the hundreds of students who had sat down around the paddy wagons, blocking their movement, and from one in particular who had chained and padlocked himself to the campus gates, in the process locking his body across the opening. Campus opposition to the war and the draft had reached a new peak.
After the arrest, my parents and grandmother were traumatized for a long time. My mother and grandmother, refugees from anti-Jewish pogroms during the civil war following the Russian revolution, were worried about my safety and my future. There was a student trip to Europe and I really wanted to go, but couldn't imagine being able to afford such a luxury. College was just about free and I had state scholarship money that had been going into the bank. My grand-mother lifted up her housedress and pulled a bunch of rolled up bills out of the elastic band of her thigh-high stockings. She berated my mother—"The rich people send their girls to Europe to get polished and become ladies. We can't do anything less for 'Suzele'." My grand-mother steadfastly refused to discuss any details of life in Bessarabia and never could imagine why anyone would want to go to the Europe she had fled, but you had to be "modern" to get ahead. So I had a ticket to Europe and money in my pocket.
First, I flew to London and went to the office of the Campaign for Nuclear Disarmament.30 There I met Sheila, the office staff person. She invited me out with two of her friends, one a Londoner and the other a black South African graduate student who was in exile. Sheila introduced me to the famous English peace activist, Peggy Duff, who invited me to go to Stockholm with her and Bertrand Russell to attend the War Crimes Tribunal.31
I decided not to go for a number of reasons. I wanted to meet radicals my own age and did not want to spend my first time away from home in a passive situation listening to other people speak, even if they were world-class thinkers. I went out to a club to hear a rock group, The Social Deviants, and met people who were squatting at a London School of Economics dorm. They turned out to be junkies, so I decided that I would travel on and try my luck in Amsterdam.
When I went to Europe, I was looking for friends and lovers. In Amsterdam, I found the anarchists after being there for two days. An art student was selling the Provo publication, _Die Witte Krant [The White Paper],_ and I volunteered to do artwork for it.32 I never did more than hand-letter an ad, but the group immediately took me in and introduced me around to their friends. A few of the people I spent large amounts of time with had been central to the political actions of Provo. Many others were in close proximity to the main instigators. The scene was much more like the traveling punk kids of today, but it is important to note that I was there when school was not in session and many activists traveled.
In Holland, art students were much more experimental. Art was much more integrated into daily life: be-ins, where people dressed up in costume, chalk drawing on the sidewalk, installation art, poetry readings on the streets.33 The same people who were radical political theorists were participating in public art. Now, a lot of the actual art was more countercultural than about the war and imperialism. The Dutch were involved in their last colonial war in New Guinea and AWOL soldiers were being protected in our midst. There was a lot of humor as well as sex-and-drugs-related content: much more "hippie" than "angry anti-war," but certainly presenting other possibilities for how art could operate in a community.
Would it be fair to say you became more anarchistic during your time with Provo?
I think the answer is the opposite. I already had anarchist ideas and was seeking out people who were politically and culturally compatible. Politically and artistically, I was already inspired by the _détournement-style_ tactics of Jerry Rubin in a revolutionary War of Independence outfit at the 1966 HUAC hearing handing out copies of the Constitution, and by the anti-nuclear weapons sunbathing action of the early 1960s at Brooklyn College. But I also had my heart set on going to Amsterdam because of what I had heard about the Provos.
I had heard about their very militant demonstrations against the royal wedding. On March 10, 1966, Princess Beatrix of Holland married Claus van Amsberg, a German noble who had been in Hitler youth and the Wehrmacht. The slogan for the Provo demonstration was, "I want my bicycle back," a reference to the fact that German soldiers had confiscated Dutch bicycles during the Nazi occupation. But the real lure was the Provo's "white bikes" campaign—free bicycles painted white left throughout the city for anyone to use and then leave on the street for the next person to use. This was a compelling model of visionary communalism for those few in SDS like me who biked to Brooklyn College—our sensible and free form of transportation. So I anticipated finding like-minded people in Amsterdam who would welcome me—I had to check it out.
I met a wide-ranging social network of students and street youth—gay and hetero. It was summer vacation and organized activism was in an ebb. A lot of people were traveling, but people still in Amsterdam had a lot of time on their hands. After a brief session of "imperialist American" baiting—half in jest, but to see how I would respond—I was adopted and cared for. When they found that I was in SDS and had been to the Free University in New York, they quickly warmed up to me. I was introduced to a Provo, Martijn, and he in turn introduced me to another in the group, Barand. They took me to stay at the house of a leading member, Roel van Duijn, who was on vacation in Lapland.34 Barand showed me a news photo of himself looking fierce in the front line of demonstrators in one of the Provo "White Riots." I later stayed with Johannes van Dam, who was Jewish, gay, and not really a political activist. At the time my hair was very short—think Mia Farrow in the film _Rosemary's Baby_ —and I wore jeans and work shirts and no makeup, so people often took me to be lesbian. This was reinforced by my directness and aversion to the repressive female behavioral roles of the time.
_Susan Simensky Bietila, View 1: 7" × 6"; View 2: 8" × 6"; Be-In, Vondelpark, Amsterdam, July 1967. Gelatin silver prints_.
The Provo scene sounds like a diverse one—with free transportation.
Unfortunately, the white bicycles weren't readily available. Mainly we walked around in groups, talking about politics, philosophy. Everyone in the political and gay circles I had joined spoke English and more. I was taken to meet "queens" and to private social clubs—kind of like basement punk shows today. It was common for gay and hetero youth to socialize together.
I also went to Provo be-ins at Vondelpark, Amsterdam and was asked to work with an underground network smuggling AWOL Dutch soldiers to safety. They were refusing to fight in New Guinea, where the Dutch were trying to hold on to their last colonial possessions in the region. The soldiers did not speak English, so pretending to be out on a date to escort them from one safe house to another required lots of fake conversations.
The group encouraged me to be a traveling companion with Adinka, the art student I first met hawking _Die Witte Krant._ There is a photo of her from that period where she is kneeling on the ground and drawing in chalk.
Adinka was going on a trip to visit the family of her brother's fiancé in Barcelona, Spain, then still a fascist dictatorship ruled by Franco. From there it was off to the island of Ibiza to rendezvous with more of the crowd.35 I went along and hitchhiked from Barcelona back to Amsterdam later in the summer with Barand.
_Adinka Tellegen, taking part in a continuous chalk drawing from England to the Netherlands, 1967. Newspaper clipping._
Vacationing in fascist Spain was a bit of a contradiction—but certainly it was an education for me. Adinka's brother was engaged to a working-class girl in Spain and we stayed in a blue collar suburb of Barcelona. I knew very little about the Spanish Civil War at that time. Ibiza was a destination spot for northern Europeans, and the vacationers were largely gay men—a very safe spot for girls at night. I was underage for going to clubs and when Barand and some of the other boys arrived, they made sure to sneak me in through windows or back doors.
Hitchhiking back from Barcelona, Barand and I were welcomed in Antwerp by artist friends of his and we spent the night on the floor of an art gallery which had a display of kinetic sculptures that smoked joints. My friends also told me about street performances, poetry recitations by Simon Vinkenoog, "the Allen Ginsberg of the Netherlands," although I did not see these. It was obvious, anyway, that the Provos were actively engaged in a massive European counterculture.
Despite spending time with lots of other boys, I was in a very tentative romantic relationship with a guy named Zeno. He offered to marry me so that I could stay in Amsterdam and be able to go to art school there for free. But I sensed that the relationship would be a very rocky one, conflicted as it would be with a "real" marriage—when neither of us was ready. I also felt a pull to return to New York with a whole new understanding of artistic and political possibilities.
I would say that I learned about anarchist culture from my friends in Amsterdam, but not identifiable anarchist political theory per se. Provo culture influenced me to look for a political underground news-paper to join when I got back to the US. And it led to my involvement down the line in street theater and street art.
Your return to Brooklyn College in the fall of 1967 was short-lived.
Reinhardt had died that August. When I returned to school, members of the art faculty were shocked and depressed. By that time, I had come to realize that I wasn't learning what I wanted to learn about art in college. I had no workable theory of art and politics, but knew that con-tent, narrative, communication with the average person—art as part of social discourse—was what I wanted to learn. But I was blocked: I didn't understand how to use images to communicate the ideas I felt were important. It wasn't a technical question, it was a philosophical one. The art history classes touched on some of the issues, but as I said, they were heavy on slide recognition and rote memorization.
What I did decide is that if all political art was "propaganda," then I would make propaganda. I had already seen Rubin's guerrilla theater at the HUAC Committee hearing and photographed Bread and Puppet's performances at antiwar demonstrations. When I returned from Amsterdam, I began to look for an apprenticeship situation with an underground publication, where I would have a structure to produce political work. I dropped out of Brooklyn College in November and ended up going to the west coast, attempting to join the staff of _The Movement,_ a political underground paper in San Francisco.
What was going on at _The Movement_ in terms of artistic production?
There was quite a bit of sophistication. There was an artist, Frank Cieciorka, who did a lot of beautiful work, although I never met him. His clenched fist was iconic. I was invited to some social activities, but was not included in political discussions or invited to try my hand at an art assignment.
You weren't in San Francisco very long—I understand you were back in New York in the spring of 1968 to finish your degree.
[Back in New York] I set off to look for a "propaganda art" job where I could really learn how to put together an underground publication. I went to the offices of _Rolling Stone_ and I told the editor Jann Wenner that I could completely revolutionize his publication, but he booted me out the door. I ended up going to a political underground paper called the _Rat_ in the Lower East Side. At that time, the _Rat_ had just been started by people I knew from SDS, the "Texas anarchists" and some native New Yorkers. One of the anarchist founders, Jeff Shero, picked the name because a rat was an appropriate image to represent the paper—a tough little city animal, resilient and dangerous. The _Rat_ covered the period's political and counterculture movements vividly. But it degenerated rapidly, becoming sensationalist and relying on sex ads for revenue, and publishing demeaning pictures of naked women. It started to look like the _Los Angeles Free Press,_ the _East Village Other,_ and other non-political underground newspapers as they moved away from radical politics into exploitative trashiness.
I had questions about whether I really belonged there, but asked about opportunities for doing drawing and artwork. I got the sense that women were welcome to do the typing, but not the writing or artwork. So I went further up the street to the _Guardian,_ which was pretty much the mainstream leftist newspaper in New York, if you could call it that, a weekly that had been in existence since the 1940s. It was originally the newspaper of the electoral-oriented Progressive Party, which was a "third party" slightly left of the Democratic Party. They hired me immediately, and I told them that I wanted to do drawings, illustrations, and political artwork. I was taught layout, but it only took a few months to convince the editors that I could do original art for articles and the front page.
The best cover I remember doing is the one about the moon landing. I drew Mount Rushmore on the moon [featuring faces of] Walter Cronkite; Monkey Bonnie, one of the animal "astronauts" who died in space; Wernher von Braun, the Nazi missile scientist [von Braun, who died in 1977, went on to develop ballistic missiles for the United States]; and Richard Nixon. I also did a presidential election cover in 1968 when Hubert Humphrey was running against Richard Nixon. I have a collage where there is a body sitting in the presidential chair [of the oval office]. Some anonymous CIA-type is unscrewing the head of one president and putting a new president's head on in its place.
How long were you at the _Guardian?_
I was hired at the _Guardian_ in the spring of 1968 during the height of the Vietnam War and quit during a "purge"—driven out by management in August 1969. Right after I had started at the paper, it was redesigned by a graphic designer, Harry Driggs. The masthead was changed from "Progressive" to "Independent Radical Weekly." The editors wanted to appeal to "youth culture" without losing their traditional readership. I was the only artist in the art department; the others did production. They contributed ideas for my artwork and offered insightful critiques with a supportive and collaborative spirit. They were a wonderful group, but were powerless in the _Guardian's_ hierarchy.
The former SDS people were not all in the same camp politically and there were arguments about hierarchy and class among the staff. Even though the [support] staff were all radical activists, they were treated with condescension; their political ideas were discounted be-cause they were not high in the organizational hierarchy compared to the editors, writers, and financial backers. Toward the end, there were big arguments at staff meetings—that's where I found out that management had a different pay scale than other staff while I was literally going hungry working there. During the purge, seven staff were fired and twelve others, including me, walked out in protest.
_Susan Simensky Bietila, Moon Landing cover, The Guardian, Aug. 2, 1969. Pen and ink collage._
Then you got a call to join the women's takeover of the _Rat._ The first women-only issue was published in January 1970—with the headline, "Women Seize Rat! Sabotage Tales!" It lists the collective as follows: "Jill Boskey—valiant typesetter for _Rat_ for unheralded decades, Jane Alpert, Larelei B., Ruth Seller, Pam Booth, Valerie Bouvier, Naomi Clauberman, Carol Crosberg, Sharon Krebs, Robin Morgan, Jacye Pelcha, Doria Price, Judy Robison, Miriam Rosen, Barbara Rothkrug, Judy Russell, Lisa Schnaeidr, Martha Shelley, Sue Simensky, Brensa Smiley, Christine Sweet, Judy Walento, Cathy Werner, and Mark, Jan, Anton, and Neil"—male staff who stayed on to help out for a while with production until they were asked to leave. Tell me more about the takeover.
The women who had been working at the _Rat_ all along had been in SDS and other student groups. They were amazingly intelligent and articulate radicals who had been doing all these menial jobs. One day, they got together and invited their friends to come by and put out a special "women's" version of the paper. The issue was so good that we decided that the right thing to do was to continue. It was one of the first feminist newspapers in what's now characterized as the "second wave" of feminism in the United States. The takeover was kind of interrelated with the street theater going on at the time: people involved in the _Rat_ had been involved in the feminist demonstration at the Miss America Pageant in Atlantic City [September 7, 1968], where the pageant was picketed by women and items of female oppression were symbolically discarded. Nobody actually burned bras, but that's where the whole fictive media image of women burning their bras came from. There was another demonstration that I was actually part of—a takeover of a Bridal Fair [February 1969] at the Felt Forum in Madison Square Garden. We took over the stage and auctioned off a bride! The _Rat_ women attended meetings held by a range of feminist radical "consciousness-raising," activist-oriented groups, as well as others—it was all one interlocking network.
_Susan Simensky Bietila,_ Rat _cover, May 8-21, 1970. Pen and ink._
This was the same period that the anarchist Yippies were active in New York. There were lots of great political stunts going on, like in August 1968, when Yippies threw dollars off the balcony at the Stock Exchange and watched the stockbrokers scramble over each other, groveling on the floor. It was influential. The theatrical presentation of political ideas was a shared aesthetic.
The overlap between radical feminism and anarchism on the level of organizational tactics and artistic protest strategies is interesting. I recall that "WITCH"—Women's International Terrorist Conspiracy from Hellwas the key initiator of the Bridal Fair action and there was at least one crossover from the _Rat_ collective—Robin Morgan. In her memoir _Going Too Far,_ Morgan characterizes WITCH—disparagingly—as a "proto-anarchist" Yippie-influenced group, so the connection has at least been acknowledged.36 What was it like to participate in such an outrageous action?
It was tremendously liberating. It was absolutely a celebration of freedom from the soul-binding of the female submissive role. I was freeing myself as compared with being a crusader against other people's oppression, no matter how just, no matter how linked to my own situation. Disrupting a Bridal Fair was certainly outrageous enough to be unanticipated by Madison Square Garden security. But it was very logical, quite a clear target. What was outrageous was not being allowed into the college library wearing pants, not being allowed to go out at night on your own; if you danced without a male partner, it was unacceptable. It was outrageous to be judged, despite your talent and intellect, by your marriageability. I was really angry at it all.
I was not part of the group who came up with the idea for the action, but I was invited to participate in the planning. One of the women who worked at a publication had access to free tickets. The action was well-planned ideologically, with delineated strategic and theatrical roles. The demonstration confronted common cultural assumptions, such as "Every girl dreams of being a bride." The modern wedding was exposed as a romanticization of women as property—the transfer of a woman from the father to her husband. In addition, it directly questioned consumer culture, because the Bridal Fair was, after all, about buying bridal gowns, flowers, china, and silver. I still think that it was an excellent action and disagree with Morgan and others who have gone more mainstream and think that the critique of marriage consumerism was an attack on women.
Thinking further about the often-ignored anarchist influence, the shared prankster-style activism of the Yippies, WITCH, and the Provos was no accident. In his 1979 memoir, Abbie Hoffman recalls that during the period of 1965-68, the Yippies in New York were in contact with their counterparts in Europe and elsewhere.37 Hoffman mentions Fritz Teufel, Karl Pawla, and Kimmune #1 in Berlin, and Jean-Jacques Lebel in Paris, who came to the United States and played a role linking "the anarchism of the Left Bank [Paris] to the street culture of Haight-Ashbury [San Francisco] and the Lower East Side." He also emphasizes the importance of the Provos:
_Susan Simensky Bietila, Yippee Be-In Poster, Rat, March 7-21, 1970. Cut paper, pen and ink_.
In Amsterdam ... street players dubbed themselves the Provos (short for provocateurs). Practicing the politics of 'free' they opened parks to free concerts, established crash pads [free housing], and ladled out soup to moneyless hungry customers. Their symbol became the white bicycle. Second-hand wheelies were painted white and left around the city. Whoever needed one could take one, pedal away, and leave it at another location for the next.... [They] established a community ambience that would be held up as a model by all of us. Dana Beal picked up on the Provos and founded a chapter on the Lower East Side.
That's Hoffman's recollection—can you say more about anarchism in New York?
In my exploration of the political world, I came across the Free University in 1965-66. This school was started by Allen Krebs and then run by Sharon Krebs, and classes were taught by a member of the anarchist Fugs rock band and by Murray Bookchin. I never attended classes, but I went there to hang out and it fueled my imagination.
Here is what Roy Lisker, one of the instructors, wrote about the Free University when you were attending:
The people that Allen Krebs engaged to set up the Free University of New York represented every shade of opinion across the New Left: poets and writers, disaffected scholars, union organizers, activists, free-lance journalists, and publishers, creative individuals of every sort. Our goal from the beginning was to establish a forum in which every direction of contemporary political activism would be represented. Courses were to be taught by persons actually involved in bringing about the changes they were advocating.
The curriculum for the first two terms contained, in addition to those on leftist politics, courses ranging from hallucinatory drugs to sexual liberation to astrology. Important courses were offered that were not available, or even imaginable, at many main-stream universities: History of the American Left (Staughton Lynd); History of the Labor Movement (Stanley Aronowitz); Cuba Today; Training in non-violent tactics; History of the National Liberation Front. Paul Krassner [a Yippie], editor of the scathing and satiric political magazine, _The Realist,_ gave a course entitled "Why the _New York Times_ is funnier than _Mad Magazine."_ The enthusiasm that prevailed in the first term of the Free University of New York, from November '65 to February '66, carried over into the spring. It was an inspiring time for all concerned.38
The Free University was one place where anarchists made themselves known. I have been told that anarchists were also a real presence at anti-Vietnam War demonstrations.
In terms of actions, anarchist affinity groups for street demonstrations were into satire and self-satire. Wearing motorcycle helmets and leather jackets, our fists in the air, we played the role of militant demonstrator, but knew that it was more theatrical, in opposition to other segments of the anti-war movement who were in essence begging, pleading, and lobbying those same politicians responsible for the war in the first place. Demonstrators were relegated to being, in essence, little more than numbers used as lobbying capital by mainstream liberal leaders—a futile and depressing strategy. Networks of anti-imperialist affinity groups went on "vote in the streets, vote with your feet" split-offs from every major demonstration. While the big antiwar coalitions were demanding the gradual withdrawal of American troops from Vietnam—"Support our troops, Bring them home"—we were calling for "victory to the Vietnamese!" and admission that the war was wrong in the first place. I remember one such split-off from a very large picket of a Democratic Party event at an upscale midtown hotel. Word was passed to disperse and converge on Wall Street, the power behind the war-makers. Thousands of demonstrators went downtown by subway and ran a gauntlet ahead of mounted police to the sound of crashing plate glass. This kind of action declared, "no to businessas-usual." Radical scholars kept tabs on the war-profiteering corporations and made the locations of the military industrial complex common knowledge, opening them up to exposure for complicity during demonstrations.
I should also mention there were a lot of street anarchists at the _Guardian._ They weren't only the writers and editors; they were the de-livery people, people in the art department, and the typists—young, like me. It was pretty much an anarchist youth culture. There were also plenty of what would now be called anarchist affinity groups and collectives that were active on the Lower East Side.
More importantly, I had become part of a movement that was becoming "articulate"—working in groups which operated by consensus and where ideas were developed collectively. Every organization I was involved with had a "bottom up" (anarchist) ethic—decision by consensus, encouraging participation in decision-making on an equal basis—as a matter of principle. My politics were clearly anarchist, but not identified as such during the 1960s, because the political dividing line in the movements of that time, anti-Vietnam War, black liberation, women's liberation, etc., was between anti-imperialist radical social change and reformism. I was also becoming more class-conscious and identified anarchism as mainly influential in terms of cultural expression—despite reading Emma Goldman's autobiography, _Living My Life,_ as well as _Labor's Untold Story_ [by Richard Boyer and Herbert Morais] during that time.
People I knew in SDS who identified as anarchist included Shero from Austin SDS (the "Texas anarchists"), who turned me away from the _Rat_ when I asked to do artwork there. Unfortunately, other self-identified anarchists in New York, all of them male, came across as in-tensely chauvinist. Many of them were associated with the 1950s Beat poets who were as infamous as the male artists among the abstract expressionists for treating women like dirt. So I did not really identify with anarchism as a contemporary movement.
So sexism, a generational disconnect, and your own prioritizing of "class struggle" over "culture"—in retrospect a false dichotomy, obviously—were factors: was there anything else?
At the _Guardian,_ how people labeled themselves politically often bore little connection to how they behaved. Pockets of anarchist ideas about decision-making existed in various departments. Work styles and networks within the staff defied political self-definitions as anarchist, socialist, new left, feminist.
The "anarchists" at the _Guardian_ were in affinity groups that carried National Liberation Front flags at anti-war demonstrations and actual pigs' heads on pikes labeled with the names of prominent liberals, like Bobby Kennedy.39 I marched with them on more than a few occasions, but kept my distance because of their uncritical hero worship of the Black Panther Party and loyalty to "fearless leaders" like Walter Teague and his Committee to Aid the National Liberation Front.40 eague was a one-man "organization" who recruited younger, blindly loyal kids to work for him. There was no consciousness of working collectively there whatsoever. My position was that the best thing that people could do was to continue to build a powerful movement against the government and to limit its ability to conduct unjust wars like the one in Vietnam in the future. Idolizing the Vietnamese "liberation fighters" was not the way to actually help people in Vietnam.
Not the typical activist position—there was a lot of messy thinking back then that, in the absence of an anarchist critique, led to some serious contradictions. Marxists and the movements influenced by them argued for social liberation via authoritarian party organizing and the establishment of state dictatorships.
"Dictatorships of the proletariat" to overthrow capitalism—dictatorships which were supposed to eventually wither away. At the _Guardian,_ most of the staff, including the outspoken anarchists, were uncritical toward the hierarchical structures of the Black Panther Party and the NLF. I certainly supported both, but not as a "true believer" looking for a perfect leader. Many among the affinity groups were just as eager as the Weathermen to prove to the Panthers that they were "heavies" and "down" for the revolution which they expected to happen momentarily.41I was vocal in criticizing these ideas as being out of touch with reality, but few were listening. A lot of "revolutionaries" were driven by guilt about their own "white privilege" and did not want to recognize [that] class oppression existed among white people. I had a few close friends who also came from blue-collar backgrounds who had similar criticisms of these trends and had less patience than I did, or less incentive to engage in the argument at all. We were often a distinct affinity group at demonstrations.
What about the _Rat_ during this time?
Working on _Rat_ became more and more contentious by the week. The collective members had diverse political views. Some had been in SDS, and among them were women aligned with opposite sides of the divide when SDS split into the Revolutionary Youth Movement factions, some aligned with the Weathermen and others with the groups looking to organize the working class. Then there were women who were in New York Radical Women and Redstockings—the feminist theorists involved in "consciousness-raising" groups.42 There were lesbian activists who were to soon be part of the formation of the Gay Liberation Front [founded in New York in 1969] and women from Third World support movements, anti-imperialist and radical movements. Many of us were in anti-imperialist circles as well as feminist consciousness-raising groups.
My disaffection with the _Rat_ came with the first wave of identity politics. By August 1970, there was pressure to have an editorial quota system—so many pages of the paper devoted to women of color, so many to lesbians, etc. I believed that there ought to be a unified movement to fight against all oppression and saw identity politics as divisive and depressing. Many of the women at the _Rat_ came to believe that working exclusively within the women's movement was the only revolutionary path, and accused women who gravitated toward activism in other movements as lacking adequate political consciousness and being traitors, rather than simply having a different analysis of how to change society. So I left the _Rat_ collective before the divisiveness became even more demoralizing.
Returning to art, how was visual art looked upon during that time in relation to radical social change?
That's a whole other can of worms, I would say. On the one hand, there were artists who were doing political work, but in the main-stream of the political movement, art was an afterthought. The big olitical debates were about the war and US imperialism, that sort of thing. The general values enacted in the dynamics of SDS and the publications, even feminist publications, were that days and sleepless nights were spent haggling over wording.
During your last year at college, you exhibited your political artwork, but it was not well-received.
I recall having no real preparation for my thesis show, which took place in late 1968. It seemed a spur-of-the-moment thing. I was working at the _Guardian,_ with a weekly thirty-six-hour marathon to produce each issue, carrying a full course load of very facile education classes with the idea of teaching high school art, and dating a jazz musician who was playing clubs until two a.m. There was no one to advise me about how to mount a show.
The exhibition was in La Guardia Hall. Art was hung on movable display structures with panels. It was dimly lit—the only light source came from the ceiling towering above. I had one or two of these panels on which to hang my work. I deliberately chose work from the _Guardian._ I also exhibited my prints, which were figurative but without obvious narrative. All these prints would be published in the _Guardian, Rat,_ or _Liberation Magazine_ over the next three years.
Most prominently displayed was the photo-collage and painted _Guardian_ cover with the presidential heads. This was obviously an intentional protest against the separation of politics and art, but even more directly a defiance of the prohibition against exactly the kind of work which the faculty defined as _not_ fine art—art with topical narrative, art with intent to communicate, art with an obvious political message. The quality of art I displayed didn't matter; the cant was, "political art isn't fine art"—any art which commented on contemporary events would be obsolete the next day whereas "real" art, "fine art," "high art"—is eternal.
To my best recollection, the most hostile reaction to my art was from Jimmy Ernst, who refused to talk to me at the exhibition opening.He looked very angry. I came across several documents which shed light on Ernst's thoughts on art and politics at the time. There are two articles, one titled "A Letter to Artists of the Soviet Union," published in the _Art Journal_ after Ernst was sent on a tour of the Soviet Union by the State Department in 1961.43 His mission was to condemn socialist realism and promote the virtues of abstraction—abstraction is equated with "freedom."44 Quite the Cold Warrior. He also wrote a manifesto for UNESCO, "Freedom of Expression in the Arts," published in a 1965 issue of _the Art Journal._45
He repeats many principles with which I would strongly agree in theory, but in his context are highly questionable. He is against artists being "forced to serve a 'revolution' which was lost long ago to those who fear the open mind and find comfort only in the various practices of anti-intellectualism." His statement is a thinly veiled dia-tribe against the Soviet Union at a time when identical criticisms of McCarthyite censorship in America were long overdue. He writes:
Art is indeed a means of communication which knows no border and is above the barrier of the linguistic.... No society or state has ever been able to hide its own shortcomings behind the screen of a carefully nurtured and directed culture.... A state that fears and represses its own intellectual minority can ill afford to stand before the world as a champion of international peace.
He advocates a "world community" of artists and asserts that "Art as a cohesive core of culture ... [is] at all times the open enemy of political or intellectual intolerance." He writes this but then a few years later, he is fuming over art which takes aim at the very forces he claims to oppose. The party line at Brooklyn College—the denigration of politically charged art and the elevation of so-called "fine art" devoid of socially-engaged import—strikes me as an ethical dead end, riddled with inconsistencies.
Reinhardt and Ernst had both painted themselves into very conflicted positions, no pun intended. And there I was, supposedly the "star" pupil, seeing the hypocrisy and stifled by it. The thesis show was a declaration of my own identity as well as an attempt to force some sort of truth out into the open. Compulsory abstraction in art and the separation of "fine art" from "poster art" was the opposite pole of the same stupidity dominating the arts in the Soviet Union—the Cold War in art theory. I was looking for a way to express my anger and make art which spoke to the present world situation. At the time, I knew little of the history of the betrayal of the anarchist movement in Spain by the Communists and its effect on the intellectual left in New York, which was of course a major influence on the politics of abstract expressionism. But I knew a lot about McCarthyism and felt that this was the cause of the Art Department's extreme narrowness of discourse. The _Guardian_ artwork in my senior thesis was no doubt art serving the revolution, albeit an altogether different kind of "revolution." The line had been drawn in the sand and I crossed it.
So that was the treatment you received from the Brooklyn College-based art establishment. How was your art treated at the _Guardian,_ the _Rat,_ and by activists generally?
When I started doing art for publications, there was little understanding that there was any importance to it, and no understanding of my idea that you could have political discussions about imagery. Art work itself came way after the debate of the issues and the politics, rather than being part of a single fabric, whereas I had an image in my mind of a real synthesis of politics and art—that there could be a language of imagery that was meant to communicate. Using metaphor, the his-tory of images, referencing the history of art, I would come up with powerful art that could be read and understood.
There was no sophisticated discussion of your art in the radical scene?
It never happened at the _Guardian_ or at the _Rat._ There were groups who were thinking about literature and theater critically, but visual art was a kind of stepchild of it all.
Your work for the _Rat_ is really distinctive and has an incredible energy to it. I'd like to learn more about some of your specific illustrations. In the "Conspire-In" poster for the Yippie "Be-In" at the Sheeps Meadow, Central Park gathering of Easter Sunday, 1970, reproduced in one of _Rat's_ March 1970 issues, I see various symbols with text incorporated into it, along with an interesting negative/positive dichotomy involving clenched fists and open hands. Then there are the "Trading Cards" fea-turing political "outlaws" of the era, which were reproduced in the _Rat_ and intended to be cut out and passed around like trading cards.
It's really hard to remember what was going through my mind when I was doing the "Conspire-In" poster or the "Trading Cards" page. What I can say with certainty was that I was asked to do the Yippee's poster for their Central Park Be-In by one of the guys in the loose affinity group network on the Lower East Side. There was no particular request as to the imagery, and I thought that cut-outs would provide a stark eye-catching device. The demonstrators' posture and dress was an accurate rendering of how we looked when we went to a demonstration. The tepee emblem [which later served as a squatters' symbol in the 1980s] was likely lifted from whatever sheet of information I was given with the text needed on the poster. It was not my own creation.
The "Outlaws of America Trading Cards" was a group project at the _Rat._ I did most of the drawings, but the text and selection of characters was the result of free-ranging discussion into the late night by any and all participants during the layout of the issue. The general process was that the collective would meet to decide on the stories for each issue. The _Rat_ women and some male friends would return with their articles at the arranged time, which they would type out on manual typewriters. Each woman doing layout would be given the blank layout boards for two facing pages and the columns of the articles clipped to them. She was free to design the pages and to choose or create the artwork. Headlines were sometimes provided by the author and at other times decided on by whoever was interested in participating in the decision; these were either hand-lettered or placed down on pages with press type. The office was one large room with tilted tables built along the walls to work on the pages. It was therefore possible to stroll around the room and glance at the entire paper as it was being created. Once the copy was assigned to each page and space was allotted for the articles, the size dimensions of the artwork and which articles they were to accompany became apparent.
