text:
Get an overview of this hotelBeach Manggis hotel in Candi Dasa with free parking
Popular property highlights
Located in Candi Dasa, this beach hotel is 1.6 mi (2.6 km) from Pura Candidasa and 2.4 mi (3.9 km) from Balina Beach. Tanah Ampo and Pasir Putih Beach are also within 9 mi (15 km).
Hotel Features
This smoke-free hotel features a restaurant, an outdoor pool, and massage/treatment rooms. Free WiFi in public areas and free self parking are also provided. Additionally, a 24-hour front desk, tour/ticket assistance, and multilingual staff are onsite.
Room Amenities
All 18 rooms feature comforts like premium bedding and pillow menus, while conveniences include dining areas and coffee makers. Free WiFi and patios are standard, as are TVs with satellite channels.Information missing or incorrect? Tell us!Opens in a new window
The Nirwana Resort & Spa, Candidasa
Hotel Amenities
Hotel Amenities
The hotel offers a restaurant. Wireless Internet access is complimentary. Event facilities measure 1076 square feet (100 square meters). For a surcharge, a shuttle from the airport to the hotel during limited hours is offered to guests. This beach hotel also offers an outdoor pool, tour/ticket assistance, and multilingual staff. Limited complimentary onsite parking is available on a first-come, first-served basis.
The Nirwana Resort & Spa, Candidasa is a smoke-free property.
- 24-hour front desk
- Airport transportation - pickup (surcharge)
- Total number of rooms - 18
- Number of floors - 1
- Breakfast available (surcharge)
- Hair salon
- Tours/ticket assistance
- Free WiFi
- Luggage storage
- Spa treatment room(s)
- Multilingual staff
- Porter/bellhop
- Restaurant
- Safe-deposit box at front desk
- Conference space size (meters) - 100
- Year Built 2005
- Conference space size (feet) - 1076
- Free self parking
- Smoke-free property
- Garden
- Outdoor pool
- Television in common areas
- Wedding services
- Parking (limited spaces)
Family Friendly Amenities
- Family Hotel
- Outdoor pool
- Free Wi-Fi
- Separate dining area
- Cribs/infant beds (complimentary)
- DVD player
Available in all rooms: Free WiFi
Available in some public areas: Free WiFi
Free self parking, Parking (limited spaces)
Room Amenities
- Private bathroom
- DVD player
- Premium bedding
- Patio
- Air conditioning
- Phone
- Minibar
- Coffee/tea maker
- Free bottled water
- Daily housekeeping
- Free toiletries
- Hair dryer
- Shower/tub combination
- Television
- Desk
- Separate dining area
- Satellite TV service
- Free WiFi
- In-room massage available
- In-room safe
- Pillow menu
- Free cribs/infant beds
- Rollaway/extra beds (surcharge)
Where to Eat
The Nirwana Resort & Spa, Candidasa has a restaurant on site.
Nearby Things to Do
Recreational amenities at the hotel include an outdoor pool.
The recreational activities listed below are available either on site or nearby; fees may apply.
- Rafting nearby
- Hiking/biking trails nearby
- Scooter/moped rentals nearby
- Mountain biking nearby
Spa Services
- Body treatments
- Facial treatments
Hotel Policies
Check-in time starts at 2 PM
Minimum check-in age is 17
Check-in time ends at 11 PM
Special check-in instructions:
Check-out time is noon
Payment types
Children and extra beds
- Children are welcome.
- Kids stay free! Children 5 years old and younger stay free when using existing bedding.
- Rollaway/extra beds are available for USD 35 per night.
- Free cribs (infant beds)!
- No pets or service animals allowed
You need to know
Extra-person charges may apply and vary depending on property policy.
Government-issued photo identification and a credit card or cash deposit are required at check-in for incidental charges.
Special requests are subject to availability upon check-in and may incur additional charges. Special requests cannot be guaranteed.
- Minimum Spring Break check-in age is 17 years old.
Minimum guest age is 17
No elevators
No pets or service animals allowed
Optional extras
- Airport shuttle fee: USD 45 per vehicle
- Rollaway bed fee: USD 35 per night
Hotel Name
- Nirwana Manggis
- Nirwana Resort Manggis
- Nirwana Resort Candidasa
- Nirwana Resort
- Nirwana Candidasa
- The Nirwana Resort Spa
- The Nirwana Resort Spa Candidasa
We should mention
This property offers transfers from the airport (surcharges may apply). Guests must contact the property with arrival details before travel, using the contact information on the booking confirmation. Children 5 years old and younger stay free when occupying the parent or guardian's room, using existing bedding. Only registered guests are allowed in the guestrooms. No pets and no service animals are allowed at this property.


text:
Melbourne, Oct 19: Putting cricket first, Australia Test vice-captain Michael Clarke is gearing up to play non-stop for the next five months, which includes all single Tests, ODIs and Twenty20 matches on Australia's agenda.
That includes two ODIs in India, a Twenty20 against Sri Lanka in Perth on Oct 31, three ODIs against Sri Lanka, five Ashes Tests after a possible two Sheffield Shield games for NSW, seven ODIs and two Twenty20s against England and then a return to the subcontinent for a maximum of nine matches at the World Cup.
Clarke may be cricket''s most eligible bachelor but he only has room in his life for one thing at the moment.
"I don''t let anything get in the way of my cricket these days. Media commitments, sponsorship commitments, my personal life, my family, my friends, that all has to wait. If I''ve got training at a certain time, I''ve got training," FOX Sports quoted him, as saying.
Cricket Australian has rested Ricky Ponting, Mitchell Johnson and Shane Watson ahead of a grueling summer, but Clarke is yet to receive any instructions from the board about having a break.
