Everyday appliances like washing machines could play a significant role in the future of charitable giving as smart technology and online transactions become more prevalent, a new report argues.
The paper, Giving Unchained – Philanthropy and the Blockchain, published today (Wednesday) by the Charities Aid Foundation, explores the implications blockchain technology may have for how charities fundraise and operate in future.
This includes setting out how 'smart' household appliances connected via the blockchain could not only generate money for their owners, but even help them decide which good causes to support with it.
The Blockchain is a shared, decentralised public record which makes virtual currencies like Bitcoin possible by providing a record of all transactions and determining who owns what at any given time. It is owned and maintained by all users of the system, who contribute some of their computers' processing power to perform calculations, thus doing away with the need for expensive third parties when making transactions.
While the technology is still in its infancy, a number of high profile financial services firms are exploring ways in which it could increase the efficiency and security of online transactions. Earlier this year, the Bank of England's chief economist said that blockchain-based digital currency issued by central banks could one day replace cash. And as greater numbers of appliances become 'smart' – connected online – it is envisaged that they could also become part of the blockchain, contributing their processing power in exchange for money.
Today's report is believed to be the first to explore what the implications may be for charitable giving. It follows a CAF paper from earlier this year on cryptocurrency – Giving a Bit (Coin) – which set out how virtual currencies could be the answer to charity transparency.
The latest report suggests that blockchain could pave the way for a revolution in how people give to charities and how charities go about addressing social problems.
"While still very much in its infancy, this technology has the potential to revolutionise the way charities work.
"Banks and financial organisations are already actively exploring the very large potential of blockchain to change the way they do business, and overhaul how people and markets interact in future. Charities and non-profit organisations will need to be alive to the huge implications this may have for fundamentally changing the nature of social action too.
"The answer to some of the biggest challenges charities face today – raising money, increasing transparency and trust and improving their effectiveness in supporting their beneficiaries – could lie in a technology which is likely to become increasingly mainstream in the not-too-distant future.
"While a world in which washing machines may play a key role in charitable giving may seem like science fiction to some, many of these technologies already exist.
Increasing transparency and openness, while reducing costs. Any user would be able to view details of any transaction. Because the blockchain is owned and maintained by all users, it removes the need for third parties and the transaction costs that come with them.
Smart contracts for giving. Smart contracts are computer protocols underpinned by blockchain technology. While traditional contracts require a third party to manage them and execute terms, smart contracts are designed to be independent and self-executing – for instance by triggering payments when certain conditions are met. This could be a significant boost for international charities working in countries where reliable third parties are difficult to come by. Such contracts could also be used by businesses to fulfil pledges on their charitable contributions.
An 'internet of things' which automates philanthropy. With everyday household appliances like energy meters, fridges, toasters and washing machines increasing being connected to the internet, blockchain technology could provide a mechanism to allow these objects to connect seamlessly to each other and to users. They would be able to sell spare processing capacity to the blockchain in exchange for money. Using smart contracts, they could even decide how to spend the money, and which charities to give to.
Opening up news ways for people to give. Experts believe that blockchain technology could lead to a fundamental rethink of how we think about asset ownership and lead to a growing 'sharing economy' as typified by services like ZipCar, Uber and AirBnB. Donations of spare physical capacity could provide new ways of addressing social problems. For example, smart buildings could be used to create new systems for identifying and allocating unused accommodation, which could be used to address the challenge of rough sleeping.


text:
Gaius Iulius Verus Maximus (c.217/220 - aprilie 238), uneori numit greșit Gaius Iulius Verus Maximinus sau Maximinus cel Tânăr, a fost fiul împăratului roman Maximin Tracul și al Caeciliei Paulina.

Maximus a fost ridicat la rangul de Caesar de către tatăl său, Maximin Tracul, în jurul anului 236. Cu toate acestea, el nu a deținut nicio putere reală.

După Asediul de la Aquileia, în aprilie 238, Maximin și fiul său Gaius Iulius Verus Maximus au fost asasinați la Aquileia de soldați din Legiunea a II-a Parthica, nemulțumiți de anarhia ce stăpânea imperiul.

Referințe și note

Împărați romani
Împărați romani uciși
Nașteri în anii 220
Decese în 238

redpajama_set_name: RedPajamaWikipedia


text:
Signed Gallery Store
Home › Signed Back to the Future Original Animation Cel - no.1701
Signed Back to the Future Original Animation Cel - no.1701
An original hand paint animation cel from the fab Back to the Future animated series! This item was signed by Christopher Lloyd at London Film and Comic Con 2017!
Back to the Future is an American animated science fiction comedy adventure television series for television based on the live action Back to the Future movie trilogy. The show lasted two seasons, each featuring 13 episodes, and ran on CBS from September 14, 1991, to December 26, 1992, and reran until August 14, 1993 on CBS. The network chose not to renew the show for a third season. It later reran on FOX, as a part of the Fox Box block from March 22 to August 30, 2003. It was the very first production of Universal Cartoon Studios.
This cel is 10" x 12", mounted and framed for protection. All Celbase animation cels have unique limited edition codes for future referencing. Please be aware some photography may show reflections from the protective packaging.
All our cels are bought from international galleries or directly from the production studios.
Back to the Future Original Animation Cel - no.0019
New Website Launches!
So excited to finally see a new Celbase website launched! We hope everybody enjoys all the amazing artwork we have on our site and thank you to everybody who has...
Copyright © 2019, Celbase. Powered by Shopify

redpajama_set_name: RedPajamaCommonCrawl


text:
Rotzpipn ist eine Band aus dem Wiener Gemeindebezirk Simmering. Der Name der Band ist österreichisch, bedeutet etwa "Rotznase" und ist ein in erster Linie in Bezug auf Kinder verwendetes Schimpfwort.
Die Band schreibt sich auch "Rotźpıpn" – mit Diakritikon auf dem z und dem aus den Turksprachen entnommenem Hinterzungenvokal /ɯ/ - ein Tribut an Bands wie Motörhead und Spın̈al Tap, die ebenfalls Heavy-Metal-Umlaute verwenden.

Die Gruppe definiert sich laut eigenen Angaben als "Simmeringer Milieuvarieté" und behandelt in ihren Liedern, Videos und Liveshows das Leben der Wiener Arbeiterschicht humorvoll und meist mit kritischem Unterton. Rotzpipn werden mitunter der Neuen Volksmusik zugeordnet.

Geschichte
Ende 2008 wurde die bis dahin nur als zwei Personen Projekt existierende Gruppe von der Band Alkbottle eingeladen, beim alljährlichen "Fett Wia A Christkindl" Konzert im Vorprogramm aufzutreten.
Obwohl Rotzpipn damals nur zwei Lieder spielten, brachten die positiven Reaktionen der Zuschauer die Musiker Fetz N. Schädl und Zacharias Umpferl dazu das Projekt auszubauen und nach weiteren Musikern zu suchen um künftig als vollwertige Rockband zu spielen.

2011 erschien das erste Album ...is a wos wert! (Deutsch: "...ist auch etwas wert!"), welches in Eigenregie in der "Fleischerei Mitschke" in Simmering aufgenommen und unter dem Label "Harlots Music" veröffentlicht wurde. Das Album enthält zum größten Teil Songs die dem klassischen Wienerlied zugeordnet werden können aber auch Lieder mit Anleihen aus dem Punk oder Metalgenre.
2012 erschien das zweite Album Im Pfusch auf Rough Trade. Produzent war Didi Baumgartner (Alkbottle). Im Pfusch ist im Vergleich zum ersten Album rauer und härter ausgefallen.
2015 veröffentlichte die Band ihr drittes Album Das Dümmste Gericht. Die Lieder wurden wieder etwas härter und Metal-lastiger. Dennoch lag auch hier das Hauptaugenmerk auf den Texten.

Konzept
Die Gruppe verfolgt bei all ihren Tätigkeiten ein künstlerisches Konzept. So verwenden zum Beispiel alle Darsteller von Rotzpipn einen Künstlernamen und Pseudoidentitäten. Sie sprechen sich höflich mit "Herr" an, während sie sich sonst nur duzen. Häufig kommt es auch zu Beschimpfungen zwischen Musikern, Publikum und Fans wie auch zu körperlichen Übergriffen innerhalb der Band.

Immer wieder wird der Bezirk Simmering übertrieben positiv dargestellt, um satirisch auf die Relativität von Patriotismus hinzuweisen. Die Statements der Darsteller wechseln immer wieder zwischen Understatement und Megalomanie.

Viele Klischees der Rockmusik werden immer wieder in satirisch überspitzter Form dargestellt, der Rider der Band beschreibt weniger die technischen Anforderungen der Band; er ist vielmehr ein weiterer satirischer Seitenhieb auf die Wünsche mancher Rockstars.

Stil

Texte
Die Texte sind häufig in vulgärem Wienerisch und behandeln neben sozialen Tabuthemen auch die österreichische Politik und Prominenz. Sie bedienen sich verschiedener rhetorischer Stilmittel wie Ironie, Sarkasmus, Zynismus, Archaismen und Vulgarismen.

Mit ihren Liedern Würschtlpolka (einer kritischen Auseinandersetzung mit der Politik diverser österreichischer Rechtsparteien) und Die Moritat von der Schwarzen Maria (einer musikalischen Reaktion auf die Abschiebung von Kindern) nahmen Rotzpipn 2010 und 2011 beim Protestsongcontest teil und erreichte jeweils den dritten Platz.

2012 bewarb sich die Band zum dritten Mal in Folge und konnte mit der Hymne 2.0 (ein bissiges Lied darüber, wie die Welt Österreich wahrnimmt) schließlich den neunten Protestsongcontest für sich entscheiden.

Musik
Die Gruppe bedient sich unterschiedlicher musikalischer Stile und Stilmittel. Meist dominieren klassische Rockriffs oder aus der Volksmusik abgeleitete Wechselbasslinien. Innerhalb eines Liedes kommt es oft zu Takt- oder Tonartwechseln. Viele Gesangslinien werden mehrstimmig gesungen, bei Studioaufnahmen experimentiert die Gruppe mit ausgefallenen Instrumenten oder Gebrauchsgegenständen wie Bierflaschen, Boomwhackers, Glockenspiel oder kaputten Akkordeons. Die Musiker bezeichnen sich selbst als Dilettanten und legen bei Livedarbietungen nur geringen Wert auf die korrekte Wiedergabe ihrer Musikstücke. Die Band verwendet fast ausschließlich billige Musikinstrumente und unterdimensionierte Verstärker. Häufig gibt es in Text und/oder Musik Anspielungen auf andere Bands oder Musiker. So spielt zum Beispiel auf dem ersten Album der Band der Gitarrist bei einem Großteil der Songs eine nur leicht abgewandelte Variation desselben Solos, das stark an das Gitarrensolo des Guns'n'Roses-Gitarristen Slash aus Knockin' on Heaven's Door erinnert. Die Besetzung der Musiker ist variabel, neben der "Kernband" (Gitarren, Bass, Schlagzeug, Gesang) werden oft Gastmusiker zur Teilnahme an den Konzerten eingeladen.

Diskografie
 2011:  …is a wos wert (CD, Harlots Music)
 2012: Im Pfusch (CD, Rough Trade)
 2015: Das Dümmste Gericht (CD, Harlots Music)
2018: Pudl di ned auf (CD, Harlots Music)

Weblinks
 Offizielle Website
 Rotzpipn im SRA-Band-Archiv

Einzelnachweise

Österreichische Band
Band der Neuen Volksmusik


text:
=============
Release notes
=============

Release 0.8.3
=============

* samtools command now accept the "catch_stdout" option.

* get_aligned_pairs now works for soft-clipped reads.

* query_position is now None when a PileupRead is not aligned
  to a particular position.

* AlignedSegments are now comparable and hashable.

Release 0.8.2.1
===============

* Installation bugfix release.

Release 0.8.2
=============

* Pysam now wraps htslib 1.2.1 and samtools version 1.2.

* Added CRAM file support to pysam.

* New alignment info interface.
   * opt() and setTag are deprecated, use get_tag() and set_tag()
     instead.
   * added has_tag()
   * tags is deprecated, use get_tags() and set_tags() instead.

* FastqFile is now FastxFile to reflect that the latter permits
  iteration over both fastq- and fasta-formatted files.

* A Cython wrapper for htslib VCF/BCF reader/writer. The wrapper
  provides a nearly complete Pythonic interface to VCF/BCF metadata
  with reading and writing capability. However, the interface is still
  incomplete and preliminary and lacks capability to mutate the
  resulting data.

Release 0.8.1
=============

* Pysam now wraps htslib and samtools versions 1.1.