Moving on to present-day, you've been busy. Your art has appeared in _World War 3_ — _Illustrated,_ a political graphic arts magazine, and you contributed to _Wobblies!,_ an immensely popular illustrated history of the anarchist-syndicalist Industrial Workers of the World union; your photographs have appeared in a number of anarchist publications; you co-curated _Drawing Resistance,_ a traveling exhibition of anarchist/activist art that toured across Canada and the United States between 2001 and 2005 (see color plate 12); and you've been involved in puppet-making and some very innovative demonstrations. It seems to me there are a whole range of opportunities for an artist within the contemporary anarchist scene. Is it fair to say that current activism is healthier, from an artistic point of view, than it was in the 1960s?
Yes! The current scene is much healthier, and everything I've been doing has led to even more exciting possibilities. There are so many more ways for my art to get into the world—so many ways to collabo-rate, so many ways for art to find its way to people who are interested and appreciative. I have continued to do artwork in collaboration with activist groups and for publication and exhibition continuously, but the past ten years are far better than anything I have experienced before.
But my purpose is not to do art within the anarchist scene, al-though I am certainly nurtured by it in the broad sense. Anarchism needs to be more than a self-limited subculture. Art should inspire critical thinking and operate in the public sphere—be seen, understood, and embraced by a much wider audience—by people who agree and are inspired as well as those who disagree. Actually, I see a great danger in making art which is meant primarily for insiders within a "scene," especially when that scene is largely a self-segregated youth counterculture, which is the case where I live.
_Susan Simensky Bietila and others, Outlaws of Amerika Trading Cards, Rat, March 7-21, 1970. Pen and ink._
To clarify, when I refer to "the contemporary anarchist scene," I have in mind the larger anarchist community, including people like yourself in Milwaukee or the _World World 3_ artists in New York, for example—which is strong, dynamic, intergenerational, and definitely activist, and outward looking, not "subcultural," as you describe it. But tell me more about your recent art-making.
I was involved in lot of street theater in the late 1980s—performance with puppets and stilt-walkers protesting against American imperialism in Central America, against CIA recruitment on a college campus, for women's reproductive rights and more—and in the '90s against the destruction of urban green space and against threats by mining companies taking over indigenous people's land and resources. So the flowering of political performance worldwide in recent years, involving giant puppets and floats, especially inspired me. I've been photographing political street theater and treasure the opportunities to document it. I'm someone who lives very much in the present, and only with much time have I come to the realization of how important it is to preserve the history of radical movements.
I started graduate school during this period, and approached print-making and photography with greater sophistication in imagery and metaphor. I started to do art which worked on multiple levels, with art historical and philosophical references instead of straightforward agitational pieces. I found that even straightforward political work was no longer excluded dogmatically from gallery shows. But what really made a difference for me was going to the Active Resistance gatherings in 1996 and 1998. I went at the urging of one of my children, who was involved and thought that I would meet people I really liked there, and he was right. It was through these gatherings thatI met many of the political artists with whom I continue to collabo-rate. In addition, I was invited to photograph these gatherings and the photographs continue to ap-pear in wonderful publications.
_Susan Simensky Bietila,_ Self Portrait, _2006. Pen and ink wash._
It was at the Chicago AR gathering that I met David Solnit, and people from the _Fifth Estate_ journal, the Beehive Collective of artists, the A-Zone anarchist social center ... and I began to learn about the new wave of anarchist activism and art. At the 1998 AR gathering in Toronto, things only got better.
In 1998, I was in New York visiting family and went to the art opening of Seth Tobocman's show of work from "War in the Neighborhood." I had visited some of the artists from _World War 3_ in the early 1980s, soon after it was started, when I was part of a poetry and art zine called _The Stake,_ but had never met Seth before. He introduced me to other artists who drew political comix and invited me to do art for the magazine. I had been working almost exclusively with photography and was really into photo-collage, but he insisted that drawn narrative was the only format for the publication. I was surprised when he said that he was familiar with my work from the past and on that basis knew that I would be able to do story-board work. I was really delighted, and more than a bit surprised at the invitation. I continue to hold the artists who draw for _World War 3_ in great esteem.
At this same time also, I was active at home in Wisconsin opposing the Crandon Mine, inspired by the amazing diversity of the groups involved—from indigenous communities and environmentalists to duck hunters and fishermen. We were all part of a coalition against Exxon, Rio Algom, and then Billeton and their attempts to build a zinc and copper mine along the pristine Wolf River, next to where the Mole Lake Chippewa harvest wild rice, in the midst of beautiful national forests and directly upstream from the Menominee Reservation. In addition to the usual graphics, posters, banners, and flags for demonstrations and photo documentation, there were more innovative projects. After David Solnit and Alli Shagi Starr visited Milwaukee, we built a giant puppet of Tommy Thompson, the pro-mining Governor, and dressed him in a fool's cap. The puppet is still around ten years later, and has been passed around from group to group. After that I made a movable installation—thirty gravestones dedicated to rivers poisoned by mining around the world. Several wonderful anti-mining activists provided the research and helped conceive the project. The tombstones were only cardboard mounted on the wires used for election yard signs, but the show traveled for years to roadsides near sites threatened by mining in Wisconsin. It was important to me because it was effective for rural and reservation settings, places where installation and political art are not common.
My family and I had lived communally for many years, and I met [graphic artist] Nicolas Lampert when he and [film artist] Laura Klein answered our posting for housemates when they first moved to Milwaukee. We collaborated on a block print, "I need community," shortly before he went to the 1999 anti-World Trade Organization demonstration in Seattle and returned full of ideas. We decided that a traveling art show was an immediate possibility. We wanted to put together a show of all of our favorite artists and at the same time to make the point that political art is quite diverse in "look" and strategy of communication. We named the show, _Drawing Resistance,_ stealing Emily Abendroth's phrase to "Celebrate Communities of Resistance," and wanted to bring it to people who would never ordinarily be ex-posed to art with this sort of content. We decided that the show's tour must be compatible with the politics it displayed, and Nicolas's experiences touring with his band Noisegate provided the model for an art show as DIY punk-band-on-tour. Almost every artist we invited agreed to lend work, despite no assurance that the art would return from tour intact. The hosts for the show in each city had to transport the works to the next stop on the tour. There was no funding other than collections taken up at the door to provide gas money, etc. _Drawing Resistance_ had thirty-three exhibitions across the United States and Canada and traveled for four years.
_Installation of "Tombstones" at Wisconsin Capitol. Monuments to rivers poisoned by mining. April 2000._
It was a great deal of work assembling the show, but what is important to me is that it happened. The show helped build networks of artists and communities as well as get excellent art out to people who had enthusiasm for it. It tapped potentials for collaboration and articulated our politics in practice. I wish that more people were thinking this way.
For me, making art is driven by collaboration with political movements. Invite me to be part of a worthwhile project where there is real collaboration and I'm ready to do my part and more. What inspires me is knowing that the art will be seen and travel to places that I will never go. It is part of me, but takes on a life of its own, going out into communities where it seeks out people ready to engage. So my strategies include making art which is easy to reproduce as opposed to work designed to be site-specific or function primarily in a gallery space, even though I enjoy curating gallery shows as well as collaborating on shows with provocative themes. But, no surprise, this involves working pretty much outside the "art as commodity" system. So I work with alternative galleries and other community spaces.
_Drawing Resistance opening at the Babylon in Minneapolis, March 2003._
Any thoughts for anarchist artists starting out today?
My experience has proven to me beyond my wildest expectations that artwork about contemporary issues does not become tomorrow's trash. The topical artwork that I did between 1968 and 1970—work that was supposedly due for "next day disposal" according to the Brooklyn College Art Department—has not disappeared. It has been reprinted and exhibited over and over since then with no effort on my part. Most important is the obvious: you don't need to choose between your activism and your art. Make art as part of the discourse. Create your own opportunities.
### NOTES TO CHAPTER 7
1 Murray Bookchin, "The Youth Culture: An Anarchist-Communist View," _Hip Culture: 6 Essays on its Revolutionary Potential_ (New York: Times Change Press, 1970): 57.
2 Ibid., 59.
3 The Student Peace Union (1959-1964) was an intercollegiate group organized by socialist, pacifist, and other anti-war students that was critical of the foreign policy of both the United States and the Soviet Union, and protested against the arms race, nuclear weapons testing, racial segregation, the American government's position during the Cuban Missile Crisis (October 1962), and its involvement in Vietnam.
4 In October 1960, a wave of African-American-led sit-ins protesting segregation in restaurants, public parks, and government institutions culminated with the founding of the Student Non-Violent Coordinating Committee. Adopting non-violent direct action as its credo, SNCC played a leading role in the civil rights movement which sought to end segregation in the United States. In 1966, black power advocate Stokley Carmichael, who was critical of the organization's non-violent tactics, was elected chairman. Thereafter, SNCC fell into rapid decline and was officially disbanded in 1970.
5 Students for a Democratic Society was founded in 1960 as an outgrowth of stu-dent participation in the civil rights movement. SDS membership mushroomed as a result of the Vietnam War. The turn towards more militant tactics by many leading figures in SDS, coupled with internal infighting, led to dissolution in 1969.
6 The Free Speech Movement, which began at the University of California, Berkeley in 1964, sought to establish academic and student freedom of speech and expression on campuses across the United States.
7 Freedom House was founded in 1941 to promote electoral "democracy" and the capitalist "free market" system throughout the world. It gets two-thirds of its funding from the American government.
8 _Détournement_ is a term coined by the French situationists (see note 13) to describe the appropriation of mass media imagery to create new work with a politically subversive message.
9 Trotskyists is a loose term for numerous political parties claiming allegiance to Russian Marxist Leon Trotsky (1879-1940).
10 Founded in 1961 by former members of the Communist Party USA, the Progressive Labor Party advocated armed revolution under PLP leadership. During the 1960s, it sought to infiltrate and manipulate the militant student movement, primarily by gaining control of SDS.
11 The House of Un-American Activities Committee was created under the mandate of Public Law 601, passed by the United States Congress in 1946. A committee of nine representatives investigated suspected threats of subversion or propaganda that "attacks the form of government guaranteed by our Constitution." In 1969, the committee's name changed to the Committee on Internal Security.The House abolished the committee in 1975 and its functions were transferred to the House Judiciary Committee. At the hearing witnessed by Simensky Bietila, Rubin showed up in an American War of Independence costume and attempted to hand out copies of the American Constitution while mocking the committee: this widely publicized guerrilla theater tactic effectively ended HUAC's reign of intimidation. Yippie—short for "Youth International Party"—was founded in New York in 1968 as a loosely affiliated network of anarchist direct action/guerrilla theater oriented collectives.
12 On October 2, 1968, a student uprising in Mexico City protesting the dictator-ship of the ruling Institutional Revolutionary Party was brutally put down by the Mexican military.
13 The situationists were a small Paris-based organization led by the theorist Guy Debord. Formed in the late 1950s and active through the 1960s, the organization was officially disbanded by Debord in the early 1970s.
14 The Bread and Puppet Theater is an activist group founded in the early 1960s by Peter Schumann in New York. During the Vietnam War, the group gained notoriety for staging dramatic puppet pageants at demonstrations.
15 The petitions are discussed in Francis Frascina, _Art, Politics, and Dissent: Aspects of the Art Left in Sixties America_ (Manchester: Manchester University Press, 1999): 23-24.
16 McCarthyism is a term referring to the social repression of the early to mid-1950s, when American Senator Joseph McCarthy spearheaded a campaign to criminalize political radicals in every walk of life, from the civil service to the film industry.
17 The German-based Bauhaus was an innovative arts and design school, founded in the 1920s, which disbanded after the Nazis came to power in 1933.
18 Arno Hauser, _The Social History of Art_ (New York: Alfred A. Knopf, 1951).
19 Ad Reinhardt cited in Frascina, 82.
20 Angry Arts was a New York-based group formed in 1967 to stage a week of art events protesting the war in Vietnam.
21 Jeanne Siegel, _ArtWords: Discourse on the 60s and 70s_ (Ann Arbor, MI: UMI Research Press, 1985): 105.
22 Ibid., 112.
23 Ibid.
24 Ibid., 116.
25 Ibid., 113.
26 Ibid., 112.
27 Ibid., 105.
28 Ibid.
29 The draft was a system of selection for mandatory service in the United States military. During the 1960s every able-bodied male over the age of eighteen had to register for the draft. College and university students were given draft deference until they graduated or reached age twenty-four. In 1966, the military instituted a "Selective Service College Qualification Test"; those students who scored low lost their draft exemption.
30 The British-based Campaign for Nuclear Disarmament (CND) organization, founded in 1958, was one of the most important activist movements of the 1960s. The CND called for the unilateral disarmament of Britain's nuclear arsenal.
31 Journalist Peggy Duff was a leading activist in the British peace movement, as was academic Bertrand Russell, who helped found the War Crimes Tribunal in 1966 to examine the conduct of the American army in Vietnam.
32 Provo (1965-67), short for provocateurs, was an Amsterdam-based anarchist group known for imaginative tactics of social disruption.
33 Be-ins were mass gatherings to celebrate counter-cultural values. The name was a variation on the civil rights "sit-in" and, like the sit-ins, be-ins were a peaceful form of protest against the status quo.
34 Roel van Duyn, author of _Message of a Wise Kabouter_ (1972), was the group's leading theorist and tactician.
35 General Francisco Franco (1892-1975) was the self-appointed "supreme leader" of a fascist Catholic regime that ruled Spain from 1939 until his death. He led the insurrection against the Spanish Republican government during the civil war.
36 Robin Morgan, _Going Too Far: The Chronicle of a Feminist_ (New York: Vintage, 1978): 72.
37 Abbie Hoffman, _Soon to Be a Major Motion Picture_ (New York: Perigee, 1980): 120.
38 "The Antiwar Movement in New York City 1965-67," updated and revised from _Les Temps Modernes_ (September 1968), <http://www.fermentmagazine.org/Bio/newleftl.html>
39 The National Liberation Front for the Liberation of Vietnam, also known as the "Vietcong," was an alliance of religious and political groups formed in 1960 to fight a guerilla war against the United States-supported government of South Vietnam.
40 The Black Panther Party, founded in California in 1966, was a militant political organization that propagated the right to self-defense and self-determination for African-Americans. Police and FBI covert actions, including assassinations, destroyed the Black Panthers by the late 1970s.
41 The Weathermen, formed in 1968 during the breakup of the SDS, operated as an underground organization dedicated to "bringing the war home" through street protests and the bombing of symbolic and operational targets in the American political and military infrastructure. Active into the 1970s, the Weatherman underground collapsed in large part due to numerous arrests, internal dissention, and the waning of American radicalism after the war in Vietnam ended.
42 New York Radical Women was the city's first feminist liberation group, founded in the fall of 1967 by Shulamith Firestone and Pam Allen. It dissolved in the winter of 1969. Redstockings was founded in 1969 by Firestone and Ellen Wallis as a more militant successor to NYRW. It folded in the fall of 1970. See Alice Echols, _Daring to be Bad: Radical Feminism in America_ , 1967-1975 (Minneapolis: University of Minneapolis Press, 1989): passim.
43 Jimmy Ernst, "A Letter to Artists of the Soviet Union," _Art Journal_ 21, no. 2 (1961): 66-71.
44 Codified in Russia during the 1930s, socialist realism combined a realist style with "socialist" content that reflected the leading role of the Communist Party and the positive transformation of society under its guidance.
45 Jimmy Ernst, "Freedom of Expression in the Arts II," _Art Journal_ 25, no. 1 (1965): 46-47.
## CHAPTER 8
WITH OPEN EYES _Anarchism and the Fall of the Berlin Wall_
_I think the current era has ominous portent—and signs of great hope. What result ensues depends on what we make of the opportunities._
—Noam Chomsky, 19951
On November 9, 1989, Communist Party rule in eastern Europe and the Soviet Union came to a dramatic end with the breaching of the Berlin wall, leaving many leftists in the capitalist west perplexed and disoriented.2 Some argued that a "genuinely new and emancipa-tory" form of socialism—to quote Nancy Fraser—might miraculously emerge from "the wreak," while others framed the historical moment as a stark choice between "socialism or barbarism"—with barbarism coming out on top in the absence of a "left opposition" calling for "workers' democracy, a working class state, and a political revolution that would restore the possibility of Communism."3 The prevailing position was summed up by Fred Halliday in _New Left Review:_ "An essential precondition for any viable socialism in the West," he wrote, "is a degree of combativity towards the very system it's challenging. Whatever their other faults, the traditional Communist parties em-bodied that quality"—but now, with the collapse of Communism in the east, they had ceased to do so.4 The solution? Return to Communism's "point of origin," namely "the critique of, and challenge to, capitalist political economy" initiated by Marx.5 Of course, from an anarchist perspective, going back to Marx was hardly the place from which to build a libertarian movement. In the years leading up to the fall of the wall, anarchists were otherwise engaged in the development of strategies that were distinctly anti-authoritarian and true to a critique of oppression in all its guises.
Anarchist-pacifism was one such strategy, expressed first and fore-most in the art of Gee Vaucher, illustrator for the punk rock band Crass. Originating out of the British music scene in the mid-1970s, punk was loosely associated with anarchism from its inception, thanks to the Sex Pistols. Songs like "Anarchy in the UK" (1976), which the group's manager Malcolm McLaren described as "a statement of self-assertion, of ultimate self-rule, of do-it-yourself," inspired thousands of fans to explore anarchist politics.6 However, the Sex Pistols' propagation of anarchism was more a publicity stunt than a political act, and their eagerness to sign on with commercial record companies was generally viewed as a betrayal of anti-capitalist principles.7 This left Crass to point the way forward.8
Formed in 1977 by Penny Rimbaud and run as a collective, Crass (the core membership of nine was occasionally augmented by others) created its own not-for-profit record company and committed itself to promoting anarchism in whatever way it could. The band financed activist publications, organized benefits for squatters and rape crisis centers, raised money for anarchists accused of plotting a bombing campaign, and helped found an Anarchist Centre in London.9 The group's resident artist, Vaucher, was a talented illustrator, collage artist, and former member of the London-based performance art collective, EXIT (1968-1972).10 Between 1978 and 1984 (the year the group disbanded), Vaucher produced artwork for a slew of Crass record covers, five issues of an illustrated political journal entitled _International Anthem,_ and numerous posters.
The cornerstone of Crass's anarchism was a combination of class war and pacifism. In a March 1981 interview for the punk journal _Flipside,_ band members condemned statist politics in the capitalist and Communist spheres on the grounds that both were violent and coercive. "Anarchism," they related, "is the only form of political thought that does not seek to control the individual through force.... Anarchy is the rejection of state control and represents a demand by the individual to live a life of personal choice, not one of political manipulation. ... By refusing to be controlled you are taking your own life into your own hands and that is, rather than the popular idea of anarchy as chaos, the start of personal order."11 However, the rejection of state violence was not a blanket renunciation of the right to resist. The members of Crass were careful to emphasize that they stood "against organized militarism, believing that the use of power to control people is a violation of human dignity"; however, when "power threatened to directly violate" an individual, one had the right to "stand against it in whatever way was necessary to prevent it," including "the possibility of force."12
When performing, Crass displayed three banners: the circled A anarchy symbol; the peace symbol; and the Crass logo, which attacked the pillars of oppression in Britain—the patriarchal family, church, and state. "Part national flag, part cross, part swastika," writes Rimbaud, the logo's "circular design broke on its own edges into serpents' heads, suggesting that the power it represented was about to consume itself."13 An insert poster for the Crass single "Nagasaki Nightmare" (1980) (see color plate 13) illustrates the band's stance towards state power in its Communist and capitalist guises. Vaucher depicts leaders of the world's five great nuclear powers (China, Britain, France, Russia, and the United States) and aspiring members to this elite "club" (Cuban dictator Fidel Castro, for example), grouped together as if for a "photo op" amid the flattened ruins of Nagasaki. Three Japanese atomic bomb victims and the bomb itself exploding in the distance add to the horror. The politicians' shared willingness to "do it again"—as the song relates—is symbolized by the jovial demeanor of American President Ronald Reagan and Soviet leader Leonid Brezhnev as they reach out to shake hands over the charred corpse of a child.14
The visceral impact of "Nagasaki Nightmare" was matched by Vaucher's productions during the Falklands War. In spring 1982, an Argentinian military dictatorship went to war with Britain over the Falklands, a small grouping of islands off the coast of South America (the islands were claimed by Britain in 1690 and formally designated as a colonial holding in the early twentieth century). The war lasted from March to June, when Argentinian forces surrendered unconditionally after suffering heavy losses at the hands of the British navy and air force. In Britain, the unpopular Conservative Party government of Margaret Thatcher successfully renewed its prospects for reelection the following year by rallying the population behind prowar jingoism. This development stunned Crass, given the actions of the government up to that point. As Rimbaud relates, after coming to power in 1979, Thatcher had "set about dismantling everything that was worthwhile about the Welfare State. In the new 'enterprise culture' compassion would become a dirty word as entrepreneurs licked their lips and Thatcher's arse alike in their rush to buy up the country's assets. Just as working-class values would be derided, so the poor would be accused of having brought about their own fate.... To Thatcher and her cronies they were wasters, layabouts, and good-for-nothings who deserved no more than the blow of her iron fist or the toe of a policeman's boot."15 And yet once Thatcher declared war, the masses had fallen in lock-step.
In a post-war essay adapted into lyrics for Crass's 1983 album, "Yes Sir, I Will," Rimbaud underlined the paradox: "Tory policies required massive unemployment, but it was they who demanded that 'we should support our lads in the Falklands,' those very same lads, who if they were at home, would be jobless in the streets."16 "Yes Sir, I Will" ends with a plea for the working class to wake up to their own oppression and rise up against the state and the ruling class it served: "It is you, the passive observer, who has given them their power. It is you, and only you, who can withdraw it. You are being used and abused, and will be discarded as soon as they've bled what they want from you. You must learn to live with your own conscience, your own morality, your own decisions, and your own self. You alone can do it. There is no authority but yourself."17 This is followed by an addendum: "A squaddy [slang for soldier], horrifically burnt in the Falklands War, was approached by Prince Charles during a presentation. 'Get well soon' said the Prince, to which the squaddy replied, 'Yes Sir, I will.'"
_Gee Vaucher, Yes Sir, I Will album cover, 1983. Collage, 32 cm × 25 cm._
_Gee Vaucher,_ Yes Sir, I Will _insert poster, 1983._
Vaucher's poster for the album, "Yes Sir, I Will," reproduced the text of this exchange and a tabloid newspaper photograph of the inci-dent. The spectacle of the horribly disfigured soldier spoke volumes about the grotesque inanity of a dutiful working-class war victim acquiescing to authority by indulging "his" Prince in the comforting fiction that he would ever "get well." The theme of self-sacrifice and submission also graced the album cover, where Vaucher collaged the figure of an emaciated soldier, war medal pinned to his chest, crucified against a bleak industrial backdrop.
The soldier's face—a gaping wound, half ripped away, revealing his teeth and jaw bone—was taken from an anti-war photo book, _War Against War!,_ published by the German anarchist-pacifist Ernst Friedrich in 1924.18 In this book, World War I-era photographs of self-satisfied officers, kings, and politicians, destroyed towns, rotting corpses, mass hangings, rape victims, and mutilated war veterans were captioned with statements of outrage, satirical commentaries, bellicose pro-war declarations, patriotic songs, and militaristic slogans. Friedrich was one of Germany's most outspoken radicals whose opposition to militarism was part-and-parcel of his opposition to the state. In fact, the class-based call to consciousness at the end of "Yes Sir, I Will" echoed the preface of _War Against War!,_ in which Friedrich argued:
... it is not the state power and force alone that compels all "subjects" to protect the throne and the money-bags, and to die for them. Capital has not only economic power in its hands; it has, equal measure and with equal power, subjected the proletariat also intellectually. This fact is easily overlooked and there still remains, therefore, so much bourgeois ideology in the proletariat! I, therefore, always say to my brothers, the proletarians, I say to the class-war fighters: "Free yourselves from bourgeois prejudices! Fight against capitalism within yourselves! In your thoughts and in your actions there still lurks unspeakably much of the philistine and the soldier, and almost in every one there is hidden a drilled subaltern, who wishes only to dominate and command, even if it be over his own comrades and over his wife and children in his family!" But I also say to those bourgeois pacifists, who seek to fight against war by mere hand caresses and tea-cakes and piously up-turned eyes: "Fight against capitalism—and you fight against every war! The battle-field in the factories and the mines, the hero's death in the infirmaries, the mass graves in the barracks, in short, the war, the apparently eternal war, of the exploited against the exploiters!"19
Incorporating Friedrich's masterful satire of World War I-era propaganda into her ironic tribute to the Falklands adventure, Vaucher signaled that Crass's anarchism was firmly rooted in history, with all that implied by way of a well-thought out, socially engaged perspective.20
While Vaucher and Crass attacked the institution of the state, both Soviet and capitalist, from an anarchist-pacifist stance, in the United States, anarchists associated with the journals _Fifth Estate_ and _Anarchy: A Journal of Desire Armed_ developed an equally thorough-going critique of the two systems. To quote David Watson, writing in the Detroit-based _Fifth Estate:_ "Ideology East and West has reasons to deny it, but the truth is that to focus on juridical property relations and terms by which hierarchically organized societies named them-selves is to commit a grave, formalistic error. Modern state socialism was only a manifestation of the capitalism it claimed to supersede."21
Whether Communist Party-run or democratically-elected, human labor and the natural world were exploited under the aegises of the state for the benefit of a ruling class—in democracies, the property owners and technocrats, and under Communism, state and party bureaucrats.22 Saturated by an ideology that valued hierarchy, domination, alienation, and production above all else, industrial societies were in the grip of the "megamachine"—a term first coined by Lewis Mumford to describe the mode of social organization and production through which the civilization-building elites of ancient post-hunter-gatherer societies secured control over the societies they ruled.23 This ancient system of social organization renewed itself in Europe in the first half of the twentieth century, when Communist and fascist state-capitalist dictatorships came to power. World War II accelerated its emergence in the democracies, which set the stage for the era of the Cold War, when the Communist megamachine faced off against its democratic-capitalist counterpart. The megamachine's spread was aided and abetted by industrialism, which empowered its technological capacities on a qualitatively different basis. The fall of the Berlin wall had brought no real liberation; it merely signaled that a less efficient version of the megamachine could not sustain itself in the face of the competition. (Avoiding the Soviet debacle, the Chinese Communist Party managed China's transition voluntarily.)24
In sum, in the early 1990s the megamachine stood triumphant. Hovering over the twentieth-century like a baleful spirit, it had spread its power across the globe, with cataclysmic results for humanity:
[The megamachine] demands that humans conform to laws implicit in the technology itself.... Modern technologies require hierarchical and authoritarian forms of social organization in order to function.... Technological systems require a dependence of humans on these systems, and on the experts to develop and run them.... Industrial technologies are inherently damaging to the environment: outcomes are not foreseeable; there are not solutions to all problems; mistakes are inevitable; contamination is an inevitable part of the industrial system.... The ways in which humans view the world, their imaginations and perceptions, become adapted to the technological world. Humans begin to think and act in terms of the machine.25
By way of countering this state of affairs, Watson and others in his milieu argued that the "renewal of the sacredness of nature, its interrelatedness, and our connectedness to it" is central.26 "If we cannot see the spirit that resides in the natural world," Watson has stated, "we cannot fully envision the ineffable human spirit of liberty that has motivated the anarchist project—before it was called 'anarchist'—from the beginning of class societies."27 In this regard, "primal" societies have an important role to play in the struggle to reconnect with the natural world and our true nature as sentient beings. They represent an alternative to draw on in the quest to shed the "technological way of life."28 As Watson puts it, pre-technological societies are the key to _"learning to live in a different way_.29
Artistically, Watson's closest collaborator has been Freddie Baer, a collage artist and past contributor to the _Fifth Estate._ Born in Chicago in 1952, Baer was active in a number of Chicago-based anarchist organizations before moving to San Francisco in 1978.30 She began donating collage work to the anarchist press in the early 1980s, and her illustrations appeared frequently in the _Fifth Estate_ throughout that decade.31 Politically, her outlook can be deduced from an interview conducted in 1992, in which she summarizes humanity's collective situation as unsustainable. "The world is changing so fast," Baer related, "the collapse of Communism in the Soviet Union and eastern Europe and the corresponding rise of nationalism on the one hand, the re-placement of governance by the rise of Capital on the other; total and complete eco-disaster that looms on the horizon; the dehumanization of the individual; the turn to the right by the United States and Great Britain. Things can't go on the way they have been."32 Searching for a way forward, Baer told her interviewer she had found inspiration in "the theoretical growth taking place within parts of the anarchist/anti-authoritarian community, especially in the pages of the _Fifth Estate_ and _Anarchy: A Journal of Desire Armed"_33
Her most arresting depiction of the megamachine thesis is an un-titled collage based on Pieter Bruegel the Elder's oil-on-panel depiction of _The Tower of Babel_ (c. 1563). Created as the cover illustration of Watson's definitive compendium, _Against the Megamachine: Essays on Empire & Its Enemies_ (1999), Baer's untitled collage merges the ancient Biblical tower with massive hydroelectric pylons connected to transformers jutting out of the uncompleted edifice (see color plate 14).
The "Tower of Babel" story in the Old Testament's book of Genesis is a tale of humanity's downfall at the dawn of ancient civilization. Settling on the plains of Mesopotamia (Shinar), people build a great city and begin constructing a structure that climbs into the heavens. However, God does not look kindly on the enterprise and sows chaos by taking away humanity's common language.34
Baer's collage is a brilliant encapsulation of the _Fifth Estate_ critique: the hubris of a mythological megamachine prefigures the folly of present-day humanity's quest to transcend our earthly limitations through technology, with equally disastrous consequences for all concerned.
I began this chapter suggesting anarchists were well prepared to respond to the fall of the Berlin wall and its aftermath. In this regard, the Persian Gulf War (January 15-February 28, 1991), which overlapped the wall's dismantling (completed in November 1991), proved to be a bellwether event. If the Cold War between the Soviet Union and United States in its capacity as capitalism's anointed leader had actually been a convergence of competing powers jockeying over re-sources in their respective spheres of influence, then it stood to reason that with one of the powers disintegrating, the other would wax expansive in its ambitions.35 In short, there would be no "peace dividend," as many hoped, in the wake of the wall's destruction.36 Instead, in the words of anarchism's most well-known political commentator, Noam Chomsky, it would be "more of the same."37 And so it was.