"I''m lucky that I''ve got so much support from family, friends, sponsors, everyone. It''s like doing an interview. I''m happy to talk but it''s last on the list. Once training is done, I''m fine. But preparing for our next match, every day, is what comes first," he said.

subset: commoncrawl


text: This article is a stub. You can help the City of Heroes wiki by expanding it. Proximo Contents[show] Overview Edit Proximo is a trainer in Cimerora. Background Edit Need data History Edit Need data Character Stats Edit Need data See Also Edit Need data Retrieved from "http://cityofheroes.wikia.com/wiki/Proximo?oldid=91865" Ad blocker interference detected! Wikia is a free-to-use site that makes money from advertising. We have a modified experience for viewers using ad blockers Wikia is not accessible if you've made further modifications. Remove the custom ad blocker rule(s) and the page will load as expected. Categories: Stubs Hero Trainers Villain Trainers Add category Cancel Save


text:
You have Adblock enabled. Adblock has been known to cause issues with site functionality. If you experience any difficulties, please try disabling Adblock. Close
Ajouter à la liste de vos favoris:
Créer un nouveau dossier:
First selfvideo jerking off
Recognize a pornstar in this video?
Help make pornstars easier to find on Tube8 by telling us who is in this video.
Thanks! The pornstar names will be added to this video when enough users submit the same names.
You must enter a pornstar name that already exists in our database.
- Catégorie: Amateurs
- Vues: 614
- Ajoutée: Il y a 4 ans
- Description: First video I ever took of myself jerking off. LOL I know I came real quick, I'll try to do better next time Was so excited by idea of jerking off for all to watch that I couldn't help it. If you like are innear northern Delaware USA, send me a msg
- Tags: homemade, jerking, stroking, masturbation, real, amatuer
- Stars De La X: Rajouter une star du X
- Favoris par: 0
- Signaler Cette Vidéo
Commentaires Vidéo
Commentaires sur toutes les vidéos (0)
Résultats de Vidéos Gay pour:
Sort by:
Sorry, we couldn't find any pages containing
Watch some of our most popular Tube8.com porn videos: Homepage or Category page


text:
New Western Energy Corporation has entered into a non-binding letter of intent (LOI) to acquire all of the issued and outstanding common shares of Legend Oil and Gas Ltd. Under the terms of the LOI, New Western Energy proposes to exchange shares of Legend Oil valued at $0.09 per share for shares of New Western Energy based on the average of the closing bid prices of New Western's common stock over the ten days prior to the Closing Date, but at a price that will be no less than $0.21 per share and no more than $0.27 per share, unless mutually agreed to by the parties.
According to Javan Khazali, President of New Western Energy, "The strategic acquisition of Legend Oil not only increases our holdings in Kansas but also allows us to diversify our geographic portfolio outside of the U.S. and into the diverse energy region of Western Canada. From an operating strategy point of view, we will immediately accelerate revenue growth through our combined production rate while having opportunities to aggressively develop oil & gas properties in historically proven zones in the near term. We are extremely excited to join forces with the Legend Oil management team that has shown in-depth technical expertise and tremendous exploration and operation capabilities. We expect this value creation to reflect on our strong post - transaction balance sheet with substantial increase in pro forma liquidity that will further enable us to continue organic growth and fund future acquisitions."
New Western Energy Corp.
Please be civil. Job and promotion will not be added into the comment page.
Leave this field blank
Your name*
Your e-mail*
Your website
Your comment*
(Use Markdown for formatting.)
Leave this field blank
This question helps prevent spam:
What color is the sky on a clear day?*
Remember me
your browser.
Schlumberger has announced an agreement to acquire a majority (51%) equity interest in Eurasia Drilling Company Limited.
Mobile news
Our news
your website
Do you have any
tips to us
About us | Contact | Privacy | Terms |
Editor: Michael Rogers
Copyright 1973-2016. Scandinavian Oil-Gas Online. ISSN 1500-709X.
Tel: +47 22 44 72 70, Fax: +47 22 44 72 87


text:
When Cthulhu calls, you pick up the phone — in this case, to take up the challenge to participate in Project Cthulhu, and pen a pastiche to the master of the monster story, H.P. Lovecraft. Also, to give Max the sequel he requested. And so it gives me unflinching horror to bring you….At the Moundville of Madness (The Unauthorized Sequel to Mudville, by the Author of Mudville)
One thought on "At the Moundville of Madness"
Also, using the monstrous blind penguins as umps is, if I may say so, brilliant. That takes some ballallallalls!


text:
forgot to cc to this list ... wendy
--- Begin Message ---
This is not exciting as Jeff Garzik's new NFSv4 server but a painful chore needs to be done. Will start to push the patches for NLM lock failover code next week. However, instead of doing large amount of patches all at once, would like to do it one by one, if they can be functionally separated. This is to avoid the tedious re-base works I have been doing for the past, (oh no), two years ??!!!
- From: Wendy Cheng <wcheng redhat com>
- To: NFS list <linux-nfs vger kernel org>
- Subject: [PATCH] Fix NLM reference count panic
- Date: Fri, 04 Jan 2008 17:48:49 -0500First one should be simple enough without too much explanation .... -- WendyThis fixes the incorrect fclose call inside nlm_traverse_files() where a posix lock could still be held by NFS client. Problem was found in a kernel panic inside locks_remove_flock() (fs/locks.c:2034) as part of the fclose call due to NFS-NLM locks still hanging on inode->i_flock list. Also see: http://people.redhat.com/wcheng/Patches/NFS/NLM/001.txt svcsubs.c | 3 +-- 1 files changed, 1 insertion(+), 2 deletions(-) --- gfs2-nmw/fs/lockd/svcsubs.c 2007-04-10 11:59:09.000000000 -0400 +++ linux/fs/lockd/svcsubs.c 2007-04-18 10:01:23.000000000 -0400 @@ -250,8 +250,7 @@ nlm_traverse_files(struct nlm_host *host mutex_lock(&nlm_file_mutex); file->f_count--; /* No more references to this file. Let go of it. */ - if (list_empty(&file->f_blocks) && !file->f_locks - && !file->f_shares && !file->f_count) { + if (!nlm_file_inuse(file)) { hlist_del(&file->f_list); nlmsvc_ops->fclose(file->f_file); kfree(file);
--- End Message ---


text:
Muscles are like puppies—they need repetitive lessons in running and fetching before becoming well behaved enough to show off in a tank top. Flex a well-trained body with this Groupon.