* Bugfixes, most notable:
  * issue #43: uncompressed BAM output
  * issue #42: skip tests requiring network if none available
  * issue #19: multiple iterators can now be made to work on the same tabix file
  * issue #24: All strings returned from/passed to the pysam API are now unicode in python 3
  * issue #5:  type guessing for lists of integers fixed

* API changes for consistency. The old API is still present,
  but deprecated.
  In particular:

  * Tabixfile -> TabixFile
  * Fastafile -> FastaFile
  * Fastqfile -> FastqFile
  * Samfile -> AlignmentFile
  * AlignedRead -> AlignedSegment
     * qname -> query_name
     * tid -> reference_id
     * pos -> reference_start
     * mapq -> mapping_quality
     * rnext -> next_reference_id
     * pnext -> next_reference_start
     * cigar -> cigartuples
     * cigarstring -> cigarstring
     * tlen -> template_length
     * seq -> query_sequence
     * qual -> query_qualities, now returns array
     * qqual -> query_alignment_qualities, now returns array
     * tags -> tags
     * alen -> reference_length, reference is always "alignment", so removed
     * aend -> reference_end
     * rlen -> query_length
     * query -> query_alignment_sequence
     * qstart -> query_alignment_start
     * qend -> query_alignment_end
     * qlen -> query_alignment_length
     * mrnm -> next_reference_id
     * mpos -> next_reference_start
     * rname -> reference_id
     * isize -> template_length
     * blocks -> get_blocks()
     * aligned_pairs -> get_aligned_pairs()
     * inferred_length -> infer_query_length()
     * positions -> get_reference_positions()
     * overlap() -> get_overlap()

  * All strings are now passed to or received from the pysam API
    as strings, no more bytes.

Other changes:
   * AlignmentFile.fetch(reopen) option is now multiple_iterators. The
     default changed to not reopen a file unless requested by the user.
   * FastaFile.getReferenceLength is now FastaFile.get_reference_length

Backwards incompatible changes

* Empty cigarstring now returns None (intstead of '')
* Empty cigar now returns None (instead of [])
* When using the extension classes in cython modules, AlignedRead
  needs to be substituted with AlignedSegment.
* fancy_str() has been removed
* qual, qqual now return arrays

Release 0.8.0
=============

* Disabled features
   * IteratorColumn.setMask() disabled as htslib does not implement
     this functionality?

* Not implemented yet:
   * reading SAM files without header

Tabix files between version 0.7.8 and 0.8.0 are
not compatible and need to be re-indexed.

While version 0.7.8 and 0.8.0 should be mostly
compatible, there are some notable exceptions:

* tabix iterators will fail if there are comments
  in the middle or the end of a file.

* tabix raises always ValueError for invalid intervals.
  Previously, different types of errors were raised
  (KeyError, IndexError, ValueError) depending on
  the type of invalid intervals (missing chromosome,
  out-of-range, malformatted interval).

Release 0.7.8
=============

* added AlignedRead.setTag method
* added AlignedRead.blocks
* unsetting CIGAR strings is now possible
* empty CIGAR string returns empty list
* added reopen flag to Samfile.fetch()
* various bugfixes

Release 0.7.7
=============

* added Fastafile.references, .nreferences and .lengths
* tabix_iterator now uses kseq.h for python 2.7

Release 0.7.6
=============

* added inferred_length property
* issue 122: MACOSX getline missing, now it works?
* seq and qual can be set None
* added Fastqfile

Release 0.7.5
=============

* switch to samtools 0.1.19
* issue 122: MACOSX getline missing
* issue 130: clean up tempfiles
* various other bugfixes

Release 0.7.4
=============

* further bugfixes to setup.py and package layout

Release 0.7.3
=============

* further bugfixes to setup.py
* upgraded distribute_setup.py to 0.6.34

Release 0.7.2
=============

* bugfix in installer - failed when cython not present
* changed installation locations of shared libraries

Release 0.7.1
=============

* bugfix: missing PP tag PG records in header
* added pre-built .c files to distribution

Release 0.7
===========

* switch to tabix 0.2.6
* added cigarstring field
* python3 compatibility
* added B tag handling
* added check_sq and check_header options to Samfile.__init__
* added lazy GTF parsing to tabix
* reworked support for VCF format parsing
* bugfixes

Release 0.6
===========

* switch to samtools 0.1.18
* various bugfixes
* removed references to deprecated 'samtools pileup' functionality
* AlignedRead.tags now returns an empty list if there are no tags.
* added pnext, rnext and tlen

Release 0.5
===========

* switch to samtools 0.1.16 and tabix 0.2.5
* improved tabix parsing, added vcf support
* re-organized code to permit linking against pysam
* various bugfixes
* added Samfile.positions and Samfile.overlap

Release 0.4
===========

* switch to samtools 0.1.12a and tabix 0.2.3
* added snp and indel calling.
* switch from pyrex to cython
* changed handling of samtools stderr
* various bugfixes
* added Samfile.count and Samfile.mate
* deprecated AlignedRead.rname, added AlignedRead.tid

Release 0.3
===========

* switch to samtools 0.1.8
* added support for tabix files
* numerous bugfixes including
* permit simultaneous iterators on the same file
* working access to remote files

redpajama_set_name: RedPajamaGithub


text:
Japan Cuts 2017 Preview
Once again, always dependable Japan Cuts, a festival of excellent, current crop of Japanese cinema takes place at Japan Society 7/13 - 7/23.
Festivities start off with a bang on Thursday, July 13th, ushered in by special guest director and JAPAN CUTS veteran Yoshihiro Nakamura (Fish Story), who introduces the Opening Night Film MUMON: The Land of the Stealth, making its U.S. Premiere. Known for his masterful genre blenders, MUMON is Nakamura's modern take on the traditional jidaigeki (period drama), full of fantastical ninja moves that uphold genre standards, yet imbued with a unique sense of eccentricity and playfulness. Director Nakamura appears in a post-screening Q&A, followed by a rollicking Opening Night Party held in Japan Society's historic theater and waterfall atrium.
As previously announced, JAPAN CUTS is proud to present this year's recipient of the CUT ABOVE Award for Outstanding Performance in Film to Joe Odagiri, the remarkably talented box office golden boy and matinee idol of Japan. Odagiri receives the award before the Centerpiece Presentation screening: the East Coast Premiere of Nobuhiro Yamashita's critically acclaimed drama Over the Fence. A baseball themed Home Run Party follows the screening in celebration of the film and the performance that anchors it. Demonstrating the breadth of his talent and penchant for taking on difficult roles, Odagiri also participates in a Q&A following the U.S. Premiere screening of Kohei Oguri's FOUJITA, about the life of the complex titular painter.
The festival's Closing Film offers a poignant and indelible deviation from traditional Japanese war dramas: Sunao Katabuchi's In This Corner of the World, winner of the Japan Academy prize for Animation of the Year. A deeply moving coming-of-age story about a persevering young woman, In This Corner of the World captures civilian life under the catastrophic tide of World War II with a tone that is at once mournful, optimistic, and enchantingly heart-swelling. The film's prolific producer (having previously worked on Patlabor, Satoshi Kon's masterpieces, and many other titles) Taro Maki attends for the post-screening Q&A.
Here are some of the films I got to watch from the lineup. Please visit Japan Society website for tickets and more info.
ANTI-PORNO
Can Sono Sion, known for his crazy movies with ultra violence and upskirt photography, have a cake and eat it too, in tackling a subject like misogyny in Japanese society? In his Nikkatsu studio commissioned Roman Porno Revival, Anti-Porno, Sono goes for it with mixed results.
Tomite Ami is Kyoko, who wakes up in her primary colored loft, prances around naked. She is supposed to be a famous author, painter and all around creator. She has a somewhat of an emotionally/physically abusive sadomasochistic relationship with her middle aged, mousy assistant Noriko (Tsutsui Mariko). Kyoko's adoring entourage arrives for a photo shoot and she makes Noriko naked on all fours barking like a dog in chain. Suddenly, director calls cut and reveals that the loft is a movie set and the roles are reversed - Noriko is the dominant one and Kyoko is the submissive, self-doubting, all around much abused young ingénue playing (terribly, she is told) the dominant role in a movie. During the course of the movie, their roles change many times.
In multiple flashbacks, Kyoko grows up with her parents' hypocrisy on sex - their puritanical education on morality didn't match their voracious sexual appetite. So she very much longed to be a whore in high school. But yet, she seems deeply confused about what it means to be a whore. She seems incapable of escaping from being 'a woman' in a male dominant Japanese society. It's a vicious cycle- being a whore equals free equals being a men's plaything.
In this meta movie-within-a-movie thing, Sono even tries to assert artistic superiority of roman porno over straight up porno: when high school student Kyoko auditions for a roman porno project, claiming she wants to be a whore, ripping her sailor school uniform off, the director scoffs at her, "This is roman porno. Do you even know what it means to be a whore?"
Satire is one thing, but Sono doesn't seem to get the spirit of the roman porno. To be honest, this Nikkatsu revival of the genre is as dated and tiresome as James Bond franchise. The revivals are supposed to be a harmless titillation with certain restrictions - no depiction of underage sex, etc. It is sad that in depicting sex act, Sono can only resort to sailor uniforms. The movie is hardly sexy or titillating. Sono falters when he tries to be serious. At the end, he just needs to cover up his muddy messages with bucket of colorful paints thrown all over his female protagonist. I like his comedies more.
ONCE UPON A DREAM
Just static images of quiet empty rooms, subway rides, cityscape at dusk consist Once Upon a Dream, a cult film that has been gathering followers over the years, including Kiyoshi Kurosawa who compared it to Jean-Luc Godard's. Indeed, the film's carefully composed exterior shots remind me of many of 80's Godard.
It tells a barely seen protagonist, a soft-spoken school teacher whose somnambulist existence and her weird demeanor are affecting her day to day life. We hear her thoughts in her voice over and her conversations with others, without us seeing anybody on screen. We hear the rustles of bed sheets, footsteps, breathing, moving of locomotives, idle, unintelligible conversations of others in the shops and cafes. Perhaps Once Upon a Dream is the most intimate eavesdropping film as opposed to the most voyeuristic.
Director Shichiri Kei, on the film's 10 years anniversary, added crisp new footage and redid sound design, giving this avant-garde classic a new life. The result is hushed contemplation of urban loneliness that is beautifully, uniquely realized visually and aurally. Quiet yet deeply affecting, Once Upon a Dream is a truly one of a kind experience.
HARUNEKO
It seems that people who want to die end up in a cafe in the middle of nowhere, deep in the forest. It is run by a young man known only as 'manager' with the help of an old woman who sits in a rocking chair, knitting, and a boy name Haru.
The glimpse of the lives of would be dead are projected on a make shift screen, followed by sing-along, accompanied by an indie rock group with a white papier-mâché of a cat on their heads. After that, they are driven through a tunnel of no return deep into the forest where they disappear slowly and become the part of soundscape in the air.
Misty forest settings are highly evocative. It recalls Tarkovsky and the idea of a station between two realms reminds me of Koreeda's affecting Afterlife. Strange and beautiful, Haruneko is a somber, haunting and highly ambitious whatsit that falters a little in its own preciousness.
YAMATO (CALIFORNIA)
Yamato, a city 35km south of Tokyo, hosts the largest US military base in Asia. A Yamato native director, Miyazaki Daisuke, tackles the influence US still has on its inhabitants in his indie feature, Yamato (California), and he has some pointy things to say about it. Sakura (Kan Kanae), plays a wayward high school dropout, first seen rapping in the city's garbage dump. She lives in a tiny house with her busy single mom and her nerdy brother. She works part-time at a traditional Japanese restaurant that serves grilled eels, but she spends most of her time alone practicing rap inside an abandoned van or tit-tatting with her brother over sharing a computer or their tiny living space (divided only by a curtain).
She hasn't quite found herself yet and her rebellion and thuggish attitude are not winning hearts and minds of her surroundings. In order to pursue her music career, she needs to get a computer and a smart phone. She finds out that there is a singing contest at a local cultural center with $2,000 US dollar prize.
Things get a little more interesting when Rei (Endo Nina), an American-Japanese daughter of mom's unseen American boyfriend comes to stay with them. At first, Sakura is downright hostile, but Rei's sunny disposition wins her over and they start hanging out. Sakura takes her to her drab local attractions - mostly shopping malls, cheap stores, the van and a rap music venue, which they get kicked out of, for being underage. They share beers and bad pizza. But Rei's insistence of hearing her rap pushes Sakura to re-evaluate her directions.
Yamato (California) treads the same path as Irie Yu's indie hit 8000 Miles trilogy in terms of disaffected youth with their shot at success theme. The interesting subtexts of US cultural dominance and the rise of jingoism are also there. It's also in the casting of Kan, a Japanese-Korean actress, playing downbeat young woman with cultural identity crisis.
There are so much potential in Yamato, especially with two beautiful, striking actresses- Kan who is very charismatic and has a good screen presence and Endo, a short haired gamine. I just wish Miyazaki concentrated more on the characters and details of their lives instead of awkward, obligatory actions that only serve to move the story forward and take up too much screen time.
Posted by Dustin Chang at 1:00 PM 0 comments
Labels: Japan Cuts 2017, Japan Society, Preview
No Washing Blood Off Hands Necessary
Lady Macbeth (2016) - Oldroyd
Exquisitely acted, framed and paced, William Oldroyd's Lady Macbeth is perhaps the most accomplished debut feature I've seen in years.
Based on Nicolai Lestov's Lady Macbeth of Mtsensk District, a 19th century Russian novel which was adapted and scripted by a well regarded feminist playwright Alice Birch, Oldroyd sets out to tell a Victorian era tale of adultery and murder with a twist. The result is a riveting movie watching experience. Lady Macbeth totally does justice to its title.
Katherine (radiant Florence Pugh), a young, bright-eyed, newlywed bride, quickly learns that a marriage in Victorian era England and at large is a life of submission, humiliation and being a captive.
The Father-in-law and husband team, Boris (Christopher Fairbank) and Alexander (Paul Hilton), represents a patriarchal world so rigid, hostile and loveless, it resembles a prison run by abusive guards - Katherine is not permitted to go outside the castle. She is told that she's not in need to ask the nature of her husband's businesses nor to know his whereabouts when he's supposedly away on business for days.
When Alexander returns, the bedroom business is anything but tender -- he orders her to strip and stand against the wall, looking away from him, while he jerks off. There is no touching, no tenderness, no passion. It's just a humiliating power game. But Katherine is no damsel in distress by any means. Far from it.
When the two brutes are out on business, Katherine is finally, temporarily, free to go outside and walk around the misty moors at the expense of horrified female servants.
One of the half-breed male servants, Sebastian (Cosmo Jarvis), working in the stable and horsing around with other stable hands, catches her eye and soon they become lovers. It starts out sensual and passionate. But soon the sexual power dynamic is reversed and Katherine dominates Sebastian both intellectually and physically.
Soon Boris discovers her secret, divulged by Anna (Naomi Ackie), a terrified black servant. Boris and Katherine's confrontation becomes violent, ending in death by a premeditated poisoning. With no witnesses around (except for Anna), Boris's death is ruled as a massive coronary. When her husband comes home while she is in bed with Sebastian, Katherine daringly manipulates the men to fight, plunging Sebastian deeper into the guilty party and deeper into her grasp.
Katherine's acts are so shocking to Anna, who witnesses her lady's immoral deeds, that she soon stops talking altogether.
Oldroyd orchestrates his theater trained, extremely measured, fitting visual style reminiscent of Haneke: static long takes, visual symmetry and only a few closeups. Ari Wegner's cinematography captures the beautiful, picturesque expanse of the moors and contrasts that with the cold, confounding interiors of the castle.
Even though Katherine is a murderess and not a likeable person, it's her youthful defiance that makes us root for her. She is radiating ball of energy that can't be contained by societal restrictions or codes or morality. She does whatever she can to survive, even using her perceived femininity and her status as a lady-of-the-house, to the fullest extent.
The real victims of Lady Macbeth are the servants. Oldroyd reminds you that in England, injustices are done and seen less as a racial issue and more of a class issue.
Pugh's central performance is nothing short of phenomenal. She wears her arrogant beauty and uses it so well in both being an offender and a victim. Her turn as a wide-eyed, untamable young woman becoming a calculating femme fatale is a sight to see. We are seeing an emergence of another important British actress and a heiress apparent to Kate Winslet.
With a crackling, loaded script by Birch, Lady Macbeth is an exhilarating, radical and timely examination of being a woman in a man's world.
Lady Macbeth opens in New York and Los Angeles on July 14.
Labels: Katherine Pugh, Lady Macbeth, movie review, ND/NF 2017, UK, William Oldroyd
Kékszakállú (2016) - Solnicki
Gáston Solnicki's take on Béla Bartok's Bluebeard's Castle (A kékszakállú herceg vára) is a concise observation of lives of a group of loosely interconnected young, upper-class women in various ages in Argentina. We track girls swimming in a pool in summer months then through fall and winter, as they struggle through growing out of childhood - working at industrial factories to avoid living under their parents and afford their own place, deciding what to major in school, struggling with self image, dealing with young motherhood and men. Instead of the straight up narrative storytelling, Solnicki's chooses to show episodic, small moments of these young women's lives - eating ice cream in bed and being chastised by dad, cooking octopus and being grossed out by it, being casually sexually harassed in a sausage factory (get it?), getting a car wrecked while parking, struggling to fit in the kid's play tent, getting locked out while on the roof, having a moment with a younger man, etc. He moves from one moment to another with such nonchalance and abandon, but there is undeniable cohesiveness to Kékszakállú. His formalist approach- static, picturesque framing, impressive sound design and minimal dialog, conveys much more than what the film doesn't actually tell or show. Great film.
Posted by Dustin Chang at 12:09 AM 0 comments
Labels: Argentina, Bluebeard, Gáston Solnicki, Kékszakállú, movie review