The Gulf War has been characterized as a "desert holocaust" for good reason.38 Following the invasion and annexation of Kuwait by Iraq in August 1990, US President George Bush (Sr.) mobilized an international coalition under the auspices of the United Nations with a mandate to push the Iraqi forces out. Through the fall of 1990 into 1991, the United States and its key allies blocked any attempts to re-solve the issue peacefully while the armed forces of America, Britain, and a host of lesser allies gathered in Saudi Arabia. Bush declared war on January 15 (American civil rights leader Martin Luther King's birthday), and an air campaign began two days later. From then through the duration of the conflict, over 1,000 sorties a day bombed Iraqi infrastructure and pulverized its military forces. By the time the coalition ground campaign began on February 23, Iraqi forces in Kuwait were demoralized and fleeing in disarray. War ended a few days later, when Bush announced a unilateral ceasefire following the complete expulsion of Iraqi forces from Kuwait.39
During the war, coalition forces committed numerous atrocities. Among them: the wide use of depleted uranium shells on the battlefield, leaving behind toxic radioactive residues; the live burial of thousands of Iraqi troops by tanks equipped with plows; the deliberate bombing of Iraqi civilians, including those in air raid shelters; the targeting of Iraqi infrastructure—power plants, dams, sewage and water treatment facilities—of no military value; and the slaughter of thousands of Iraqi troops fleeing Kuwait along the so-called "highway of death." Last but certainly not least, the coalition left behind epic ecological devastation. Massed troops, bombings, and tank battles chewed up the fragile desert, and the destruction of oil facilities resulted in thick black toxic clouds that blanketed the region for months, while oil spilled freely into the Gulf waters and across vast desert expanses.40
Death against life—that was the issue that gripped print artist Richard Mock in the war's immediate aftermath. The result was four linocut prints—the "Gulf War" series (1991). At the time he created them, Mock's socially critical linocuts were a regular feature in the op-ed pages of the _New York Times,_ where he had been active as an editorial illustrator since 1980. Additionally, he was a painter and sculptor with a substantial exhibition record, including one-man shows at the Houston Contemporary Art Museum, the Bronx Museum of Art (New York), the Albright Knox Museum (Buffalo), and the Centro Cultural Arte (Monterrey, Mexico).41 Anarchism was a less well-known aspect of his work, but a vital one nonetheless. In a 2001 interview, Mock attributed his political orientation to an empathy with the planet going back to his childhood years in California during the 1950s (born in 1944, he grew up in Long Beach, just north of the Baja peninsula). In the profoundest sense, he had "always been an anarchist." His first exposure to anarchism came during his college years at the University of Michigan (1961-1965), when he read British art critic and political philosopher Herbert Read's essays, notably _Anarchy and Order_ (1954), an exposition of anarchist-communism along the lines of Kropotkin (see Chapter Two) which had a lasting impact on Mock's outlook.42
Asked in 2001 what an anarchist society would entail, Mock responded: "We would create harmony between man and nature. And we would discover, in an anarchist society, new dimensions of being human. We would take down our armor and be revelatory, revelatory in allowing the growth of collective attachments to the earth and to other people."43 Artistically, he encapsulated his social-ecological vision best in his abstract paintings— _Untitled_ (1995), for example—in which rhythmic flecks of bright color unfold organically in a dynamic interplay that finds resolution in the whole (see color plate 15). Mock called these paintings "cosmic" and "transcendent" because they create a visual field that expands beyond the picture plane, and this was his metaphor for the open structure of a harmonizing anarchist social order, as "natural" as nature itself.44
_Richard Mock,_ Oil Spill Kill, 1991. _Linocut print, 20" × 15¾."_
How, then, did the Gulf War prints figure in his politics? Referring to his linocuts, Mock once observed, "It's [in] the nature of my being to attack the enclosing power structures that are out to suck the planet dry."45 Here we have the key. "Oil Spill Kill" (1991) presents the conflict as an assault on nature in the raw—Arabian Gulf wildlife is shown struggling and dying in a pool of black oil. "Victim" (1991) depicts the remains of a camel, body parts askew, whose skull has been crushed by an artillery shell. One eye pops out of the fleshy rot like a jack-in-the-box, making the abjectness of its plight almost comical, were it not for the red blood pooled underneath. Power and inequality are the theme of "Raper Vapor" (1991), in which a spiny-tailed desert lizard cautiously approaches a discarded gas mask, flicking its tongue in a futile attempt to understand it in its own terms. And lastly, Mock presents the Gulf War as death turned in on itself, transforming the entire ecology, humans included, into a "Target" (1991). Responding as nature will, the desert's most lethal creatures surround the threat and prepare to strike. Seemingly, the natural order is attempting to combat death in the guise of a species gone awry, so disconnected from the planetary order of things it is bent on destroying itself along with everything else.
_Richard Mock,_ Raper Vapor, 1991. _Linocut print, 20"_ × _15¾"._
Mock's prints critiqued the Gulf War as an extension of an authoritarian power structure: an ecologically catastrophic, socially oppressive power structure at war with nature and in denial of the consequences. And so I return to the question of how to overcome this state of affairs. In one of the most searching responses to the demise of Communism, _Dreamworld and Catastrophe: The Passing of Mass Utopia in East and West_ (2000), prominent leftist Susan Buck-Morris lamented the passing of Marxism in the Communist East as a critical means of understanding capitalism while characterizing Marx's understanding of socialism as a failure: as if the two—Marx's critique and his vision of the future—had nothing to do with one another.46 Anarchism is never raised in her book, which ends with an appeal for leftists to strive, like Lenin, to be "as radical as reality."47 This evocation of Lenin, whose certainty regarding the nature of reality led him to found one of the twentieth-century's most oppressive regimes, stands in stark contrast to the libertarian attitude of the artists discussed in these pages. To paraphrase Mock, they strive to be revelatory; that is, self-revealing politically, socially, and ecologically. Critiquing oppression while calling attention to the anarchic potentialities within society, they prefigure a world of possibilities in which each and every one of us are the index of reality's radicalism.
### NOTES TO CHAPTER 8
1 Noam Chomsky, _Chomsky on Anarchism,_ Barry Pateman, ed. (Edinburgh: AK Press, 2005): 189.
2 Susan Buck-Morris, _Dreamworld and Catastrophe: The Passing of Mass Utopia in East and West_ (Cambridge, MA: MIT Press, 2000): 238. In a 1995 interview, Noam Chomsky also noted this reaction and expressed amazement that "people who had considered themselves anti-Stalinist and anti-Leninist—were demoralized by the collapse of the tyranny." See Chomsky, 166.
3 Nancy Fraser, "Postcommunist Democratic Socialism?" _After the Fall: 1989 and the Future of Freedom,_ George Katsiaficas, ed. (London: Routledge, 2001): 200; Brian Palmer, "Socialism or Barbarism in Eastern Europe?" _Between the Lines_ (November 9-23, 1989): 3.
4 Fred Halliday, "The Ends of Cold War," _New Left Review_ 180 (March-April, 1990): 21.
5 Ibid., 23.
6 Malcolm McLaren cited in Neil Nehring, _Flowers in the Dustbin: Culture, Anarchy, and Post-War England_ (Ann Arbor: University of Michigan Press, 1993): 312.
7 For a period compendium of the Sex Pistols' exploitive courting of establishment outrage and the ensuing media frenzy, see Fred and July Vermorel, _Sex Pistols: The Inside Story_ (London: Ominibus Press, 1978).
8 Craig O'Hara, _The Philosophy of Punk: More Than Noise!_ (Edinburgh: AK Press, 1999): 81.
9 Penny Rimbaud, _Shibboleth: My Revolting Life_ (Edinburgh, AK Press, 1998): 99-100;117-122.
10 Gee Vaucher, _Crass Art and Other Pre Post-Modernist Monsters_ (Edinburgh, AK Press, 1999): 6-7.
11 Crass cited in O'Hara, 81.
12 Crass cited in O'Hara, 88.
13 Rimbaud, 90-91.
14 Crass, "Nagasaki Nightmare" (1980). www.plyrics.com/lyrics/crass/nagasakinightmare.html
15 Rimbaud, 110-111. See Crass, "Yes Sir, I Will" (1983). _www.plyrics.com/lyrics/crass/yessiriwill.html_
16 Ibid., 236
17 Rimbaud, 239.
18 Ernst Friedrich, _War Against War!_ (Seattle: Real Comet Press, 1987).
19 Ibid., 25.
20 Crass's anarchist-pacifist politics are clearly indebted in the first instance to the position of the War Resisters International (WRI). Founded in 1921, the WRI's leading theorists, Bart de Ligt and H. Runham Brown, were both anarchists, and they played a key role in the formulation of the WRI's program. For the WRI, capitalism and authoritarian institutions such as the state must be abolished be-fore a lasting peace can be achieved. Resistance to authority is part of the class struggle pitting the working class against its class oppressors. Pacifism is a tactical strategy for overthrowing authoritarian power on a qualitatively different basis. For a definitive statement of the WRI platform, see Bart de Ligt, _The Conquest of Violence: An Essay on War and Revolution_ (London: George Routledge & Sons, 1937).
21 David Watson, "The Fall of Communism and the Triumph of Capital" (1992), re-printed in _Against the Megamachine: Essays On Empire and Its Enemies,_ (New York: Autonomedia, 1999): 95.
22 Ibid., 96.
23 On Mumford and the megamachine thesis, see Steve Millet, "Technology as capital: _Fifth Estate's_ critique of the megamachine," _Changing Anarchism: Anarchist Theory and Practice in a Global Age,_ Jonathan Purkis and James Bowen, eds. (Manchester: Manchester University Press, 2004): 82.
24 Watson, "The Fall of Communism and the Triumph of Capital," 106-108.
25 Ibid., 90.
26 David Watson, "Anarchy and the Sacred," _Against the Megamachine,_ 167. This chapter is a compilation of excerpts from articles and responses to letters published in the _Fifth Estate_ between 1987 and 1989.
27 Ibid.
28 Ibid., 174.
29 Watson, "Against the Megamachine," _Against the Megamachine,_ 144. This chapter is based on articles published in the _Fifth Estate_ in 1981-84 and 1997.
30 Thomas Murray Sate, "Questions for Freddie Baer," _Ecstatic Incisions: The Collages of Freddie Baer_ (Stirling, Scotland: AK Press, 1992): 3–4.
31 Ibid., 6.
32 Ibid., 9. _Anarchy: A Journal of Desire Armed_ published extensively on the prospects for anarchism in eastern Europe and Baer's comments suggest she was following the discussion closely. Many authors were cautionary, noting a rise in nationalism and fascism alongside anarchist currents. See, for example, Laure A., "At the Berlin Wall: A Personal Report on Eastern Europe," _Anarchy: A Journal of Desire Armed_ 24 (March-April 1990): 4, 10. Others pointed out that Soviet-style industrialism had left ecological devastation in its wake and now a "massive penetration of Western capital" was reducing the East to "essentially Third World status as markets and sources of materials and cheap labor." See Will Guest, "Ecocide on the East Side: The Environmental Crisis in Eastern Europe," _Anarchy: A Journal of Desire Armed (Summer_ 1991): 21.
33 Ibid., 11.
34 "Tower of Babel." _www.biblegateivay.com/passage/?search=genesis%2011; &version=31_
35 Noam Chomsky, _Deterring Democracy_ (New York: Hill and Wang, 1991): 27–28. On Chomsky's anarchism, see _Chomsky on Anarchism,_ Barry Pateman, ed. (Edinburgh: AK Press, 2005).
36 William Blum, _Killing Hope: U.S. Military and CIA Interventions Since World War II_ (Monroe, ME: Common Courage Press, 2004): 320.
37 Ibid., 320.
38 Chomsky, _Deterring Democracy,_ 59.
39 "Gulf War." http://en.wikipedia.org/wiki/Gulf_War (accessed November 7, 2006).
40 See Blum, 335–337 and Doublas Keller, _The Persian Gulf TV War_ (Boulder, CO: Westview, 1992): 410–414.
41 Allan Antliff, "Ecological Anarchy: Richard Mock," _Alternative Press Review_ 10 no. 2 (2006): 6.
42 Ibid., 7.
43 Allan Antliff, "Richard Mock," _Quivers: Twenty Linocuts_ (Omaha, NB: Gallery 72, 2002): 10.
44 "Richard Mock: Interview with Allan Antliff" (June 25, 2001).
45 Ibid.
46 Buck-Morris, 239.
47 Ibid., 278.
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## INDEX
291 , –7,
291 art gallery , ,
abstract expressionism ,
abstraction and abstractionists –5, –4, , –7, , –40, –3, , , –8,
_Action d'art_
Amsterdam –54,
_Anarkiia_ , –84, –9,
Angrand, Charles
anti-obscenity laws. _See_ censorship
Arbeiter Ring
_Ark_ ,
Armory Show, the –3, –5
_Art Journal_
Artistocrats, the
Askarov, German ,
Baer, Freddie –90
Bakunin, Michael –14,
Barand –3
Barr, Alfred
Bataille, George
Belgium –42
Bellmer, Hans
Bergson, Henri , , –9,
_Creative Evolution_
_Laughter_
_Time and Free Will_
Berlin wall –95
Biomechanics –6
Black Guards –4. _See also_ Moscow Federation of Anarchists
Black Panther Party, the
Bloch, Anna
Bookchin, Murray –4,
Bradberry, George
Bread and Puppet Theater ,
Breton, André ,
Britain , –5, , , –4, –1
Brooklyn College, City University of New York –49, , –8,
Bruegel the Elder Pieter
_The Tower of Babel_
Bruno, Guido
Buck-Morris, Susan –5
_Dreamworld and Catastrophe: The Passing of Mass Utopia in East and West_ –5
Burliuk, David
Bush, George, Sr.
Cage, John
California School of Arts –7
_Camera Work_ ,
Cantine, Holly –16,
capitalism –14, –18, –46, –3, , , , –1, –195
censorship –67, , ,
Central Institute of Labor
Cheka, the –7, , ,
Chernyi, Lev ,
Chomsky, Noam , ,
City University of New York. See Brooklyn College
Cold War, the , –7, , ,
collage and collage artists –8, –7, , , –5
Collins, Jess –9
_If All the World Were Paper and All the Water Sink_
_Mouse's Tale_ –8
Communism (general) –3, , –5
Communism and the Communist Party (Russian) –7, –9, –105,
Communism (US) and the American Communist Party , –8
Comstock, Anthony –7
constructivism and constructivists –109
Cooney, Blanche –21
Cooney, James –21
Courbet, Gustave , –33
_Burial at Ornans_
_Painter's Studio: A Real Allegory, The_
_Return from the Conference_ –3,
_Stonebreakers_
Crass –7
Cross, Henri Edmond , –6
_The Wanderer_ –6
_Cubism_ ,
cubism and cubists –64, , ,
Dada, advent of –67
Dadaism and Dadaists ,
de La Fresnaye, Roger
de Zayas, Marius , –7,
di Prima, Diane
drawing. _See_ illustrations and illustrators
_Drawing Resistance_ , –6
Dubois-Pillet, Albert
Duchamp, Marcel , –4
Duchamp-Villon, Raymond
Duff, Peggy
Duncan, Robert –29,
egoism and anarchism –5, , –8
England _See_ Britain
FalklandsWar –7
Federation of Artists, the –3
feminism and feminists , –9,
Feuerbach, Ludwig , –1, –8
_Fifth Estate_ , –90
Filippov, Nikolai ,
Fini, Leonor
_Flipside_
Ford, Charles Henry –18, –5
France. _See_ Paris; Paris Commune
Free Speech League
freedom and liberty –14, –33, –6, –6, , , , –2, –8, –8, , –7,
Friedrich, Ernst –7
futurism and futurists –83
galleries , , –7, –6, , , ,
Gan, Alexei , , ,
_Constructivism_
Gastev, Alexei –5, ,
gay and lesbian, people and issues –29, , , –3, . _See also_ homosexuality
Gay Liberation Front
_Gazeta Futuristov_ –7
Ge, Aleksandr
Germany and Germans –21, , –3, , , , –7. _See also_ Berlin wall
Gérôme, Jean-Léon ,
_Greek Interior_
Gideonse, Harry –8
Gleizes, Albert –2,
Goldman, Emma –3, ,
Golub, Leon –5
Greenberg, Clement
Greenwich Village ,
Grün, Karl –18,
_Guardian,_ the –6, –9
Gulf War, Persian –5. _See also_ Iraq
Hapgood, Hutchins –2,
harmony and social order –6, –3. _See also_ freedom
Haviland, Paul ,
Havel, Hippolyte –5
_Saint Anthony, Guardian of Morals_ –5
Hegel, Wilhelm –21
Hegelian theory and Hegelians –21
Higher State Artistic and Technical Workshops, The (Vkhutemas) –99
Hoffman, Abbie –61
Holland –50. See also Amsterdam
homosexuality and homosexuals , –29
House of Anarchy, the ,
House of Free Art, the
illustrations and illustrators , –6, 56–60–5, , , , , , –56, –72, –3, –92
industrialism and industrialists –46, , –60, –106, –20, –9
_International Anthem_
Iraq . See also Gulf War
Jacobus, Harry
Jarry, Alfred –1,
_Supermale_
journals, magazines, and newspapers. See 291; _Action d'art; Anarchy: A Journal of Desire Armed; Anarkiia; Ark; Art Journal; Camera Work; Fifth Estate; Flipside; Gazeta Futuristov; Guardian; International Anthem; L'Assiette Au Beurre; L'en dehors; La Plume; Le Pere Pinard; Les Temps Nouveaux; Liberation Magazine; Movement, The; Politics; Rat; Retort; Revolutionary Almanac, The; Rogue; Vanguard, The; View; World War. See also_ pamphlets
Kamenskii, Vasilii
Kant, Immanuel –2
KingUbu –9
Khlebnikov, Velimir
Kropotkin, Peter –14, ,
Kruchenykh, Alexei , ,
Kuwait . _See also_ Gulf War
_L'Assiette Au Beurre_
_L'en dehors_
_La Plume_ –40
laborers. _See_ workers
Lamantia, Philip ,
_Le Père Pinard_
Lebasque, Henri –1
_Provocation_ –1
Léger, Fernand
_Les Temps Nouveaux_ –41, –6
Lenin, Vladimir Ilyich –3, –3,
Libertarian Circle, the , ,
liberty. _See_ freedom and liberty linocuts –4
Living Theatre collective
Luce, Maximilien , –3
_Factory Chimneys: Couillet Near Charleroi, The_ –3
Lukashnin, A.
Macdonald, Dwight –15,
magazines. See journals, magazines, and newspapers
_Magnanimous Cuckold_ , –9
Magraw, Roger
Magritte, René
Makhno, Nestor
Malevich, Kazimir , –7
_White on White_ –4
Marx, Karl –18, , –2,
Marxism and Marxists , , , , , , –4, , ,
Maximov, Gregorii –5
Mayaskovskii, Vladimir
McCarthyism –29
megamachine, the –90
metaphysics –2, –9, , –5, , , , –8
Metzinger, Jean ,
Meyer, Agnes Ernst
Meyerhold, Vsevolod –101, –9
Millman, Paul
Mirbeau, Octave
Mock, Richard –5
Gulf War series
"Oil Kill Spill"
"Raper Vapor" –4
"Target"
_Untitled_
"Victim"
modernism and modernists –66
Morozov mansion ,
Moscow Federation of Anarchists –9
_Movement, The_ –5
Mumford, Lewis
Napoleon III, Louis –5,
nature, issues and depictions of , –8, , , , –9, –5
neo-impressionism and neo-impressionists –46
New York –67, , , , , –6, –40, , –5, –65, ,
New York Society for the Repression of Vice . _See also_ Comstock, Anthony
newspapers. _See_ journals, magazines, and newspapers
Nicholas II, Tsar
Nin, Anaïs ,
non-objectivism and non-objectivists , –4, –7
Norton, Alan
nudes , –6
obscenity laws. _See_ censorship
October Revolution, the –2,
pacifism and pacifists , , , –8
painting and painters , –30, , –5, –6, , , , –91, –8, , , –7, , –6, ,
pamphlets –12, , . _See also_ journals, magazines, and newspapers
Paris –33, , , –6
Paris Commune, the –33, ,
performing arts and performers. See theater and performance
Peters, Yakov
photography and photographers , , –43, –2, , , , , , , , , , –4,
Picabia, Francis –67
_Dances at the Spring_ , ,
_De Zayas, De Zayas!_
Edtaonisl ( _Ecclesiastic_ ) ,
_Girl Born Without a Mother_
_New York Perceived Through the Body_
object portraits , , , ,
_Portrait of a Young American Girl in a State of Nudity_ , , , –61, –7
_Saint of Saints-This is a Portrait About Me, The_
_Voila Haviland The Poet as He Sees Himself_
Picasso, Pablo: Guernica
Pissarro, Camille –40,
_Apple Picking at Eragmy-sur-Epte_
_Social Turpitudes_
Pissarro, Lucien
Plamen, Baian , ,
poetry and poets , , , , , , , –18, , , , , , ,
Poetry Conference, the
Poetry Forum, the
_Politics_ , ,
Popova, Luibov –9
Proudhon, Pierre-Joseph , , –33
Provo –54, –60
punk music and musicians , , , . _See also_ Crass
Rainer, Dachine –16
_Rat_ –60, –71
realism –9
reason. _See_ Hegel; Hegelian theory and Hegelians; Proudhon
Reclus, Elisée –4
Red Army, the , , ,
Reinhardt, Ad –5, ,
_Retort, The_ ,
Rexroth, Kenneth ,
_Revolutionary Almanac, The_ –5
Rimbaud, Penny –4
Rodchenko, Aleksandr , , , –91, –9,
(pseud. Anti) –5, ,
_Black on Black_ –8, ,
_Oval Hanging Spatial Construction no_.
"Rodchenko's System" –7, –1
_Ten Years of Uzbekistan_
_Rogue_
Rosenblum, Walter
Rozanova, Olga
Russia and Russians , –89, , –3, . _See also_ Soviet Union
Russian Revolution, The –89
San Francisco , , –5, ,
sculpture and sculptors , , , , –8, ,
sexuality, issues and representations of, –61, –7, , –29, –4
Seurat, Georges
scientific management –7
Signac, Paul
Snyder, Gary
Simensky Bietila, Susan –77
Sobey, James Thrall
socialism and socialists , , , , –4, , , , , ,
Soviet Union , , , , , , –8, . _See also_ McCarthyism; Russia
Spain , , ,
Spanish Civil War , ,
spirit and spirituality , –20, , , , , , ,
"world spirit" –20
State Higher Theater Workshops
Steiglitz, Alfred , , –7,
Stepanova, Varvara , –9, –9, –9
Still, Clyfford –8
Stirner, Max –5, , , –7,
street theater , –4, ,
Students for a Democratic Society (SDS) –9, –7, –2, –7, –6
suprematism and suprematists , –7,
surrealism and surrealists –18, –3,
Tatlin, Vladimir , –81, , –8, ,
Taylorism. See scientific management
Thatcher, Margaret
theater and performance. –9, , –101, , –9, . _See also_ Biomechanics; street theater
Tice, Clara –6
Trotsky, Leon –3,
Udalt'sova, Nadezhda ,
United Kingdom. _See_ Britain
United States of America and Americans –14, –67, –2, –29, –77, , –1. _See also_ Greenwich Village; New York; San Francisco
Vaucher, Gee –7
Van Rysselberghe, Theo –40
_The Wanderers_ –40
_Vanguard, The_
Verhaeren, Emile –40
Vesnin, Alexandr
Vietnam War and the anti-Vietnam War movement , –41, , –4
_View_ , the –19, –5
Villon, Jacques
WITCH (Women's International Terrorist Conspiracy from Hell) –9
_War Against War!_ (Friedrich) –7
Watson, David –90
Wilde, Oscar: monument of –5
workers, issues and representations of –5, , –46, –3, , , , –7, ,
World War I , , , , , , –7
World War II , , , , ,
_World War_ , –4
Yakulov, Georgii
Yippies , –61,
Zola, Emile , –30,
Activist and art critic ALLAN ANTLIFF, Canada Research Chair in Modern Art at the University of Victoria in Victoria, BC, Canada, edited _Only a Beginning: An Anarchist Anthology_ (Arsenal Pulp Press, 2004) and is author of _Anarchist Modernism: Art, Politics, and the First American Avant- Garde_ (University of Chicago Press, 2001). He is art editor for the UK-based journal _Anarchist Studies_ and a frequent contributor to _Canadian Art_ and _Galleries West_ magazines.
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La crise de la sidérurgie dans le bassin lorrain, est une crise industrielle dans le domaine de la sidérurgie qui a lieu en Lorraine française, en Lorraine belge, dans la Sarre et dans la région des Terres Rouges, au Grand-Duché de Luxembourg, soit dans ce que l'on appelle le bassin lorrain.
Elle a débuté dans les années 1960, lorsque la minette lorraine est devenue moins compétitive, avant d'atteindre son paroxysme en 1973-1974, à l'instar de toute la sidérurgie mondiale qui traverse alors d'énormes difficultés. Elle a été résorbée progressivement au cours des années 1980, lorsque les restructurations successives ont amené à l'abandon des usines dont la stratégie était fondée sur la proximité du minerai de fer, dont l'extraction fut définitivement arrêtée en 1997.
La principale conséquence industrielle de cette crise est la disparition de nombreuses usines sidérurgiques dans toute la région. À cela s'ajoutent les conséquences économiques et sociales sur la population. Si on relève encore quelques fermetures postérieures aux années 2000 (aciéries de Gandrange, Rodange et Schifflange, hauts fourneaux de Hayange), il s'agit exclusivement d'unités modernisées mais qui s'avèrent mal configurées pour être durablement rentables.
Origine de l'activité sidérurgique dans la région
On note les premières traces d'exploitation du fer dans la région au avec, notamment, la création d'une forge à Herserange en 1551. Pour faire fonctionner une forge, il faut de la fonte, celle-ci est coulée dans un fourneau. Celui-ci a besoin de deux éléments pour fonctionner: du minerai de fer et du charbon de bois afin de faire monter le four jusqu'à une température suffisante pour faire fondre le fer (celle-ci est variable selon ce que l'on veut comme qualité de fer, mais, est bien souvent supérieur à 1000 degrés). Le charbon de bois se fabrique en brulant du bois d'une certaine manière. Ce qui ne pose pas de problème en termes de ressource puisque la région abonde en forêts (elle sera même appelée le « département des forêts » lorsque la France l'envahit après la révolution). Quant au minerai, il est extrait dans des mines, qu'elles soient à ciel ouvert ou souterraines. Pour ce faire, il faut donc que le sol soit riche en fer, ce qui est le cas dans la région puisque la teneur en fer de la minette lorraine est d'environ 30 à 40 %.
Les premières grosses industries apparaitront lors de la révolution industrielle au .
Causes
Causes économiques
Les causes de cette crise peuvent parfois varier en importance d'une usine à l'autre, mais sont généralement les mêmes. Celles-ci sont structurelles et non pas conjoncturelles :
l'essoufflement de l'industrialisation avec notamment la fin de la reconstruction après la Seconde Guerre mondiale ;
la sidérurgie sur l'eau de Fos-sur-Mer, construite à grand frais par les Lorrains, est plombée au moment de son inauguration par le premier choc pétrolier ;
la limitation des prix de l'acier, en haut de cycle, décidée en France pour soutenir la construction automobile, a limité les capacités d'investissement autofinancé ;
la faiblesse du franc favorise les exportations et les stratégies opportunistes. Mais cette stratégie, fondée sur des produits à faible valeur ajoutée, n'est pas viable dans le long terme. En effet, le développement rapide de la sidérurgie des pays d'Europe ou d'Amérique du Sud perturbe le marché mondial… sans parler du Japon qui produit 6,3 % de l'acier mondial en 1960 et 16,5 % en 1974 ;
la consommation des produits plats (tôles, plaques, tubes roulés,…) augmente alors que celle des produits longs (rails, fils, poutrelles…) stagne. Mais la capacité optimale d'un laminoir à large bande atteint en 1970, tandis que la demande nationale pour les produits plats ne dépasse pas . Ce basculement est mal anticipé par les sidérurgistes français qui continuent de croire au marché des profilés lourds et sous-estiment l'avenir de la tôle ;
le lien admis entre développement économique et production d'acier cesse d'être vrai, l'économie se recentre sur d'autres fondamentaux. Au début des années 1970, le PIB français par habitant atteint et la production d'acier . Cette production est un record qui précède une baisse de la consommation individuelle, masquée uniquement par la croissance de la population.
Ces causes se traduisent par une stagnation voire une baisse des volumes et du prix de vente de l'acier, et plus particulièrement des produits longs. Les sidérurgistes tentent de résister avant d'admettre que le secteur est pris dans une crise structurelle… et non pas conjoncturelle.
Causes techniques
Des causes techniques bouleversent également le métier :
l'apparition de nouveaux procédés : le laminage à chaud en continu repousse la taille minimale d'un complexe sidérurgique rentable entre annuelles de produits plats, ou à de produits longs. Les hauts fourneaux suivent cette évolution grâce à la généralisation de l'agglomération des minerais, tandis que le convertisseur à l'oxygène et la coulée continue se généralisent. Or l'« usine modèle » fondée sur les procédés de la génération antérieure (convertisseurs Thomas ou Martin, coulée en lingot ou en gueuses,…) avait une capacité optimale de ;
le four à arc électrique permet la production à faible coût de beaucoup d'alliages ;
l'apparition de nouveaux matériaux, comme les plastiques ou le béton précontraint : les usines spécialisées sur les aciers bas de gamme ou sur la fonte en gueuses voient leurs marchés disparaitre progressivement.
L'acier lui-même évolue dans ses caractéristiques comme dans sa mise en œuvre : pour construire un pont on utilise, à caractéristiques techniques égales, deux fois moins d'acier en 1979 qu'en 1959.
Enfin, la faible teneur en fer du minerai local, la minette lorraine (entre 30 et 40 %), est un handicap insurmontable dès que le coût du transport maritime baisse. La faible teneur en fer de la minette entraîne en effet une consommation accrue de coke au haut fourneau, pour fondre une gangue de faible intérêt économique (la valorisation de cette gangue sous la forme de laitiers n'étant pas rentable).
Le manque d'investissement pour maintenir certaines entreprises à la pointe de la technologie est aussi patent. Le coût de la main d'œuvre qui augmente rend incontournable l'évolution vers des usines géantes et automatisées. L'ouverture des frontières européennes élimine rapidement les sidérurgistes ayant mal anticipé ces changements.
Histoire de la crise
Années 1960 - 1973 : premiers chocs
La crise de la sidérurgie débuta lors de la seconde moitié du , dans les années 1960. En 1961, une première crise secoue le secteur, avec un ralentissement de la demande et un effondrement des prix. À compter de cette époque, les mines feront l'objet de restructurations avec des conséquences sur leurs effectifs. La « Convention générale État-sidérurgie » de 1966, suivie l'année suivante par un plan professionnel programmant suppressions de postes sur cinq ans. Puis, en 1971, un plan de conversion des sites du groupe Wendel-Sidelor concernera suppressions d'emplois.
La pauvreté en fer de la minette lorraine pénalise durement les mines lorraines, alors que les sidérurgistes importent de plus en plus de minerais riches pour sauver leurs marges. En 1955, les mines de fer lorraines emploient quelque . De 1960 à 1967, les effectifs des mines de fer lorraines passent de . La production lorraine se stabilise ensuite de 1968 à 1974 autour de annuelles, la productivité permet de diminuer dans le même temps les effectifs qui passent de .
Pour autant, la situation ne semble pas désespérée, surtout pour les sidérurgistes : . La production régionale d'acier passe de en 1966 à en 1970. Quelques unités très modernes, comme la coulée continue d'aciers spéciaux ou le procédé Kaldo illustrent les ambitions des sidérurgistes.
1973 - 1975 : le cataclysme
En 1973, le premier choc pétrolier, enclenche une mutation économique mais l'aspect pétrolier masque la profondeur de la crise. De 1960 à 1973, la croissance française est en moyenne de 6 %. De 1974 à 1979, elle n'est que de 3 %.