The Deal
- $49 for a one-month membership, with enrollment and processing fees included (a $292 value)
- Members gain unlimited access to fitness and mixed martial arts classes, cardio, free weights, and state-of-the-art gym equipment.
- Click here to view the class schedule.
UFC Gym Honolulu
ou don't have to be an Ultimate Fighting Championship fighter to train like one at UFC Gym. That's because instructors at the fitness haven teach the training techniques and mixed martial arts of the UFC during a wealth of classes and personal training sessions. Members can learn kicks, punches, and jabs during cardio box and combat challenge classes, and build formidable abs in Killer Core. In doing so, they'll develop the strength, muscle tone, and endurance to finally outrun the neighborhood bull.


text:
Timeline for the alternative universe in which the 22nd Amendment was never passed.
The 22nd. Amendment is passed in the Senate but fails in the House.
President Eisenhower announces that he will not seek a third term in 1960. Although he wishes to run, Mamie and his doctor convence him that a third campaign would kill him.
Ronald Reagan is elected Governor of California with a 23% margin over Pat Brown.
After a strong showing as a write-in in the Wisconsin primary, Governor Ronald Reagan announces his candidacy for the presidency.
Reagan wins primaries in Pennsylvania, Oregon, California, New Jersey, South Dakota, and Illinois.
At the Republican National Convention, Reagan wins the nomination with 782 delegate votes to Richard Nixon's 292 and Nelson Rockefeller's 177. Reagan chooses Michigan Governor George Romney as his running mate.
In November, the Reagan/Romney ticket defeats the Humphrey/Muskie (Democrat) ticket and the Wallace/Chandler (American Independent) ticket.
President Reagan pledges in his inaugural address to win the Vietnam War in one year or withdraw. Reagan also nominates Richard Nixon to be Chief Justice of the Supreme Court. Reagan also nominates Warren Burger as Associate Justice, both Nixon and Burger are easily confirmed.
Congress passes the Economic Recovery Act of 1969 which cuts the top income tax rate from 70% to 55%.
In June, U.S. and Allied forces invade North Vietnam and within a month capture Hanoi.
Apollo 11 lands on the Moon.
Republican Senate Leader Everett Dirksen dies and is replaced (thanks to Reagan's support) by Roman Hruska.
In July, North Vietnam surrenders and is joined with the South to form the Republic of Vietnam (ROV).
With the victory in Vietnam, Republicans retake control of Congress, Gerald Ford becomes Speaker of the House and Roman Hruska becomes Senate Majority Leader.
President Reagan announces that the U.S. will begin construction of a new missile and space defense system.
Congress passes the Balanced Budget Act of 1971, which cuts the top income tax rate to 40% and provides for spending cuts that will balance the budget by 1977.
Reagan nominates William Rehnquist and Robert Bork Antonin Scalia to the Supreme Court, both are confirmed.
65% of U.S. troops are out of Vietnam.
President Reagan wins renomination unopposed.
Democrats engage in a fierce battle and nominate George McGovern and Tom Eagleton (Eagleton would later be replaced by Sargent Shriver).
President Reagan wins 50 states and over 62% of the popular vote.
The Supreme Court rules (5-4) in favor of Wade in the Roe vs Wade case and declares that the unborn have constitutional rights under the 14th. Amendment.
80% of U.S. troops are now out of Vietnam.
With success in Vietnam and the domestic front, the Republicans continue their majority in Congress.
Reagan nominates Antonin Scalia to the Supreme Court, his is confirmed fairly easily.
Having achived great success on the domestic front, Reagan now turns to the Soviet Union and decides to seek a third term.
Republicans re-nominate Reagan and Texas Senator George H.W. Bush for VP (George Romney had decided not to seek a third term as Vice President).
Mo Udall defeats Jimmy Carter, Scoop Jackson, and others in the Democrat primaries and becomes the party nominee. Udall chooses Carter as his vice-presidential nominee.
Congress balances the budget for the first time since 1957.
In November, Reagan defeats Udall in a landslide and the Republicans also retain both houses of Congress. Gerald Ford retires as Speaker and House Majority Leader John Rhodes is elected as Speaker. Roman Hruska retires from the Senate and Barry Goldwater is elected Senate Majority Leader.
President Reagan continues in the conservative tradition and pushes the Economic Freedom Act of 1977 through Congress which provides for a three-bracket tax system of 10%, 20%, and 30%.
The U.S. build a permanant station on the Moon and have nearly completed the SDI program.
Reagan and Soviet Dictator Leonid Brezhnev hold their first sumit at Helsinki, Finland
The Iranian revolution is defeated by the Shah and Iran remains an U.S. ally until the present day.
President Reagan is renominated by the Republicans over liberal Republican Congressman John Anderson.
The Democrats nominate Massachusetts Senator Ted Kennedy and Mississippi Governor Cliff Finch for vice-president.
Illinois Congressman John Anderson launches a bid for the presidency as an independent.
Reagan/Bush wins re-election over Kennedy/Finch and the Republicans retain control in the House and Senate.
President Reagan is shot but soon recovers. Reagan also nominates Sandra Day O'Conner as the first female Justice of the U.S. Supreme Court.