text:
Home » News » Sports » Angelique Kerber Begins Reign as Number One With US Open Win
Angelique Kerber Begins Reign as Number One With US Open Win
Angelique Kerber. (Getty Images)
Germany's Angelique Kerber began her reign as world number one in imperious style battling past a determined Karolina Pliskova 6-3, 4-6, 6-4 to win the U.S. Open final on Saturday and claim her second grand slam title of the year.
Last Updated:September 11, 2016, 08:56 IST
Edited by: Amit Kumar
New York: Germany's Angelique Kerber began her reign as world number one in imperious style battling past a determined Karolina Pliskova 6-3, 4-6, 6-4 to win the US Open final on Saturday and claim her second grand slam title of the year.
Kerber, who officially takes over from Serena Williams as the world's top ranked player on Monday, underlined her number one status displaying the heart of a champion as she fought off a fierce challenge from Czech 10th seed Pliskova on a steamy afternoon at Arthur Ashe Stadium.
The 28-year-old began the grand slam year beating Serena Williams to lift the Australian Open title and celebrated again on Saturday when she became the first German to win the U.S. Open in 20 years.
Kerber recovered from a break down in the third set, pouncing on her wilting opponent with two breaks in the third set to end a compelling, high-quality battle of attrition that stretched for two hours and seven minutes.
"It's just amazing," a beaming Kerber, told an adoring crowd after being handed the winner's cheque of $3.5 million. "I won my second grand slam in one year. That's the best of my career.
"Actually just incredible.
"When I was a kid I was always dreaming to be the number one player in the world and win grand slams and today is the day I won my second grand slam here, especially in New York. I'm the number one player on Monday.
"All the dreams came true this year."
Pliskova, who arrived at Flushing Meadows with a reputation of grand slam under-achiever having never before been beyond the third round of the major, shed that label by making it all the way to the final and taking on back-to-back world number ones.
After beating top ranked Serena Williams in the semi-finals to end the American's record-equalling 186 consecutive weeks stay at the top of the rankings, the big-hitting Pliskova, who has led the WTA Tour in aces the last two season, attacked her successor in the final but could not break down the tireless German's defence.
"You cannot compare those two (Serena Williams and Kerber)," explained Pliskova. "The game is totally different. The persons
as well. Serena is going for every shot.
"With Angie, you cannot wait for mistakes. She doesn't give you anything. I have to be the one who is aggressive.
"I beat very good players. It was always my goal to pass the third round and I made it to the final."
Kerber admires compatriot Steffi Graf, but is blazing her own trail.
Comparisons, though, are inevitable after Kerber became the first German to win the U.S. Open since Graf claimed the last of her five Flushing Meadows crowns in 1996.
"She was always my idol and I told her so many times," said Kerber, adding that Graf had sent her a text wishing her luck. "For me it's really important to go on my own way."
The match opened with Kerber breaking a jittery Pliskova and ended the same way, though for most part it was a very even contest.
Kerber's loss of the second set was the first time the entire championship that she had dropped a set.
But she fought back in the third set and wore down the towering Czech, who started to wilt in the punishing conditions, hunching over her racket at the end of almost every point.
Kerber, sensing an opening, stepped up the pressure and pace getting triple break point at 5-4 and closed out the contest on the first match point.
"I think I was just believing in myself in the third set and just going for it," said Kerber.
"I think that was the key at the end of the third set, that I was trying to take the chance, just go for it."
US Open 2016
first published:September 11, 2016, 08:56 IST


Description: Players stand in dice five shape. Ball passing goes: red-white, white-green, green-red; and simultaneously orange-green, green-blue, blue-orange. The middle player (green) is turning around. Variations: – Alternative stick holding – Different ways of passing – Include passes between players red-orange, white-blue Don't forget to change the drill rotation.


text:
Het tennistoernooi van Nottingham van 2015 werd van maandag 8 tot en met zaterdag 27 juni 2015 gespeeld op de grasbanen van het Nottingham Tennis Centre in de Engelse stad Nottingham. De officiële naam van het toernooi was Aegon Open Nottingham.

Het toernooi bestond uit twee delen:
 WTA-toernooi van Nottingham 2015, het toernooi voor de vrouwen (8–15 juni)
 ATP-toernooi van Nottingham 2015, het toernooi voor de mannen (21–27 juni)

Toernooikalender

2015


the handcrafted 15" tall snow leopard by hansa is an amazingly realistic and brilliantly designed stuffed animal that will last a lifetime. hand cut fabrics, hand sewn stitches, and a hand finished face are just a few of the things that go into the thoughtful and highly skilled creation of this snow leopard.
the stuffed animal's custom designed coat is carefully hand cut by skilled craftsmen for a distinctive look and unique personality. every piece comes with a teaching tag describing, in detail, the animal's habitat, lifestyle, gestation period, care of young and eating habits.


Yes. It would be comforting for me to pretend that the choice of which poet to feature in this instalment of Mycroft's Poetry Corner was a simple one, however, I assure you that it was not.
How the pressure of this selection moved me, with incessant weighing of the existential pros versus the subjective cons of one poet over another; I declare now it was exacting an unspecified toll on my balding pate. Indeed, it is my hypothesis that the inner turmoil was increasingly manifesting itself through my interrupted sleep, my reduced appetite(s) and most worrisome of all - uncharacteristically meagre bowel-movements. This simply would not do.
Therefore, when these perturbations were combined, with the acquisition by my housekeeper, of a persistent dry cough which meant that even my last refuge of an afternoon snooze was repeatedly interrupted by her hacking, I took action. I packed a moderately-sized valise and powered up my vintage Mercedes and decided to spend a number of days travelling around the South-west of Ireland. And what honest Joy! I say now that I feel I have a greater understanding of why the Germans invaded Poland & Czechoslovakia; if it was not to feel the satisfaction of putting precision-engineered machinery to its intended use, then I Sir, am a Dutchman.
My love of motoring aside, the primary purpose of this brief holiday was to look at my fair country through the other end of the telescope, hopefully gaining a welcome change of perspective and to clear the maelstrom within my head.
How the small coastal villages and snaking roads of West Cork and Kerry slipped by, as if in some sort of beautiful slideshow; each colourful hamlet, each grassy roadway, each sprawling Atlantic vista wound wistfully by to an aged cassette recording of Boccherini's La Musica Notturna Delle Strade Di Madrid. It is almost a cliché but the beautiful scenery of that area really hit its mark and the fresh Atlantic air did this traveller the power of good.
Ironically though, despite all the cathartic scenery, it was a labyrinthine boarding house in the back-streets of Dingle that would provide the required inspiration. I had taken a small but clean room in the attic; and this unlikely "Old Maid in the Garret" was compelled to keep his wits about him at all times to avoid cracking a head on low door frames and even lower light fittings.
It was after dawn on the first morning and lying in the strange gloom I was set to thinking about all the previous occupants of that room and what their lives had been like. Whose black knickers were those under the pouffe? Why was there sand at the bottom of the bin? Whose cigarette butts were clogging the lead flashing outside the window? And burying my head into the thin pillow to try for another forty winks, I could not help but laugh at the recollection of a poem where similar questions were asked and in that instant I knew I had my man - Philip Larkin.
Philip Larkin was born in Coventry in 1922 to middle class parents and enjoyed a comfortable early life at the heart of little England, attending Grammar School in Coventry and later he studied English at St John's College, Oxford. He spent the majority of his working life as a University librarian, chiefly at the University of Hull. A job which afforded him ample time to write and also uniquely positioned him to watch the world and youth pass him by with a jaundiced leer.
Reputed to be deeply anti-social, Larkin was still affectionately known as the "Hermit of Hull" and often imagined by his early readers as a modern-day Eeyore. In the later part of this life, he enjoyed a combination of critical success and wide public readership that other poets could only envy. However, in my opinion, the question of how a poet with such a gloomy outlook could become one of the best loved English poets of the second half of the twentieth century is still largely unanswered.
A few year's after his death in 1985, the publication of his "Selected Letters" dealt a considerable blow to the reputation of the poet (and the man) when many right-wing, sexist ideals were found in much of his personal correspondence. In the eyes of the QMBC, this storm in a tea-cup does nothing to damage the man's reputation and truthfully even adds to the emotional honesty and candour of his poetry in places.
I have included 3 of his poems below to give some small impression of the man's work but I invite the reader to discover more themselves.
PS - A hearty congratulations to Mr & Mrs McGavin on the birth of their wee bairn - the sentiments of the final poem here almost certainly do not apply in this instance.
Whose window shows a strip of building land?