En 1975, la chute brutale de la demande d'acier, engagée en 1973 au moment du premier choc pétrolier . La production d'acier passe de plus de en 1974, à environ en 1975.
1975 - années 1980 : les restructurations
Les premières fermetures concernent essentiellement des unités obsolètes, mal configurées ou mal gérées. Elles ne suffisent pas. En 1979, le plan du gouvernement est celui du « sauvetage de la sidérurgie française ». Il implique d'autres très lourdes conséquences sociales, avec suppressions d'emplois en quelque 18 mois concentrées sur les bassins industriels de Longwy, mais aussi dans le Valenciennois, à Denain et Trith-Saint-Léger. Alors qu'en 1930, le bassin de Longwy avait produit, à lui seul, plus du tiers de la production sidérurgique française avec 26 hauts fourneaux en activité, ce plan condamne ses sites historiques, comme l'usine de la Chiers qui passe, en 1976, de 5 hauts fourneaux et à 140 emplois en 1980.
En France, l'élection présidentielle française de 1981 amène à la suspension du plan Barre de 1978 destiné à restructurer le secteur. La nationalisation ne change pas fondamentalement la gestion d'une industrie gérée de facto par l'État. Le gouvernement Pierre Mauroy doit trouver un équilibre entre les attentes de l'électorat et la réalité de la situation, qu'il n'appréhende que lentement. Finalement, le Plan acier de 1984 annonce une restructuration douloureuse. Mais .
Au Luxembourg, dont l'industrie sidérurgique a été pendant longtemps la première source d'emploi et de richesse du pays, jusqu'à représenter 45 % du PIB national, l'Arbed ferme son dernier haut fourneau à Esch-Belval en 1997. Les usines se réorganisent sur des produits longs fabriqués dans des aciéries électriques. Les grosses poutrelles fabriquées à Differdange, et surtout les palplanches de Belval, sont des marchés stables et très rentables.
La vallée de la Fensch s'organise finalement autour de son laminoir à chaud, un des plus vieux d'Europe, mais aussi un des plus performants, pour produire des tôles. Une nouvelle aciérie dotée de convertisseurs à l'oxygène de type LWS et de deux coulées continues est construite en 1977-1978. En 1984, la construction de l'usine de Sainte-Agathe à Florange oriente le site vers les aciers à haute valeur ajoutée pour l'automobile.
Le laminoir à rails de Hayange, en reprenant le carnet de rails de chemin de fer fabriqués au Luxembourg, atteint des niveaux de production satisfaisants sur un marché très rentable.
La vallée de l'Orne abandonne progressivement ses hauts fourneaux pour la production de produits longs. Le complexe sidérurgique de Rombas-Gandrange bénéficie des investissements au détriment de l'usine d'Hagondange qui est sacrifiée. Mais cette stratégie est victime d'une succession d'échecs techniques (convertisseur Kaldo, filière blooming, four à arc électrique à courant continu) qui désorganisent l'usine. L'arrêt de la production de rails et de palplanches, au bénéfice de ses voisins de Hayange et Belval, prive le site d'un carnet rentable.
La sidérurgie sarroise se restructure également. Dillinger Hütte se réoriente sur des produits de niches ou à très forte valeur ajoutée, en investissant massivement et avec succès dans son outil industriel, notamment sur la production de tôles fortes. Libre d'acheter son minerai ailleurs qu'en France, elle opte très tôt pour le minerai importé et s'organise pour regrouper sa production de fonte sur quelques gros hauts fourneaux.
En Meurthe-et-Moselle, l'usine historique de Pont-à-Mousson conserve 3 hauts fourneaux (sur 5 avant 1964) de petite taille. Après la fermeture de l'usine de Florange, ce sont les tout derniers hauts fourneaux en activité dans le nord-est de la France. L'usine se concentre sur la fabrication de canalisation en fonte coulée par centrifugation, ainsi que sur des éléments d'assainissement urbain.
Toutes ces usines sont soutenues par l'Institut de recherche de la sidérurgie, le plus important centre de recherche français sur la sidérurgie, installé à Maizières-lès-Metz. Ce centre de recherche privé mène des études et des développements sur les procédés de fabrication ainsi que les applications de l'acier.
Années 1980 : une industrie moderne au sein d'un secteur en crise
À la fin des années 1970, la cause essentielle avancée pour expliquer la crise de la sidérurgie lorraine reste la vétusté des entreprises sidérurgiques de la région. Celles-ci se modernisent donc, ou ferment. Mais les difficultés, à partir des années 1980, qui s'étendent aux sites littoraux de Fos-sur-Mer (l'usine Solmer) et Dunkerque, démontrent que l'ensemble de la sidérurgie est menacée : la crise ne se résume alors plus au seul bassin lorrain.
En Allemagne, malgré la construction d'une aciérie et d'une cokerie neuves en 1982-84 sous l'égide de l'ARBED, Saarstahl, présente sur le marché plus concurrentiel du tréfilage et de la forge, périclite, victime des dissensions entre ses actionnaires : Usinor-Sacilor acquiert une participation majoritaire pour 1 Deutsche Mark symbolique en 1989, qu'il revend cinq ans après, à la suite de pertes continues.
Au sud de Nancy, l'usine de Neuves-Maisons achève sa mutation en 1986, en devenant une aciérie électrique spécialisée sur les produits sidérurgiques « bas de gamme », marché où elle a la taille critique pour être rentable (capacité de 1,1 million de tonnes par an).
Pendant ce temps, les dernières mines de fer ferment. La dernière en fonctionnement est celle des Terres Rouges à Audun-le-Tiche, qui ferme définitivement en 1997. Elle n'emploie alors que alors qu'en 1955, les mines de fer lorraines employaient quelque .
Conséquences
Situation après les restructurations
La fermeture des mines et des usines a entrainé une catastrophe sociale. Car .
Cependant, la sidérurgie française rescapée de la crise constitue un ensemble bien plus cohérent et solide que le secteur disparate des années 1970. Les années 1990 et 2000 voient prospérer de nombreuses usines cohérentes et très rentables :
Les grosses poutrelles fabriquées à Differdange, et surtout les palplanches de Esch-Belval, sont des marchés stables et très rentables ;
Les tôles fortes de la Dillinger Hütte sont réputées dans le monde entier. L'aciérie et le laminoir, alimentés par les hauts fourneaux modernes de la , disposent d'installations uniques au monde. L'usine reste bénéficiaire pendant toute la crise.
Quelques autres survivent difficilement, dont notamment :
le complexe sidérurgique de Gandrange se restructure laborieusement : de mauvais choix technologiques et la perte d'une production très rentable de rails et de palplanches, le contraint à se spécialiser sur le , un fil en acier très fin et difficile à réaliser ;
les usines de Rodange et Schifflange, trop petites pour développer une excellence technique qui soit synonyme de rentabilité ;
l'aciérie et la forge d'Ascométal à Hagondange, pressées par leur puissants clients de l'industrie automobile ;
l'usine à chaud de Florange, quoique performante, souffre d'un sous-investissement chronique et d'une absence de visibilité stratégique. En effet, pour ses dirigeants, il s'agit d'une usine « fusible » en cas de restructuration importante…
Saarstahl, même profondément modernisée, peine à être rentable, pénalisée par les faibles prix pratiqués sur ses marchés du tréfilage et de la forge.
La crise économique mondiale des années 2008 et suivantes remet en cause la pérennité de ces sites, survivants de la restructuration menée dans les années 1970…
Liste des fermetures
Bon nombre d'entreprises ont fermé, parmi celles-ci:
En :
L'usine d'Athus
L'usine d'Halanzy
L'usine de Musson
La mine d'Halanzy
La mine de Musson
En :
L'usine d'Aubrives à Villerupt fermée par le groupe Pont-à-Mousson en novembre 1968.
L'usine de Mont-Saint-Martin.
L'usine sidérurgique de Knutange, une des premières sacrifiées au début des années 1970
L'usine de Pompey, célèbre pour sa participation à la construction de la tour Eiffel, en 1986
La Providence Réhon, fermée en 1987
l'usine sidérurgique d'Uckange de Lorfonte (anciennement HFRSU), fermée en 1991
l'usine Lorfonte, site de Rombas, arrêt du secteur des hauts-fourneaux (le R7 le et le R5 le ). Rattachement de l'agglomération de minerai à l'usine de Florange (Usinor)
l'usine Lorfonte, site de Jœuf
l'usine sidérurgique d'Hagondange, aussi appelée UCPMI, fermée pour l'essentiel en 1979
Au :
L'usine d'Esch-Schifflange
Les mines de Lasauvage
Actions
La fermeture des usines s'accompagne d'un traumatisme important, bien au-delà de la disparition des emplois. En effet, .
En :
En 1976, plusieurs manifestations d'ouvriers de l'usine d'Athus à Bruxelles à Luxembourg Ville et à Athus même.
En :
En 1966, le « plan professionnel » (Convention entre l'État et l'industrie de la sidérurgie) prévoit des restructurations dans le Nord et en Lorraine. Pour s'opposer aux suppressions d'emplois, la CGT et la CFDT mènent une grève générale dans la sidérurgie lorraine du au . Cette grève permet la signature de la première convention sociale le , avec des garanties individuelles et collectives, ainsi que la préretraite à 60 ans.
En 1978 :
Le , la CFDT de Longwy monte un SOS lumineux à Longwy,
Le Usinor-Chiers-Chatillon annonce la suppression de 12 000 emplois,
Le naissance de « Radio SOS Emploi »,
Le , journée « Ville Morte » et manifestation de 25 000 personnes,
Du 26 au , blocage des routes de la région de Longwy à l'appel de l'Intersyndicale (CGT,CFDT, FO, CGC, FEN)
En 1979 :
Manifestations « Les flammes de l'Espoir » à Longwy.
Le , des rouleaux de feuillards bloquent tout le centre de Longwy et des radios clandestines font leur apparition comme Lorraine cœur d'acier.
Le , le journal télévisé d'Antenne 2 se fait en direct de Longwy.
Le , occupation du relais TV du Bois de Châ.
Le , occupation de la tour Eiffel.
Le , manifestation à Paris, avec le soutien des syndicats et des mouvements de gauche, le reportage du Monde parle de 60 000 personnes selon la police, et plusieurs centaines de milliers selon la CGT. De violents affrontements ont lieu avec les forces de l'ordre.
Le , blocage du Tour de France lors de l'étape de Tellancourt.
Le , vol de la coupe de France de football dans les locaux du FC Nantes.
Au :
Usines toujours en activité
Certaines entreprises sont toujours en activité. Elles ont, pour beaucoup, été reprises par le groupe sidérurgique Arcelor Mittal. Parmi elles:
En :
Plus aucune en Lorraine belge ni en Gaume mais encore quelques-unes dans le Hainaut et la région de Liège.
En :
Ascometal, site d'Hagondange
ArcelorMittal, site de Florange
Saint-Gobain PAM, site de Pont-à-Mousson
SAM, groupe Riva SpA, site de Neuves-Maisons
Akers Yard, site de Thionville
Au :
ArcelorMittal, site de Differdange
ArcelorMittal, site de Rodange
ArcelorMittal, site de d'Esch-Belval
L'après-sidérurgie
Certaines localités ont plus ou moins bien réussi leurs reconversion post-sidérurgique avec la mise en œuvre de certains projets. Citons notamment :
Le parc Eiffel Énergie qui dès 1990 a reconverti l'ensemble des usines de Nancy-Pompey.
Le site luxembourgeois d'Esch-Belval.
Le Terminal conteneurs d'Athus
Le site transfrontalier du Pôle européen de développement.
Héritage
Musées
Certains sites sont aujourd'hui transformés en musée, parfois à ciel ouvert.
Le Fond-de-Gras et la ligne Pétange - Fond-de-Gras - Doihl accueillant le Train 1900, à Lasauvage
Le site haut fourneau d'Esch-Belval, à Esch-sur-Alzette.
Le musée Athus et l'acier, à Athus.
Le musée des mines d'Halanzy et de Musson, à Musson.
Le U4 à Uckange.
La Völklinger Hütte, à Völklingen (Patrimoine mondial de l'UNESCO)
Mais les traces de la sidérurgie se voient également dans l'architecture et l'urbanisation des villes. De nombreux quartiers ouvriers aux maisons typiques sont encore sur pied de nos jours dans beaucoup de ces localités.
Les crassiers
Ces endroits ou la partie inutilisable du minerai était entassée en monticules sont aujourd'hui toujours visibles. D'une région à l'autre on les appellera crassiers ou terrils. Ils ont des formes différentes.
Cinéma, téléfilms et souvenirs
La répression par Valéry Giscard d'Estaing des radios libres "Lorraine cœur d'acier" et "Radio-Quinquin'", sur fond de désindustrialisation forcée a été la trame du téléfilm policier de 2020 Les Ondes du souvenir, succès populaire avec une audience leader de 5,4 millions de téléspectateurs.
Notes et références
Notes
Références
(voir dans la bibliographie)
Voir aussi
Bibliographie
Où va la sidérurgie ? auteurs J.Y.ROGNANT - C.ROMAIN - F.ROSSO, Éditions Syros - 1977.
Espace et luttes , 1980, CPPAP 743, Édité par Association Espace et Luttes - Paris
Articles connexes
Histoire de la métallurgie et de la sidérurgie dans le bassin lorrain
ArcelorMittal
Industrie lourde
Les mains d'or (chanson de Bernard Lavilliers évoquant la fermeture des sites industriels).
Médias
De nombreux documentaires évoquent le passé sidérurgique :
« Sous le Gueulard la Vie » (Emmanuel Graff et Isabel Gnaccarini, VPS Lausanne prod)
« L'Héritage de l'Homme de Fer » (Emmanuel Graff et Stéphane Bubel, La Bascule prod., Nancy)
« La Trace des Pères » (Emmanuel Graff, Textes de Hamé/La Rumeur, Faux Raccord prod, Metz)
Industrie sidérurgique en France
Histoire de la métallurgie en France
Sidérurgie en province de Luxembourg
Histoire de la Lorraine | {
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Funeral stopped after family discovered that their late son's manhood is missing
by AlreadyViral Editor February 23, 2022, 7:15 am 8k Views
A funeral at Daveyrown, South Africa, was stopped after a family discovered that their late son's manhood was missing.
The family who claimed that their son's body was mutilated and violated at the funeral parlour, made the bizarre discovery when they went to wash their 18-year-old son's body.
It was also gathered that while the grieving parents, Portia and Bellium Tjale refused to bury their disfigured son, relatives and neighbours also blocked funeral parlour workers from taking the tent, chairs and other items. They declared that they would hold the property as security until the owner of the funeral parlour refunded them their R30 000.
39-year-old Portia told Daily Sun;
"My son's body was collected from home with every body part intact, but some parts are now missing. Why is that?"
Portia alleged that the funeral parlour which first collected their son's body did not have cold storage. As a result, they transferred the body to another parlour. She later discovered that his manhood was missing at the second parlour.
She went on to say that the owners of the second parlour tried to hide the issue by telling her not to come.
The grieving mother said;
"The same funeral parlour owner and his wife then even contacted me again to convince me not to come because I suspect they saw something was not right. My son deserves to be buried with dignity and not to be subjected to such disrespect"
Portia's husband, Bellium, (48) added;
"My son suffered and the next thing he was disrespected. Mpho was epileptic and had autism."
The grieving parents reported the matter to the Daveyton police, who have since opened a case of violation of a corpse.
However, the two parents allege that the police told them to bury their sons' body despite the missing manhood.
FamilyfuneralManhoodSon
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Drunk ashawo lady removes her 'piato' at night club, inserts Smirnoff bottle in 'horny pot' and dances crazily in front of her colleagues | {
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Q: Euler method for ODE with noisy derivative I am interested in numerically integrating a noisy differential equation:
$\frac{dx}{dt} = f(x,t) + \epsilon(t)$ where $\epsilon(t) \sim \mathcal{N}(\mu, \sigma^2)$. Is this a RODE or SODE?
How does this relate to numerically integrating $\frac{dx}{dt} = f(x,t)$ using Euler and adding Gaussian noise after each step?
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{"url":"https:\/\/puzzling.meta.stackexchange.com\/questions\/6287\/desktop-vs-mobile-formating\/6288#6288","text":"# Desktop vs mobile formating\n\nI recently asked a question: Ruin, has come to our family\n\nIn this the answer depended on seeing the words formated in a certain way so they looked like steps and occurred n number of words into the line. I didn't think whilst creating this about how mobiles would format the text differently. Is there a tag for this, or a way of letting people know without giving too much away about the puzzle?\n\nThe first thing I'd like to note is that when Stack Exchange changes the design, especially the width of the main bar (now set to a fixed value of 728 pixels wide), your puzzle may break (even on desktop). Therefore, it might be better to include an image of the letter (which you can use to add more 'feeling'\/'texture' to your puzzle as well).\n\nIf you choose a font which is legible but not 'optimized' for reading text, you might include a transcript of the letter into your puzzle as well. That transcript doesn't need to be formatted as the text in your puzzle. An example of this can be found here, though I haven't checked if the formatting is vital to solving that puzzle as it is in yours.\n\nIf all you need is to have line breaks in certain places, you can accomplish that in several ways:\n\n1. End a line with 2 spaces, then a carriage return\n2. Include a <BR> tag at the end of the line (HTML for line break)\n3. Use fixed width text (indent everything with 4 spaces)\n\nThe last option might be particularly relevant, because it will prevent \"soft\" line breaks (breaks that happen just because the screen isn't wide enough).\n\nSo you can\narbitrarily decide how long each line is\nAnd if a line is too long to fit in the available space, you will get a scroll bar instead of having it wrap to the next line","date":"2022-01-23 17:47:51","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.4585936367511749, \"perplexity\": 916.1060396775524}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-05\/segments\/1642320304309.5\/warc\/CC-MAIN-20220123172206-20220123202206-00008.warc.gz\"}"} | null | null |
The Sabine-Southwestern War was a military conflict in the United States from 1836 to 1837. It was a war with Native Americans in Louisiana along the Sabine River, the border between the Republic of Texas and the United States.
References
Wars between the United States and Native Americans
History of United States expansionism | {
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\subsection{Multiplayer AR/VR Gaming}
Today's multiplayer AR/VR gaming systems, do not support features that have bearing to players' location in the physical coordinate space. Given
the growing demand for `location-based entertainment'~\cite{locentertainment},
recent solutions make use of visual SLAM (using VIO) to localize players with respect to their individual reference frames~\cite{game2}. However, for a collaborative multiplayer setting, a global coordinate system is essential.
This is accomplished today using either visual markers or anchors that do not offer a smooth multiplayer experience, or expensive/extensive installation of IR cameras and laser tags that do not offer a cost-effective, on-demand deployment (e.g. consumer homes).
\begin{figure}[!htb]
\centering
\includegraphics[width=0.9\linewidth]{figs/section-5-plots/dynosoccer}
\includegraphics[width=0.55\linewidth, height=75pt]{figs/section-5-plots/missedkicks}
\includegraphics[width=0.43\linewidth, height=73pt]{figs/section-5-plots/gamesched}
\caption{ {\em Top:} \textsc{DynoLoc}\xspace enables location-based multiplayer gaming. {\em Bottom:} Mobility-aware range scheduling distributes time slots heterogeneously across nodes, thus improves overall interactivity of the game.}
\vspace{-0.2 cm}
\label{fig:app-1}
\end{figure}
We create a simple Android based multiplayer VR game called \texttt{DynoSoccer} to demonstrate the value that \textsc{DynoLoc}\xspace can bring to such gaming systems. \texttt{DynoSoccer} transforms ordinary physical spaces, like a living room
that is not particularly well-lit or textured
, into a gaming arena, where players can interact with each other based on their {\em real} physical locations. The game consists of a virtual ball that is bounced around by the players. Each player needs to adjust their position to be in the proximity of the ball in order to `kick' it. However, this requires the system to be responsive to the players' and the ball's movement, otherwise resulting in a `missed kick'. We show (Fig.~\ref{fig:app-1}) that even for high ball speeds, \textsc{DynoLoc}\xspace results in 50\% to 90\% less `missed kicks' compared to the \texttt{H-Agnos} baseline, increasing the VR experience significantly. \textsc{DynoLoc}\xspace adaptively schedules the relevant ranges (compared to round-robin in \texttt{H-Agnos}) in the topology based on the mobility of individual players.
We plan to fuse VIO with \textsc{DynoLoc}\xspace to overcome VIO's challenges in less-favorable visual environments (Fig.~\ref{fig:vio-indoor-exp}), especially in the multi-player context.
%
\subsection{Active Shooter Scenario}
\begin{figure}[!htb]
\centering
\includegraphics[width=0.96\linewidth]{figs/section-5-plots/geofenceerror.pdf}
\caption{\textsc{DynoLoc}\xspace enables real time geofencing in an active shooter scenario helping in safe evacuation.}
\label{fig:app-2}
\end{figure}
We demonstrate in Fig.~\ref{fig:app-2} how \textsc{DynoLoc}\xspace can create a realtime geofence for safe evacuation of trapped victims in a chaotic situation like spotting an active shooter.
A person mimicking an active shooter runs in a specified path. Four volunteers (unaware of the shooter's path) equipped with \textsc{DynoLoc}\xspace tags and body camera scout the general area (corridors, hallways etc.). If the shooter is detected in the video frame, we mark the respective location as unsafe and update the geofence (a polygon connecting the unsafe points). We show in Fig.~\ref{fig:app-2} ({\em right}), how localization error can lead to inaccurate geofencing resulting in `exposed areas' or zones that are potentially dangerous but marked safe. Even a 2\,m median localization error can lead to a few hundred sq. ft. of exposed area, compared to the \textsc{DynoLoc}\xspace's limited exposure.
\begin{comment}
\subsection{Hazmat Localization}
Finally, we demonstrate how \textsc{DynoLoc}\xspace is useful in tracking hazardous materials (chemical spills, radioactive emissions etc.) in industrial plants or warehouses. When such accidents happen, emergency responders or SWAT team members carry sensors that detect intensity of the emission or fumes. We show an example, where such sensors when integrated to \textsc{DynoLoc}\xspace tags, can create an intensity map without any extra effort, as well as identify sources of such emissions.
\vspace{-0.52 cm}
\begin{figure}[!htb]
\centering
\subfloat[Smoke emission map]
{
\hspace{-15pt}
\includegraphics[width=0.5\linewidth]{figs/section-5-plots/hazmat_heatmap}
\label{fig:app-3-a}
}
\subfloat[Total time for source localization]
{ \hspace{-8pt}
\includegraphics[width=0.6\linewidth, height=90pt]{figs/section-5-plots/hazmatsensors}
\label{fig:app-3-b}
}
\caption{\textsc{DynoLoc}\xspace in an emulated Hazmat scenario, where multiple mobile agents scout an area to create an intensity map or localize the Hazmat source.}
\vspace{-0.35 cm}
\label{fig:app-3}
\end{figure}
We deploy a smoke source in an indoor environment with an area of about 400 sq. meters. We use smoke sensors(\texttt{MQ-2}) along with \textsc{DynoLoc}\xspace tags to record location tagged intensity values. Fig.~\ref{fig:app-3} shows that additional sensors decrease the time proportionately by cooperatively scanning different regions of an area in tandem, increased topology size (of sensors to be tracked), contributes to additional error.
\end{comment}
\subsubsection{Challenge 1: Lack of Dedicated Anchors}
All nodes in our case are potentially mobile, thus lacking fixed locations. Therefore, there are no ``anchor" nodes or fixed nodes with known locations in our setting.
There are two implications of this - i) some of the time-efficient localization techniques such as Time Difference of Arrival (TDOA) \cite{tdoa} etc. cannot be used as it requires fixed anchors and wired connection between them for fine grained clock-synchronization. ii) localization depends on pairwise distance-measurement or ranges between nodes. In this case, without any information about the topology resulting the wireless connectivity, the simplest approach would be to range a node with rest of the nodes, and then relatively localize the nodes using the geometry resulting from the set of pairwise ranges and repeat the operation to continuously localize each node. However, for $n$ nodes, it would require $^nC_2$ ranges.
Each ranging operation is time-consuming and cannot be arbitrarily minimized. For example, due to the broadcast nature of the wireless medium, each pair of nodes has to take turn in Time Division Multiple Access (TDMA) fashion for interference-free accurate ranging. Each ranging operation may take tens of milliseconds depending on the modality (WiFi, Ultra Wide Band etc.), data-rate of frame-exchange (110kbps, 64Mbps etc), frame-exchange protocols (single-sided and double-sided Two-way ranging \cite{twr} etc ), sampling rate and latency of the underlying electronics \cite{deca-chip-perf} and so on. For example, in our case, we are using double-sided two-way ranging protocol which requires 4 frame exchanges between each pair. We are using UWB with a datarate of 110kbps, which requires a TDMA slot time is 8msec for a maximum 110Bytes of payload. Therefore, each ranging takes 8$\times$4 = 32 msec. We are collecting all ranges using UDP frame via WiFi channel. The request and response alone takes about 2$\times$4=8msec. Therefore, each operation of range collection takes minimum 32+8 = 40msec. If there are 16 nodes, according to the above simplest method, we need to gather $^{16}C_2=120$ ranges to determine the geometry of relative localization. Therefore, the gap between successive localization of each node is at least 120$\times$40= 4800msec. However, if we had 3 static anchors covering the whole area of interest, each node could be localized in 3$\times$40=120msec after 3 ranging with 3 anchors, and for 16 nodes, collection of all possible ranges would take 16$\times$120 = 1920msec. The localization delay for the case without any dedicated anchors and with 3 anchors are \textit{quadratic} and \textit{linear} respectively in the \# of nodes. Therefore, the biggest challenge here is to minimize the localization delay by scheduling only a subset of all possible $^nC_2$ ranges, taking into consideration factors such as link or range quality, node mobility, effect on the relative geometry etc. Therefore, the lack of dedicated anchors requires complex orchestration of ranging operations among nodes, to scale localization with the \# of nodes and to minimize the delay.
\end{comment}
\subsection{Geometric Rigidity}
\begin{figure}[!tbh]
\centering
\includegraphics[width = 0.95\linewidth ]{figs/trans-rot-flip.png}
\caption{Translation, Rotation and Flip}
\label{fig:node-trans-rot-flip}
\end{figure}
\vspace{0.1in}
As discussed before, nodes can be relatively localized with respect to each other using pairwise ranges.
These ranges (edges) along with nodes (vertices) form a graph representing the relative geometry.
By rotating, translating and flipping the local coordinate of this graph\footnote{using additional information from IMU, floor-plan etc.}, we can derive the absolute locations of all the nodes as shown in Figure \ref{fig:rel-graph}. Therefore, in our problem, localization involves relative localization followed by absolute localization.
However, the nodes can be mobile. Therefore, localization is done in real time and continuously.
If all possible ranges are collected in orderly fashion, the approach is deterministic and the resulting relative geometry (a clique) is unambiguous (See Figure \ref{fig:rigidity}(a)).
However, there are two issues:
\noindent i) \textit{Connectivity constraint}: Due to limited ranging capability, ranges for all pairs of nodes are not always available.
\noindent ii) \textit{Time constraint}: Collection of ranges for all pairs of nodes may require prohibitively high amount of time as discussed before.
As a result of the above two constraints, only a subset of edges are available to form the relative geometry. However, a random subset of the edges won't guarantee unambiguous graph. To this end, we introduce the notion of graph ``rigidity". As shown in Figure \ref{fig:rigidity}, equal \# of edges are taken to form two subgraphs as in (b) and (c) from the complete graph in (a). The graph in (b) is not rigid, as node 1 can freely move around. But graph in (c) is rigid discounting the effect of translation, rotation and flip. If the resulting graph is not rigid, relative localization will fail. Therefore, it requires careful selection of edges in a given time budget. For example, the graph in (a) and (c) are both rigid and will result in the same relative localization. Yet, (a) uses 28 edges and (c) uses 16. Therefore, the orchestration to collect only sufficient number of ranges to relatively localize the nodes, is challenging.
\begin{comment}
\subsubsection{Challenge 4: Geometric Requirements}.
Without at least three (in 2D) fixed anchors with known location, we need to first derive the relative locations of the nodes from the pairwise ranges. To this end, we derive relative locations by forming a ``rigid graph" with all the nodes on it. This graph can be translated and/or rotated to derive the absolute location of each node. The requirements of this rigid graph formation put additional constraints on the ranging operation as described below.
To elucidate the inherent geometric challenges of relative localization, let us first introduce some relevant terms from computational geometry.\footnote{Unless otherwise stated, the scenario is for 2D space, although it can be generalized for 3D.}
\uterm{Distance Geometry Problem (DGP)} A \textit{realization} is a function that maps a given set of vertices to 2D Euclidean space. Given a simple, connected, weighted graph $G=(V, E, d)$, where $V$ is the set of vertices, $E$ is the set of edges and $d(u, v)$ is the Euclidean distance or range between nodes $u, v \in V$, the DGP problem is to find whether there is a realization $x:V \rightarrow{} \mathbb{R}^2$ such that each range is preserved i.e $\forall (u,v) \in E, \; ||x(u) - x(v)|| = d(u, v)$ where $||.||$ is the Euclidean norm.
As shown in \textcolor{red}{Figure \ref{fig:dgp-explanationi}}, given a set of inter-node distances, we can construct a quadrilateral of a specific shape which can be translated, rotated, or flipped resulting in infinite number of realizations with the same Euclidean congruence. However, in relative localization, only the shape matters. Therefore, in our case, the DGP problem is to find whether there is any realization, or unique or countable or infinite number of realization discounting above three effects. Based on this notion, we can define rigidity as follows:
\uterm{Rigid Graph Formation Problem (RGFP)} A simple, connected, weighted graph $G=(V, E, d)$ is ``rigid" if there is only one realization discounting translation, rotation and flip. As shown in \textcolor{red}{Figure \ref{fig:rigid-graph}}, case (a) is rigid, it has only one realization, case (b) is not rigid, it has two realizations, case (c) is not rigid either, it has infinitely many realizations. In our case, determining graph rigidity has two benefits:
\begin{enumerate}[noitemsep, topsep = 0em]
%
\item Identify nodes that require additional information besides ranges for relative localization
%
\item Besides initial and periodic ranging on all available edges (formed due to wireless connectivity of the ranging modality), schedule ranging on a subset of the edges to ensure maximum benefit (relative localization) with minimum cost (\# of edges).
\end{enumerate}
%
Note that, DGP is NP-hard (see Saxe's proof \cite{dgp-proof}) for any dimensional Euclidean space. As such, determining rigidity of arbitrary graph is also NP-hard. However, using graph property, we can formulate efficient signatures for rigidity as described in Section \ref{sec:method}. In our case, each edge represents a time-consuming ranging operation. Therefore, to minimize localization time, it is imperative to find a maximal (highest number of vertices) rigid sub-graph, if any, using minimum number of edges. We can define the problem as follows.
\uterm{Minimally Rigid Maximal Sub-graph (MRMS) Problem (MRMSP)} Find a (vertex induced) rigid subgraph $G_r = (V_r, E_r)$ of a given simple, connected, weighted graph $G=(V, E, d)$, such that i) adding any new vertex $v \in (V-V_r)$ (along with the incident edges) to the sub-graph will make the sub-graph non-rigid and ii) removing a single edge from $E_r$ will render the sub-graph non-rigid.