Brezhnev dies and Mikhail Gorbachev becomes leader of the USSR.
Vice President George H.W. Bush is elected over Walter Mondale.
The Soviet Union desolves and the Russian Federation is formed.
Bush nominates Clarence Thomas to the Supreme Court.
Democrats retake control of the House of Representatives after 16 years of Republican control.
Saddam Hussien's Iraq invades Kuwait.
U.S., Iranian, and other allied forces invade Iraq and Kuwait and depose and kill Saddam.
Vice President George H.W. Bush is re-elected over Massachusetts Governor Michael Dukakis, and the Republicans retain the Senate.
With U.S. help, the Northern Alliance defeats the Taliban and forms a U.S. freindly democratic government.
Bush appoints Edith Jones to the Supreme Court.
Bush appoints J. Clifford Wallace to the Supreme Court, Wallace will become the first Mormon to sit on the Court.
Democrats take control of both houses of Congress for the first time in two decades.
Bush appoints David Souter to the Supreme Court.
President Bush retires and Vice President Jack Kemp wins the Republican nomination over several challengers. Kemp picks Bob Dole as his running mate a choice which costs him many conservative votes.
Arkansas Governor Bill Clinton wins the Democrat nomination and chooses Tennessee Senator Al Gore as his VP.
That November, Clinton defeats Kemp by a narrow margin thanks to Perot's vote splitting.
Clinton appoints Ruth Bader Ginsburg to the Supreme Court.
Republicans write the 'Contract with America' and retake the House and Senate.
Former Senator Pat Buchanan defeats Senator Bob Dole for the Republican nomination and chooses former Defense Secretary Dick Cheney as his running mate.
President Clinton is renominated unopposed.
That November, the Buchanan/Cheney ticket defeats the Clinton/Gore ticket by a close margin.
Republicans retain the majorites in Congress.
President Buchanan is renominated by the Republicans unopposed.
Vermont Governor Howard Dean wins the Democrat nomination and chooses Dick Gephardt as his VP.
President Buchanan defeats Dean in a landslide that November.
North Korean Terrorists attack the World Trade Center on September 11. President Buchanan responds by invading North Korea.
Under heavy international pressure, the People's Republic of China allows free elections.
President Buchanan is renominated unopposed by the Republicans.
North Carolina Senator John Edwards wins the Democratic nomination and chooses Massachusetts Senator John Kerry as his running mate.
President Buchanan defeats Edwards 52% to 46% in the popular vote. Buchanan also wins 366 electoral votes to Edwards' 172
The People's Republic of China is dissolved.
President Buchanan appoints Edith Jones as Chief Justice and Sam Alito as Associate Justice.
China, Mongolia, and Taiwan unite to form the Democratic Republic of China (DRC).
Florida Governor Jeb Bush is nominated by the Republicans and chooses former Arkansas Governor Mike Huckabee as his running mate. In the Democratic primaries, Barack Obama wins the Democratic nomination and chooses Delaware Senator Joe Biden as his running mate.
Early October polling shows Governor Bush leading Senator Obama by 4%.
In November, Bush narrowly defeats Obama and wins the election by a 3 point margin.
2009 John Ellis Bush is sworn in as the 41st. President of the United States.


text:
Hot Links: Big Surf, Prison Books, and NAU's Porn Lawsuit
Big Surf water park in Tempe will close at the end of Labor Day, perhaps for good. Golfland Entertainment Centers, which has operated the park since 1991, announced yesterday it will cease operations of the park to focus on its other ventures. The El Paso-based company that owns Big Surf has not announced its plans for the park...Prison inmates in Eloy are being denied books, according to Prison Legal News, a Seattle-based, non-profit publication that provides reading materials to prison inmates nationwide. PLN filed a lawsuit in U.S. District Court against the Corrections Corporation of America and officials at the Saguaro Correctional Center in Eloy, claiming the prison prohibited at least six inmates from receiving books...Two Arizona men have started a company called Out West Family Films. Based in the old mining town of Chloride, the company allows tourists to star in their own gunfight films in the middle of downtown. The company hopes to begin operations this weekend...
One man is dead and another wounded after an early morning shootout in Maryvale. According to police, several people stormed a house near 59th avenue and Encanto and started shooting. They are looking for suspects, described as two Hispanic men between the ages of 30 and 40, who were seen fleeing in a dark blue or dark green Chevy S10 pickup truck...A Pima County Sheriff's deputy has been fired after he rolled his patrol vehicle and showed signs of impairment after the crash. The former deputy, Daniel M. Williams, was off duty and wearing civilian clothing when the crash occurred...Northern Arizona University has dropped its lawsuit against California pornography company La Touraine Inc., which operates a site called Naughty American University (NAU). La Touraine has agreed to stop using Northern Arizona University's trademark.
Get the ICYMI: Today's Top Stories Newsletter Our daily newsletter delivers quick clicks to keep you in the know
Catch up on the day's news and stay informed with our daily digest of the most popular news, music, food and arts stories in Phoenix, delivered to your inbox Monday through Friday.
Recommended for You
Powered by SailThru
Upcoming Events
Arizona Diamondbacks vs. San Francisco Giants
TicketsFri., Aug. 25, 6:40pm
Phoenix Rising Football Club vs. Sacramento Republic FC
TicketsSat., Aug. 26, 8:00pm
Arizona Diamondbacks vs. Los Angeles Dodgers
TicketsTue., Aug. 29, 6:40pm
Comedian Mitch Fatel
TicketsThu., Aug. 10, 8:00pmpowered by goldstar
Sponsor Content


text:
Frankies Bikinis Camilla | Black One Piece Swimsuit
Frankies Bikinis
If you like the Poppy, you will LOVE the Frankies Bikinis Camilla black one piece! This seamless swimsuit features a handful of seriously stylish details to spice up your look this season.