text:
\section{Introduction}
\setcounter{equation}{0}
Our starting point is a second order Gaussian stationary-increment process $\left\{X(t),\,\, t\in\mathbb R\right\}$, over the probability
space $\mathbf L_2(\Omega,\mathscr F,\mathbb P)$, with spectral measure $\Delta$. Let $\mathcal G$ denote the Gaussian Hilbert space generated by this process, and $\mathscr F_A$ the sigma field induced by $\left\{X(t),\,\, t\in A\right\}$, where $A$ is a Borel set. Given $Y\in\mathbf L_2(\Omega,\mathscr F_{\mathbb R}, \mathbb P)$, we present a new way to obtain the conditional expectation $\mathbb E\left[Y|\mathscr F_A\right]$
which is based on the Wiener chaos written in terms of a special basis for $\mathcal G$ and the Hermite polynomials. Writing
\begin{equation}
\label{eq:Y_chaos}
Y(t)=\sum_{\alpha\in \mathcal J}y_\alpha(t) H_\alpha
\end{equation}
we have
\begin{equation} \label{eq:general_idea}
\mathbb E\left[Y(t)|\mathscr F_A\right] = \sum_{\alpha\in \mathcal J_0}y_\alpha(t) H_\alpha,
\end{equation}
where $\mathcal J_0$ depends only on $A$ and $\Delta$. That is, each chaos element $H_\alpha$ is either measurable with respect to $\mathscr F_A$ or independent with respect to it.\\

Surprisingly, this natural decomposition of the space $\mathcal G$, with respect to a variety of prediction problems, has received little treatment in the literature, if any. \\

If $Y\in\mathcal G$, the problem of computing $\mathbb E\left[Y|\mathscr F_A\right]$ reduces to the problem of orthogonal
projection onto the closed linear span of the functions $\left\{X(s),\,\, s\in A\right\}$, denoted by $\mathcal G_A$. When $Y=X(t)$ with $t>0$ and $A=(-\infty,0]$, this is the classical Wiener-Kolmogorov prediction problem. If $t>T$ and $A=[-T,T]$, this is the finite horizon prediction problem, which was solved by Krein; see \cite{Dmk}. In this work, we consider these two cases and employ similar methods to obtain an orthonormal basis for the space $\mathcal G_A$. We also note that another case of interest is the interpolation problem, when $t\in(-T,T)$ and $A=(-\infty-T]\cup[T,\infty)$. This problem was solved by Dym and McKean \cite{Dmk}. We also refer to this book for background material on these various problems. \\

In the more general case where $Y$ is of the form $Y=f(U)$ where $U\in \mathcal G$ and $f:\mathbb R \rightarrow \mathbb R$ measurable function such that $f(U) \in \mathbf L_2\left(\Omega,\mathscr F(\mathcal G),\mathbb P\right)$, it follows from see \cite{Howland1979} that
\begin{equation} \label{eq:Howland}
\mathbb E \left[ f(U)| \mathscr F_A\right]=\int_{\mathcal G} f\left( \mathbb E[U| \mathscr F_A](w)+(I-P)(w)  \right) d\mu(w),
\end{equation}
where $d\mu$ is the standard Gaussian measure on $\mathcal G$ and $P$ is the orthogonal projection from  $\mathcal G$ onto $\mathcal G_A$. If $\mathcal G_A$ is one dimensional, then \eqref{eq:Howland} reduces to
\[
 \mathbb E \left[ f(U)| \mathscr F_A\right]=f_M\left( \mathbb E[U| \mathscr F_A] \right),
\]
where $f_M(U)$ is the Mehler transform of $f$; see \cite[Ex. 4.18]{MR99f:60082}. In general, formula \eqref{eq:Howland} does not lead to easy computations because of the Gaussian integral. \\

A more simple case of non-linearity is $Y=U^n$ for $U\in \mathcal G$ and natural $n$, in which $\mathbb E\left[Y|\mathscr F_A\right]$ is the $n_{th}$ moment of the the random variable $U$ with respect to the conditional Gaussian distribution
\[
f_{U|\mathcal F_A}(u)=\frac{1}{\sqrt{2\pi \sigma_{MSE}^2}} \exp\left\{-\frac{(u-\mathbb E\left[U|\mathscr F_A \right])^2}{2\sigma_{MSE}^2} \right\},
\]
where $\sigma_{MSE}^2=\mathbb E\left[(U-\mathbb E\left[U|\mathscr F_A \right])^2\right]$.
Thus
\begin{equation} \label{eq:simple_non_linear}
\mathbb E\left[U^n|\mathscr F_A\right]= h_n^{[-\sigma_{MSE}^2]}\left(\mathbb E\left[U|\mathscr F_A \right] \right),
\end{equation}
where $h_n^{[\alpha]}(x)$ is the $n_{th}$ Hermite polynomial with parameter $\alpha$:
\begin{equation}\label{eq:hermite_poly_param}
h_n^{[\alpha]}\left(x\right)\triangleq  n! \sum_{m=0}^{\lfloor n/2 \rfloor}  \frac{\alpha^{m}x^{n-2m} \cdot \left(-1/2\right)^m}{m!(n-2m)!}.
\end{equation}
Relation \eqref{eq:simple_non_linear} can be reformulate as
\begin{equation} \label{eq:simple_non_linear2}
\mathbb E\left[ h_n(U)|\mathscr F_A \right]=h_n^{[\sigma_{MSE}^2]}\left( E\left[U|\mathscr F_A \right] \right),
\end{equation}
where we denote $h_n=h_n^{[1]}$. In a way, formulas \eqref{eq:simple_non_linear} and \eqref{eq:simple_non_linear2} may be seen as transforming the non-linear problem of finding $\mathbb E\left[Y|\mathscr F_A \right]$ into the linear problem of finding  $\mathbb E\left[U|\mathscr F_A \right]$, since the latter is an element of the linear space $\mathcal G$. This approach can be generalized for any $Y\in \mathbf L_2\left(\Omega,\mathscr F,\mathbb P\right)$ if we decompose it as a sum of polynomials in elements of $\mathcal G$. This is the celebrated Wiener chaos decomposition.\smallskip

By introducing a special family of chaos elements $H_\alpha$, we can write the chaos expansion of $f(U)$ and obtain from \eqref{eq:general_idea} the chaos expansion of $\mathbb E\left[f(U)|\mathscr F_A\right]$. The resulting expansion allows us to recover the statistics of $\mathbb E\left[f(U)|\mathscr F_A \right]$, and moreover to obtain a representation of its elements directly from the path $\left\{ X(t),\quad t\in A \right\}$. Thus we replace the non-linear problem of finding \eqref{eq:Howland} by the linear problem of expressing an element of $\mathcal G$ by the samples $\left\{ X(t),\,\, t\in A\right\}$. This will be discussed in greater details in Section \ref{sec:application}.\\

 If $\left\{E_k,k\in  J \right\}$, where $J\subset \mathbb Z$, is an orthogonal basis for the Gaussian Hilbert space $\mathcal G$, then a standard recipe for obtaining a basis for $\mathbf L_2\left(\Omega,\mathscr F(\mathcal G),\mathbb P\right)$ is as follows \cite{MR99f:60082, Holden, HidaKuo}: let $\mathcal J$ be the set of multi-indexes over $J$, i.e. the set of functions $J \rightarrow \mathbb N$ with compact support. For $\alpha=\left(...\alpha_{j_1},\alpha_{j_2},... \right)\in \mathcal J$, define
\[
H_{\alpha}(\omega)=\prod_{j \in J} h_{\alpha_j}\left( E_j(\omega) \right),
\]
where $\left\{ h_n ,n\geq 0\right\}$ are the Hermite polynomials
\begin{equation}\label{eq:hermite_poly}
h_n\left(x\right)\triangleq  n! \sum_{m=0}^{\lfloor n/2 \rfloor}  \frac{x^{n-2m} \cdot \left(-1/2\right)^m}{m!(n-2m)!}.
\end{equation}

Assume moreover that $\left\{E_j,j\in J_0\right\}$, $J_0 \subset J$ is an orthogonal basis for $\mathcal G_A$  and denote by $\mathcal J_0$ the subset of multi-indexes whose support is contained in $J_0$.
Our underlying observation is given by the following theorem.

\begin{theorem} \label{th:main}
For every $\alpha \in \mathcal J$, $H_\alpha$ is measurable with respect to $\mathscr F_A$ if and only if $\alpha \in \mathcal J_0$, i.e. the support of $\alpha$ is contained in $J_0 \subset \mathbb Z$.
\end{theorem}
Theorem \ref{th:main} can be implicitly found in \cite[Ch. 7]{MR99f:60082}, and an explicit proof will be given in Section 2. Theorem \ref{th:main} might have been useless unless we could obtain some explicit orthogonal bases for the space $\mathcal G_A$, this is the content of Sections 4 and 5 in which we review some methods to do so in two cases of interest for the time index set $A$. The setting for Sections 4 and 5 is given in Section 2. In Section 6 we discuss on some application of these chaos elements, together with some examples for the case where $X$ is the fractional Brownian motion.

\section{Proof of Theorem \ref{th:main} \label{sec:proof_main}}

\begin{theorem} \label{th:main2}
Let $\left\{E_k,k\in  J \right\}$ be an orthonormal basis for $\mathcal G$ such that $\left\{E_j,j\in J_0\right\}$ span $\mathcal G_A$ where $J_0 \subset J$. Then for every $\alpha \in \mathcal J$, $H_\alpha$ is measurable with respect to $\mathscr F_A$ if and only if $\alpha$ is contained in $\mathcal J_0 \subset \mathbb Z$.
\end{theorem}

\begin{proof}
Let $\Gamma(\mathcal G)$ be the symmetric Fock space of $\mathcal G$. Recall that we have \cite[p. 18]{MR99f:60082}
\[
\Gamma(\mathcal G)=\bigoplus_{n=0}^\infty \mathcal G^{:n:}=\mathbf L_2(\Omega,\mathscr F_{\mathbb R},\mathbb P),
\]
where $\mathcal G^{:n:}$ is the $n_{th}$ symmetric tensor power of $\mathcal G$. We also denote by
 \[
 :X_1,\ldots,X_m:
 \]
the Wick product of the elements $X_1,\ldots,X_m$ of $\mathcal G$.

Let $P$ denote the orthogonal projection onto $\mathcal G_A$, i.e. for an element $X\in \mathcal G$ we have
\[
PX=\mathbb E\left[ X|\mathscr F_A \right].
\]

Since $\|P\|=1$, $\Gamma P$, the second quantization of $P$, is a bounded linear operator on $\Gamma(\mathcal G)$ \cite[Theorem 4.5]{MR99f:60082}, and by \cite[Theorem 4.9]{MR99f:60082} we have that $\Gamma P: \mathbf L_2\left(\Omega,\mathscr F_{\mathbb R},\mathbb P\right) \rightarrow \mathbf L_2(\Omega,\mathscr F_A,\mathbb P)$ equals the conditional expectation
\[
Y \rightarrow \mathbb E\left[Y|\mathscr F_A \right],\quad Y\in \mathbf L_2(\Omega,\mathscr F_{\mathbb R},\mathbb P).
\]
We have
\begin{equation} \label{eq:thm1_proof}
\begin{split}
\mathbb E\left[ H_\alpha |\mathscr F_A\right] & = (\Gamma P) H_\alpha \\
& =  \Gamma P \prod_{j \in J}  h_{\alpha_j} \left(E_j \right)\\
& \stackrel{a} = \Gamma P : \prod_{j \in  J}  E_j^{\alpha_j}: \\
& \stackrel{b} = ~ : \prod_{j \in J}  P E_j^{\alpha_j} :\\
& = \prod_{j \in  J}  h_{\alpha_j} \left(P E_j \right)\\
& = \prod_{j \in J_0}  h_{\alpha_j} \left(P E_j \right) ~\prod_{j \in  J \setminus J_0}  h_{\alpha_j} \left(P E_j \right).
\end{split}
\end{equation}
where $(a)$ follows from \cite[Theorem 3.21]{MR99f:60082} and (b) follows from the definition of the second quantization of $P$ (see \cite[Theorem 4.5]{MR99f:60082}). Since $E_j \in \mathcal G_A^\bot$ for $j\in J\setminus J_0$, if $\alpha_j \neq 0$ for some entry of $\alpha$ then $h_{\alpha_j}(PE_j)=h_{\alpha_j}(0) = 1$. In this case $\mathbb E\left[H_\alpha|\mathscr F_A \right]=0$, which means that $H_\alpha$ is independent of $\mathscr F_A$. The other option is that $\alpha_j=0$ for all $j\in J\setminus J_0$. Since for $h_{\alpha_j}(x)\equiv 1$ by definition if $\alpha_j=0$, this case leads to
\[
\mathbb E\left[H_\alpha|\mathscr F_A \right]=\prod_{j \in J_0}  h_{\alpha_j} \left(P E_j \right)= \prod_{j \in J_0}  h_{\alpha_j} \left(E_j \right) = H_\alpha.
\]
\end{proof}