As shown in \textcolor{red}{Figure \ref{minimally-rigid}}, the input graph has 8 nodes, however it has a sub-graph (5-clique) that is rigid. In the first step (see (b)), we can prune the nodes that cannot be part of any rigid sub-graph. In the next step (see (c)), we can prune some edges making it minimally rigid. The resulting graph in (c) is MRMS. In our ranging operations, the MRMS edges are of utmost priority to form the rigid graph and hence compute the relative localization. We can add additional edges to MRMS, if the time budget permits, to meet the requirements of the other challenges discussed so far. Regardless, if the MRMS requirement is not carefully followed in the ranging operation, the relative localization will fail. For example in Figure \ref{fig:minimally-rigid}(d), the same number of edges as in (c) has been selected for ranging without considering their utility for MRMS formation, but the resulting sub-graph, although has all the vertices, still fails to be rigid. Therefore, ranging on the sequence of edges in (c) is always preferred to on those in (d) even though both sequences have equal \# of edges and hence require equal time. MRMS problem is NP-hard due to NP-hardness of DGP and Rigidity problem.
\uterm{Euclidean Distance Matrix (EDM) Completion and De-noising Problem (EDMCDP)} If all pairwise ranges ($^nC_2$ for $n$ nodes) are available, we can represent the edge-weight function $d$ as an n$\times{}$n matrix $D$, which is known as EDM. We can derive a $n$-node clique (which is a rigid graph \cite{edm}) from this EDM in polynomial time using standard algorithm \cite{cmds}.
However, in our case i) all pairwise ranges are not always available due to limited ranging capability (i.e. as low as 10m for NLOS UWB in our case) and sprawling indoor area (i.e 50m $\times{}$50m in our case).
ii) the derived ranges (i.e. EDM entries) can be extremely noisy to the point of violating simple triangular inequality as illustrated in the previous challenge. Unfortunately, standard algorithm cannot be used for this EDM with incomplete and noisy entries.
Therefore, the EDMCDP is to de-noise the ranges and complete the EDM before using standard algorithm to form the rigid graph.
\uterm{Absolute Localization Problem (ALP)} Once the rigid graph is formed, the next challenge is to derive absolute localization by translation and rotation operation\footnote{We mitigate the flip or reflection by following a specific vertex ordering and origin assignment, resulting from node ID's as discussed in Section \ref{sec:method}}. Determining the required translation $x_t$ and rotation $\theta$ from additional information such as IMU-heading, historical data of track of each node, floor-map etc. as shown in \textcolor{red}{Figure \ref{absolute-loc}} is also challenging.
\begin{figure}
\centering
\includegraphics[width = 0.65\linewidth ]{figs/geom-problem.png}
\caption{Hierarchy of Geometric Challenges}
\label{fig:geom-problem-hiearchy}
\end{figure}
\vspace{0.1in}
We can summarize the geometric challenges in our setting as shown in Figure \ref{fig:geom-problem-hiearchy}.
\end{comment}
\subsection{Node Mobility}
A primary challenge in maintaining the relative geometry of the nodes as updated as possible stems from the mobility pattern of such nodes. The relative motion among the nodes determine the rate at which the inter-nodal distance varies. Let's call such inter-nodal distance an \textit{edge}. We demonstrate a scenario where multiple nodes move at uniform speeds on a building floor. Figure XX(a) shows the evolution of such edges with time. Note that every edge evolves at a different rate depending on the respective nodes' individual speed as well as the direction of their motion.
\begin{figure}[!tbh]
\centering
\includegraphics[height=1.4in, width = 0.45\linewidth ]{figs/challenge_mobility_1.jpg}
\includegraphics[height=1.4in, width = 0.45\linewidth ]{figs/challenge_mobility_2.jpg}
\caption{Impact of node mobility on ranging uncertainty}
\label{fig:node-mobility}
\end{figure}
\vspace{0.1in}
This poses a severe challenge in orchestrating the range measurements. As shown in figure XX(b), the frequency of such ranging measurements itself might bring different utilities when looking at individual edge uncertainties. The edges with a high degree of change need to scheduled more often compared to edges that have a limited change within a certain period of time. Also note that the total number of ranging measurements that can be scheduled within a given time is finite. Hence bundling the \textit{most updated} set of edges while performing the measurements is crucial.
\subsection{Channel Characteristics}
Due to node mobility, the link characteristics of each pair of nodes may change drastically. This adversely affects the ranging accuracy over time. When both nodes of a link are static, they are close by and they have line of sight (LOS) between them, we have the best scenario wherein, the channel characteristics remains unchanged, the received signal is strong and the effect of multi-path is minimal \cite{wireless-prop}. Hence, the ranging accuracy is high. However, we can have the following scenarios:
\begin{enumerate}[noitemsep, topsep = 0em]
\item The inter-node distance can be high
\item There is no line of sight (NLOS)
\item one or both nodes are moving
\end{enumerate}
Any or a combination of the above scenarios affect ranging accuracy. In particular, for our case of time of flight-based ranging using UWB, with LOS, the maximum range turns out to be around 50 meter. However, in presence of obstacles such as walls, furniture etc, this range decreases drastically form 50 to 20 meters or so. In other words, the range is related to the number of obstacles in between the ranging nodes. When the nodes are moving in case of NLOS, we have the worst scenario and the ranging accuracy are the poorest.
\\
\begin{figure}
\centering
\includegraphics[height=2.8in, width = 0.7\linewidth ]{figs/labeled-floor-map.png}
\caption{Indoor map with a pair of nodes in different settings}
\label{fig:indoor-map}
\end{figure}
\vspace{0.1in}
\begin{figure*}
\centering
\fbox{\includegraphics[width=0.95\textwidth, ]{figs/dummy/dummy-back.png}}
\caption{Ranging accuracy for diverse indoor scenarios}
\label{fig:indoor-cdf}
\end{figure*}
\vspace{0.1in}
To substantiate our claim, we create diverse scenarios with two nodes in an indoor environment as shown in Figure \ref{fig:indoor-map}. We then generate the CDF of ranging errors for all cases as shown in Figure \ref{fig:indoor-cdf}. First, we keep both the nodes static in LOS and change the distance from 1 meter up to 40 meters. We observe that the resulting ranging error and its variance is very small and always under 0.25 meters. Now, we put the two nodes in NLOS locations again varying the distance from 1 meter up to 40 meters and also with varying number of walls (wall-count = 1, 2, 3 up to 12) between them. As shown in the figure, NLOS has worse accuracy than LOS but it only worsens with the increase of the number of walls. Finally, we move the nodes linearly with a relative speed between them from 0.5 and 2 m/sec for all the above scenarios. This time, the accuracy gets worse compared to the corresponding static scenarios. Out of all the cases, the scenario with the highest node mobility, NLOS with maximum wall/obstacle count has the poorest range and the worst accuracy of up to 10 meter.
We now show that, one or more low-accuracy ranges drastically reduces the localization accuracy. It requires at least three edges in the relative geometry to localize a node. The core of the solution for the localization without dedicated anchors is the selection of edges for ranging. To illustrate the impact of edge-selection, we create a scenario with 7 static nodes as shown in \textcolor{red}{Figure \cite{fig:edge-loc-quality}}. We want to localize node 1 which is incident with 6 edges marked as $e_{1k}$ for $k=1,\ldots,6$. We first collect the range values for all the six edges for long enough time. Then using ground truth of the pairwise distances between node 1 and the rest of the nodes, we rank the edges in terms of the median ranging error as follows: $e_{11} < e_{12} < e_{13} < e_{14} < e_{15} < e_{16}$ ($e_{11}$ has the least ranging error). Also note that the 3 best accurate edges are LOS links, the rest are NLOS links and 2 of the LOS edges have a length more than those of NLOS edges. Therefore, unless LOS link, short range does not necessarily imply high ranging accuracy.
Let us consider the scenarios of edge-selection and the resulting localization error using multilateral as shown in Table \ref{tab:accuracy}.
\begin{table}[h!]
\centering
\begin{tabular}{||p{0.73in} | p{0.8in} | p{1.3in}||}
\hline \hline
\textbf{Selected Edges} & \textbf{Localization Accuracy} & \textbf{Comment}\\ \hline
$e_{11}$, $e_{12}$, $e_{13}$ & 20 cm & Best edges ensure least error \\ \hline
$e_{14}$, $e_{15}$, $e_{16}$ & 200 cm & Worst edges result in the worst error\\ \hline
$e_{11}$, $e_{12}$, $e_{13}$, $e_{14}$, $e_{15}$, $e_{16}$ & 210 cm & No guarantee of accuracy despite all edges \\ \hline
$e_{11}$, $e_{12}$, $e_{13}$, $e_{16}$ & 175 cm & A Single ``bad" edge ($e_{16}$) affects accuracy\\ \hline \hline
\end{tabular}
\caption{Localization accuracy involving diverse edges}
\label{tab:accuracy}
\end{table}
The observation is as follows- multilateration does not take into account the accuracy of the edges. So, using more edges in it, does not guarantee improvement of localization accuracy. Rather, we need to rank the edges according to their ranging accuracy and only select the best ones for multilateration. Ranging on all incident edges, including the low-accuracy ones rather deteriorate the localization accuracy. However, these edge-rankings are not permanent, rather changing due to node mobility. Therefore, computing edge-ranks in the presence of arbitrary node mobility and selecting a subset of them for localization to increase accuracy is indeed a challenging task. We use Channel Impulse Response (CIR) of the link to quantify its quality as discussed in Section \ref{sec:method}.
\subsection{Absolute Localization}
Once the relative geometry is determined, the next step is to absolutely localize all the nodes by determining the translation and rotation required to affix the rigid graph as a whole. For this purpose, at least one node with known location (beacon node) is required from which the distance and direction of the rigid graph is determined. To determine the direction or orientation of the rigid graph, the IMU, which gives the heading of the nodes relative to earth's north-south axis, is useful. Note that, without anchor nodes, dead-reckoning by IMU is a natural choice for localization via tracking, as it does not require any external set-up at all. However, displacement computed by IMU using the state-of-the-art methods \cite{IMU-survey}, can only be reliably used for few seconds due to run-away error as shown in Figure \ref{fig:imu-exp}. On the other hand, orientation or heading computed by the IMU is reliable over time as shown in Figure \ref{fig:imu-exp}. Additionally, indoor floor-map may be available, which further restricts the translation and rotation of the rigid graph. In summery, deriving absolute locations from relative ones fusing diverse set of information such as from IMU, floor-pan, path-history of the nodes etc. is very challenging in a dynamic environment and requires thoughtful design of systme and methods.
\begin{comment}
All nodes in our case are potentially mobile, thus lacking fixed locations. Therefore, there are no ``anchor" nodes or fixed nodes with known locations in our setting.
There are two implications of this - i) some of the time-efficient localization techniques such as Time Difference of Arrival (TDOA) \cite{tdoa} etc. cannot be used as it requires fixed anchors and wired connection between them for fine grained clock-synchronization. ii) localization depends on pairwise distance-measurement or ranges between nodes. In this case, without any information about the topology resulting the wireless connectivity, the simplest approach would be to range a node with rest of the nodes, and then relatively localize the nodes using the geometry resulting from the set of pairwise ranges and repeat the operation to continuously localize each node. However, for $n$ nodes, it would require $^nC_2$ ranges.
Each ranging operation is time-consuming and cannot be arbitrarily minimized. For example, due to the broadcast nature of the wireless medium, each pair of nodes has to take turn in Time Division Multiple Access (TDMA) fashion for interference-free accurate ranging. Each ranging operation may take tens of milliseconds depending on the modality (WiFi, Ultra Wide Band etc.), data-rate of frame-exchange (110kbps, 64Mbps etc), frame-exchange protocols (single-sided and double-sided Two-way ranging \cite{twr} etc ), sampling rate and latency of the underlying electronics \cite{deca-chip-perf} and so on. For example, in our case, we are using double-sided two-way ranging protocol which requires 4 frame exchanges between each pair. We are using UWB with a datarate of 110kbps, which requires a TDMA slot time is 8msec for a maximum 110Bytes of payload. Therefore, each ranging takes 8$\times$4 = 32 msec. We are collecting all ranges using UDP frame via WiFi channel. The request and response alone takes about 2$\times$4=8msec. Therefore, each operation of range collection takes minimum 32+8 = 40msec. If there are 16 nodes, according to the above simplest method, we need to gather $^{16}C_2=120$ ranges to determine the geometry of relative localization. Therefore, the gap between successive localization of each node is at least 120$\times$40= 4800msec. However, if we had 3 static anchors covering the whole area of interest, each node could be localized in 3$\times$40=120msec after 3 ranging with 3 anchors, and for 16 nodes, collection of all possible ranges would take 16$\times$120 = 1920msec. The localization delay for the case without any dedicated anchors and with 3 anchors are \textit{quadratic} and \textit{linear} respectively in the \# of nodes. Therefore, the biggest challenge here is to minimize the localization delay by scheduling only a subset of all possible $^nC_2$ ranges, taking into consideration factors such as link or range quality, node mobility, effect on the relative geometry etc. Therefore, the lack of dedicated anchors requires complex orchestration of ranging operations among nodes, to scale localization with the \# of nodes and to minimize the delay.
\end{comment}
\subsection{Background on Related Works}
The rich literature in the area of active (locating and identifying) indoor localization
can be broadly categorized as (i) Anchor-based, and (ii) Infrastructure-free approaches, as shown in Fig.~\ref{fig:lit-trade-off}.
\mypara{Anchor-based:}
These approaches often surpass their infra-free counterparts in accuracy
at the expense of a-priori deployed infrastructure for localization -- a tradeoff captured in Fig.~\ref{fig:lit-trade-off}.
Here, beacons are deployed at known locations and serve as reference points or anchors. A node estimates its distances (also called ranges) from
three such anchors, which are then combined with the anchors' locations to estimate its own location by a technique typically known as multilateration. Given a technology to perform accurate ranging, localization can be seen as a {\em trivial} extension. Hence, most of the prior works in this space has focused on the accurate estimation of such ranges,
particularly using WiFi access points as beacons, while some have also leveraged ultrasonic beacons~\cite{ultrasound-loc-survey}. The WiFi-based works leverage signal information across multiple dimensions -- frequency (~\cite{tonetrack-15}, ~\cite{chronos-16}), antenna arrays (~\cite{spotfi-15}, ~\cite{monoloco-18}), or both (~\cite{md-track-19}), to improve accuracy in the face of limited WiFi bandwidth and multipath. Some of them~\cite{locap} adopt a finger-printing approach (using RSSI, CSI, etc.) to calibrate the environment a-priori that is later used for for real-time location inference. Optical tracking systems (e.g. HTC Vive~\cite{htc-vive-benchmark}) that are popular in the AR/VR industry, employ multiple IR beacons (LEDs/ cameras) to provide mm-level tracking accuracy, but are restricted to line-of-sight and expensive to deploy.
The fundamental dependence on pre-deployed anchors (mostly static, but sometimes mobile -- e.g. outdoor drones~\cite{dhekne2019trackio}), prevents such approaches from catering to our target environment.
\mypara{Infrastructure-free:}
Works in this category are more amenable to our target environment, but exhibit a different tradeoff between accuracy and robustness, as captured in Fig.~\ref{fig:lit-trade-off}.
Inertial sensor-based solutions~\cite{bo2013smartloc, zhang2012inertial, romit-imu-16, romit-imu-18} are inherently local to a node (no ranging needed), and hence popular.
However, with only dead-reckoning of nodes, errors accumulate significantly over time~\cite{intertial-loc-survey}, especially in case of pedestrian mobility.
State-of-the-art AR/VR solutions (e.g. ARCore~\cite{arcore-cap}) leverage visual inertial odometry (VIO) that combines both cameras and IMUs to provide accurate cm-level tracking~\cite{vio-dev-survey} in favorable conditions. However, they often require anchors \cite{rowe-ipin-19} and their performance suffers significantly in poorly illuminated and/or poorly textured environments~\cite{opto-survey}, in presence of motion-blur \cite{opto-blur} and/or multiple moving objects in the video-frames \cite{opto-obj}, as shown in Fig.~\ref{fig:vio-indoor-exp}. Such practical conditions result in various errors relating to drift, loop-closure, scale ambiguity etc. (for SLAM-based approaches) \cite{vslam-ov}, and errors related to projection, parameterization etc. (for optical-flow based approaches) \cite{opt-flow-ov}.
\\
\sfrtxt{
\cite{lifi-14} uses CSI of WiFi to determine LOS. \cite{p2ploc-18} uses P2P UWB ranging for localization of mobile node considering dilution of precision. \cite{calib-free-ipsn-2017} uses Gaussian Mixture and MDS to solve indoor localization using UWB ranges.
}
\\
Hence,
dynamic environments, particularly those in first responder scenarios can significantly benefit from an alternate RF modality that can deliver good accuracies (sub-1-2m), and which is robust to the lacking of such favorable conditions.
In
AR/VR gaming applications, such a modality can be complementary in helping to eliminate the accumulating errors faced by VIO, with periodic absolute location fixes.
\subsection{Role of RF in Infra-free Localization:}
The recent popularity of ultra wide-band (UWB~\cite{decawave}) technology, and its ability to span a wide 500-1000 MHz bandwidth with superior multipath suppression (owing to its impulse transmissions), has made it a popular candidate for sub-m localization~\cite{warehouseWSN,fernandez2007application,gowda2017bringing, uwb_pos}, albeit with the help of infrastructure anchors.
Existing works in this space are largely concerned with scalable ranging (SurePoint~\cite{kempke2016surepoint}, SnapLoc~\cite{grobetawindhager2019snaploc}) and tracking of individual mobile nodes (e.g., indoor drone, PolyPoint~\cite{kempke2015polypoint}).
However, in the absence of reference anchors, UWB's two-way-ranging (TWR~\cite{sahinoglu2006ranging}) mechanism can enable the nodes to only range with each other. Hence, localization in dynamic environments presents a different
challenge, which, beyond the estimation of accurate ranges, needs to translate the ranges to an accurate localization solution.
Indeed, the key focus of infra-free RF localization comes down to bridging this gap between estimated ranges and node localization that arises in the absence of anchors.
\mypara{Primer on Relative Localization:}
In contrast to anchor-based approaches, where nodes are {\em absolutely} localized in the coordinate space defined by the anchors, localization in infra-free set-up is a two-step process. First, the nodes are {\em relatively} localized among themselves, following which some meta information (e.g. orientation of the nodes, or floorplans) is leveraged to transform such relative localization to the absolute coordinate space. Relative localization refers to the geometry or a topology among the nodes, where the pairwise distance between nodes as well as their relative orientation are preserved.
In determining such a relative localization, the construct of a {\em rigid body} comes handy.
\begin{figure}[!thb]
\centering
\includegraphics[width = 0.7\linewidth ]{figs/rigidity-example.png}
\caption{(a) Clique (rigid) (b) non-clique (rigid) (c) 2 choices for node 5 (not rigid) }
\label{fig:rigid}
\end{figure}
Let $G=(V, E, d)$ be a weighted graph, where $d$ is the set of range measurements (weights) for the |E| edges defined on the |V| nodes.
A \textit{realization} is a function $x:V \rightarrow{} \mathbb{R}^2$ that maps the set of vertices $V$ to the 2D Euclidean space such that each range value is preserved i.e $\forall (u,v) \in E, \; ||x(u) - x(v)|| = d(u, v)$ where $||.||$ is the Euclidean norm. The graph $G=(V, E, d)$ is ``rigid" if there is only one realization, discounting any translation, rotation and flip. Thus, a rigid topology gives us a unique relative localization solution. As shown in the Figure \ref{fig:rigid}, the complete graph or clique in (a) is rigid because given the 10 ranges between all pairs of nodes, this graph is the only realization, although it can be rotated, translated and/or flipped. Similarly, given the 9 ranges, the graph in (b) is rigid. However, (c) is not rigid, since given the 8 ranges, there are two choices to fix node 5 relative to the edge (2,3) resulting in two potential realizations.
\begin{figure}[!htb]
\centering
\subfloat[]
{
\includegraphics[width=0.4\linewidth]{figs/section-2-plots/var-refresh-rate}
\label{fig:var-refresh-rate}
}
\subfloat[]
{
\includegraphics[width=0.4\linewidth]{figs/section-2-plots/var-node-count}
\label{fig:var-node-count}
}
\vspace{-10 pt}
\caption{Localization error for
(a) various refresh rates
(b) various no. of nodes
}
\vspace{-7 pt}
\end{figure}
\begin{figure}[!htb]
\centering
\subfloat[]
{
\includegraphics[width=0.4\linewidth]{figs/section-2-plots/var-topologies}
\label{fig:var-topologies}
}
\subfloat[]
{
\includegraphics[width=0.4\linewidth]{figs/section-2-plots/var-configurations}
\label{fig:var-configurations}
}
\vspace{-10 pt}
\caption{Localization error after using various low-rank matrix completion techniques for(a) various topologies and (b) various set of edges from same topology
}
\vspace{-13 pt}
\end{figure}
\begin{figure}[!htb]
\centering
\subfloat[]
{
\includegraphics[width=0.4\linewidth]{figs/section-2-plots/var-percent-noisy-link}
\label{fig:var-noisy-link}
}
\subfloat[]
{
\includegraphics[width=0.4\linewidth]{figs/section-2-plots/var-mobility}
\label{fig:var-mobile-node}
}
\caption{Localization error for
(a) various levels of link noises
(b) various fractions of mobile nodes}
\end{figure}
If {\em all} possible ranges between nodes are available, computing the relative localization is straight-forward in a {\em static} environment. The edge weights of the graph are maintained in the form of an adjacency matrix, \texttt{EDM} (a.k.a. Euclidean Distance Matrix), where each entry represents a measured range between two nodes. Then, an approach called Multidimensional Scaling (MDS) is applied on the \texttt{EDM} matrix, whereby an EVD (Eigen Value Decomposition) results in an embedding of the nodes (i..e. relative localization) in a 2D Cartesian space.
While such a framework of relative localization is appropriate for our dynamic environments,
the theoretical approaches in this space are built on several assumptions\footnote{e.g. static nodes, accurate range estimates of all node pairs at no cost, etc.} that do not hold in practice.
\subsection{Challenges in a Dynamic Environment}
\label{subsec:challenges}
To accurately localize/track mobile nodes, the localization solution needs to be computed and refreshed at a granularity finer than node mobility. For instance, a 1 Hz refresh rate is appropriate to track nodes with speeds of 1-1.5 m/s, targeting a 1-2 m error.
However, the refresh rate automatically enforces a latency-bound (cost) for the whole process of relative localization, which involves both the range estimation/collection as well as the solution computation. This results in a {\em latency-bounded} version of the infra-free localization problem (referred to as LB-IFL) that has not been addressed before.
To understand the impact of such a latency cost on existing approaches in practice, we conduct an experimental study (details of testbed described in Section 4) comparing a genie/anchor-aided localization solution (with all possible range estimates available instantaneously, TDOA) with the one described above, i.e., MDS applied on EDM constructed with a random set of edges (called Base), whose ranges are measured within the latency budget offered by the refresh rate.
\mypara{Latency vs. accuracy:}
Every ranging operation takes a finite amount of time to complete. For instance, with a popular UWB hardware~\cite{decawave}, it takes approximately 40\,ms to complete a single range estimate, i.e. a two-way-ranging (TWR) operation. This inherently limits the number of ranges that can be estimated/collected per second to 25 to support a localization update rate of 1\,Hz.
The results in Fig.~\ref{fig:var-refresh-rate} and ~\ref{fig:var-node-count} show that when budget restrictions increase for a given topology size or vice versa, the accuracy degrades by several folds, clearly exposing a trade-off between latency (cost) and accuracy. This can be attributed to the lack of intelligent topology estimation (edge selection) and robust relative localization schemes that are needed to work with limited latency budgets and hence incomplete range estimates respectively -- aspects that have not been addressed thus far. We now further dissect the specific factors contributing to this performance degradation.
\mypara{Incomplete range estimates:} In practice, the physical communication range between nodes will limit the topology from being complete.
This is further compounded by the limited number of edges (ranges) that can be estimated due to the latency budget.
Since an incomplete EDM (i.e. estimated topology is not a clique) can lead to localization inaccuracies, matrix completion methods (e.g., \texttt{SDR}~\cite{keshavan2010matrix}, \texttt{OptSpace}~\cite{alfakih1999solving}) are used to complete the EDM before the nodes can be relatively localized. However, the latter are not designed keeping in mind the geometrical implications relevant to a localization problem. This can lead to large localization errors as shown in Fig.~\ref{fig:var-topologies}, thereby advocating the need for relative localization schemes that are robust to incomplete range estimates.
\mypara{Impact of geometry:} The interesting result in Fig.~\ref{fig:var-configurations} further indicates that the specific set of edges selected, albeit incomplete, has a large impact on the localization solution as well. This indicates that the geometry of the topology (particularly its rigidity) associated with the edges measured, has a direct impact on the solution and must be factored into the edge and hence topology selection process.
\mypara{Inaccurate range estimates:} Inaccurate estimates of even a small set of ranges can lead to degraded accuracy for the entire topology. Here, two key environmental factors, namely LOS blockages (due to body, concrete, etc.), and node mobility, can significantly affect the accuracy of the range estimates. Given the limited budget for range estimation, it is clear that when the edges are picked randomly without taking into account their channel or mobility characteristics, the performance degrades quite rapidly even with a small set of affected edges, as seen in Fig.~\ref{fig:var-noisy-link}.
Thus, characterizing the nature of the edges with respect to their channel and mobility is essential for improved localization accuracy.
\subsection{Design Requirements}
From the above discussions, it is clear that the combination of ``infrastructure-free" and ``node mobility" in practical, dynamic indoor environments, makes the {\em latency-bounded} version of the localization problem, highly challenging.
In addressing these challenges, two key design requirements emerge for \textsc{DynoLoc}\xspace,
(a) Support a reasonable number of nodes in a practical deployment setting ($\approx$ few 10s), many of them being mobile ($\leq$ 2 m/s), and (b) Offer a location update rate ($\ge$1\,Hz), tolerable to the underlying location based service that eventually consumes such information.
\section{Introduction}\label{sec:introduction}}
\section{Introduction}
\input{introduction.tex}
\section{Motivation \& Challenges}
\input{background.tex}
\section{{\bf \Large \textsc{DynoLoc}\xspace}: Design}
\input{method.tex}
\section{Implementation and Evaluation}
\input{results.tex}
\section{Applications}
\input{applications.tex}
\section{Conclusion}
\input{conclusion.tex}
\IEEEPARstart{T}{his} demo file is intended to serve as a ``starter file''
for IEEE Computer Society journal papers produced under \LaTeX\ using
IEEEtran.cls version 1.8b and later.
I wish you the best of success.
\hfill mds
\hfill August 26, 2015
\subsection{Subsection Heading Here}
Subsection text here.
\subsubsection{Subsubsection Heading Here}
Subsubsection text here.
\section{Conclusion}
The conclusion goes here.
\appendices
\section{Proof of the First Zonklar Equation}
Appendix one text goes here.
\section{}
Appendix two text goes here.
\ifCLASSOPTIONcompsoc
\section*{Acknowledgments}
\else
\section*{Acknowledgment}
\fi
The authors would like to thank...
\ifCLASSOPTIONcaptionsoff
\newpage
\fi
\section{Introduction}
\input{introduction.tex}
\section{Motivation \& Challenges}
\input{background.tex}
\section{{\bf \Large \textsc{DynoLoc}\xspace}: Design}
\input{method.tex}
\section{Implementation and Evaluation}
\input{results.tex}
\section{Applications}
\input{applications.tex}
\section{Conclusion}
\input{conclusion.tex}
\bibliographystyle{IEEEtran}
\subsection{Background of Solution}
In absence of fixed anchor nodes, we localize all the nodes in two step i) \textit{relative localization}: we construct an unambiguous geometric shape called ``rigid graph" by measuring distances (called ranges) between the nodes ii) \textit{absolute localization}: using additional information such as heading from IMU etc., we rotate, translate or flip the rigid graph to derive absolute locations of the nodes with respect to a global coordinate.
In the relative localization step, based on the range measurements, we generate an $n\times{}n$ matrix called Euclidean Distance Matrix (EDM) for $n$ nodes. In EDM, the entry in the $i$-th row and $j$-th column is the range between node $i$ and $j$ (nodes are labeled 1 through $n$). Given all the pairwise ranges between the nodes i.e the EDM, we can form an unambiguous geometric shape called the rigid graph \cite{edm-graph} discounting rotation, translation and flip. However, each ranging operation takes tens of milliseconds which cannot be arbitrary reduced (out of scope of this work) for a given system. Given a fixed ranging time (say 40 msec), forming the EDM for even a small number of nodes (say 16) requires high number of measurements ($^{16}C_2 = 120$) and thus significant amount of time ($120\times{}40/1000=4.8$ sec). On the other hand, nodes are potentially mobile causing temporal inconsistency during formation of the EDM. Consider a range measured first during the computation ($(120-1)*40/1000 = 4.76$ secs prior to the EDM completion). Due to the mobility of this node (say 1$ms^{-1}$), the first range will be way off the true value (at most $(1ms^{-1}\times{}4.76$ sec)$= 4.76$ meter). As such, the location of the node will also be off (by at least few meters) the true location. Therefore, although a complete EDM guarantees an unambiguous rigid graph, it does not minimize localization error. Now consider the case, where we take subset of all possible ranges (say 32 ranges out of 120). Here, all the ranges will be fresher and more accurate (first measured range will be off by at most $(31\times{}40/1000)=1.24$ meter) than the previous case. However, taking an arbitrary subset of ranges will not guarantee rigid graph (32 ranges will never form a rigid graph for 16 nodes). Also ranging between a pair of nodes will not be possible if they are out of each others wireless range.
We want to find a ``sweet spot" between these two following cases - i) a full EDM rendering erroneous relative locations due to temporal inconsistency and ii) incomplete EDM rendering no rigid graph despite fresher set of ranges. Therefore, the relative localization step entails ``edge selection" step wherein, we select a subset of all possible edges for ranging such that i) the rigid graph formation is guaranteed and ii) the localization error (due to the temporal inconsistency) is minimized. In other words, we want to form the rigid graph using minimum number of ranges. However, the underlying problem is NP-hard \cite{maximal-core}. Even for a small number of nodes ($16$), the number of choices is combinatorially explosive
($2^{^{16}C_2}$, trillions of trillions of choices). Only a very tiny fraction of these choices actually result in a rigid graph (out of 1 million random choices, only XX\% resulted in rigid graph for $n=16$, this ratio decreases drastically with the increase in the number of nodes, see Figure \ref{fig:comb-edge-selection}). On top of that, only a tiny fraction (see Figure \ref{fig:comb-edge-selection}) of these choices rendering a rigid graph actually result in high-accuracy localization. This indeed is the proverbial ``needle in the haystack" search.
However, we do not have endless time to compute a single snapshot of locations, due to constant mobility of the nodes. Therefore, finding the desired sweet spot between a fully complete and an almost incomplete EDM is indeed very challenging yet rewarding task for high-accuracy localization.
Once we derive an incomplete EDM, the next task is to fill in the missing values. There are many algorithms \cite{edm-completion} for this purpose. However, two incomplete EDMs with the same number of missing ranges may result in two different rigid graphs after application of the same EDM completion algorithm. See Figure \ref{fig:edm-ambiguity}, where one incomplete EDM renders correct rigid graph whereas the other one does not, although both will take equal amount of time due to equal number of edges. This signifies the importance of edge selection f
or ranging to ensure fast and accurate localization.
\begin{comment}
\subsection{Overview of Solution}
The localization problem is to determine the location (coordinate in 2D) of each entity in a global coordinate system, given the distances (ranges) between them, measured using wireless modality. We represent the entities as nodes in a graph. Each of $n$ nodes has a unique node ID labeled 1 through $n$. The length of the edge between two nodes correspond to the measured range between the two entities.