The knotted macrame accents can be seen from the front, side, and back which simply means you'll be getting nods of approval from all around! Tie treatment in back can be worn your way. Tie as a halter, tie across back, or simply tie over the shoulder. Don't be surprised if your designer swimsuit gets the attention of that hot lifeguard either.
The cheeky coverage of the bottom keeps this classy one piece trendy and offers a body flattering cut.
2017 Frankie's Bikinis One Piece
- Macrame side detail
- Seamless one piece
- Halter straps tie across back
- Multi-way Straps
- Cheeky coverage.
- 79% Nylon 29% Spandex


text:
Koigu Dropped Stitch Scarf by String Yarns
Welcome! You are browsing as a guestWould you like to see 18 projects made from this pattern and much more? create a free account What am I missing?
Koigu Dropped Stitch Scarf
KOIGU THREE WAYS -- a simple three-skein drop stitch scarf project.
Can be completed in Koigu Cashmere, Mori, or KPPPM.
For more information, see: http://www.stringyarns.com/searchresult.php?shop/des...


text:
Group Invariant Scattering
- Chastity Harrington
- 2 years ago
- Views:
1 Group Invariant Scattering STÉPHANE MALLAT Centre de Mathématiques Appliquées de l École Polytechnique (CMAP) Institut des Hautes Études Scientifiques (IHÉS) Abstract This paper constructs translation-invariant operators on L 2.R d /, which are Lipschitz-continuous to the action of diffeomorphisms. A scattering propagator is a path-ordered product of nonlinear and noncommuting operators, each of which computes the modulus of a wavelet transform. A local integration defines a windowed scattering transform, which is proved to be Lipschitz-continuous to the action of C 2 diffeomorphisms. As the window size increases, it converges to a wavelet scattering transform that is translation invariant. Scattering coefficients also provide representations of stationary processes. Expected values depend upon high-order moments and can discriminate processes having the same power spectrum. Scattering operators are extended on L 2.G/, where G is a compact Lie group, and are invariant under the action of G. Combining a scattering on L 2.R d / and on L 2.SO.d// defines a translation- and rotation-invariant scattering on L 2.R d /. 22 Wiley Periodicals, Inc. Introduction Symmetry and invariants, which play a major role in physics [6], are making their way into signal information processing. The information content of sounds or images is typically not affected under the action of finite groups such as translations or rotations, and it is stable to the action of small diffeomorphisms that deform signals [2]. This motivates the study of translation-invariant representations of L 2.R d / functions, which are Lipschitz-continuous to the action of diffeomorphisms and which keep high-frequency information to discriminate different types of signals. We then study invariance to the action of compact Lie groups and rotations. We first concentrate on translation invariance. Let L c f.x/ D f.x c/ denote the translation of f 2 L 2.R d / by c 2 R d. An operator ˆ from L 2.R d / to a Hilbert space H is translation-invariant if ˆ.L c f / D ˆ.f / for all f 2 L 2.R d / and c 2 R d. Canonical translation-invariant operators satisfy ˆ.f / D L a f for some a 2 R d that depends upon f [5]. The modulus of the Fourier transform of f is an example of a noncanonical translation-invariant operator. However, these translation-invariant operators are not Lipschitz-continuous to the action of diffeomorphisms. Instabilities to deformations are well-known to appear at high Communications on Pure and Applied Mathematics, Vol. LV, (22) 22 Wiley Periodicals, Inc.
2 332 STÉPHANE MALLAT frequencies []. The major difficulty is to maintain the Lipschitz continuity over high frequencies. To preserve stability in L 2.R d / we want ˆ to be nonexpansive: 8.f; h/ 2 L 2.R d / 2 kˆ.f / ˆ.h/k H kf hk: It is then sufficient to verify its Lipschitz continuity relative to the action of small diffeomorphisms close to translations. Such a diffeomorphism transforms x 2 R d into x.x/, where.x/ 2 R d is the displacement field. Let L f.x/ D f.x.x// denote the action of the diffeomorphism on f. Lipschitz stability means that kˆ.f / ˆ.L f /k is bounded by the size of the diffeomorphism and hence by the distance between the and, up to a multiplicative constant multiplied by kf k. Let j.x/j denote the euclidean norm in R d, jr.x/j the sup norm of the matrix r.x/, and jh.x/j the sup norm of the Hessian tensor. The weak topology on C 2 diffeomorphisms defines a distance between and over any compact subset of R d by (.) d.; / D sup j.x/j C sup jr.x/j C sup jh.x/j: x2 x2 x2 A translation-invariant operator ˆ is said to be Lipschitz-continuous to the action of C 2 diffeomorphisms if for any compact R d there exists C such that for all f 2 L 2.R d / supported in and all 2 C 2.R d / (.2) kˆ.f / ˆ.L f /k H C kf k sup x2r d jr.x/j C sup x2r d jh.x/j : Since ˆ is translation invariant, the Lipschitz upper bound does not depend upon the maximum translation amplitude sup x j.x/j of the diffeomorphism metric (.). The Lipschitz continuity (.2) implies that ˆ is invariant to global translations, but it is much stronger. ˆ is almost invariant to local translations by.x/, up to the first- and second-order deformation terms. High-frequency instabilities to deformations can be avoided by grouping frequencies into dyadic packets in R d with a wavelet transform. However, a wavelet transform is not translation invariant. A translation-invariant operator is