\section{Hilbert spaces associated with a Gaussian stationary increment process}
\label{sec:3}
In this section we review standard ideas from the literature on continuous time Gaussian stochastic processes. We describe two additional Hilbert spaces isomorphic to $\mathcal G$, using the notions of the \textit{Wiener integral} and the \textit{trigonometric isomorphism}.
This sets the frameworks for sections 4 and 5 in which we obtain a basis for $\mathcal G$ that satisfy the conditions in Theorem \ref{th:main2}. \\

Assume first we are given a Gaussian stationary process $\dot{X}(\cdot)\triangleq \left\{\dot{X}(t),t\in \mathbb R\right\}$. The spectral measure $\Delta(\gamma)$ is determined by Bochner's theorem through
\begin{equation} \label{eq:stationary_cov}
\mathbb E \left[\dot{X}(t_1)\dot{X}(t_2)\right]=\int_{-\infty}^\infty e^{i\gamma(t_1-t_2)}d\Delta(\gamma).
\end{equation}
This defines the so called \textit{trigonometric isomorphism} between the Gaussian Hilbert space $\mathcal G$ generated by $\dot{X}(\cdot)$, i.e. the close linear span of $\left\{\dot{X}(t),t\in \mathbb R \right\}$ in $\mathbf L_2\left(\Omega, \mathscr F_{\mathbb R},\mathbb P\right)$, and the space $\mathbf L_2\left(d\Delta \right)$, given by
\[
\dot{X}(t)\longrightarrow e^{i\gamma t}.
\]

If $\dot{X}(\cdot)$ is path-wise integrable then $X(t)\triangleq\int_0^t \dot{X}(s)ds$ is a Gaussian stationary increment process, with covariance function
\begin{equation} \label{eq:cov}
\mathbb E\left[X(t_1)X(t_2)\right]=\int_{-\infty}^\infty  \frac{1-e^{i\gamma t_1}}{i\gamma} \frac{1-e^{-i\gamma t_2}}{-i\gamma} d\Delta(\gamma),
\end{equation}
so that $\frac{e^{i\gamma t}-1}{i\gamma }$ is the image of $X(t)$ under the trigonometric isomorphism. In the setting of distributions, we can write the following correspondences
\begin{equation} \label{eq:correspondance}
\begin{split}
\dot{X}(t)  \longleftrightarrow & \quad e^{i\gamma t}  \quad \longleftrightarrow  \delta(t-\cdot),  \\
X(t)  \longleftrightarrow   & ~~ \frac{e^{i\gamma t-1}}{i\gamma}  ~~ \longleftrightarrow  \mathbf 1_t,
\end{split}
\end{equation}
where the left relation is the trigonometric isomorphism and the right relation is the Fourier transform. In \eqref{eq:correspondance} we used $\delta(t)$ to denote the Dirac delta distribution concentrated at the origin and
\[
\mathbf 1_t(x) \triangleq \mathbf 1_{[0,t]}(x) \triangleq \begin{cases} 1, & x\in [0,t] \\
0, & x\notin [0,t],
\end{cases}.
\]
We see that for a given $t\geq 0$, $X(t)$ may be interpreted as the stochastic integral of the deterministic function $\mathbf 1_t$ \cite[p. 87]{MR99f:60082}, and can be extended to $t<0$ by setting
\[
\mathbf 1_t(x) = \begin{cases} 1 & 0< x \leq t, \\
      -1 & -t\leq x < 0, \\
   0 & {\rm otherwise} \end{cases}.
 \]

For $f\in \mathbf L_2(\mathbb R)$ we denote by $\widehat{f}$ its Fourier transform
\[
 \widehat{f}(\gamma)=\int_{-\infty}^\infty f(t)e^{i\gamma t}
\]
and by $\widecheck{f}$ its inverse. \\

In general, for $f\in \mathbf L_2(\mathbb R)$ subject to
\begin{equation} \label{eq:hilbert_space_condition}
\int_{\mathbb R} |\widehat{f}(\gamma)|^2d\Delta(\gamma) < \infty,
\end{equation}
we can define its stochastic integral with respect to the process $X(\cdot)$ as the zero mean Gaussian random variable with variance $\int_{\mathbb R} |\widehat{f}(\gamma)|^2d\Delta(\gamma)$. The set of functions in $\mathbf L_2(\mathbb R)$ which satisfy \eqref{eq:hilbert_space_condition} constitute a pre-Hilbert space, and we denote its completion by $\mathbf L_\Delta$. The map $I:\mathbf L_\Delta \longrightarrow \mathcal G$ that carries an element of $\mathbf L_\Delta$ into its stochastic integral is an Hilbert space isomorphism, and $\mathcal G$ can be regarded as the Gaussian Hilbert spaces associated with the Hilbert space $\mathbf L_\Delta$ \cite{MR99f:60082}, in the sense that for each $f_1,...,f_n \in L_\Delta$, $I(f_1),...,I(f_n)$ have a joint central normal distribution with covariance matrix $Q$,
\[
Q_{j,i}=\left(f_i,f_j\right)_{\Delta}=\int_{-\infty}^\infty \widehat{f}_i(\gamma)  \overline{\widehat{f}_j}(\gamma)d\Delta(\gamma),
\]
where $\left(\cdot,\cdot\right)_\Delta$ is the inner product in $\mathbf L_\Delta$ induced by the norm \eqref{eq:hilbert_space_condition}. Using these notations, the covariance function \eqref{eq:cov} can be written as
\[
\mathbb E\left[X(t) X(s)\right]=\left(\mathbf 1_t,\mathbf 1_s\right)_\Delta.
\]
In the case of $d\Delta=d\gamma$, $\mathbf L_\Delta$ reduces to $\mathbf L_2(\mathbb R)$ and the image of $f\in L_2(\mathbb R)$ under $I$ is called the Wiener stochastic integral of $f$ \cite[Chapter 9]{doob1953stochastic}. \\

\begin{Rk} \label{rk:pathwise_integral}
We note that for $f\in \mathbf L_\Delta$, one can define its stochastic integral with respect to $X$ in the usual way by first setting
\[
\int_{\mathbb R} f(t) dX(t)=\sum_i \alpha_i \left(X(t_{i+1})-X(t_i)\right)
\]
for a simple function $f(t)=\sum_i \alpha_i \mathbf 1_{[t_{i+1},t_i]}$, and then take the limit in $\mathbf L_\Delta$ for a general $f\in \mathbf L_\Delta$. It can be shown that we obtain
\begin{equation} \label{eq:Wiener_integral}
\int_{\mathbb R}f(t)dX(t)= I(f)
\end{equation}
in $\mathbf L_2\left(\Omega,\mathscr F_{\mathbb R},\mathbb P\right)$, that is, both definitions coincide.
\end{Rk}

In many practical cases, almost every sample path of the stationary increment process $X(\cdot)$ is nowhere differentiable. This happens for example in the case of the Brownian motion or the fractional Brownian motion. However under the condition
\begin{equation} \label{eq:measure_growth_condition}
\int_{\mathbb R} \frac{d\Delta(\gamma)}{1+\gamma^2} <\infty,
\end{equation}
it is easy to see that the indicator function $\mathbf 1_t$ still belongs to $\mathbf L_\Delta$. Since both spaces $\mathcal G$ and $\mathbf L_\Delta$, as well as the isometric map between them, are determined exclusively by the spectral measure $\Delta$, starting with $\Delta$, we may use the representation
\[
X(t)=I(\mathbf 1_t),\quad t\in \mathbb R
\]
as the definition of the process $X$. We set
\[
z_t=T(\mathbf 1_t)=\frac{e^{i\gamma t-1}}{i\gamma},\quad t\in \mathbb R.
\]
Under the condition \eqref{eq:measure_growth_condition}, each $z_t$ belongs to $\mathbf L_2(d\Delta)$. Denote by $\mathbf Z$ the close linear span of $\left\{z_t,~t\in \mathbb R\right\}$ in $\mathbf L_2(d\Delta)$ and by $\mathbf Z_A$ the close linear span of $\left\{z_t,~t \in A\right\}$ in $L_2(d\Delta)$. It is well known (see for example \cite{Dmk} that $\mathbf Z=\mathbf L_2(d\Delta)$. \\

We have obtained the following isomorphic Hilbert spaces
\[
\mathbf L_\Delta \xrightarrow{ \quad I \quad } \mathcal G \xrightarrow{ \quad T \quad } \mathbf Z.
\]
Note that in the sense of distributions,  $T\circ I: \mathbf L_\Delta \rightarrow \mathbf Z$ is the Fourier transform. \\
The importance of the above Hilbert spaces isomorphism is that it allows us to exchange the problem of obtaining an orthogonal basis for $\mathcal G$ and $\mathcal G_A$ with the problem of doing so in $\mathbf Z$ and $\mathbf Z_A$. Our benefit comes from the fact that now the theory of orthogonal projections into spaces of analytical functions is at our disposal.

\section{Prediction with respect to the entire past}
\label{sec:entire_past}
In order to be in the setting of Theorem \ref{th:main}, we first need to  find an explicit orthogonal basis for the space $\mathcal G_A$. In this section we will show how to do so in the case that $A=\left(-\infty,0\right]$ which corresponds to the Wiener-Kolmogorov prediction problem. The case where $A=\left[-T,T\right]$ for some $T>0$ which corresponds to the problem solved by Krein is treated in the next section.\\

Recall that in the case of prediction with respect to the entire past, Szeg\"{o} theorem provides us with a criterion whether the prediction is perfect or not, i.e. when
\[
\mathbb E\left[X(t)| \mathscr F_{\left(-\infty,0\right]}\right]=X(t),\quad \forall t\in \mathbb R,
\]
is in $\mathbf L_2(\Omega,\mathscr F_{\mathbb R},\mathbb P)$ or not. Or in trigonometric language: whether $z_t\in \mathbf Z_{(-\infty,0]}$ or else
\[
\mathbb E\left[\left(X(t)-\mathbb E\left[X(t)| \mathscr F_{\left(-\infty,0\right]}\right]\right)^2 \right]>0.
\]
Szeg\"{o} criterion says that if
\begin{equation} \label{eq:szego1}
\int_{-\infty}^\infty \frac{\log\Delta'(\gamma)}{\gamma^2+1}d\gamma>-\infty,
\end{equation}
then $\mathbf  Z_{(-\infty,0]} \neq \mathbf Z$, and in particular $z_T \notin \mathbf Z_{(-\infty,0]}$ for any $T>0$.
The other option
\begin{equation}
\int_{-\infty}^\infty \frac{\log\Delta'(\gamma)}{\gamma^2+1}d\gamma = -\infty,
\end{equation}
implies $\mathbf Z_{(-\infty,0]} =\mathbf  Z$, i.e., the future is completely determined by the past.