When all the nodes are mobile, we solve the localization problem on a snapshot basis as follows.
Given $n$ nodes, we have $^nC_2$ possible ranges, from which we construct a unique geometric shape called rigid graph. The location of a node in this rigid graph is called relative location and the process is called relative localization. This rigid graph, however, can be rotated, translated or flipped with respect to any arbitrary coordinate system.
So, using additional information, we fix the rigid shape with respect to a global coordinate system and thus derive unique absolute location of each node. This is called absolute localization. In other word, the solution in absence of fixed static anchor locations, involves relative localization followed by absolute localization.
In relative localization step, the objective is to determine the rigid graph as accurately as possible using the measured ranges. If all the $^nC_2$ ranges are measured accurately, we do derive a unique rigid graph \cite{rigid-geometry}. However, each measurement takes some finite amount of time, which cannot be minimized due to system constraint arising out of the wireless modality. For applications dependent on the real-time location of potentially mobile nodes, the objective is to localize each node repeatedly as fast as possible. However, even for a small number of nodes, say $n=16$, and for the ranging time of say 40 msec, relative localization involving all possible ranges would take $^{16}C_2\times40/1000 =4.8$ sec. Within these few seconds, some highly mobile nodes will move to new locations, which would cause high localization error and would result in unsatisfactory performance of the dependent application. Moreover, some pairs of nodes may be out of each others wireless range, so range measurement is not possible for them. Either way, this ``all pair" range measurement is not practical. Therefore, the main challenge of relative localization is to determine the rigid graph as fast as possible by using least number of measurements (say 3$\times$16=48 ranges in 48$\times$40/1000 $=$ 1.92 sec instead of $^{16}C_2=$120 ranges in 4.8 sec). The natural question is - which edges to select for ranging.
To concertize the notion of edge selection as described above, we present the all-pair ranges between $n$ nodes, as an $n \times n$ matrix $\mathbf{D}$ called Euclidean Distance Matrix (EDM), where entry for row $i$ and column $j$ is the range between node $i$ and $j$. A compete EDM matrix gives unambiguous rigid graph. However, if we measure a subset of all-pair ranges, the resulting EDM will be incomplete i.e. the EDM will have missing values. An incomplete EDM does not guarantee a rigid graph. In literature, there are many ways to fill in the missing values of a given incomplete EDM \cite{edm-completion}. However, in our case, the controller can choose which ranges to measure and thereby also the incomplete EDM. Note that, even if we apply the same matrix completion algorithm, two different incomplete EDM with the same number of missing entries will result in two completely different rigid graphs as shown in Figure \ref{fig:incomplete-edm}. This ambiguity in rigid graph, defeats the whole purpose of relative localization, wherein we intend to determine the rigid graph uniquely for unambiguous localization. Therefore, an important challenge of relative localization is to determine which entries in the EDM are crucial for determining unambiguous rigid graph formation. We also need to take into account the geometric dilution of precision (GDOP) \cite{gdp-paper} in determining this importance. Even if the measurements errors are low, the relative localization can incur huge error due to GDOP resulting from poor choice of the edges as shown in Figure \ref{fig:gdp}.
Apart from the geometric requirement,
noisy entries of the EDM can also affect accuracy of rigid graph formation. This noise manifests in two ways i) node mobility and ii) channel characteristics. As for node mobility, between successive measurements, the range between two static nodes remain unchanged whereas the range between two highly mobile nodes may change drastically. This change will result in temporal inconsistency in localization as shown in Figure \ref{fig:temporal-inconsistency}, where mobile nodes will contribute more to the localization error than static nodes. Therefore, it is desirable to range edges incident with highly mobile nodes more frequently than those incident with static nodes over time. In other word, the edge selection should also incorporate the expected node mobility. As for the channel characteristics, due to multi-path effect, some of the NLOS links will result in poor ranging accuracy compared to the LOS links. Therefore, existing channel characteristics should also be taken into account when selecting edges for ranging.
In summary, accurate and fast localization in our case is tied to accurate and fast formation of the rigid graph in the relative localization step. This involves i) deriving an incomplete EDM satisfying the geometric requirements (rigidity and GDOP) and using high-accuracy ranges (based on node-mobility and channel characteristics), ii) deriving the complete EDM from the incomplete one and solving it for relative localization. \\
\end{comment}z
We start with an overview of our solution and then discuss each component of it.
\begin{figure}
\centering
\includegraphics[width = 0.99\linewidth ]{figs/flow.png}
\vspace{-0.1in}
\caption{Overview of our algorithm}
\label{fig:flow}
\vspace{-0.1in}
\end{figure}
Figure \ref{fig:flow} shows an overview of the steps involved in our solution. The steps are i) Edge Selection ii) Ranging iii) Missing Range-estimation iv) Relative Localization and v) Absolute Localization. These steps are continuously repeated and the locations are fed into the applications as discussed in Section \ref{sec:applications}. Additionally, time-varying information about existing link quality and node mobility are collected periodically and used in these steps. Absolute localization also requires information beyond ranging. We discuss each step in detail in the order they are executed.
\subsection{Edge Selection}
As discussed in Section \ref{sec:background}, the ranging time cannot be arbitrarily minimized due to system constraint. Therefore, the goal is to repeatedly update the locations using least number of ranges. The edge selection step involves deciding which of the all possible links ($^nC_2$ for $n$ nodes) to range at the current instant for localization.
In particular, we want to schedule ranging only for i) currently existing physical links, ii) links that are important for localization iii) links that are incident with highly mobile nodes rather than static nodes, and iv) links with better channel characteristics and thus potentially better ranging accuracy. We now discuss measures involved in this step to ensure each of the above four objectives.
\subsubsection{Link Discovery}
The controller periodically broadcasts a special discovery frame. Upon receiving this frame, each node sends out a beacon frame in the $i$-th TDMA slot where $i=1, 2,\ldots, n$ is the node ID. Each node upon receiving the beacon frame, adds the source node to its list of neighbors. The controller then collects the neighbor list from each node to build physical connectivity graph $G(t) = (V, E(t))$ at instant $t$, where $V$ is the set of nodes and $E(t)$ is the set of existing links. In the following steps, ranging is only scheduled for the links in $E(t)$ instead of all $^nC_2$ possible ones.
\begin{figure}
\centering
\includegraphics[width = 0.8\linewidth ]{figs/rigidity-example.png}
\vspace{-0.1in}
\caption{(a) Clique (rigid) (b) non-clique (rigid) (c) 2 choices for node 5 (not rigid) }
\label{fig:rigid}
\vspace{-0.1in}
\end{figure}
\subsubsection{Determining Essential Links}
In this step, we determine the links that are important for localization. The importance is determined from two perspectives i) relative geometry- wherein the edges which are required to form unambiguous geometric shape (which we term rigid graph) from the ranges, are prioritized, and ii) node mobility- wherein the edges incident with relatively mobile nodes are prioritized. First, we describe how to resolve the geometric shape from scratch, which is required when the system starts up. We then present algorithm to incrementally update the shape using node mobility. To this end, we introduce the notion of rigid graph.
\vspace{0.01in}
\\ \noindent
\uline{Rigid Graph:}
Given a weighted graph $G=(V, E, d)$, where $d$ is the set of ranges of the edges, a \textit{realization} is a function $x:V \rightarrow{} \mathbb{R}^2$ that maps the set of vertices $V$ to 2D Euclidean space such that each range is preserved i.e $\forall (u,v) \in E, \; ||x(u) - x(v)|| = d(u, v)$ where $||.||$ is the Euclidean norm.
The graph $G=(V, E, d)$ is ``rigid" if there is only one realization discounting translation, rotation and flip. As shown in the Figure \ref{fig:rigid}, the complete graph or clique in (a) is rigid because given the 10 ranges between all pairs of nodes, this graph is the only realization, although it can be rotated, translated and/or flipped. Similarly, given the 9 ranges, the graph in (b) is rigid. However, given the 8 ranges, as shown in (c), there are two choices to fix node 5 relative to the edge (2,3) resulting in two realizations, therefore, it is not rigid.
We can define the following recursive definition of a rigid graph in 2D:
\begin{enumerate}[noitemsep, topsep = 0em]
%
\item A complete graph or clique is a rigid graph.
%
\item If $G=(V,E)$ is a rigid graph and a node $i$ has a set of edges $E_i$ to at least three non-collinear nodes in $G$, then the graph derived after adding node $i$ to the graph along with the connecting edges, is also rigid, i.e. $G^{'}=(V^{'},E^{'})$ is rigid, where $V^{'}=V\cup \{i\}$ and $E^{'}=E \cup E_i$.
\end{enumerate}
For example, the subgraph containing nodes 1 through 4 in Figure \ref{fig:rigid}(c) is rigid and all nodes are non-collinear. If we add an extra range (4,5), then node 5 becomes part of the rigid graph as shown in (b).
Using the above definition, we can incrementally build a rigid graph from the connectivity graph derived in the previous step, except we need to know the ranges to determine collinearity. Therefore, we have a cyclic dependency as follows- the minimal rigid graph minimizes the required \# of ranges, yet determining the rigid graph requires knwoing the ranges. To this end, we introduce the notion of k-core graph to opportunistically range as the rigid graph is being built.
\vspace{0.05in}
\begin{figure}[htb!]
\centering
\includegraphics[width = 0.8\linewidth ]{figs/k-core-example.png}
\vspace{-0.1in}
\caption{3-core Decomposition}
\label{fig:k-core-examples}
\vspace{-0.1in}
\end{figure}
\vspace{0.01in}
\\
\noindent \uline{k-core Decomposition}: A k-core is a connected graph in which every node has a degree of at least k. For, $k\leq2$, the graph cannot be rigid in 2D. As shown in Figure \ref{fig:k-core-examples}, the connectivity graph has 7 nodes, however, only nodes 1 through 4 can form a rigid graph.
Therefore, we try to find the rigid graph only on the $3$-core graph. In other words, the edges in the $3$-core graph are more important to range than the rest.
We specially handle the nodes in the $2$-core or $1$-core later.
Partitioning a graph into k-cores for $k=1,2,3,\ldots$ is called ``k-core decomposition". It works as follows. First the nodes with the lowest degree are removed one by one and are included in the k-core, where k is the degree during its removal. Such node-removal affects degrees of other nodes, so the operation is repeated. The intra-core edges are preserved. As shown in Figure \ref{fig:k-core-examples}(a), nodes 6 and 7 have the minimum degree of 1, so both belong to 1-core. After removing them, we derive the 2-core subgraph in Figure \ref{fig:k-core-examples}(b). We then remove the 2-degree node 2, to derive the 3-core subgraph in Figure \ref{fig:k-core-examples}(c).
%
The 3-core decomposition does not require any ranging. It requires the connectivity matrix which is already determined in the previous step. Therefore, this step precedes rigid graph formation which requires ranging. We present two versions of rigid graph formation- i) bootstrap rigid graph formation: we compute the rigid graph from scratch when the system starts up ii) rigid graph update: based on the update in connectivity, 3-core graph and node mobility model, the existing rigid graph is updated. We briefly discuss them in the following.
%
\\
\noindent \textbf{1. Bootstrap Rigid Graph Formation: }
To maximize the chance of being part of the rigid body, we first rank the $m$ nodes in the 3-core subgraph according to the non-increasing value of their degree (degree in 3-core, not in the connectivity graph). Let the ordering be $n_1, n_2, \ldots, n_m$, where node $n_1$ has the highest degree. We start with the triangle (rigid graph) $<n1, n2, n3>$. It requires 3 ranges for the 3 sides of the triangle. From the connectivity matrix, we check whether node $n_4$ has edges with the above 3 nodes. If so, we measure 3 more ranges between them $n_4$ and the above 3 nodes. If the rigidity conditions are satisfied, we add $n_4$ to the triangle to form the new 4-node rigid graph. Then we try the next node $n_5$ and so on. The maximum number of ranging in this step equals the number of edges in the 3-core subgraph.
%
\\
\noindent \textbf{2. Rigid Graph Update: }The new rigid graph $G_R(t+1)$ is derived from the existing rigid graph $G_R(t)$ by node deletion and addition. If there is a change in the connectivity graph (which is also continuously updated), the 3-core is recomputed. We first remove the nodes which fail the rigidity test in $G_R(t)$, due to the change in their connectivity. Removal of one or more nodes may cause other nodes to be removed from $G_R(t)$. This way, we derive a smaller rigid graph $G^{'}_R(t)$ from $G_R(t)$. We then make a set of candidate nodes $V_c(t+1)$ for addition to $G^{'}_R(t)$ after rigidity test. The set $V_c(t+1)$ includes two types of nodes i) nodes which are in the updated 3-core graph but not in $G^{'}_{R}(t)$ and ii) nodes selected by the mobility model discussed later. We start with a node $n_i \in V_c(t+1)$ which has 3 edges to any three non-collinear nodes in $G^{'}_{R}(t)$, we measure the 3 ranges, make $n_i$ part of the rigid body by updating $G^{'}_{R}(t)$ and try another node in $V_c(t+1)$. After all nodes are tested, we derive the new rigid graph $G_R(t+1)$ from $G^{'}_{R}(t)$.
%
The maximum \# of ranging done in this step is $3\times{}|V_c(t+1)|\leq$ \# of ranges required for rigid graph formation as in the bootstrap step.
%
\begin{figure}[htb!]
\centering
\includegraphics[width = 0.99\linewidth ]{figs/mobility-model.png}
\vspace{-0.2in}
\caption{Modeling of Uncertainty for a static and a mobile node}
\label{fig:mobility-model}
\vspace{-0.1in}
\end{figure}
\vspace{0.01in}
\\
\noindent \uline{Mobility Model}: In the above, we handle the change of the rigid graph when nodes break away from it or new nodes become part of it due to change in connectivity. However, without change in connectivity, the rigid graph can also change due to the mobility of its existing nodes. In this section, we describe our approach to handle change in rigid graph due to node mobility. To this end, we define an uncertainty function $u_i(t)$ for each node $i \in V$ at instant $t$ which gives a distance measure of the uncertainty due to the elapsed time since the last localization of node $i$. We also set a threshold called tolerance of uncertainty denoted by $\Delta u$ and try to ensure that at any instant $t$, $\forall i \in V, \; u_i(t) < \Delta u$. The nodes are queried occasionally for their instantaneous linear velocity reading from the IMU. If the velocity is 0, the node is static and we define $u_i(t)$ by an exponential function $ab^{c\Delta t}$ where $\Delta t$ is the elapsed time since the last query for velocity and parameters $a,b$ and $c$ are chosen in accordance with $\Delta u$. In Figure \ref{fig:node-mobility}, the curve for the function $u_i(t)$ is shown for static node. The value of $u_i(t)$ keeps increasing until node $i$ is localized (at 500msec and 1000msec marks) and we reset it to zero and then again increase it exponentially. As for a mobile node, when we derive a non-zero velocity $v_1$ from the IMU at instant $t_1$, we update the function as $u_i(t) = u_i(t_1)+v_1(t-t_1)$ (See 500msec mark).
If at $t_2$ node $i$ is localized and velocity is determined to be $v_2$, we reset the function as $u_i(t) = v_2(t-t2)$ (See 650msec mark). In other words, the uncertainty is modeled on the observed node velocity which is not used in the localization, rather is used as a measure for localization priority for nodes. The uncertainty keeps increasing until the node is ranged and localized. We maintain a priority queue for all nodes based on their current uncertainty values. We always choose the highest priority node for the rigidity test and subsequent computation for localization. In this way, locations for relatively mobile nodes are more frequently updated than those for the static ones and our system is able to handle heterogeneous node mobility.
%
\subsubsection{Link Quality Metric}
%
\begin{figure}[htb!]
\centering
\includegraphics[width = 0.95\linewidth ]{figs/linkq.png}
\vspace{-0.2in}
\caption{Factors of link quality metric shown in amplitudes of IQ-samples}
\label{fig:linkq}
\vspace{-0.1in}
\end{figure}
%
\begin{figure}[htb!]
\centering
\includegraphics[width = 0.8\linewidth ]{figs/linkq-validate.png}
\vspace{-0.2in}
\caption{LQ values and ranging errors for varying true distances}
\label{fig:linkq-validate}
\vspace{-0.1in}
\end{figure}
When all the nodes are closer to each other, the available links for ranging are often more than the required 3 links per node for the rigid graph formation. As discussed in Section \ref{sec:background}, one or more ``poor quaity" links can deteriorate the accuracy of the ranging and thereby, the accuracy of the resulting localization. Therefore, these links should be ordered according to a quality metric which reflects the probability of ranging error, and the best 3 links should be selected.
In our system, each node maintains a list of Link Quality (LQ) metric value, one for each other node. Each LQ value is initialized as zero (indicates no wireless link). The LQ value is updated whenever a frame is received due to either overhearing or the node being the destination. We assume that this LQ value holds momentarily due to potential node moblity, therefore each LQ value is reset to zero in every few seconds after the last update. We now discuss the formulation of the LQ values.
In our system, the UWB chip Decawave DW1000\cite{deca1000}, calculates the range based on the finegrained timestamp of the firstpath detection. The ranging error (always a positive bias) originates mostly due to the erroneous detection of the firstpath.
The detection algorithm is proprietory, however, the chip saves the set of IQ samples on which the anaysis is done after each frame reception. Depending on the operating mode selected for the UWB chip, the number of 4-byte IQ samples is about between 1K and 4k.
Collecting kilobytes of IQ sample from the UWB chip in submillisecond after every frame-reception is not feasible due to relatively slow microcontroller (72MHz STM32F series \cite{stm32}). It can also be disruptive to ongoing ranging operation as the UWB chip has to switch between TX and RX mode on demand without getting stuck in the LQ calculation. Therefore, we compute the LQ metric based on the few frame quality values saved in UWB chip registers after every frame reception in a manner non-disruptive to the ongoing ranging operation. The LQ value for the link between node $i$ and $j$ is defined as follows:
\begin{equation}
LQ(i, j) = f_1 \times{} f_2 \times{} f_3 \times{} f_4
\label{eq:linkq}
\end{equation}
Here, the 4 factors are defined as follows:
\begin{enumerate}[noitemsep, topsep = 0em]
\item \textit{Inverse of avg. peak-count in the preceeding window ($f_1$)}: This is the inverse of the average number of peaks (defined by a rising value followed by a falling value) in a small (typically 6 samples) window of amplitudes preceding the the first-path (FP). Each peak must be $>$ the Standard Deviation (STD) of the noise (constantly calculated by the chip). As shown in Figure \ref{fig:linkq}, FP is marked by red and the std noise is marked by the dotted horizontal line. Here, the peak count is 1, window size is 6 (3 peaks possible), hence avg. peak count is 1/3 and $f_1=\frac{1}{0.005+1/3}=2.955$. Here, a small constant $0.005$ is used to circumvent divide-by-zero error. The lower the number of peaks, the higher the chance of accurate FP detection, and the higher the value for $f_1$.
\item \textit{FP to STD-noise amplitude ratio ($f_2$)}: The UWB chip updates the standard deviation of noise (STD-noise) during each frame reception. We take $f_2$ to be the ratio between FP amplitudue and STD-noise amplitude. This value is in dB. The higher the ratio i.e. $f_2$, the more prominent the FP is and the higher probability of accurate FP detection.
\item \textit{ FP to peak amplitude ratio ($f_3$)}: This is the ratio between the FP amplitude and the peak or maximum amplitude in the entire sample. In most LOS cases (high quality ranging), the FP is often the peak amplitude. Therefore, the higher the ratio i.e $f_3$, the more accurate is the ranging value.
\item \textit{ Ratio between the FP power and the total received power ($f_4$)}: This ratio is also indicative of the prominance of the FP. In LOS cases, the ratio in dB is significantly higher than the NLOS cases. Hence, higher the value for $f_4$, the the more accurate is the range value.
\end{enumerate}
All the four factors defined above are ratios i.e. unit-less values. Therefore, LQ is unitless and its high value is indicative of less erroneous ranges. The computation is simple and requires only few tens of bytes to be read from the registers of the UWB chip, therefore can be done in the microcontroller attached to the UWB chip in submillisecond.
To validate the LQ metric, we put a pair of UWB transceivers in LOS and NLOS conditions at various distances and collect the true distance, computed range and the corresponding LQ value. We take the error to be the difference between the true distance and the computed range. We then plot the average error and the avg. LQ values against corresponding true distances for diverse scenarios as shown in Figure \ref{fig:linkq-validate}. Here, the LQ value is inversely proportional to the ranging error. Also notable is the gap between NLOS and LOS both for ranging error and metric value. For NLOS case, nodes could not be ranged beyond 20 meters in extreme indoor multipath environment.
\subsubsection{Summary of Edge Selection}
\begin{algorithm}[htb!]
\caption{Edge Selection in Controller}
\label{algo:edge-selection}
\begin{algorithmic}[1]
%
\STATE Initialize rigid graph $G_R = \varnothing$, link matrix $\mathbf{L}$, mobility matrix $\mathbf{U}$
%
\FOR{each node $i\in V$}
\STATE Command node $i$ to transmit beacon frame
\ENDFOR
\WHILE{\textbf{True}}
%
\FOR{each node $i\in V$}
\STATE Collect link quality and velocity from node $i$
\STATE Update matrix $\mathbf{L}$ and $\mathbf{U}$
\ENDFOR
%
\STATE Purge nodes from $G_R$ after rigidity test
\STATE Compute 3-core graph $G_3$
\STATE $V_c \xleftarrow{} \{i | i \in G_3 \mbox{ and } i \not\in G_R\}$
\STATE $V_c \xleftarrow{} V_c \cup \{ {\underset{i}{\arg\max} \; \mathbf{U}} \}$
\REPEAT
\STATE $i \xleftarrow{}$ highest degree node from $V_c$
\STATE $E_i \xleftarrow{}$ edges between $i$ and $G_R$
\IF{Node $i$ and graph $G_R$ is rigid }
\STATE $E^{'}_{i} \xleftarrow{}\{ {\underset{e_1, e_2, e_3 \in E_i}{\arg\max} \; \mathbf{L}(E_i)} \} $
\STATE \textbf{Collect ranges} for edges in $E^{'}_i$
\STATE Add $i$ to $G_R$ by edges in $E^{'}_i$
\STATE $V_c \xleftarrow{} V_c - \{ i \}$
\ENDIF
\UNTIL{no node $i \in V_c$ can be added to $G_R$}
\STATE \textbf{compute relative locations}
\ENDWHILE
\end{algorithmic}
\end{algorithm}
%
We can summarize the steps in edge selection in Algorithm \ref{algo:edge-selection}. Line 1 through 4 are part of the bootstrap step. We start with empty rigid graph $G_R$, initialize the $n\times{}n$ zero matrix as link quality matrix $\mathbf{L}$ and initialize mobility model $\mathbf{U} = [u_1(t) \;\; u_2(t)\;\; \ldots u_n(t)]^T$. Operations from Line 5 through 25 are repeated continuously for relative localization. In each round, link quality and mobility matrices are updated (See Line 5 through 9), then rigid graph is updated based on their updated values (See Line 10 through 23). Finally, relative locations are calculated in Line 24 which is discuss later. Note that, we only measure ranges in Line 19. The purpose of this algorithm is to minimize this measurement yet achieve high accuracy localization. We now present the details of this ranging operation.
%
\subsection{Ranging}
%
\begin{figure}[htb!]
\centering
\includegraphics[width = 0.99\linewidth ]{figs/twr.png}
\vspace{-0.1in}
\caption{Different Ranging Protocols}
\label{fig:twr}
\vspace{-0.1in}
\end{figure}
In this section, we describe the ranging operation and related protocol in detail. We start by briefly describing the two-way ranging protocol and its optimization. We then discuss frame format, its payloads and other system parameters associated with ranging operation.
\\ \noindent \textbf{Two-way Ranging (TWR):} We can calculate the range between two nodes from the TOF of a single frame between them. However, it requires both their clocks to be precisely synchronized which is not practical in our setting. Two-way ranging protocol as shown in Figure \ref{fig:twr}, does not require clock synchronization. The simplest and quickest version of TWR is the Single Sided TWR (SS-TWR) where two frame named ``poll" and ``response" are exchanged to calculate TOF. However, there are a number of sources of error due to clock drift and frequency drift \cite{twr}. To minimize the error, one extra frame named ``final" is exchanged and it is called Double Sided TWR (DS-TWR) as shown in Figure \ref{fig:twr}. Calculating the intervals $D1$, $R1$, $D2$ and $R2$ from the tx/rx timestamps of these three frames, we can calculate the TOF as $\frac{D1\times{}D2-R1\times{}R2}{2\times{}(D1+R1)} = \frac{D1\times{}D2-R1\times{}R2}{2\times{}(D2+R2)}$. Both values should be close and an average of them is taken as TOF. When node 1 ranges with more than 2 nodes, it can run DS-TWR sequentially with them (SEQ-DS-TWR in Figure \ref{fig:twr}). We introduced an extra frame called ``Init" (for initialization) that triggers the ranging. The Init frame also carries additional information about the parameters of ranging that we describe later. Also, due to Init frame, the pairwise ranging operation finishes with node 1 instead of node 2 or 3 and the initiator node 1 is able to calculate the range. Note that, the tx timestamp of the poll and final (pre-calculated \footnote{Once the frame is scheduled for tx, the tx-timestamp can be calculated from the antenna delay even before the acutal tx}) frames and the rx timestamp of the response frame embedded in the Final frame as data so that the initiator node can calculate the TOF without any further frame exchanges.
Altogether, this operation takes 4 TDMA slots per ranging pairs. However, we can save slots by making the common node 1 to transmit all frames (Init and Response frames) only once. This way, two slots are saved per nodes. For example if a node ranges with $n$ nodes, instead of $4n$ slots, it will require $2+2n$ slots (6 slots instead of 8 in the example in Figure \ref{fig:twr}). We term this protocol Optimized Double Sided TWR (OPT-DS-TWR). Here, the initiator node will embed the slot time, response order of the nodes etc in the Init and Response frames, and the participating nodes follow the order in sending the Poll and Final frames.
%
\begin{figure}[htb!]
\centering
\includegraphics[width = 0.99\linewidth ]{figs/forwarding.png}
\vspace{-0.1in}
\caption{Alternate Data Forwarding Methods}
\label{fig:forwarding}
\vspace{-0.1in}
\end{figure}
\\ \noindent \textbf{Data Forwarding:} Collecting ranges, mobility and link quality information etc. from the nodes also adds up to the localization delay. We have two choices- in-band and out-of-band data forwarding as shown in Figure \ref{fig:forwarding}. In in-band strategy,
we use the UWB links to transfer the data via a forwarding tree rooted at the controller node as shown in Figure \ref{fig:forwarding}(a). However, due to the limited ranging of UWB, some of the nodes may be disconnected form the controller (no path between Node 1 and 3 etc). Also, this does not scale up well with increasing \# of nodes and increasing node mobility due to the constant overhead of maintaining the forwarding tree. Note that, there may be rigid subgraphs which otherwise be relatively localized yes does not have a path to the controller node. The other strategy is out-of-band modality such as WiFi, LoRa etc. that fully covers the area of interest. This way, each node can directly forward data frame to the controller node, as shown in Figur \ref{fig:forwarding}(b).
In our system, we used WiFi-long-range (range of 100 meter indoor) since WiFi is ubiquitous, inexpensive, has higher data-rate and can itself be used for ranging in case UWB links fail for sustained period of time, although WiFi ranging is less accurate than UWB (discussed in Section \ref{sec:fail-safe}).
%
%
\begin{figure}[htb!]
\centering
\includegraphics[width = 0.9\linewidth ]{figs/frame.png}
\vspace{-0.1in}
\caption{UWB Frame Format}
\label{fig:frame}
\vspace{-0.1in}
\end{figure}
\\ \noindent \textbf{Data Frame:} The controller directs a particular node $i$ for ranging with a list of nodes via WiFi. This WiFi packet also contains UWB slot-time which is calculated precisely using UWB data-rate and maximum UWB frame-size. By not using a fixed-slot time for all cases, our system can dynamically adapt to the existing \# of nodes. After reception of the WiFi frame, the node initiates the OPT-DS-TWR protocol immediately with the list of nodes. In particular, the Init frame contains the list of nodes plus the slot time. After ranging is done, Node $i$ sends the ranging data along with i) velocity and heading data (discussed in Section \ref{sec:abs}) from IMU and ii) current link quality values of existing links. Moreover, all participating nodes in the ranging, also piggybacks their velocity, heading and link quality values over the UWB frame i.e via Poll and Final frames. We re-purposed IEEE 802.15.4-2011-UWB frame for Init, Poll, Response and Final frames which carry the information, as shown in Figre \ref{fig:frame}. This way, the controller can get mobility and link quality data as part of continuous ranging, which eliminates Step 6 through 9 in Algorithm \ref{algo:edge-selection}. Since, 2-way WiFi communication takes at least few milliseconds, for tens of nodes, the saving is hundreds of milliseconds in the main loop of the algorithm. In the frame format (See Figure \ref{fig:frame}, the Frame Control (FC) byte distinguishes our custom IEEE-802.15.4 frame. It is followed by sequence-\#, Personal Area Network (PAN ID), source and destination ID\footnote{1-byte source and destination ID supports 256 nodes, which we assume, is sufficient for all indoor deployments}, Function Code (FCode) (indicates frame type- i.e whether Init, Poll etc.), variable-byte payload and and 2-byte Frame Control Sequence (FCS). Mobility, link quality and timestamp information (40 bit each) for ranging are embedded onto the payload. Due to limited data-rate of UWB link, this simplified and reduced frame-format minimizes required slot time and thereby, the ranging time.
%
\subsection{Missing Range Estimation \& Relative Localization}
%
%
%
%
Once the rigid graph has been formed for $n$ nodes, we derive $n\times n$ the Euclidean Distance Matrix (EDM) $\mathbf{D}$ where each entry in row $i$ and column $j$ is the distance between node $i$ and $j$. We then try to solve it for relative locations using a widely known technique known as Classical Multi-Dimensional Sampling (CMDS) that we now briefly introduce. We also justify our choice in the following.