constructed with a scattering procedure along multiple paths, which preserves the Lipschitz stability of wavelets to the action of diffeomorphisms. A scattering propagator is first defined as a path-ordered product of nonlinear and noncommuting operators, each of which computes the modulus of a wavelet transform [3]. This cascade of convolutions and modulus can also be interpreted as a convolutional neural network []. A windowed scattering transform is a nonexpansive operator that locally integrates the scattering propagator output. For appropriate wavelets, the main theorem in Section 2 proves that a windowed scattering preserves the norm: kˆ.f /k H D kf k for all f 2 L 2.R d /, and it is Lipschitz-continuous to C 2 diffeomorphisms. When the window size increases, windowed scattering transforms converge to a translation-invariant scattering transform defined on a path set xp that is not
3 GROUP INVARIANT SCATTERING 333 countable. Section 3 introduces a measure and a metric on xp and proves that scattering transforms of functions in L 2.R d / belong to L 2. xp ; d/. A scattering transform has striking similarities with a Fourier transform modulus but a different behavior relative to the action of diffeomorphisms. Numerical examples are shown. An open conjecture remains on conditions for strong convergence in L 2.R d /. The representation of stationary processes with the Fourier power spectrum results from the translation invariance of the Fourier modulus. Similarly, Section 4 defines an expected scattering transform that maps stationary processes to an l 2 space. Scattering coefficients depend upon high-order moments of stationary processes and thus can discriminate processes having the same second-order moments. As opposed to the Fourier spectrum, a scattering representation is Lipschitzcontinuous to random deformations up to a log term. For large classes of ergodic processes, it is numerically observed that the scattering transform of a single realization provides a mean-square consistent estimator of the expected scattering transform. Section 5 extends scattering operators to build invariants to actions of compact Lie groups G. The left action of g 2 G on f 2 L 2.G/ is denoted L g f.r/ D f.g r/. An operator ˆ on L 2.G/ is invariant to the action of G if ˆ.L g f / D ˆ.f / for all f 2 L 2.G/ and all g 2 G. Invariant scattering operators are constructed on L 2.G/ with a scattering propagator that iterates on a wavelet transform defined on L 2.G/, and a modulus operator that removes complex phases. A translation- and rotation-invariant scattering on L 2.R d / is obtained by combining a scattering on L 2.R d / and a scattering on L 2.SO.d//. Available at is a package of software to reproduce numerical experiments. Applications to audio and image classification can be found in [, 3, 4, 8]. NOTATION. kk WD sup x2r d j.x/j, k k WD sup.x;u/2r 2d j.x/.u/j, krk WD sup x2r d jr.x/j, and khk WD sup x2r d jh.x/j where jh.x/j is the norm of the Hessian tensor. The inner product of.x; y/ 2 R 2d is x y. The norm of f in a Hilbert space is kf k and in L 2.R d / kf k 2 D R jf.x/j 2 dx. The norm in L.R d / is kf k D R jf.x/jdx. We denote the Fourier transform of f by f y.!/ WD R f.x/e ix! d!. We denote by g ı f.x/ D f.gx/ the action of a group element g 2 G. An operator R parametrized by p is denoted by RŒp and RŒ D frœp g p2. The sup norm of a linear operator A in L 2.R d / is denoted by kak, and the commutator of two operators is ŒA; B D AB BA. 2 Finite Path Scattering To avoid high-frequency instabilities under the action of diffeomorphisms, Section 2.2 introduces scattering operators that iteratively apply wavelet transforms and remove complex phases with a modulus. Section 2.3 proves that a scattering is nonexpansive and preserves L 2.R d / norms. Translation invariance and Lipschitz continuity to deformations are proved in Sections 2.4 and 2.5.
4 334 STÉPHANE MALLAT 2. From Fourier to Littlewood-Paley Wavelets The Fourier transform modulus ˆ.f / D jf y j is translation invariant. Indeed, for c 2 R d, the translation L c f.x/ D f.x c/ satisfies Lc bf.!/ D e ic! f y.!/ and hence j Lc bf j D jf y j. However, deformations lead to well-known instabilities at high frequencies []. This is illustrated with a small scaling operator, L f.x/ D f.x.x// D f.. s/x/ for.x/ D sx and krk D jsj <. If f.x/ D e ix.x/, then scaling by s translates the central frequency to. s/. If is regular with a fast decay, then (2.) kj b L f j j b f jk jsj jj kk D krk jj kf k: Since jj can be arbitrarily large, ˆ.f / D jf y j does not satisfy the Lipschitz continuity condition (.2) when scaling high frequencies. The frequency displacement from to. s/ has a small impact if sinusoidal waves are replaced by localized functions having a Fourier support that is wider at high frequencies. This is achieved by a wavelet transform [7, 4] whose properties are briefly reviewed in this section. A wavelet transform is constructed by dilating a wavelet 2 L 2.R d / with a scale sequence fa j g j 2 for a >. For image processing, usually a D 2 [3, 4]. Audio processing requires a better frequency resolution with typically a 2 =8 []. To simplify notation, we normalize a D 2, with no loss of generality. Dilated wavelets are also rotated with elements of a finite rotation group G, which also includes the reflection with respect to : x D x. If d is even, then G is a subgroup of SO.d/; if d is odd, then G is a finite subgroup of O.d/. A mother wavelet is dilated by 2 j and rotated by r 2 G, (2.2) 2 j r.x/ D 2dj.2 j r x/: Its Fourier transform is y 2 j r.!/ D y.2 j r!/. A scattering transform is computed with wavelets that can be written (2.3).x/ D e ix.x/ and hence y.!/ D y.! /; where.!/ y is a real function concentrated in a low-frequency ball centered at! D whose radius is of the order of. As a result, y.!/ is real and concentrated in a frequency ball of the same radius but centered at! D. To simplify notation, we denote D 2 j r 2 2 G, with jj D 2 j. After dilation and rotation, y.!/ D. y! / covers a ball centered at with a radius proportional to jj D 2 j. The index thus specifies the frequency localization and spread of y. As opposed to wavelet bases, a Littlewood-Paley wavelet transform [7, 4] is a redundant representation that computes convolution values at all x 2 R d without subsampling: (2.4) 8x 2 R d W Œ f.x/ D f?.x/ D f.u/.x u/du:
5 Its Fourier transform is GROUP INVARIANT SCATTERING 335 2W Œ f.!/ D y f.!/ y.!/ D y f.!/ y.!/: If f is real then f y.!/ D f y.!/, and if y.!/ is real then W Œ f D W Œ f. Let G C denote the quotient of G with f ; g, where two rotations r and r are equivalent. It is sufficient to compute W Œ2 j r f for positive rotations r 2 G C. If f is complex, then W Œ2 j r f must be computed for all r 2 G D G C f ; g. A wavelet transform at a scale 2 only keeps wavelets of frequencies 2 j > 2. The low frequencies that are not covered by these wavelets are provided by an averaging over a spatial domain proportional to 2 : (2.5) A f D f? 2 with 2.x/ D 2 d.2 x/: If f is real, then the wavelet transform W f D fa f;.w Œ f / 2ƒ g is indexed by ƒ D f D 2 j r W r 2 G C ; 2 j > 2 g. Its norm is (2.6) kw f k 2 D ka f k 2 C 2ƒ kw Œ f k 2 : If D then W f D fw Œ f g 2ƒ with ƒ D 2 G C. Its norm is kw f k 2 D P 2ƒ kw Œ f k 2. For complex-valued functions f, all rotations in G are included by defining W f D fa f;.w Œ f / ; 2ƒ g and W f D fw Œ f g ; 2ƒ.The following proposition gives a standard Littlewood-Paley condition [7] so that W is unitary. PROPOSITION 2.. For any 2 or D, W is unitary in the spaces of real-valued or complex-valued functions in L 2.R d / if and only if for almost all! 2 R d (2.7) ˇ j D r2g j y.2 j r!/j 2 D and jy.!/j 2 D ˇ j D r2g where ˇ D for complex functions and ˇ D 2 for real functions. j y.2 j r!/j 2 ; PROOF. If f is complex, ˇ D, and one can verify that (2.7) is equivalent to (2.8) 8 2 jy.2!/j 2 C j y.2 j r!/j 2 D : j > ;r2g Since W 3 Œ2 j r f.!/ D f y.!/ y 2 j r.!/, multiplying (2.8) by jf y.!/j 2 and applying the Plancherel formula proves that kw f k 2 D kf k 2. For D the same result is obtained by letting go to.
6 336 STÉPHANE MALLAT Conversely, if kw f k 2 D kf k 2, then (2.8) is satisfied for almost all!. Otherwise, one can construct a function f where f y has a support in the domain of! where (2.8) is not valid. With the Plancherel formula we verify that kw f k 2 kf k 2, which contradicts the hypothesis. If f is real then jf y.!/j D jf y.!/j so kw Œ2 j r f k D kw Œ 2 j r f k. Hence kw f k remains the same if r is restricted to G C and is multiplied by p 2, which yields condition (2.7) with ˇ D 2. In all the following, y is a real function that satisfies the unitary condition (2.7). It implies that y./ D R.x/dx D and jy.r!/j D jy.!/j for all r 2 G. We choose y.!/ to be real and symmetric so that is also real and symmetric and.rx/ D.x/ for all r 2 G. We also suppose that and are twice differentiable and that their decay as well as the decay of their partial derivatives of order and 2 is O.. C jxj/ d 2 /. A change of variable in the wavelet transform integral shows that if f is scaled and rotated, 2 l g ı f D f.2 l gx/ with 2 l g 2 2 G, then the wavelet transform is scaled and rotated according to (2.9) W Œ.2 l g ı f / D 2 l g ı W Œ2 l g f: Since is invariant to rotations in G, we verify that A commutes with rotations in G: A.g ı f / D g ı A f for all g 2 G. In dimension d D, G D f ; g. To build a complex wavelet concentrated on a single frequency band according to (2.3), we set y.!/ D for! <. Following (2.7), W is unitary if and only if (2.) ˇ j y.2 j j!j/j 2 D and jy.!/j 2 D ˇ j 2 j D j y.2 j j!j/j 2 : If z is a real wavelet that generates a dyadic orthonormal basis of L 2.R/ [4], then y D 2 y z!> satisfies (2.7). Numerical examples in the paper are computed with a complex wavelet calculated from a cubic-spline orthogonal Battle-Lemarié wavelet z [4]. In any dimension d 2, y 2 L 2.R d / can be defined as a separable product in frequency polar coordinates! D j!j, with in the unit sphere S d of R d : 8.j!j; / 2 R C S d y.j!j/ D y.j!j/./: The one-dimensional function y.j!j/ is chosen to satisfy (2.). The Littlewood- Paley condition (2.7) is then equivalent to 8 2 S d j.r /j 2 D : r2g
7 2.2 Path-Ordered Scattering GROUP INVARIANT SCATTERING 337 Convolutions with wavelets define operators that are Lipschitz-continuous under the action of diffeomorphisms, because wavelets are regular and localized functions. However, a wavelet transform is not invariant to translations, and W Œ f D f? translates when f is translated. The main difficulty is to compute translationinvariant coefficients, which remain stable under the action of diffeomorphisms and retain high-frequency information provided by wavelets. A scattering operator computes such a translation-invariant representation. We first explain how to build translation-invariant coefficients from a wavelet transform, while maintaining stability under the action of diffeomorphisms. Scattering operators are then defined, and their main properties are summarized. If U Œ is an operator defined on L 2.R d /, not necessarily linear but which commutes with translations, then R U Œ f.x/dx is translation invariant if finite. W Œ f D f? commutes with translations but R R W Œ f.x/dx D because.x/dx D. More generally, one can verify that any linear transformation of W Œ f, which is translation invariant, is necessarily. To get a nonzero invariant, we set U Œ f D M Œ W Œ f where M Œ is a nonlinear demodulation that maps W Œ f to a lower-frequency function having a nonzero integral. The choice of M Œ must also preserve the Lipschitz continuity to diffeomorphism actions. If.x/ D e ix.x/, then.x/ D e ix.x/, and hence (2.) W Œ f.x/ D e ix.f?.x// with f.x/ D e ix f.x/: The convolution f? is a low-frequency filtering because y.!