Karhunen(1950) has showed that under the conditions \eqref{eq:szego1} and $\Delta(\infty)=\int_{-\infty}^\infty d\Delta(\gamma) <\infty$, the spectral density can be decomposed as
\begin{equation} \label{eq:spectral_dec}
\Delta'(\gamma)=h(\gamma)\overline{h}(\gamma),
\end{equation}
where $h$ is an \textit{outer function} in the Hardy space $H^{2+}$ (see \cite{MR0027954}). An outer function $h\in H^{2+}$ satisfies the property that the span of $e^{i\gamma t}\overline{h}(\gamma)$, $t<0$, in $\mathbf L_2(d\gamma)$ equals $H^{2-}$,
or equivalently, that the
span of $z_t \overline{h}$, $t<0$, in $\mathbf L_2(\mathbb R)$ equals $H^{2-}$. To see this equivalence we note that, for $k\in\mathbf L_1(\mathbb R)$
\[
\int_{\mathbb R}e^{i\gamma t}k(\gamma)d\gamma=0,\forall t>0\,\,\iff\,\,
\int_{\mathbb R}\frac{e^{i\gamma t}-1}{\gamma}k(\gamma)d\gamma=0, \forall t>0
\]
as is seen by differentiation and integration with respect to $t$.\\

Throughout this section we assume the spectral measure $\Delta$ satisfies both condition \eqref{eq:measure_growth_condition} and Szeg\"{o} criterion for optimal prediction with respect to the entire past. We further assume that $d\Delta$ is absolutely continuous with respect to the Lebesgue measure, namely $d\Delta=\Delta'(\gamma)d\gamma$. In view of the discussion in \cite[Section 4.3]{Dmk}, this assumption does not limit the generality of our approach. Since these assumption does not yet guaranty $\Delta(\infty)<\infty$, we look instead at the measure $\frac{\Delta'(\gamma) d\gamma}{1+\gamma^2}$. We have
\[
\int_{-\infty}^\infty \frac{\log\left(\frac{\Delta'(\gamma)}{1+\gamma^2} \right)}{1+\gamma^2} d\gamma < \infty,
\]
so that we may decompose $\frac{\Delta'(\gamma) d\gamma}{1+\gamma^2}$ as
\begin{equation} \label{eq:def_h}
\frac{\Delta'(\gamma) d\gamma}{1+\gamma^2} = h(\gamma)\overline{h}(\gamma)
\end{equation}
with $h$ outer, and thus obtain the decomposition:
\[
d\Delta=\Delta'(\gamma)d\gamma=|\left(\gamma-i\right)h(\gamma)|^2d\gamma.
\]
Since the process $X$ is real, $\Delta$ is always even and we can also impose the condition $h(-\gamma)=-\overline{h}(\gamma)$ \cite[Exercise 2.7.4]{Dmk}, so that the inverse Fourier transform of $(\gamma-i)h$ is a real distribution.\\

\begin{lem} \label{lem:span}
The closed linear span of the functions $\left\{(\gamma+i)z_th,\,\,t\leq 0 \right\}$ in $\mathbf L_2(d\gamma)$ equals $H^{2-}$.
\end{lem}
\begin{proof}
Denote by $\mathbf K$ the close linear span of $\left\{z_t(\gamma+i)h,\,\,t\leq 0 \right\}$ in $\mathbf L_2(d\gamma)$.
Let $\mathbf L$ be the set of functions $f\in H^{2-}$ such that $(\gamma-i)f(\gamma)$ still belongs to $H^{2-}$. It has been noted above that for $h\in H^{2-}$ outer, the closed linear span of $\left\{z_th,\,\,t \leq 0 \right\}$ in $\mathbf L_2(d\gamma)$ is all $H^{2-}$, so if a function $f\in \mathbf L$ satisfies
\[
\int_{-\infty}^\infty z_t(\gamma)(\gamma+i)h(\gamma)\overline{f}(\gamma)d\gamma=0,
\]
for all $t\leq 0$, we conclude that $(\gamma+i)f \equiv 0$, so $f=0$, and we have that $\mathbf K^\perp \bigcap \mathbf L=\left\{0\right\}$. In order to complete the proof it is enough to show that $\mathbf L$ is a dense subset of $H^{2-}$. The Schwartz space $\mathscr S$ of smooth rapidly decreasing functions is a dense subset of $\mathbf L_2(\mathbb R)$ and is invariant under differentiation and Fourier transformation. For $s\in \mathscr S$ we denote by $\widetilde{s}$ its projection into $\mathbf L_2\left( (-\infty,0]\right
$, which is the set of functions in $\mathbf L_2(\mathbb R)$ supported in $\left(-\infty,0\right]$. Recall that $H^{2-}=\widehat{\mathbf L_2}\left( (-\infty,0]\right) = \left\{ \widehat{f},\,f\in \mathbf L_2\left((-\infty,0] \right) \right\}$, which implies that $\widehat{\widetilde{\mathscr S}}$ is a dense subset of $H^{2-}$. We prove that it is also contained in $\mathbf L$. Let $s\in \mathscr S$, then
\[
 \left(\gamma+i\right)\widehat{\widetilde{s}}= \left(\gamma+i\right)\int_{-\infty}^0 s(t)e^{i\gamma t}dt= i\int_{-\infty}^0 \frac{ds}{dt}(t)e^{i\gamma t}dt+i\int_{-\infty}^0 s(t)e^{i\gamma t}dt,
\]
and the last two terms are the Fourier transform of functions in  $\mathbf L_2\left( (-\infty,0]\right)$, hence belong to $H^{2-}$. This completes the proof.
\end{proof}

In what follows we construct an orthonormal basis for the space $\mathbf L_\Delta$ in terms of the function $h$ and the functions
\[
e_n(\gamma)=\frac{1}{\sqrt{\pi}} \frac{1}{1-i\gamma}\left(\frac{1+i\gamma}{1-i\gamma}\right)^n, n\in \mathbb Z,
\]
which constitute an orthonormal basis of $\mathbf L_2(\mathbb R)$. In addition, the family $\left\{e_n,\,\, n\geq 0\right\}$ spans the Hardy space $H^{2+}$ while the family $\left\{e_n ,\,\, n<0\right\}$ spans $H^{2-}$; see \cite[Section 2.5]{Dmk}.
We also note that for $n\geq 0$ and $x>0$, the inverse Fourier transform of the $\{e_n\}$ are the Laguerre functions:
\[
e_n^{\vee}(x)=\frac{1}{\sqrt{\pi}n!}\frac{d^n}{d\gamma^n}\left(e^{-i\gamma x}\left(i-\gamma\right)^n \right)
\]
evaluated at $\gamma=-i$. \\

We are now looking for a set of functions in $\mathbf L_\Delta$ whose images under $I$ constitute an orthogonal basis in $\mathcal G$. In view of \eqref{eq:def_h}, the function $(\gamma+i)\overline{h}s$, when $s$ belongs to the Schwartz space $\mathscr S$ of smooth rapidly decreasing functions, belongs to $\mathbf L_2(\mathbb R)$. Moreover
the linear span of the functions $(\gamma+i)\overline{h}s$ with $s\in\mathscr S$ is dense in $\mathbf L_2(d\gamma)$. Indeed, let $g\in\mathbf L_2(d\gamma)$ be such
that
\[
\int_{\mathbb R}s(\gamma)(\gamma+i)\overline{h}(\gamma)\overline{g}(\gamma)d\gamma=0,\quad \forall s\in\mathscr S.
\]
Then the function $(\gamma-i)h\overline{g}$ (which need not belong to $\mathbf L_2(d\gamma)$) defines the zero distribution on $\mathscr S$, and so is a.e. equal to $0$.\\
This proves that there exists a sequence $\left\{s_k\right\}:=\left\{s_k,\,k\in \mathbb N\right\}$ of Schwartz functions such that
\[
\lim_{k\rightarrow\infty}\|(\gamma+i)\overline{h}s_k-e_n\|_{\mathbf L_2(d\gamma)}=0.
\]
Therefore the sequence $\left\{s_k\right\}$ tends to $\frac{e_n}{(\gamma+i)\overline{h}}$ in $\mathbf L_2(d\Delta)$,
and so the sequence $\left\{\widecheck{s_k}\right\}$ is a Cauchy sequence in $\mathbf L_\Delta$. We denote its limit
by $\xi_n$.\\

If $\frac{e_n}{(\gamma+i)\overline{h}}$ is in $\mathbf L_2(d\gamma)$, then $\widehat{\xi_n}=\frac{e_n}{(\gamma+i)\overline{h}}$.

\begin{theorem} \label{th:orth_basis}
The set $\left\{ I(\xi_n),~n\in \mathbb Z\right\}$ forms an orthonormal basis for $\mathcal G$. Moreover,
\[
    \mathbb E\left[  I(\xi_n)|\mathscr F_{(-\infty,0]}\right]= \begin{cases}
                                                             I(\xi_n), & n<0 \\
                                                            0,  & n\geq 0
                                                            \end{cases},
\]
so that $\left\{ I(\xi_n),~n<0 \right\}$ spans the past, and $\left\{I(\xi_n),~n\geq 0 \right\}$ spans its orthogonal complement.
\end{theorem}

\begin{proof}
The fact that the $ I(\xi_n)$ are orthonormal is immediate by construction since
\[
\mathbb E\left[ I(\xi_n)  I(\xi_m)\right]=\int_{-\infty}^\infty \frac{e_n(\gamma)}{(\gamma-i)\overline{h}(\gamma)}\frac{\overline{e_m}(\gamma)}{(\gamma+i)h(\gamma)} (1+\gamma^2)|h(\gamma)|^2 d\gamma=\left(e_n,e_m\right)_{\mathbf L_2(d\gamma)}.
\]
To show that they span $\mathcal G$, let $f \in \mathbf L_\Delta$ and assume that $ I(f)$ is perpendicular to their span. Then for all $n \in \mathbb Z$,
\[
0=\mathbb E\left[ I(\xi_n)  I(f)\right]=\int_{-\infty}^\infty e_n (\gamma-i) \overline{\widehat{f}}{h} d\gamma=\left(e_n,(\gamma+i)\overline{h}\widehat{f}\right)_{\mathbf L_2(d\gamma)}.
\]
Since $\left\{e_n,\,\, n\in \mathbb Z\right\}$ is an orthonormal basis for $\mathbf L_2(d\gamma)$, it follows that $\widehat{f}~\overline{h}$ is zero in $\mathbf L_2(d\gamma)$. But it follows from condition \eqref{eq:szego1} that $h(\gamma)\neq 0$ almost everywhere, so we conclude that $f$, and thus $I(f)$, equals zero. Note that this also proves that $\left\{\xi_n,\,\, n\in \mathbb Z\right\}$ is an orthonormal basis of $\mathbf L_\Delta$.\\
Now for $t\leq 0$,
\begin{equation} \label{eq:proj}
\begin{split}
\mathbb E\left[I(\xi_n)X(t)\right] & =\int_{-\infty}^\infty \frac{e_n}{(\gamma+i)\overline{h}}(\gamma^2+1)\overline{z_t} \left|h(\gamma)\right|^2 d\gamma \\
& = \int_{-\infty}^\infty  (\gamma-i)e_n \overline{z_t}h d\gamma \\
& = \left(e_n,(\gamma+i)z_t\overline{h} \right)_{\mathbf L_2(d\gamma)}.
\end{split}
\end{equation}
From Lemma~\ref{lem:span} we know that the span of $\left\{ z_t(\gamma+i)\overline{h},\,\,t \leq 0  \right\}$ in $\mathbf L_2(\mathbb R)$ equals $H^{2-}$, so the last term in \eqref{eq:proj} vanishes for $n\geq0$. In order to calculate $\widetilde{I(\xi_n)}\triangleq \mathbb E\left[I(\xi_n)|\mathscr F_{(-\infty,0]}\right]$ for $n<0$ we can use the trigonometric isomorphism and instead look for the projection of $\frac{e_n(\gamma)}{(\gamma+i)\overline{h}}$ onto $\mathbf Z_{(-\infty,0]}$. Let $g=c_1 z_t+ \ldots +c_nz_t$, where $t_1,\ldots t_n, \leq 0$ and $c_1,...,c_n \in \mathbb C$.
\[
\begin{split}
\mathbb E\left[|I(\xi_n)-\widetilde{I(\xi_n)}|^2\right]=\inf_{g}\int_{-\infty}^\infty|
\frac{e_n(\gamma)}{(\gamma+i)\overline{h}(\gamma)}- g(\gamma)|^2 \Delta'(\gamma) d\gamma= \\
\inf_{g}\int_{-\infty}^\infty
|e_n(\gamma)- g(\gamma)(\gamma+i)\overline{h}(\gamma)|^2 d\gamma.
\end{split}
\]
Since the span of $\left\{ z_t(\gamma+i)\overline{h},\,\,t \leq 0  \right\}$ is $H^{2-}$ and $e_n \in H^{2-}$ for $n<0$, the last projection norm is trivial, so $\widetilde{I(\xi_n)}=I(\xi_n)$ in $\mathcal G$, and hence $I(\xi_n) \in \mathscr F_{(-\infty, 0]}$.
\end{proof}

\begin{Ex}
The chaos expansion of $X(t)$ is given by
\[
 X(t)=I(\mathbf 1_t)=\sum_{n=-\infty}^\infty \left(\xi_n,\mathbf 1_t \right)_\Delta  I\left(\xi_n\right)=\sum_{n=-\infty}^\infty \left(\xi_n,\mathbf 1_t \right)_\Delta H_{\epsilon(n)},
\]
where $\epsilon(n)=\left(...,0,1,0,...\right)$ with $1$ at the $n_{th}$ place. It follows that
\begin{equation} \label{eq:conditional_expansion}
\mathbb E\left[X(t)|\mathscr F_{(-\infty,0]} \right]=\sum_{n=-\infty}^{-1} \left(\xi_n,\mathbf 1_t \right)_\Delta I\left(\xi_n\right).
\end{equation}