\\ \noindent \textbf{Classical Multi-Dimensional Sampling (CMDS) Problem: }
Given a complete $n\times{}n$ matrix $\mathbf{D^2}$ of squared distances $d^2_{ij}$'s between node $i$ and $j$, CMDS problem is to find the $n\times{}d$ coordinate Matrix $\mathbf{X}$ that gives the location $\mathbf{x}_{i}$ of each node $i$ in $d$-dimensional Euclidean Space. In matrix notation, this amounts to solving for $\mathbf{B}=\mathbf{XX^T}$ where $\mathbf{X^T}$ is the transpose of matrix $\mathbf{X}$. As shown in \cite{cdms-paper}, $\mathbf{X}$ can be calculated as follows:
\begin{equation}
\mathbf{D^2}=\mathbf{C\,1^T}+\mathbf{1\,C^T}-2\mathbf{XX^T}
\label{eq-dsquare}
\end{equation}
Here, $\mathbf{C}$ is the $n\times{}1$ matrix with only the diagonal elements of $\mathbf{XX^T}$ and $\mathbf{1}$ is the $n\times{}1$ matrix of ones. By multiplying both sides of Equation \ref{eq-dsquare} by the centering matrix $\mathbf{J}=\mathbf{I}-\frac{1}{n}\mathbf{1\,1^T}$ and some algebraic manipulation, we have
\begin{equation}
\mathbf{X} = \mathbf{Q}\sqrt{\Lambda}\mbox{ where } \mathbf{B}=\mathbf{Q}\,\Lambda\,\mathbf{Q^T}=-\frac{1}{2}\mathbf{JD^2J}
\label{eq:cmds}
\end{equation}
Therefore, CMDS algorithm involves the following 3 steps-
\begin{enumerate}[noitemsep, topsep = 0em]
\item Given square-distance matrix $\mathbf{D^2}$, compute Gram matrix $\mathbf{B}$ of $\mathbf{X}$ as $\mathbf{B}=-\frac{1}{2}\mathbf{JD^2J}$,
\item Find the Eigenvalue decomposition of $\mathbf{B}=\mathbf{Q}\,\Lambda\,\mathbf{Q^T}$,
\item Finally, compute $\mathbf{X}=\mathbf{Q}\sqrt{\Lambda} $ for Eigenvector matrix $\mathbf{Q}$ and diagonal Eigenvalue matrix $\Lambda$.
\end{enumerate}
\begin{figure}[htb!]
\centering
\includegraphics[width = 0.99\linewidth ]{figs/cmds-adv.png}
\vspace{-0.1in}
\caption{Advantage of CMDS over sequential multilateration}
\label{fig:cmds-adv}
\vspace{-0.1in}
\end{figure}
Due to the ``centering" assumption, each column of solution $\mathbf{X}$ sums to zero i.e, the origin of $\mathbf{X}$ coincides with the centroid of the $n$ locations.
Note that CMDS minimizes the loss function (also called \textit{strain}) $L( \mathbf{X} ) = ||\mathbf{XX^T} - \mathbf{B}||=||\mathbf{XX^T} - \frac{1}{2}\mathbf{JD^2J}||$. As such, the CMDS can end up in local minima. These local minima is more likely to occur when dimension is low (2 as in our case) \cite{cmds-book}. Therefore, cross-validate the locations using heading data of nodes discussed in Section \cite{ref:heading-validate} and also run a smoothing filter on the successive node locations over a small window. CMDS has distinct advantage over using sequential multilaterion as shown in Figure \ref{fig:cmds-adv}. Here, 5 nodes form a complete graph as shown in Figure \ref{fig:cmds-adv}(a). We can use multilateration to fix node 4 using node 1, 2 and 3, then fix node 5 using 1,3 and 4. However, nodes mulitlaterated later in the sequence will incur prohibitively high localization error (shown in shaded circle). But CMDS solve the localization problem by taking into consideration all the ranges (complete EDM) and jointly minimizing the loss function. In this way, no node ordering is required and localization error is not lumped into handful of nodes.
\begin{algorithm}[htb!]
\caption{EDM Completion Algorithm}
\label{algo:edm-completion}
\begin{algorithmic}[1]
%
\STATEx \textbf{Input:} Weighted Graph $G = (V, E, d)$ where $V$ is the set of vertices, $E$ are edges with measured ranges $d$.
\STATEx \textbf{Output:} Complete EDM $\mathbf{D}$
%
\STATE Initialize relative location matrix $\mathbf{X}$ using sequential multilateration
%
\STATE Iteration count c $ \leftarrow 1$
\WHILE{c$<$ Max. Iteration}
\STATE $E_c \leftarrow \{(i, j)|i < j \mbox{ and }i,j \in V\}$
\WHILE{$E_c \neq \emptyset$}
\FOR{ $\forall(i, j)\in E_c$}
\STATE $\Delta \mathbf{x}_{ij} \leftarrow \displaystyle{\frac{ | ||\mathbf{x}_i -\mathbf{x}_j|| - d_{ij} | }{||\mathbf{x}_i -\mathbf{x}_j||} } \; (\mathbf{x}_i -\mathbf{x}_j)$
\ENDFOR
\STATE $(i, j) \leftarrow \displaystyle{ \underset{(i,j) \in E_c}{\arg\max} \; \Delta \mathbf{x}_{ij}}$
\STATE $\mathbf{x}_i \leftarrow \mathbf{x}_i + \lambda \Delta \mathbf{x}_{ij}/2$, $\mathbf{x}_j \leftarrow \mathbf{x}_j - \lambda \Delta \mathbf{x}_{ij}/2$
\STATE $E_c \leftarrow E_c - \{(i,j )\}$
\ENDWHILE
\STATE Update each entry $(i,j)$ of $\mathbf{D}$ as $ d_{ij} = ||\mathbf{x}_i -\mathbf{x}_j||$
\STATE $\displaystyle {\lambda \leftarrow \frac{1}{1+c\Delta\lambda} }$
\ENDWHILE
%
\end{algorithmic}
\end{algorithm}
%
\\ \noindent \textbf{EDM Completion: } In the rigid graph formation, some node may have only 3 edges. Therefore, the rigid graph is not necessarily a clique and a lot of edges remain to be measured. But CMDS requires a clique i.e. a complete EDM, so we need to fill in the missing values. There are few ways to do it. We adopt EDM completion algorithm based on proximity adjustment \cite{confo-sampling} as shown in Algorithm \ref{algo:edm-completion}.
Here, we take an arbitrary triangle on nodes $<i, j, k>$ from the rigid graph, arbitrarily consider i) location of node $i$ as the origin, ii) edge $(j, k)$ aligned along the positive x-axis, iii) location of node $k$ in the positive y-quadrant. Then we resolve relative locations for the rest of the nodes using sequential multi-lateration.
Each node will get a relative location due to the already satisfied rigidity condition. Now, we want to improve EDM via successive updates based on the difference between the distance $||\mathbf{x}_i - \mathbf{x_j}||$, derived using the current locations for node $i$ and $j$ and the distance $d_{ij}$ found in the current EDM $\mathbf{D}$. This change vector $\Delta \mathbf{x}_{ij}$ is applied to locations of both node $i$ and $j$ as in Line 10 in Algorithm \ref{algo:edm-completion}. Note that $\lambda$ is the update-rate which decreases with iterations. We repeat the iterations by considering each edge once and updating $\mathbf{X}$ and then $\mathbf{D}$ and terminate the algorithm when improvement in successive iterations is negligible ($>$0.001\%) or when a max. iteration (computed from remaining time for localization) is reached. Once the full EDM is computed from the rigid graph, the CMDS algorithm is applied to derive the relative location matrix $\mathbf{X}$ according to Equation \ref{eq:cmds}.
\\ \noindent \textbf{Relative Locations for Excluded Nodes: } So far, we computed the relative locations for nodes which were part of the rigid graph. Now, we discuss technique to relatively localize nodes not part of the rigid graph. We categorize nodes in two types- nodes having i) two edges and ii) only one edge with nodes of the current rigid graph. For 2-edge case, we have two candidate relative locations by solving the two distance equations, See node 5 in Figure \ref{fig:rigid}(c). Out of the two, we choose the relative location using the link information of the candidate node and the proximity constraint. For example, in Figure \ref{fig:rigid}(c), node 5 detects links with node 2 and 3, but not with node 1 or 4. Therefore, we choose that location for which both distances $d_{1,5}$ and $d_{4,5}$ are of greater value. However, the 1-edge case is not geometrically solvable. However, we still use the single range the heading information derived from the IMU to determine the relative location of the node as discussed in the next section.
%
\begin{comment}
\subsection{Absolute Localization}
Given the relative location $\mathbf{x}$ of a node, we derive the absolute location by the transformation $\mathbf{x}_{abs} = \mathbf{r}\mathbf{x}+\mathbf{t}$, where $\mathbf{r}$ and $\mathbf{t}$ are the $2\times{}2$ rotation matrix and $2\times{1}$ translation matrix respectively indicating the required rotation and translation of the coordinate of the relative locations. First, we discuss how to find the vectors using one fixed node. Then we discuss how to embed the rigid graph on a floormap.
\\ \noindent \textbf{Using One Fixed Node: } We assume that a fixed node works as the origin of the global coordinate system. Such a node can be placed at the entrance or any other known landmark of the indoor area. If we take this particular node as the origin during rigid graph formation, then $\mathbf{t} =\begin{bmatrix}
0 \\
0
\end{bmatrix}$.
To determine the rotation angle $\theta$ for the relative coordinate,
we also assume that the earth's magnetic north-south axis aligns with the global y-axis. Under this scenario, we measure the headings (angle with earth's north-south axis) of any 2 mobile\footnote{We assume that when the nodes are mobile, the angle between the direction of motion and the earth's north-south axis is constant for the time being. } nodes $i$ and $j$ as $\theta_i$ and $\theta_j$ and set $\theta =(\theta_j - \theta_i$) and $\mathbf{r} = \begin{bmatrix}
\cos\theta & -\sin\theta \\
\sin\theta & \cos\theta
\end{bmatrix}$. Since the headings are noisy, we generally maintain a small window of past heading values for each node, take smoothed values to calculate $\theta$ and finally take average of all the $\theta$ values calculated for all pairs of mobile nodes.
\\ \noindent \textbf{Embedding on Floormap: }
\end{comment}
\clearpage
\subsection{Relative Localization}
In this section, we discuss the method to determine the relative geometry formed by the nodes. The first step is to range between pair of nodes to determine inter-node distances. As discussed in Section \ref{sec:time-constraint}, the naive approach i.e. making measurements for all possible pairs, does not take into consideration the dynamic topology of the connectivity graph, and takes prohibitively long time for localization. Therefore, we propose ``edge selection" algorithm based on factors such as channel characteristics, node mobility and geometric requirements. We discuss a novel protocol that enables this dynamic edge selection. Finally, we discuss ways to estimate the missing ranges in the above steps to unambiguously form the relative geometry.
\vspace{0.05in}
\\
\vspace{0.05in}
\begin{figure}
\centering
\includegraphics[width = 0.75\linewidth ]{figs/k-core-example.png}
\vspace{-0.1in}
\caption{K-core Decomposition Examples}
\label{fig:k-core-examples}
\vspace{-0.1in}
\end{figure}
\vspace{0.05in}
\\
\noindent \textit{1. k-core Decomposition}: A k-core is a maximal graph in which every node is connected to at least k other nodes. In our case, k-core indicates whether a subgraph can form a rigid graph or not. In particular, if k$\geq$3, the subgraph has potential to be a rigid graph in 2D based only on the range measurements, otherwise not \cite{2d-geom}. As shown in Figure \ref{fig:k-core-examples}, the connectivity graph $G(0)$ has 7 nodes, however, only nodes 1, 2, 3 and 4 can form a rigid graph (described later) required for relative localization. Therefore, including ranges such (5, 6), (2, 5) etc. will not contribute to the relative geometry. Partitioning a graph into k-cores for $k=1,2,3,\ldots$ is called ``k-core decomposition". It works as follows. First the nodes with the lowest degrees are removed one by one and are included in the k-core, where k is the degree during its removal. Such removal affects other nodes and the operation is repeated until no nodes remain. As shown in Figure \ref{fig:k-core-examples}(a), nodes 6 and 7 have the minimum degree i.e 1 and both belong to 1-core. After removing them, we derive the graph as in Figure \ref{fig:k-core-examples}(b), where there is no 1-degree node. We then remove all 2-degree nodes and then all 3-degree nodes to derive 2-core and 3-core subgraphs (c) respectively.
\vspace{0.05in}
\\
\noindent \textit{2. Rigid Graph Formation}: After formation of 3-core subgraph $G_3(0)$ out of initial connectivity graph $G(0)$, the next operation is to form a rigid subgraph $G_R(0)$ from it. However, to determine rigidity, the edge-lengths or the ranges are required. Therefore, in this step, the controller collects ranges between all pairs of nodes ($^mC_2$ for $m$ nodes) in the 3-core graph. Rigid graph is formally defined in the following.
\begin{comment}
A \textit{realization} is a function that maps a given set of vertices to 2D Euclidean space. Given a simple, connected, weighted graph $G=(V, E, d)$, where $V$ is the set of vertices, $E$ is the set of edges and $d(u, v)$ is the Euclidean distance or range between nodes $u, v \in V$, the DGP problem is to find whether there is a realization $x:V \rightarrow{} \mathbb{R}^2$ such that each range is preserved i.e $\forall (u,v) \in E, \; ||x(u) - x(v)|| = d(u, v)$ where $||.||$ is the Euclidean norm.
\end{comment}
\begin{comment}
\\ \noindent
\uline{Rigidity Graph:}
Given a connected and weighted graph $G=(V, E, d)$, where $d$ is the set of ranges of the edges, a \textit{realization} is a function $x:V \rightarrow{} \mathbb{R}^2$ that maps the set of vertices $V$ to 2D Euclidean space such that each range is preserved i.e $\forall (u,v) \in E, \; ||x(u) - x(v)|| = d(u, v)$ where $||.||$ is the Euclidean norm.
The graph $G=(V, E, d)$ is ``rigid" if there is only one realization discounting translation, rotation and flip. As shown in the Figure \ref{fig:k-core-examples}(c), subgraph $G_3(0)$ in (c) is rigid, but $G_2(0)$ in (b) is not, because given the ranges for edge (2, 5) and (4, 5), node 2 can be placed in two locations relative to edge (2, 4). These two realizations are not rotation, translation or flip of each other. Similarly, 1-core subgraph i.e. the connectivity graph $G(0)$ is not rigid. We can define the following recursive definition of a rigid graph in 2D:
\begin{enumerate}[noitemsep, topsep = 0em]
%
\item A complete graph or clique is a rigid graph.
%
\item If $G=(V,E)$ is a rigid graph and a node $i$ has a set of edges $E_i$ to at least three non-collinear nodes in $G$, then the graph derived after adding node $i$ to the graph along with the connecting edges, is also rigid, i.e. $G^{'}=(V^{'},E^{'})$ is rigid, where $V^{'}=V\cup \{i\}$ and $E^{'}=E \cup E_i$.
\end{enumerate}
Condition 2 above requires each node to have degree 3 i.e. the graph has to be 3-core. But this is not a sufficient condition, we also need to test collinearity of the 3 nodes connecting each node in the subgraph. As such, rigid graph is a subgraph of 3-core graph. The 3-core in Figure \ref{fig:k-core-examples}(c) satisfies this condition, therefore it is rigid. If we add edge (5,1) and or (5,3) to the connectivity graph, then node 5 also becomes part of the rigid graph by the above conditions. For a given 3-core, we can start with an arbitrary edge or triangle (a clique) and try to add nodes following the above conditions to form a rigid graph $G_R(0)$. Overall, after collecting all possible ranges in the bootstrap step, we can summarize the derivation as $G(0) \Rightarrow{} G_3(0) \Rightarrow{} G_R(0)$. Nodes not part of the rigid graph $G_R(0)$ requires additional information for localization discussed later. Once, remaining nodes are added to $G_R(0)$ using additional information such as heading from IMU etc, we derive augmented rigid graph $G_R^{'}(0)$ and from it, the absolute locations of each node as described in Section \ref{sec:abs-loc}.
\uterm{Incremental Update} After bootstrapping, all computations are incremental to the initial rigid graph $G_R(0)$ derived in the previous step. In particular, we derive the rigid graph $G_R(t+1)$ from $G_R(t)$ for iteration $t=1,2,\ldots$etc. There are two steps in the incremental computation as described below.
\\ \noindent \textit{1. K-core Update}: Due to the broadcast nature of wireless communication, each node overhears transmissions within its communication range and determines the source node ID from the received frame. These overheard nodes are the neighbors in the connectivity graph. Each node maintains a list of overheard neighbors along with the timestamps. The nodes that have not been overheard for long time are purged from the list.
At each iteration $t$, the neighbor list of each nodes are collected to form the connectivity graph $G(t)$. Then k-core decomposition is computed on it to derive 3-core graph $G_3(t)$. Note that, this step does not require any ranging.
\\ \noindent \textit{2. Rigid Graph Update}: This step requires ranging to update the rigid graph. To miin
\end{comment}
\subsection{DynoLoc in a Nutshell}
\textsc{DynoLoc}\xspace operates in epochs (rounds), where the locations of all nodes are estimated at the end of each epoch, the duration of which is determined by the application's refresh rate (e.g., 0.5-2 Hz).
In every epoch, the following sequence of operations is executed.
\mypara{(i) Topology estimation for ranging (Sec \ref{subseq:ranging-tech})}
Given the underlining physical topology (based on connectivity), \textsc{DynoLoc}\xspace
prioritizes edges which contribute to the resulting topology being {\em maximally rigid}, while avoiding those, whose ranges could be corrupted by multi-path; and it is {\em adaptive} in that it prioritizes edges associated with nodes that have been mobile in the recent past thereby leading to a good localization accuracy.
\mypara{(ii) Concurrent ranging (Sec \ref{subseq:concur-ranging})} The selected edges are then ranged using a concurrent ranging protocol that amortizes the overhead of ranging from a node across its neighbors, while enabling concurrent ranging in non-interfering neighborhoods, to minimize the overall latency.
\mypara{(iii) Robust relative localization (Sec \ref{subseq:relative-loc})} After the estimated topology has been ranged, \textsc{DynoLoc}\xspace's localization algorithm intelligently
identifies and applies EDM only on sub-graphs of the topology that are rigid and combine them effectively, to deliver both a robust and accurate relative localization.
\mypara{(iv) Absolute localization (Sec \ref{subseq:abs-loc})} \textsc{DynoLoc}\xspace finally transforms the relative localization solution into an absolute one with little additional meta information (contributed by floor plans or a single reference node), while still delivering the desired refresh rate for the solution.
\begin{comment}
\begin{figure}
\centering
\includegraphics[width = 1.01\linewidth, height=95pt ]{figs/section-3-plots/flowchart}
\label{fig:sys-overview}
\vspace{-0.2in}
\caption{DynoLoc System Overview}
\label{fig:sec-3}
\vspace{-0.2in}
\end{figure}
\end{comment}
\subsection{Estimation of Ranging Topology}
\label{subseq:ranging-tech}
\textsc{DynoLoc}\xspace's innovation lies in leveraging the graph theoretic construct of geometric rigidity to help identify the set of edges that would collectively contribute to the accurate localization of the topology as a whole while also adapting itself to track the topology as it evolves with node mobility and channel conditions. \textsc{DynoLoc}\xspace accomplishes this by first characterizing the links in the physical connectivity topology, followed by leveraging such a characterization for adaptive link selection.
\mypara{A. \underline{Characterizing the Connectivity Topology:}}
Every link is characterized based on the mobility of its nodes, the multipath nature of its wireless channel, as well as its contribution to the topology's rigidity.
\begin{figure}[!htb]
\centering
\includegraphics[width = 0.65\linewidth ]{figs/section-4-plots/linkq}
\vspace{-0.1in}
\caption{First Path Detection in UWB receiver}
\label{fig:l-metric}
\vspace{-0.15in}
\end{figure}
\mypara{LOS vs. NLOS:} Every node maintains a list of its neighbors, identified by overhearing their transmissions, whose channel impulse response (CIR) is also collected. A node $i$ can thus directly range with any of its neighbors $j$, whose link quality ($L_{ij}$) is estimated from its corresponding CIR. $L_{ij}$ captures the potential accuracy of ranging on the link based on the certainty of it being a LOS (direct) or NLOS (indirect) path. %
This NLOS probability is computed by as $p_{NLOS} =(f_1\times{}f_2\times{}f_3\times{}f_4)$, where $f_1 =$ avg. peak count before the detected first path (FP) in the preceding window (See Figure \ref{fig:l-metric}), $f_2=$ ratio of std-noise to FP amplitude, $f_3=$ ratio of peak to FP amplitude and $f_4=$ ratio of total received power to FP power. The link quality is the inverse of this probability.
Initially, during the bootstrapping phase, every node sequentially broadcasts a {\em beacon} packet that is heard by its neighbors. Once the system reaches a steady state, the neighborhood list is implicitly maintained by all nodes without the need for additional ranging. This helps realize a physical connectivity graph across the nodes, where every edge is a potential candidate for range estimation, and is weighted by its quality (i.e. certainty for delivering accurate ranges).
\mypara{Mobility:} In addition, every node $i$ also maintains a mobility metric $M_i$ that capture its location uncertainty since its last localization. This metric increases as a function of the time-since-localization (TsL) and is computed using the node's acceleration, $a_i$ (\sfrtxt{\uline{obtained from its IMU}}). In particular, for every IMU read (indexed by $k$), $M_i(k) \leftarrow M_i(k-1)+v_i(k)\cdot \Delta t$, where node velocity $v_i(k) \leftarrow v_{i}(k-1) + a_i\cdot \Delta t$, and $\Delta t$ is the elapsed time since the last IMU read.
$M_{i}$ and $v_i$ are reset to zero, whenever the node is localized. When $a_i$ is zero (static nodes), we assign an exponential function to $M$ as follows: $M_i \leftarrow (e^{TsL} -1 )$. This allows the node to be prioritized for ranging, even if it is static, but sufficient time has elapsed since its last localization.
\mypara{Geometric Rigidity:} Recall that a rigid graph admits a unique relative localization solution. Since the connectivity topology of nodes might not be a rigid graph in practice, \textsc{DynoLoc}\xspace aims to select edges from this underlying connectivity that ensures {\em maximal} rigidity to the resulting node topology.
It does so by identifying maximal rigid sub-graphs from the physical connectivity graph, by leveraging the construct of $k$-core sub-graphs that are used to ensure graph rigidity~\cite{eren2004rigidity}. In a $k$-core sub-graph, every vertex has a degree of at least $k$. It is known that a $k$-core sub-graph is rigid in $k-1$ dimensional space~\cite{dokmanic2015euclidean}. Hence, for rigidity in 2D, we seek to obtain 3-core sub-graphs\footnote{This allows \textsc{DynoLoc}\xspace to also be extended for 3D localization, where 4-core sub-graphs will be leveraged instead.}. Note that a 2-core sub-graph will not be rigid in 2D and will admit multiple localization solutions.
\textsc{DynoLoc}\xspace identifies the maximal 3-core sub-graphs by starting with the connectivity graph and partitioning it into $k$-core subgraphs for k = 1,2 and 3 sequentially. It starts with identifying 1-core nodes (one by one) that have a degree of 1 and removes them and their incident edges iteratively till no more 1-core nodes can be found. Then, it repeats the process for 2-core nodes with degree 2. After the removal of 1-core and 2-core nodes, we are left with maximal 3-core sub-graphs (as shown in the example in Fig.~\ref{fig:k-core-example}) that are rigid.
\begin{figure}[htb!]
\centering
\includegraphics[width = 1.01\linewidth ]{figs/section-3-plots/kcore}
\vspace{-0.15in}
\caption{Example of k-core Decomposition}
\label{fig:k-core-example}
\vspace{-0.15in}
\end{figure}
\mypara{B. \underline{Estimating the Ranging Topology:}}
We now describe
\textsc{DynoLoc}\xspace's algorithm for edge and hence topology selection that will be used for ranging.
At a high level, \textsc{DynoLoc}\xspace aims to devote its ranging resources to links, whose ranges are outdated (due to mobility), followed by those that contribute the most to the topology's rigidity, while also avoiding those with potentially inaccurate ranges (due to NLOS).
Specifically, at every iteration, \textsc{DynoLoc}\xspace picks the node (say $i$) with the highest mobility $M$ (location uncertainty) metric. If $i$ is part of the 3-core, and has more than three edges, then three of its edges with the highest $L$ metric (range accuracy) are selected. Otherwise, its incident edges ($\le$ 3) are directly selected. When multiple nodes have the same $M$ metric, the node selection is done based on the $k$-core metric, with nodes belonging to a higher core (ties broken with higher node degree) prioritized over those belonging to a lower core. The process repeats until the ranging budget is exhausted by the edges selected for ranging. Initially, when the system is bootstrapped and no information on node mobility is available, all nodes are assumed to have outdated $M$ metric and edge selection is done primarily based on their contribution to rigidity and their LOS nature.
The complete DynoLoc method is presented in Algorithm \ref{algo:dynoloc}.
\begin{algorithm}[htb!]
%
\caption{DynoLoc Algorithm}
\label{algo:dynoloc}
\begin{algorithmic}[1]
\scriptsize
\STATE Make every node send hello frame
\STATE Collect and initialize link quality metric $L$
\STATE Run core-decomposition based on $L$
\STATE Build initial rigid graph $G_R$
\STATE Initialize mobilty metric $M$
\WHILE{\textbf{True}}
\STATE Choose node $i$ with the highest metric $M(i)$
\STATE Remove $i$ (with its edges) from $G_R$ or $V^{'}$
\IF{$i$ has $\geq$ 3 edges with $G_R$}
\STATE Select 3 edges for $i$ with max. $L$
\STATE Add $i$ back to $G_R$ with selected edges
\ELSE
\STATE Put $i$ in the excluded node list $V^{'}$
\STATE Select $i$'s edge(s)
\ENDIF
\STATE Range for the above-selected edges [\textbf{\S 3.3}]
\STATE Set $M(i) \leftarrow 0$
\STATE Update $M$ \& $L$ metrics from collected data
\STATE Compute core-decomposition
\STATE Remove nodes which are not in 3-core, from $G_R$
\IF{Refresh interval elapsed}
\STATE Complete EDM for $G_R$ [\textbf{\S 3.4.A}]
\STATE Determine relative locations [\textbf{\S 3.4.B}]
\STATE Determine \& output absolute locations [\textbf{\S 3.5}]
\ENDIF
\ENDWHILE
\end{algorithmic}
%
\end{algorithm}
\vspace{-15pt}
\subsection{Aggregated and Concurrent Ranging}
\label{subseq:concur-ranging}
\input{concurrentranging}
\subsection{Robust Relative Localization}
\label{subseq:relative-loc}
\input{robustrelative}
\vspace{-1em}
\subsection{Absolute Localization}
\label{subseq:abs-loc}
\input{abslocalization}
\section{Introduction}
\input{introduction.tex}
\section{Motivation \& Challenges}
\input{background.tex}
\section{{\bf \Large \textsc{DynoLoc}\xspace}: Design}
\input{method.tex}
\section{Implementation and Evaluation}
\input{results.tex}
\section{Applications}
\input{applications.tex}
\section{Conclusion}
\input{conclusion.tex}
\balance
\bibliographystyle{abbrv}
\section{Relative Localization}
In Rigid body formation, given the set of range $R$, we form a rigid body $\mathds{B}_R$ of $n$ nodes such that the length of each edge incident with a pair of nodes equals the corresponding pairwise range in $\mathbf{D}$. To this end, we use Classical Multi-Dimensional Sampling (CMDS) \cite{cmds-paper}. This is a widely used algorithm to map $n$ nodes into Cartesian space given the pairwise distances among them. We briefly discuss the CMDS problem and algorithm here.
\\ \noindent \textbf{Multi-Dimensional Sampling (CMDS) Problem: }
Given a complete $n\times{}n$ matrix $\mathbf{D^2}$ of squared distances $d^2_{ij}$'s between node $i$ and $j$, CMDS problem is to find the $n\times{}d$ coordinate Matrix $\mathbf{X}$ that gives the location $\mathbf{x}_{i}$ of each node $i$ in $d$-dimensional Euclidean Space.
In matrix notation, this amounts to solving for $\mathbf{B}=\mathbf{XX^T}$ where $\mathbf{X^T}$ is the transpose of matrix $\mathbf{X}$. As shown in \cite{cdms-paper}, $\mathbf{X}$ can be calculated as follows:
\begin{equation}
\mathbf{D^2}=\mathbf{C\,1^T}+\mathbf{1\,C^T}-2\mathbf{XX^T}
\label{eq-dsquare}
\end{equation}
Here, $\mathbf{C}$ is the $n\times{}1$ matrix with only the diagonal elements of $\mathbf{XX^T}$ and $\mathbf{1}$ is the $n\times{}1$ matrix of ones. By multiplying both sides of Equation \ref{eq-dsquare} by the centering matrix $\mathbf{J}=\mathbf{I}-\frac{1}{n}\mathbf{1\,1^T}$, we have
\begin{equation}
\begin{split}
& -\frac{1}{2}\mathbf{JD^2J} = -\frac{1}{2}\mathbf{J}( \mathbf{C\,1^T}+\mathbf{1\,C^T}-2\mathbf{XX^T} )\mathbf{J}
\\
& = -\frac{1}{2}\mathbf{J\,C\,1^T\,J}-\frac{1}{2}\mathbf{J\,1\,C^T\,J}+\frac{1}{2}\mathbf{J}(2\mathbf{B})\mathbf{J}
\\
& \mbox{ [centering $\mathbf{1}$ by $\mathbf{J}$ yields zero matrix i.e, $\mathbf{1^T\,J}=\mathbf{J\,1}=\mathbf{0}$ ]}
\\
& = -\frac{1}{2}\mathbf{J\,C\, 0}-\frac{1}{2}\mathbf{0\,C^T\,J}+\mathbf{J\,B\,J}
\\
& = \mathbf{0}+\mathbf{0}+\mathbf{B} = \mathbf{B}
\\
\\
& \implies \mathbf{XX^T} = \mathbf{B} = -\frac{1}{2}\mathbf{JD^2J} = \mathbf{Q}\,\Lambda\,\mathbf{Q^T}
\\
& \implies \mathbf{X} = \mathbf{Q}\sqrt{\Lambda}
\end{split}
\end{equation}
Therefore, CMDS algorithm involves the following 3 steps-
\\ \noindent
I) Given square-distance matrix $\mathbf{D^2}$, compute Gram matrix $\mathbf{B}$ of $\mathbf{X}$ as $\mathbf{B}=-\frac{1}{2}\mathbf{JD^2J}$,
\\ \noindent
II) Find the Eigenvalue decomposition of $\mathbf{B}=\mathbf{Q}\,\Lambda\,\mathbf{Q^T}$,
\\ \noindent
III) Finally, compute $\mathbf{X}=\mathbf{Q}\sqrt{\Lambda} $ for Eigenvector matrix $\mathbf{Q}$ and diagonal Eigenvalue matrix $\Lambda$.
\\
Due to the ``centering" assumption, each column of solution $\mathbf{X}$ sums to zero i.e, the origin of $\mathbf{X}$ coincides with the centroid of the $n$ locations. To derive the absolute location, we multiply $\mathbf{X}$ with the transformation matrix $\mathbf{M}$ to drive the absolute location. Section \ref{absolute-loc} describes it in detail.
Note that CMDS minimizes the loss function (also called \textit{strain}) $L( \mathbf{X} ) = ||\mathbf{XX^T} - \mathbf{B}||=||\mathbf{XX^T} - \frac{1}{2}\mathbf{JD^2J}||$. As such, the CMDS can end up in local minima. These local minima is more likely to occur when dimension is low (2 or 3 as in our case) \cite{cmds-book}.
\subsubsection{Rigid-body Formation Relative Localization}
We represent the network of nodes by a undirected graph $G_c$(V, $E_c$) where V represents the set of nodes and $E_c$ represents the set of edges or connectivity links between individual nodes. Connectivity links indicate reachability between two nodes and are determined by the nodes overhearing other nodes when the latter transmit. Note that $E_c$ is dynamic and depends on the relative location of the individual nodes. Let $G_m$ denote the measurement graph, given by $G_m$(V, $E_m$), where $E_m \subseteq E_c$. For each edge, $e_i \in E_m$, we estimate a \textit{measurement-tuple}, $M_{e_i}$ = $<R_{e_i}, t_{e_i}>$. $R_{e_i}$ is the range estimate or the inter-node distance obtained at time instant $t_{e_i}$. At any time instant $t$, we construct a spatial embedding of the nodes $V$ from the graph $G_m$. We call such an embedding, a rigid body $L_{rigid}^t = \texttt{rigid}( G_m )$, that provides relative localization of the individual nodes. Our objective is to minimize $|L^t - L_{rigid}^t|$, where $L^t$ denotes the `true' relative localization of the nodes in $V$ at time $t$.