The complementary projection is given by
\[
 \sum_{n=0}^{\infty} \left(\xi_n,\mathbf 1_t \right)_\Delta I\left(\xi_n\right),
\]
so that the variance of the prediction error is
\[
\sum_{n=0}^{\infty} \left(\xi_n,\mathbf 1_t \right)^2_\Delta.
\]
\end{Ex}

\begin{Ex}
The Wick exponent of the process $X(t)$ has the chaos expansion
 \[
:e^{ X(t)}: = \exp \left\{ I(\mathbf 1_t)-\| \mathbf 1_t \|_{\Delta}^2 \right\} = \sum_{\alpha\in \mathcal J} c_\alpha H_\alpha,
 \]
 with (see \cite[Exercise 2.8 (e)]{new_sde}
 \[
 c_\alpha = \prod_{n=-\infty}^\infty \frac{\left(1_t,\xi_n\right)_\Delta^{\alpha_n}}{\alpha_n!}.
 \]
 It follows that
 \[
 \mathbb E\left[:e^{X(t)}:| \mathscr F_{(-\infty,0]}  \right]=\sum_{\alpha\in \mathcal J_0} c_\alpha H_\alpha,
 \]
 where now for $\alpha \in \mathcal J_0$,
 \[
 c_\alpha = \prod_{n=-\infty}^{-1} \frac{\left(1_t,\xi_n\right)_\Delta^{\alpha_n}}{\alpha_n!}.
 \]

From \cite{MR99f:60082} we have
\[
\mathbb E\left[ :e^{ U}:| \mathscr F' \right]=:e^{ \mathbb E\left[U |\mathscr F' \right]}:=\exp\left\{\mathbb E\left[U | \mathscr F' \right]-\frac{1}{2}\mathbb E\left[X^2 | \mathscr F' \right]\right\}
\]
and we obtain the following identity:
\begin{equation}
\sum_{\alpha\in \mathcal J_0} c_\alpha H_\alpha= \exp \left\{\sum_{n=-\infty}^{-1} \left(\xi_n,\mathbf 1_t \right)_\Delta I\left(\xi_n\right)-\frac{1}{2}\sum_{n=0}^{\infty} \left(\xi_n,\mathbf 1_t \right)^2_\Delta\right\}.
\end{equation}
\end{Ex}

\subsection{Basis Elements and Sample Path Relation}
In the classical prediction problem we are asked to find the conditional expectation with respect to $\mathscr F_{(-\infty,0]}$ expressed in term of the path $X_{(-\infty,0]}(\cdot) = \left\{X(t),\,t\in A \right\}$. By Theorem \ref{th:orth_basis} for $n<0$,
$I(\xi_n)$  is completely determined by $X_{(-\infty,0]}(\cdot)$.
Due to the trigonometric isomorphism we have
\[
I(\xi_n)=I\left(\mathbf 1_{[-\infty,0]}\xi_n\right)+I\left(\mathbf 1_{[0,\infty]}\xi_n\right),
\]
and
\begin{equation} \label{eq:xi_is_zero}
 \|1_{[0,\infty]}\xi_n\|_\Delta^2=\int_{-\infty}^\infty  \left|\widehat{1_{[0,\infty]}\xi_n}(\gamma) \right|^2 d\Delta(\gamma).
\end{equation}
Note that $\widehat{1_{[0,\infty]}\xi_n}$ is the projection of
\begin{equation} \label{eq:proj_2_H2}
\widehat{\xi_n}(\gamma)=\frac{e_n(\gamma)}{(\gamma+i)\overline{h}(\gamma)}= \frac{1}{\sqrt{\pi}} \left(\frac{1+i\gamma}{1-i\gamma}\right)^n  \frac{ 1}{(1+\gamma^2)\overline{h}(\gamma)}
\end{equation}
into $H^{2+}$ (see \cite[Ch. 2.4]{Dmk}). If $\overline{h}$ does not vanish too fast as $\gamma \rightarrow \infty$, (in general, condition \eqref{eq:szego1} does not guarantee that), then $\widehat{\xi_n}$ belongs to $H^{2-}$, in which case $\widehat{1_{[0,\infty]}\xi_n}=0$. That is, we have the following proposition:
\begin{prop} \label{prop:path}  If $\widehat{\xi_n} \in \mathbf L_2(d\gamma)$, then
\[
\mathbb E \left[ I(\xi_n) | \mathscr F_{(-\infty,0]}\right] =  I(\xi_n \mathbf 1_{(-\infty,0]}),
\]
 for all $n=-1,-2,\ldots$.
\end{prop}
If the condition in Proposition~\ref{prop:path} is met, then the $\mathbf L_2\left(\Omega,\mathscr F_{\mathbb R},\mathbb P\right)$ stochastic integral with respect to the process $X(t)$ can also be evaluated from its sample path which is defined in the standard way as in explained in \eqref{eq:Wiener_integral}. See also \cite[Sec. 2]{Gripenberg96} for a pathwise definition of the stochastic integral with respect to the fractional Brownian motion. In such case we get
\begin{equation} \label{eq:fBm_int}
I(\xi_n) = I(\mathbf 1_{(-\infty,0]}\xi_n)=\int_{-\infty}^0 \xi_n(t) dX(t).
\end{equation}
This allows us to express the $I(\xi_n)$ in \eqref{eq:conditional_expansion} in terms of the sample path, which leads to a prediction formula for a general Gaussian stationary increment process.

\section{Bounded time interval} \label{sec:bti}

In the case where $A=[-T,T]$ we are looking for an orthogonal basis for $\mathcal G_{[-T,T]}$, or equivalently,
for its image under the trigonometric isomorphism $\mathbf Z_{[-T,T]}$. For the conditions to optimal prediction in this case we refer to \cite[Section 6.4]{Dmk}. The space $\mathbf Z_{[-T,T]}$ is a reproducing kernel Hilbert space of entire functions isometrically included in $\mathbf L_2(d\Delta)$, and invariant under
the backward shift operators
\[
R_af(\gamma)=\frac{f(\gamma)-f(a)}{\gamma-a},\quad a\in\mathbb C.
\]
Therefore, by a theorem of de Branges, see \cite[Theorem 3]{dbhsaf1}, and (for instance) by an application of
\cite[Theorem 3.1]{ad1}, one sees that the reproducing kernel of $\mathbf Z_{[-T,T]}$ is of the form
\begin{equation}
\label{dbrk}
K(\gamma,\lambda)=\frac{B(\gamma)\overline{A}(\lambda)-A(\gamma)\overline{B}(\lambda)}{\gamma-\overline{\lambda}},
\end{equation}
where $A(\gamma)$ and $B(\gamma)$ are entire function of the variable $\lambda$ of finite exponential type.
A characterization of certain orthogonal sets in such spaces is given in \cite[Theorem 22]{MR0229011}. We recall the
result for completeness. Set $E(\gamma)=A(\gamma)-iB(\gamma)$. Then there exists a continuous function
$\varphi(x)$ ($x\in\mathbb R$) such that $E(x,T)e^{i\varphi(x)}\in\mathbb R$ for all $x\in\mathbb R$. Let
$\alpha\in\mathbb R$ and let $x_1,x_2,\ldots \in\mathbb R$ be such that $\varphi(x_n)\equiv \alpha$ (mod $\pi$).
The functions
\[
K(\gamma, x_n)
\]
form an orthogonal set of $\mathbf Z_{[-T,T]}$, and it is complete if and only if the function $e^{-i\alpha}E(\gamma)-e^{i\alpha}
\overline{E}(\overline{\gamma})$ does not belong to $\mathbf Z_{[-T,T]}$.\\

One can compute explicitly the functions $A(\gamma)$ and $B(\gamma)$ in some special cases. For instance
Dym and Gohberg considered in \cite{dg-80} the case where the spectral density $\Delta^\prime$ is the form
\[
\Delta^\prime(\gamma)=1-\widehat{k}(\gamma),
\]
where $k\in\mathbf L^1(\mathbb R)$ and such that $1-\widehat{k}>0$
(in fact, they consider the matrix-valued non Hermitian case). The case where
\[
\Delta^\prime(\gamma)= c_H |\gamma|^{1-2H},
\]
which corresponds to the fractional Brownian motion, was considered by Dzhaparidze and H. van Zanten in \cite{DVZ} and will be revisited again in Section \ref{sec:fbm_bounded}.
More generally, one needs to use Kreins's theory of strings, as explained in \cite{Dmk} and \cite[Section 2.8]{DVZ}, to compute the reproducing kernel of $\mathbf Z_{[-T,T]}$. \\

\section{Chaos and Prediction with respect to the Fractional Brownian Motion}
\label{sec:application}
\setcounter{equation}{0}

In this section we now specialize to the case where the spectral measure of $X$ is given by
\begin{equation} \label{eq:fBm_spectral}
d\Delta(\gamma)= C_H |\gamma|^{1-2H} d\gamma,\quad 0<H<1,
\end{equation}
where $C_H = \frac{\Gamma(1+2H) \sin \left(\pi H\right)} {2 \pi}$ and $\Gamma(x)$ is Euler's Gamma function. This measure satisfies conditions \eqref{eq:measure_growth_condition} and \eqref{eq:szego1}. The corresponding stationary increment Gaussian process $X$ is called the fractional Brownian motion with Hurst parameter $H$ and is denoted $B_H$. Its covariance function is given by
\[
\mathbb E\left[ B_H(t) B_H(s) \right]= \frac{1}{2}\left( |t|^{2H}+|s|^{2H}-|t-s|^{2H} \right),
\]
and it can be shown to have an almost surely continuous sample paths \cite{Molchan}. This process has been found useful in a host of applications, and was extensively studied in the past few decades; see for example \cite{bosw, Ustunel, MR1790083, duncan2009control}. \\

In what follows, we will evaluate the coefficients in the chaos expansion for $B_H(\cdot)$ based on an orthonormal basis with the properties of Theorem \ref{th:main2}. We start with the case of prediction with respect to the entire past.

\subsection{Prediction with respect to the Entire Past ($A=\left(-\infty,0\right]$)}
For this process $B_H(\cdot)$  with $A=\left(-\infty,0\right]$ we derive the following:

\begin{theorem}
The outer function in the Wiener-Hopf spectral decomposition of $\frac{\Delta'(\gamma)}{1+\gamma^2}$, where $\Delta'(\gamma)=C_H |\gamma|^{1-2H}$, that admits the reality condition $(-\gamma-i)h(-\gamma)=(\gamma+i)\overline{h}(\gamma)$ is given by
\[
h(\gamma) =\sqrt{C_H}\exp\left\{i \pi \frac{2H-1}{4}sign(\gamma)\right\} \frac{i-\gamma}{1+\gamma^2} |\gamma|^{1/2-H}.
\]
\end{theorem}

\begin{proof}
Using the formula \cite[(11), p. 193]{duren}, to find the outer factor $\tilde{h}$ in the decomposition \eqref{eq:def_h},
 we see that the outer factor in the factorization of \eqref{eq:fBm_spectral} is given by
\begin{equation} \label{eq:fBm_outer_integral}
\begin{split}
\phi(z) & = \exp \left\{\frac{1}{2\pi  i} \int_{-\infty}^\infty  \frac{\gamma z +1}{\gamma -z} \frac{\log(C_H |\gamma |^{1-2 H} / (\gamma^2+1) )} {\gamma^2+1} d\gamma   \right\} \\
& = \exp\left\{ -i\left(\pi/2-\tan^{-1}(z)-z\sqrt{-1/z^2} \tanh^{-1} (1+\frac{2}{z^2}) \right) \right\} \\
\times & \exp \left\{ -\frac{i}{4\pi} \left( \log(-z)-\log(z) \right)\left( (2H-1)\left(\log(z)+\log(-z) \right)-2\log(C_H)  \right) \right\}
\end{split}
\end{equation}
Taking the restriction of $\phi(z)$ to the real line we obtain
\[
\frac{i\gamma+1}{1+\gamma^2}  ~ \sqrt{C_H} \exp\left\{sign(\gamma) \frac{2H-1}{4} \pi i\right\} |\gamma|^{1/2-H}.
\]
To finish the proof we multiply the last term by $i$ in order to impose the reality condition
\[
(-\gamma-i)h(-\gamma)=(\gamma+i)\overline{h}(\gamma) \Longleftrightarrow \overline{h}(-\gamma)=-\overline{h}(\gamma).
\]
\end{proof}

The deterministic coefficients in the chaos expansion \eqref{eq:conditional_expansion} for the conditional expectation $\mathbb E\left[ B_H(t)| \mathscr F_{(-\infty,0]} \right]$ are given by
\[
\begin{split}
r^H_j(t) \triangleq \left(\mathbf 1_t,\xi_j\right)_\Delta & = \int_{-\infty}^\infty z_t(\gamma) \frac{\overline{e_j}(\gamma)}{h(\gamma)(\gamma-i)} (1+\gamma^2)|h(\gamma)|^2d\gamma \\
& =\frac{1}{\sqrt{\pi C_H}} \int_{-\infty}^\infty \frac{e^{i\gamma t}-1}{\gamma}   \left(\frac{1-i\gamma}{1+i\gamma}\right)^j \\
& \times \exp\left\{i \pi \frac{2H-1}{4}sign(\gamma)\right\}  |\gamma|^{H-0.5}  d\gamma.
\end{split}
\]
See Figure \ref{fig:Coeff} for a graphical illustration of these coefficients for a few cases of the Hurst parameter $H$.

\begin{figure}
        \begin{subfigure}[b]{0.3\textwidth}
                \centering
                \includegraphics[scale=0.3]{coeff_T1_H02.eps}
                \caption{H=0.2}
        \end{subfigure}%
        \begin{subfigure}[b]{0.3\textwidth}
                \centering
                \includegraphics[scale=0.3]{coeff_T1_H05.eps}
                \caption{H=0.5}
        \end{subfigure}
        \begin{subfigure}[b]{0.3\textwidth}
                \centering
                \includegraphics[scale=0.3]{coeff_T1_H07.eps}
                \caption{H=0.7}
        \end{subfigure}
        \caption{The value of the coefficient $r^H_j(1)=\left( \mathbf 1_{[0,1]}, \xi_j \right)_\Delta$, $-100\leq j \leq 100$, in the sum \eqref{eq:PaleyWiener_fBm} for different values of the Hurst parameter $H$ and $t=1$. $H=0.5$ corresponds to the Brownian motion.}  \label{fig:Coeff}
\end{figure}
This gives us
\begin{equation} \label{eq:PaleyWiener_fBm}
 B_H(t)=\sum_{j\in \mathbb Z} r^H_j(t) I(\xi_j),
 \end{equation}
which is a representation of the process $B_H(t)$ as a sum of mutually orthogonal Gaussian random variables independent of time, weighted by the coefficients $\left\{ r^H_j(t),\, j\in \mathbb Z\right\}$.

It can be shown that the sum \eqref{eq:PaleyWiener_fBm} converges in $\mathbf L_2 (\Omega)$ uniformly in $t\in \mathbb R$. We now split the sum in \eqref{eq:PaleyWiener_fBm}
into two sums, of elements of $j<0$ and $j\geq0$ respectively, which by \eqref{eq:conditional_expansion} corresponds to the sum of two (in general not orthogonal) processes:
\begin{equation} \label{eq:fbm_past}
\sum_{j\leq -1}r_j^H(t) I(\xi_j) =   \mathbb E \left[B_H(t) |\mathscr F_{(-\infty,0]}\right],
\end{equation}
and
\begin{equation} \label{eq:fbm_future}
\sum_{j\geq 0} r_j^H(t) I(\xi_j) = B_H(t) -  \mathbb E \left[B_H(t) \right].
\end{equation}
For $t<0$, $r_j^H=0$ for any $j\geq0$, and $B_H(t)$ coincides with \eqref{eq:fbm_past}. For $t>0$, the sum in \eqref{eq:fbm_past} represents what the past `thinks' the future looks like given a specific realization, i.e. the projection of the future on the past. The other part of the sum \eqref{eq:fbm_future} is the complementary projection which can only be determined by the future. Figure \ref{fig:sample_path} illustrates a realization of $B_H(\cdot)$ rendered according to the two sums in \eqref{eq:fbm_past} and \eqref{eq:fbm_future}. From \eqref{eq:fbm_future} we also obtain an expression for the prediction error in estimating $B_H(t)$ for $t>0$:
\begin{equation} \label{eq:error_fbm}
\mathbb E\left(B_H(t) -  \mathbb E \left[B_H(t)|\mathscr F_{(-\infty,0]} \right] \right)^2 = \sum_{j\geq 0} \left(r_j^H(t) \right)^2.
\end{equation}
A closed form expression for $\mathbb E\left(B_H(t) -  \mathbb E \left[B_H(t) |\mathscr F_{(-\infty,0]} \right] \right)^2$
was derived in \cite{Gripenberg96}, which leads to the identity
\begin{equation} \label{eq:fbm_error_exact}
\sum_{j\geq 0} \left(r_j^H(t) \right)^2 =  \frac{\sin\left(\pi(H-\frac{1}{2})\right) \Gamma\left(\frac{3}{2}-H\right)^2} {\pi\left(H-\frac{1}{2}\right) \Gamma \left(2-2H\right)}t^{2H}.
\end{equation}
Figure~\ref{fig:sample_path_square} illustrates the prediction error obtained by truncating the sum \eqref{eq:fbm_past} compared to the optimal error \eqref{eq:fbm_error_exact}.

\begin{figure}
\begin{center}
  \includegraphics[scale=0.75, trim = 0.5cm 0cm 2cm 0cm, clip=true]{PaleyWinerExpansion.eps}
  \caption{\label{fig:sample_path} One sample path of the fractional Brownian motion $B_H$ with $H=0.7$ rendered according to the two components of the sum \eqref{eq:PaleyWiener_fBm}: The `past' corresponds to the negative indices and the 'future' to the non-negative
indices.}
  \end{center}
\end{figure}

\begin{center}
\begin{figure}
\begin{tikzpicture}
  \node at (0,0) {\includegraphics[scale=0.5, trim = 0.7cm 0.7cm 0cm 0cm, clip=true]{coeff_err.eps} };
  \node at (0,-3.5) {number of coefficients};
  \node[rotate=90] at (-6.1,0) {error};
  \end{tikzpicture}

\caption{\label{fig:sample_path_square} Error in estimating $B_H(1)$ with $H=0.7$ as a function of the number of coefficients in the sum \eqref{eq:fbm_past}. The dashed curve is the exact error obtained by \eqref{eq:fbm_error_exact}. }
\end{figure}
\end{center}

The real power of Theorem \ref{th:main2} is in simplifying expressions for the condition expectation of a non-linear function of $B_H(t)$. For example, the chaos expansion of $B_H(t)^2$ is found by the Wick product identity \cite[Eq. 2.4.10]{new_sde}:
\begin{equation}
\begin{split}
B_H^2(t)- \left| t \right|^{2H}  & =  : B_H(t), B_H(t):  \\
& = \sum_{j} \left(r^H_j(t)\right)^2 h_2(E_j) + \sum_{i\neq j} r^H_i(t) r^H_j(t) h_1(E_i)h_1(E_j),
\end{split}
\end{equation}
where $i$ and $j$ go over all integers ,and we used the fact that \[
\sum_{j\in \mathbb Z} \left(r^H_j(t) \right)^2 = \mathrm{var}\left( B_H(t)\right) = \left| t \right|^{2H}.
\]
From Theorem~\ref{th:main2} we conclude
\begin{equation}
\begin{split} \label{eq:fbm_square}
\mathbb E \left[B_H^2(t)| \mathscr F_{(-\infty,0]} \right] & = \\
 \left| t \right|^{2H} +  \sum_{j\leq -1} & \left(r^H_j(t)\right)^2 h_2(E_j) + \sum_{i\neq j \leq -1} r^H_i(t) r^H_j(t) h_1(E_i)h_1(I(E_j)).
\end{split}
\end{equation}

\subsection{Prediction with respect to $A={[-T,T]}$ \label{sec:fbm_bounded}}
An orthonormal basis $\left\{\xi_n\,\,n\in \mathbb Z\right\}$ for the space $\mathbf Z_{[-T,T]}$ in the case of the fractional Brownian motion was obtained in \cite{MR2178502}. This basis is defined in terms of the zeros of $J_{1-H}$, which is the Bessel function of the first order with parameter $1-H$. Specifically
\begin{equation}
\xi_n (\gamma)=\frac{S_T(2\gamma_n/T,\gamma)}{\| S_T (2\gamma_n/T,\cdot)\|_{\Delta}},
\end{equation}
where $...<\gamma_{-1}<\gamma_0=0<\gamma_1<...$ are the zeros of $J_{1-H}$. In addition
\[
\begin{split}
S_T(2\eta,2\gamma)=S_T(0,0)(2-2H)\Gamma^2(1-H)\left(\frac{T^2\gamma\eta}{4} \right)e^{iT(\gamma-\eta)} \\
\times \frac{J_{-H}(T\eta)J_{1-H}(T\gamma)-J_{1-H}(T\eta)J_{-H}(T\gamma)}{T(\gamma-\eta)},
\end{split}
\]
for $\eta \neq \gamma$, and
\[
\begin{split}
S_T(2\gamma,2\gamma)=S_T(0,0)(2-2H)\Gamma^2(1-H)\left(\frac{T\gamma}{2} \right) \\
\times\left( J^2_{1-H}(T\gamma)+\frac{2H-1}{T\gamma}J_{-H}(T\gamma)J_{1-H}(T\gamma)+J^2_{-H}(T\gamma)\right),
\end{split}
\]
for $\gamma \in \mathbb R$.\\

\section{Concluding Remarks}
In this work we combined the Wiener chaos decomposition with the problem of linear prediction for Gaussian stationary-increment processes. This was done by considering a special basis for the Gaussian Hilbert space generated by the process, in which each basis element is either completely measurable with respect to the observations or independent of it. This special basis allows us to easily derive the chaos expansion of a random variable conditioned on the $\sigma$-field generated by the observations. The result is a chaos approach to prediction, which can be employed to derive an intuitive expression for the chaos expansion of the random variable with respect to the observations. Since the Wiener chaos has been found useful in describing solutions to stochastic differential equations \cite{new_sde, Hou06wienerchaos}, our approach has the potential of better understanding the dynamics of such stochastic systems when past observations of the sample path are given.

\bibliographystyle{plain}

redpajama_set_name: RedPajamaArXiv


text:
Pardosa pertinax is een spinnensoort in de taxonomische indeling van de wolfspinnen (Lycosidae).

Het dier behoort tot het geslacht Pardosa. De wetenschappelijke naam van de soort werd voor het eerst geldig gepubliceerd in 2000 door von Helversen.

Wolfspinnen


text:
Three Former Buckeyes Who Would Be Perfect for the Alliance of American Football League
By David Regimbal on February 14, 2019 at 10:10 am @davidreg412
Football hasn't technically been laid to rest this year thanks to the Alliance of American Football league.
The new league kicked off its inaugural season last Saturday, giving aspiring professionals another opportunity to chase their NFL dreams. There are just eight teams and five Buckeyes already on rosters in wide receiver Jalin Marshall, offensive lineman Chase Farris, tight end Marcus Baugh, defensive tackle Tracy Sprinkle and cornerback Doran Grant.
Marshall had the best Buckeye debut.
Jalin Marshall scores the first touchdown in AAF history. pic.twitter.com/pd7xZZmu5e
— Eleven Warriors (@11W) February 10, 2019
Watching Marshall do his thing got us thinking — which former Buckeyes could use a stint in the AAF to remind the NFL what they're capable of?
Cardale Jones, Quarterback
The man who commandeered Ohio State to and through the 2014 College Football Playoff deserves better than the Los Angeles Chargers practice squad.
Jones has spent his professional career in limbo behind Phillip Rivers after Buffalo traded him to the Chargers. His career stats are bleak: just six-of-11 passing for 96 yards with an interception in just one game appearance with the Bills in 2016.
He just signed a reserve/future contract with the Chargers, who may need a backup with Geno Smith's pending free agency looming. If things don't go his way, he needs the opportunity to remind NFL teams what he can do.
Devin Smith, Wide Receiver
Devin Smith, Jones' favorite target during that playoff run, needs a fresh start as well.
The ceiling seemed so high for Smith at the start of his professional career when he was drafted by the New York Jets in the second round of the 2015 NFL draft. But due to lingering injuries, the speedy wideout appeared in just 10 games over the last four seasons, totaling a meager 10 catches for 135 yards and a touchdown.
The Dallas Cowboys just signed Smith to a reserve/future contract.
The #Cowboys have signed former #Jets second-round WR Devin Smith to a futures deal, per agent @Clarity_jb. Hes battled injuries the last few years.
If he gets healthy and Dallas still lets him go, the AAF would be a perfect showcase.
Braxton Miller, Flash of Lightning
Arguably the most electrifying athlete to suit in Scarlet and Gray since Teddy Ginn, Braxton Miller's professional career has mimicked his final season in Columbus. He's showcased the tantalizing playmaking ability, but he hasn't quite settled into the wide receiver role.
After the shoulder injury that derailed his 2014 season, Miller made the switch H-back for Ohio State. He totaled 601 yards of offense and scored four touchdowns for the Buckeyes in 2015, and the following spring, he was drafted in the third round by the Houston Texans.
Like Smith, he's battled injuries and appeared in just 21 games since 2016. He was released by the Texans before the start of last season, and was then picked up and signed to Philadelphia's practice squad. He didn't play in any game for the Eagles in 2018.
It's hard to say whether Miller will ever find his footing on the perimeter, but what's certain is the football universe deserves more highlights like this.


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