We divide the time through which the system operates into slots. Each slot allows enough time to profile a single edge and update $L_{rigit}^t$. In essence, the problem boils down to a scheduling problem that intelligently selects and estimates $M_{e_i}$ at each time slot.
\smallskip
\noindent\textbf{Edge Uncertainty:} At the $T^{th}$ time slot, we define {\em staleness} of an edge $e_i$ as $s_i = (T - t_{e_i})$. Intuitively, higher the value of $s_i$, higher is the uncertainty associated with the measurements in $M_{e_i}$. For modeling such uncertainty we take a Bayesian approach. We maintain a table ($\texttt{TAB}_{edge}$) that, for any edge $e_i$, maps the time interval between two successive $M_{e_i}$s to the change in the estimated ranges. For instance, let $M_{e_i}^1$ and $M_{e_i}^2$ be two successive measurements obtained at time slots $t_1$ and $t_2$. We record the mapping $(t_2 - t_1) \rightarrow |R_{e_i}^2 - R_{e_i}^1|$ or, $\Delta t_i \rightarrow \Delta R_i$. At the $T^{th}$ time slot, we define edge uncertainty for edge $e_i$ as,
\begin{equation}
\label{eqn:edge_uncertainty}
U_{edge}^i = \texttt{Prob}( \Delta R > R_{threshold} | \Delta t = s_i)
\end{equation}
where $s_i$ is the current edge staleness and $R_{threshold}$ is the maximum $\Delta R$ the system can tolerate which is given as an input parameter to the system. Equation (\ref{eqn:edge_uncertainty}) can be easily computed from $\texttt{TAB}_{edge}$. When an edge E gets included in $E_c$, we create $\texttt{TAB}_{E}$ and maintain its history. When E is excluded from $E_c$ (i.e., E is no longer a valid connectivity link), we reset $\texttt{TAB}_{E}$.
\smallskip
\noindent\textbf{Node Uncertainty:} We need {\em at least} three edges ($\in E_m$) incident on a node to make it a part of the rigid body, $L_{rigid}$. Consider a situation where out of several edges incident on a node $n_i$, three edges have low edge uncertainties ($U_{edge}$(s)). Although the remaining edges have high uncertainties, measuring those edges might not be best use of resources as the benefit that it will bring to the node $n_i$ in question might be minimal. We define node uncertainty as,
\begin{equation}
\begin{aligned}
U_{node}^i &= \frac{u_1 + u_2 + u3}{3} \\
u_1, u_2, u_3 &= min_3 \{ U_{edge}^{i1}, U_{edge}^{i2}, \cdots, U_{edge}^{iN} \}
\end{aligned}
\end{equation}
where $i1, i2, \cdots, iN$ are the $N$ incident edges on node $n_i$. If $N \leq 2$, $u_3 = 1$ and/or $u_2 = 1$ respectively.
Initially, when adequate ranging information is not available to estimate the edge uncertainties ($U_{edge}$), we computer the node uncertainty ($U_{node}$) directly using IMU sensor data. A node's accelerometer information is used to track its total displacement, $D_i$. Under such circumstances, we define node uncertainty as,
\begin{equation}
\label{eqn:imu_uncertain}
U_{node}^i = \frac{D_i}{R_{threshold}}
\end{equation}
\subsection{Algorithm Sketch}
In the following, we describe the different steps involved in the \textsc{DynoLoc}\xspace{'s} algorithm for relative localization.
\mypara{Bootstrap Phase:}
\begin{packedenumerate}
\item \ul{\em Connectivity Graph:} Every node sequentially broadcasts a neighborhood discovery frame. This frame is overheard by the remaining nodes which helps every node determining its neighbors. Consequently, the graph $G_c$ is constructed using the neighborhood information.
\item \ul{\em Ranging:} $U_{node}^i$ is calculated for each node $n_i$. Since enough historical information is not available, equation (\ref{eqn:imu_uncertain}) is used to estimate the same. The nodes are sorted according to their uncertainty values. Let the ordering be denoted by $[n_1, n_2, \cdots, n_{|V|-1}, n_{|V|}]$ s.t., $[U_{node}^{n_1} \leq U_{node}^{n_2} \cdots U_{node}^{n_{|V|-1}} \leq U_{node}^{n_{|V|}}]$. Next the following edges are ranged: $(n_1, n_{|V|})$, $(n_2, n_{|V|-1})$ and so on, i.e., the node with the highest uncertainty is ranged with the one with the lowest and likewise.
\end{packedenumerate}
\mypara{Steady State:}
For every time instant, T:
\begin{packedenumerate}
\item \ul{\em Update:} Based on the latest measurement taken at $(T-1)$, new edges are created or removed from $E_c$. Accordingly, $G_c$ is updated. Depending on the measured edge at $(T-1)$, the corresponding table $\texttt{TAB}_{edge}$ is updated. The value of $U_{edge}$ is and $U_{node}s$ are updated.
\item \ul{\em Edge Selection:} We select the edge that induces maximum relative improvement on node uncertainty. (see Algorithm (1))
\item \ul{\em EDM Update:} Only entries corresponding to the edges that are in $E_c$ are kept in the EDM. The rest of the entries are considered missing. The missing entries are predicted by a standard matrix completion algorithm based on {\em semi-definite rank relaxation} (SDR). Our edge selection mechanism is aware of the underlying rigid body and hence helps the SDR algorithm to predict missing entries with greater accuracy.
\item \ul{\em Rigid-body Formation:} Once the EDM is complete, we run {\em multi dimensional scaling} (MDS) on it to estimate $L_{rigid}^T$. Discussion about IMU will also come here how we apply MDS.
\end{packedenumerate}
\begin{algorithm}
\caption{Edge Selection}
\begin{algorithmic}
\STATE try out every edge and see which edge creates the maximum relative benefit for the node-pair. select that edge.
\end{algorithmic}
\end{algorithm}
\subsection{Motivation for RF-based localization in dynamic infrastructure-free environments}
Indoor localization is a well-studied problem \cite{indoor-loc-survey}. Due to poor or lack of reception of GPS, the solution for indoor localization often involves a host of technologies such as optical \cite{optical-loc-survey}, visual \cite{viz-loc-survey}, inertial \cite{intertial-loc-survey}, RF \cite{loc-survey} etc. We can categorize these solutions as ``active" and ``passive" localization. In active localization, the objects are identifiable \cite{passive-loc-ov} whereas in passive case, they are not \cite{loc-survey} (here, only the count of the tracked objects matter). This work pertains to only active localization, as such the identity of each localized object is essential. In particular, each tracked object or person carries a device that enables both the identification and localization of them. We further categorize the active localization in two groups- such as localization with and without infrastructure. In the solution with infrastructure \cite{loc-survey}, some designated devices are put at pre-computed location for optimal operation. These devices with fixed locations are called ``anchors". Depending on the technology used, anchors periodically transmit RF \cite{loc-survey} or infrared \cite{lidar-survey} beacons, which in turn enables the localization or tracking via a variety of methods, such as Time Difference of Arrival (TDOA)\cite{toda-ov}, Time of Flight (TOF) \cite{tf-ov} etc. in conjunction with multi-lateration \cite{multi-lat-ov}, fingerprint-matching \cite{fingerprinting-loc-survey} etc. In these applications, the deployment involves careful placement and calibration of the anchors \cite{vr-devs}. Dysfunction of a single anchor disable the localization system in entirety. Anchor-based real-time indoor localization service provided by companies such as \cite{loc-solution-survey} has been deployed in factories, warehouses etc. for many years.
Another broad area within active localization is based fingerprinting-based localization, wherein- the area of interest is scanned beforehand using a variety of methods \cite{} and then the objects are localized by matching the fingerprints. However, any change in the environment necessitates re-collection of the fingerprints.
Our work concerns indoor localization in a ``dynamic" environment, which is unknown, unmapped and unexplored and has a lot of moving parts to the extent that- i) the anchors cannot be deployed in pre-planned way ii) the environments cannot be fingerprinted beforehand and/or the fingerprints cannot be used reliably due to constant change in the environments. Assumption of such dynamic environments often is necessary for example in the case of tracking first responders such as firefighters \cite{tackio}, SWAT team \cite{swat-app} etc; sometimes the assumption is desirable for example, in case of Augmented and Virtual (AR/VR) reality applications, which currently requires meticulous placement and calibration of the controller plus the beacons \cite{vr-dev-survey}. In summary, dynamic indoor environment requires infrastructure-free solution that does not depend on fingerprinting. Our works pertains to active localization in GPS-denied dynamic indoor environments without fingerprinting and anchor-placement.
\begin{table}
\begin{center}
\begin{tabular}{|p{1.5cm}|p{2cm}|p{4cm}|}
\hline
\textbf{Method} & \textbf{Pros} & \textbf{Cons} \\ \hline
Inertial Tracking (IT) & No anchor-deployment, inexpensive hardware & Run-away positional drift w/o periodic fix \\ \hline
Optical Tracking (OT) & Most precise (millimeter-level accuracy) & Requires LOS \& anchor-deployment, expensive hardware \\ \hline
Visual Odometry (VO) \cite{}, Visual Intertial Odometry (VIO) & accurate under favorable condition, no anchor-deployment & Suffers from runaway optical drift w/o periodic sync, less accurate in low-light, poorly-textured environments , computationally extensive, requires expensive hardware \\ \hline
\end{tabular}
\caption{Comparison of non-RF solutions for infrastructure-free indoor localization for dynamic indoor environments}
\end{center}
\label{tab:dyn-loc-sol}
\end{table}
\\
First, we turn to the candidate solutions that are not RF-based. Table \ref{tab:dyn-loc-sol} summarizes such solutions and their pros and cons in the context of dynamic environments. IT has been successfully used in automobile and robot navigation \cite{auto-nav} because of the deterministic displacement from the rotation-count of the wheels. However, pedestrian dead-reckoning \cite{ped-dr-survey} suffers from positional drifts without periodic fix. Facilitating periodic fix for each object at any arbitrary location is equivalent to dense infrastructure-deployment and is infeasible in our dynamic environment. The line-of-sight requirement rules out the OT approach for our scenario since most indoor space has walls and obstacles. Finally, the VO/VIO approach requires camera plus IMU and can render sub-centimeter-level accuracy \cite{vio-dev-survey} under favorable condition. However, it suffers from a range of limitations such as:
\begin{enumerate}
\item \textbf{Technological Limitation: }regardless of the instruments used, VIO system performs poorly (even fails completely) in poorly illuminated and/or poorly textured environments\cite{opto-survey}, in presence of motion-blur \cite{opto-blur} and/or multiple moving objects in the video-frames \cite{opto-obj} etc.
\item \textbf{Sensor/Instrumental Limitations: }Sometimes to operate in challenging environments such as poorly illuminated and textured etc., VIO requires expensive camera/sensors \cite{vio-dev-survey} and high computation power. Inexpensive cameras \cite{vio-dev-survey} and low compute power typically renders poor accuracy \cite{vio-ins-acc}.
\item \textbf{Algorithmic Limitations: }Finally, the algorithms for VIO also suffer from multitude of limitations. For example the SLAM-based algorithms \cite{vslam-ov} suffers from initialization error \cite{vlam-ov}, drift error \cite{vslam-st}, loop-closure error \cite{vslam-lce}, ``pure rotation" error \cite{vslam-rot-e}, scale ambiguity \cite{vslam-de} and so on. Other class of optical flow-based algorithms suffer from limitations related projection errors (3D-to-2D or 3D-to-3D) \cite{of-proj}, parameterization error \cite{of-overview}, vector-clustering error \cite{of-cluster-e} and so on. Finally quality of visual-feature selection, de-noising and filtering of the sensor data etc. also introduces additional errors \cite{of-feature-e}. All these types of errors accumulate over type to a point where the system breaks. Therefore, these kind of systems often incorporate periodic resetting mechanism and cross-validation \cite{of-augmentation} which is often infeasible in our dynamic unknown environments.
%
\end{enumerate}
Due to such limitations of VIO, we use RF both as a stand-alone and augmenting technology for localization in dynamic indoor environments.
\\
\begin{figure}[!htb]
\centering
\subfloat[]
{
\includegraphics[width=0.45\linewidth]{figs/rel-lit-1}
\label{fig:infra-spectrum}
}
\subfloat[]
{
\includegraphics[width=0.45\linewidth]{figs/rel-lit-2}
\label{robust-spectrum}
}
\caption{Accuracy of difference approaches in spectrum of (a) deployment overhead and (b) robustness.
}
\end{figure}
We can arrange all the works on localization on two broad categories 1) deployment overhead: precise infrastructure (anchors) placement on one end and totally infrastructure-free on the other end of the spectrum (See Figure \ref{fig:infra-spectrum} and 2) robustness: totally known/fingerprinted environment on one end and fully unknown environment on the other end of this spectrum (See Figure \ref{fig:robust-spectrum}. As evident from these figures, current literature is devoid of works in the dynamic unknown environments (shaded region of the two spectrum), let alone high-accuracy works for that region. Our work contribute to this un-addressed region of utility which encompasses a lot of applications in public safety, entertainment etc. as outlined in applications in Section \ref{sec:app}. With this clarity, we introduce our work for dynamic unknown environments.
\\
}
\subsection{System Implementation and Prototype}
\textsc{DynoLoc}\xspace system consists of a set of {\em UWB tags}, interfaced with an {\em embedded computer} (e.g., a smartphone or a Raspberry-Pi) and a {\em central controller} that orchestrates range measurements and runs the localization algorithms. Both, the controller and the embedded computer are connected to a local WiFi network for exchanging control information and application data (e.g., sensor readings, video feeds).
\begin{figure}[!htb]
\centering
\includegraphics[width = 1\linewidth ]{figs/section-4-plots/system}
\vspace{-0.05in}
\caption{Prototype of DynoLoc node}
\label{fig:system}
\vspace{-0.05in}
\end{figure}
\begin{figure*}[!htb]
\centering
\subfloat[]
{
\includegraphics[width=0.25\linewidth]{figs/section-4-plots/base-var-node}
\label{fig:base-var-node}
}
\subfloat[]
{
\includegraphics[width=0.25\linewidth]{figs/section-4-plots/base-var-percent-mobile}
\label{fig:base-var-percent-mobile}
}
\subfloat[]
{
\includegraphics[width=0.25\linewidth]{figs/section-4-plots/base-var-refresh-rate}
\label{fig:base-var-refresh-rate}
}
\subfloat[]
{
\includegraphics[width=0.25\linewidth]{figs/section-4-plots/base-var-velocity}
\label{fig:base-var-velocity}
}
\vspace{-2 pt}
\caption{Comparison of localization error for various method of localization, default values: no. of nodes = 12, velocity = 1m/s, refresh rate = 1Hz, fraction of mobile nodes =50\%}
\vspace{-5 pt}
\label{fig:base-var}
\end{figure*}
\noindent\textbf{$\blacksquare$ \textsc{DynoLoc}\xspace UWB Tag:} The tag consists of a Decawave DW1000 \cite{decawave} ultra-wideband radio (costs $\approx$10\$) that houses an extremely precise picosecond crystal (for TOF calculation), which can achieve a distance resolution as high as 2.2\,mm~\cite{decawaveUserManual}. We interface the DW1000's \texttt{SPI} pins (serial clock, master output, master input and slave select), \texttt{V$_{cc}$} and \texttt{GND} pins to a low power ARM Cortex-M based microcontroller unit STM32 NUCLEO-F042K6 (costs $\approx$10\$), where the latter acts as the \texttt{SPI} master. \textsc{DynoLoc}\xspace's optimized ranging protocol is implemented in about 4000 lines of \texttt{C} code and runs on the STM32 microcontroller. Additionally, the STM32 sends and receives specific ranging instructions or estimated ranges through its serial port from an external host device. While most commercially available UWB tags cost somewhere between \$100--200, \textsc{DynoLoc}\xspace tags show that it is feasible to keep the cost to under 20\$ (10-20\% of the COTS tags) without compromising any feature. Our tags can use the UWB permissible channels spanning from 3--7\,GHz (with 500\,MHz bandwidth). In most of the experiments, we use 3.5\,GHz as the center frequency for improved range.
\noindent\textbf{$\blacksquare$ Tag Host:} An embedded computer (e.g., a smartphone) acts as a USB host for the \textsc{DynoLoc}\xspace Tag. It sends specific ranging instructions to the tag (e.g., range with 5 specific neighboring nodes) and receives measured ranges. The tag host also keeps track of the node's mobility from the inertial sensors as well as the link quality information as obtained from the tag ($M$ and $L$ metrics). The device driver for our tag is implemented on the Android platform and runs as a background service intercepting commands from the controller and passing it on to the tag and vice versa. The tag host also houses a pressure sensor to identify the vertical elevation, i.e., floor number which is useful in a multi-storeyed deployment scenario.
\noindent\textbf{$\blacksquare$ Controller:} The controller is in charge of the overall topology estimation, gathering ranges from individual tags and running the localization engine. Depending on the application (see more in \S~\ref{sec:applications}), the controller sends the information back to the individual nodes, displays them locally in a dashboard or offload it to a cloud service for remote visualization or decision making. The controller logic is implemented in about 1000 lines of \texttt{Python} code.
\subsection{System Evaluation}
\textsc{DynoLoc}\xspace is evaluated comprehensively spanning {\em in-the-wild} deployments to controlled experiments in realistic indoor settings (supplemented by simulations only for larger topologies of over 16 nodes).
In the following, we describe our methodology followed by some key performance results.
\vspace{-0.25 cm}
\subsubsection{In-the-wild Deployment}
\label{subsec:in-the-wild}
\textsc{DynoLoc}\xspace{} has been deployed and tested in a real firefighter's drill. A total of ten firefighters, each carrying a \textsc{DynoLoc}\xspace{} tag individually, enter the test building (2 floors, each $\approx$50\,m$\times$100\,m) emulating a severe fire incident. A pressure-sensor+UWB based mechanism
\sfrtxt{(See Appendix E)}
for floor identification at each node locally, was added to \textsc{DynoLoc}\xspace for this drill. The fire chief, stationed outside the building, tracks every move of the personnel crawling through the dark and smoky passages through \textsc{DynoLoc}\xspace's real-time dashboard and instructs them accordingly through a walkie-talkie. In one specific incident, a firefighter who was lost and separated from his colleagues issued an SOS call. \sfrtxt{The chief knowing his location from the breadcrumb feature of \textsc{DynoLoc}\xspace{} app (See Fig.
\ref{fig:vertical-detection}(b)
in Appendix E) was able to intervene and immediately assist by redirecting his crew accurately towards the lost firefighter.}
We learned (from the fire chief) \textsc{DynoLoc}\xspace's true value in such challenging scenarios, which are common and often lead to firefighter fatalities.
A snapshot of the drill is shown in Fig.~\ref{fig:dynoloc-overview}.
\subsubsection{Controlled Experiment}
Here, we describe results derived from known path and speed of the mobile nodes.
\\
\textbf{Testbed:} The testbed consists of eight pre-planned navigable routes (marked with adhesive tapes on the floor) in an indoor area of about 50\,m$\times$40\,m. The collective length of the routes is $\approx$500\,meters The routes encompass various types of indoor areas: open hallways, corridors, meeting rooms, lab spaces and so on, such that we have a fair representation of typical indoor settings (both LOS and NLOS). In addition to the \textsc{DynoLoc}\xspace tag (+ smartphone host), each volunteer carries another UWB tag operating at a different frequency (6.5\,GHz) for ground truth collection.
\\
\noindent{}
\textbf{Baselines:} The ground-truth is collected using a system of densely placed, synchronized, static anchor nodes (by multilateration using TDOA information), deployed throughout the building floor, which gives a localization accuracy in the order of 10--20\,cm, thereby serving as a lower bound on performance for infrastructure-free solutions.
\sfrtxt{Further, the volunteers carrying the \textsc{DynoLoc}\xspace{} nodes followed paths of known shapes (see Fig.\ref{fig:exp-paths} in Appendix D). As such, any deviation from the target shape was considered as the error in TDOA itself, which is $\leq$10cm.}
We also consider two heuristics that are subsets of \textsc{DynoLoc}\xspace, namely \texttt{H-Agnos} and \texttt{H-Dyn}.
\texttt{H-Agnos} employs \textsc{DynoLoc}\xspace's relative localization component but adopts a naive edge selection approach (ranges every pair of possible edges through round-robin) that does not account for node dynamics and link quality. In contrast, \texttt{H-Dyn}'s edge selection accounts for node mobility by ranging on edges incident with the most dynamic nodes, but disregards the geometric rigidity requirement in its relative localization.
\sfrtxt{
Note that, SnapLoc\cite{grobetawindhager2019snaploc} and TrackIO\cite{dhekne2019trackio} are two recent works that uses UWB-range based indoor localization. However, SnapLoc is fixed anchor-based and only considers a small indoor space with constant LOS, whereas TrackIO is infrastructure-full (drone-based) yet is outperformed by \textsc{DynoLoc}\xspace{} for similar node count and mobility.
}
\\\noindent\textbf{Overall Localization Performance:} Fig.~\ref{fig:base-var} highlights the overall performance of \textsc{DynoLoc}\xspace{} as a function of various factors, namely number of nodes, fraction of mobile nodes, their velocity and the targeted location update rate. \textsc{DynoLoc}\xspace scales well for a reasonable number of nodes\,(Fig.~\ref{fig:base-var-node}) that is practical in most real life contexts. Even in challenging scenarios, where all the nodes are mobile (at 1 m/s), \textsc{DynoLoc}\xspace provides an average localization error of under 2 meters for a 1 Hz update rate,
while \texttt{H-Agnos} and \texttt{H-Dyn} incur an error of 6--7 meters and 5--6 meters respectively (Fig.~\ref{fig:base-var-percent-mobile}). Lack of \textsc{DynoLoc}\xspace's robust relative localization component (included in \texttt{H-Agnos} and \texttt{H-Dyn}), will only lead to further degradation.
While \texttt{H-Agnos} accounts for underlying graph rigidity, it is devoid of the notion of node mobility and link-quality and hence, renders poor accuracy; whereas \texttt{H-Dyn} takes into consideration the node mobility, but renders poor accuracy due to lack of enforcing rigidity requirement.
Also note that, even for a more demanding location update rate of 2\,Hz, \textsc{DynoLoc}\xspace maintains a sub-meter localization accuracy (Fig.~\ref{fig:base-var-refresh-rate}), even when half the nodes are mobile. \textsc{DynoLoc}\xspace's ranging algorithm being mobility-aware, adaptively expends the available time resources in collecting the most critical ranges. This allows it to deliver under 2m accuracy even when 4 times the update rate is desired.
\\\noindent\textbf{Benefits of \textsc{DynoLoc}\xspace's Design:}
\textsc{DynoLoc}\xspace's design components are benchmarked in isolation as follows.
\noindent\underline{\textit{Adaptive Ranging:} }
Fig.~\ref{fig:base-var} clearly shows the significant merits and usefulness of \textsc{DynoLoc}\xspace's adaptive ranging (edge selection) mechanism (compared to random selection in Rand). We now explore the merits of its mobility and link quality metrics as part of its topology estimation.
\noindent\textit{$\bullet$ Mobility Metric:}
Fig.~\ref{fig:effective-m} demonstrates that performing the edge selection (purely based on rigidity constraints) without considering mobility metric ($M$) results in a sub-optimal localization accuracy. Additionally, Fig.~\ref{fig:mobility-accuracy} shows the reactive nature of the $M$ metric in tracking node mobility through its acceleration.
\noindent\textit{$\bullet$ Link Quality Metric:} Similarly, Fig.~\ref{fig:effective-l} shows the impact of link quality metric ($L$) on edge selection. $L$ acts as a classifier for NLOS versus LOS ranges. Particularly in NLOS scenarios, $L$ plays a critical role in selecting non-noisy edges, thereby leading to a better localization accuracy. Fig.~\ref{fig:effective-l} indicates that introducing the L metric improves accuracy by 30 -- 40\% in two different NLOS scenarios (meeting rooms/office space and lab spaces denoted by NLOS1 and NLOS2 respectively).
\noindent\underline{\textit{Robust Relative Localization:}}
Fig.~\ref{fig:edm-completion-error} shows that \textsc{DynoLoc}\xspace{}'s EDM-completion approach is able to bound the range estimation (for missing entries) to within 2m in most cases. This can be attributed to its approach of targeting the rigid sub-graphs individually, while using a sequential multilateration approach that exploits the node geometry for estimating the missing ranges. Its rigid sub-graph based relative localization further contributes to a much improved accuracy over existing approaches as seen in Fig.~\ref{fig:edm-vs-multilat}.
\begin{figure}[!htb]
\centering
\subfloat[]
{
\includegraphics[width=0.4\linewidth]{figs/section-4-plots/mobility-q}
\label{fig:mobility-accuracy}
}
\subfloat[]
{
\includegraphics[width=0.47\linewidth, height=92pt]{figs/section-4-plots/linkq-perf}
\label{fig:linkq-perf}
}
\caption{(a) Consistency of Mobility Metric (b) Correlation of Link Quality metric with ranging error}
\vspace{-12 pt}
\label{fig:mobility-heading-accuracy}
\end{figure}
\begin{figure}[!htb]
\centering
\subfloat[]
{
\includegraphics[width=0.5\linewidth]{figs/section-4-plots/effective-m}
\label{fig:effective-m}
}
\subfloat[]
{
\includegraphics[width=0.5\linewidth]{figs/section-4-plots/effective-l}
\label{fig:effective-l}
}
\caption{Comparision of DynoLoc without Mobility\,(M) and Link Quality\,(L) metrics}
\label{fig:dyno-metric}
\end{figure}
\begin{comment}
\begin{figure}[!htb]
\centering
\subfloat[]
{
\includegraphics[width=0.53\linewidth, height = 1.1in ]{figs/section-4-plots/linkq}
\label{fig:linkq-example}
}
\caption{Illustration of link metric calculation and its correlation with range-error}
\label{fig:linkq}
\end{figure}
\end{comment}
\begin{comment}
\begin{figure*}[!htb]
\centering
\subfloat[]
{
\includegraphics[width=0.25\linewidth]{figs/section-4-plots/algo-var-node}
\label{fig:algo-var-node}
}
\subfloat[]
{
\includegraphics[width=0.25\linewidth]{figs/section-4-plots/algo-var-percent-mobile}
\label{fig:algo-var-percent-mobile}
}
\subfloat[]
{
\includegraphics[width=0.25\linewidth]{figs/section-4-plots/algo-var-refresh-rate}
\label{fig:algo-var-refresh-rate}
}
\subfloat[]
{
\includegraphics[width=0.25\linewidth]{figs/section-4-plots/algo-var-velocity}
\label{fig:algo-var-velocity}
}
\caption{Comparison of localization error for various features of DynoLoc algorithm, default values: no. of nodes = 12, velocity = 1m/s, refresh rate = 1Hz, fraction of mobile nodes =50\%}
\label{fig:algo-var}
\end{figure*}
\end{comment}
\noindent\underline{\textit{Concurrent Ranging:}} As seen in Fig.~\ref{fig:concurrent}(b), \textsc{DynoLoc}\xspace's aggregated and concurrent ranging directly contributes to a larger ranging budget and hence localization accuracy. \textsc{DynoLoc}\xspace's tag supports two data rates (low, 100\,Kbps and high, 6.8\,Mbps, Fig.~\ref{fig:concurrent}(a)). The high data rate results in a lower latency per ranging (2\,ms vs 8\,ms for low data rate case). While it restricts the maximum communication range (about 10 m) and allows for more concurrent ranging, it also requires a higher node density to ensure a reasonably connected topology. This is reflected in the results presented in Fig.~\ref{fig:concurrent}.
\noindent\underline{\textit{Absolute Localization:}} Conversion of relative to absolute localization incurs an additional error of 10--20\%, as shown in Fig.~\ref{fig:abs-vs-rel}. As expected, it increases as the network size grows larger. Fig.~\ref{fig:heading-accuracy} shows how the value of heading is tracked over time as the user moves (walk and sprint). While noisy heading values lead to higher additional errors, this is still restricted to just 40-60\,cm even for a user sprinting at 2\,m/s.
\\\noindent \underline{\textit{End-to-End System Latency:}} Fig.~\ref{fig:latency-effect-a} breaks down \textsc{DynoLoc}\xspace's overall latency broadly into three categories. In table~\ref{fig:latency-effect-b}, we present some battery life benchmarks for different components of the system.
\begin{comment}
\begin{figure}[!htb]
\centering
\subfloat[]
{
\includegraphics[width=0.3\linewidth]{figs/section-4-plots/aoa-example}
\label{fig:gdop-example}
}
\subfloat[]
{
\includegraphics[width=0.45\linewidth]{figs/section-4-plots/aoa-estimation-error}
\label{fig:gdop-aoa}
}
\caption{AoA estimation in GDOP calculation}
\label{fig:gdop}
\end{figure}
\end{comment}
\begin{figure}[!htb]
\centering
\subfloat[]
{
\includegraphics[width=0.4\linewidth]{figs/section-4-plots/edm-completion-error}
\label{fig:edm-completion-error}
}
\subfloat[]
{
\includegraphics[width=0.4\linewidth]{figs/section-4-plots/edm-vs-multilat}
\label{fig:edm-vs-multilat}
}
\caption{(a) EDM completion Error, (b) Relative localization error}
\label{fig:edm-related}
\end{figure}
\begin{figure}[!htb]
\centering
\subfloat[]
{
\includegraphics[width=0.3\linewidth]{figs/section-4-plots/uwb-diff-mode}
\label{fig:diff-data-rate-ranging-cdf}
}
\subfloat[]
{
\includegraphics[width=0.35\linewidth, height = 97pt]{figs/section-4-plots/concur-ranging}
\label{fig:concurrent-ranging}
}
\caption{(a) Ranging error for different modes of UWB, (b) Loc. accuracy \& node dis-connectivity for concurrent ranging, UWB datarates \& node densities}
\label{fig:concurrent}
\end{figure}
\begin{figure}[!htb]
\centering
\subfloat[]
{
\includegraphics[width=0.4\linewidth, height =90pt]{figs/section-4-plots/abs-vs-rel}
\label{fig:abs-vs-rel}
}
\subfloat[]
{
\includegraphics[width=0.45\linewidth, height=90pt]{figs/section-4-plots/heading-q}
\label{fig:heading-accuracy}
}
\caption{(a) Accuracy of absolute Localization, (b) Impact of heading error}
\label{fig:abs}
\end{figure}
\begin{figure}[!htb]
\centering
\subfloat[]
{
\includegraphics[width=0.5\linewidth]{figs/section-4-plots/latency-share}
\label{fig:latency-effect-a}
}
\subfloat[]
{
\includegraphics[width=0.45\linewidth]{figs/section-4-plots/power}
\label{fig:latency-effect-b}
}
\caption{(a) Breakdown of latency components (b) Power benchmarks for various system components}
\label{fig:latency-effect}
\end{figure}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,204 |
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