/ D y.!/ covers a frequency ball centered at! D, of radius proportional to jj. A nonzero invariant can thus be obtained by canceling the modulation term e ix with M Œ. A simple example is (2.2) M Œ h.x/ D e ix e iˆ.y h.// h.x/ where ˆ. h.// y is the complex phase of h./. y This nonlinear phase registration guarantees that M Œ commutes with translations. From (2.) we have that R M Œ W Œ f.x/dx D jf y./j j./j. y It recovers the Fourier modulus representation, which is translation invariant but not Lipschitz-continuous to diffeomorphisms as shown in (2.). Indeed, the demodulation operator M Œ in (2.2) commutes with translations but does not commute with the action of diffeomorphisms and in particular with dilations. The commutator norm of M Œ with a dilation is equal to 2, even for arbitrarily small dilations, which explains the resulting instabilities. Lipschitz continuity under the action of diffeomorphisms is preserved if M Œ commutes with the action of diffeomorphisms. For L 2.R d / stability, we also impose that M Œ is nonexpansive. One can prove [4] that M Œ is then necessarily a pointwise operator, which means that M Œ h.x/ depends only on the value of h at x. We further impose that km Œ hk D khk for all h 2 L 2.R d /, which
8 338 STÉPHANE MALLAT then implies that jm Œ hj D jhj. The most regular functions are obtained with M Œ h D jhj, which eliminates all phase variations. We derive from (2.) that this modulus maps W Œ f into a lower-frequency envelope: M Œ W Œ f D jw Œ f j D jf? j: Lower frequencies created by a modulus result from interferences. For example, if f.x/ D cos. x/ C a cos. 2 x/ where and 2 are in the frequency band covered by y, then jf?.x/j D 2 j y. / C a y. 2 /e i. 2 /x j oscillates at the interference frequency j 2 j, which is smaller than j j and j 2 j. The integration R U Œ f.x/dx D R jf?.x/jdx is translation invariant but it removes all the high frequencies of jf?.x/j. To recover these high frequencies, a scattering also computes the wavelet coefficients of each U Œ f : fu Œ f? g. Translation-invariant coefficients are again obtained with a modulus U Œ U Œ f D ju Œ f? j and an integration R U Œ U Œ f.x/dx. If f.x/ D cos. x/ca cos. 2 x/ with a <, and if j 2 j jj with j 2 j in the support of y, then U Œ U Œ f is proportional to aj. /j j.j 2 j/j. The second wavelet y captures the interferences created by the modulus, between the frequency components of f in the support of y. We now introduce the scattering propagator, which extends these decompositions. DEFINITION 2.2. An ordered sequence p D. ; 2 ; : : : ; m / with k 2 ƒ D 2 G C is called a path. The empty path is denoted p D. Let U Œ f D jf? j for f 2 L 2.R d /. A scattering propagator is a path-ordered product of noncommutative operators defined by (2.3) U Œp D U Œ m U Œ 2 U Œ ; with U Œ D Id. The operator U Œp is well-defined on L 2.R d / because ku Œ f k k k kf k for all 2 ƒ. The scattering propagator is a cascade of convolutions and modulus: (2.4) U Œp f D ˇˇjf? j? 2ˇˇ j? m j: Each U Œ filters the frequency component in the band covered by y and maps it to lower frequencies with the modulus. The index sequence p D. ; 2 ; : : : ; m / is thus a frequency path variable. The scaling and rotation by 2 l g 2 2 G of a path p is written 2 l gp D.2 l g ; 2 l g 2 ; : : : ; 2 l g m /. The concatenation of two paths is denoted p C p D. ; 2 ; : : : ; m ; ; 2 ; : : : ; m /; in particular, p C D. ; 2 ; : : : ; m ; /. It results from (2.3) that (2.5) U Œp C p D U Œp U Œp : Section 2. explains that if f is complex valued then its wavelet transform is W f D fw Œ f g ; 2ƒ, whereas if f is real then W f D fw Œ f g 2ƒ. If f is complex then at the next iteration U Œ f D jw Œ f j is real, so next-stage
9 GROUP INVARIANT SCATTERING 339 wavelet transforms are computed only for k 2 ƒ. The scattering propagator of a complex function is thus defined over positive paths p D. ; 2 ; : : : ; m / 2 ƒ m and negative paths denoted p D. ; 2 ; : : : ; m /. This is analogous to the positive and negative frequencies of a Fourier transform. If f is real, then W Œ f D W Œ f so U Œ f D U Œ f and hence U Œ p f D U Œp f. To simplify explanations, all results are proved on real functions with scattering propagators restricted to positive paths. These results apply to complex functions by including negative paths. DEFINITION 2.3. Let P be the set of all finite paths. The scattering transform of f 2 L.R d / is defined for any p 2 P by (2.6) xsf.p/ D U Œp f.x/dx with p D U Œp ı.x/dx: p A scattering is a translation-invariant operator that transforms f 2 L.R d / into a function of the frequency path variable p. The normalization factor p results from a path measure introduced in Section 3. Conditions are given so that p does not vanish. This transform is then well-defined for any f 2 L.R d / and any p of finite length m. Indeed, k k D k k so (2.4) implies that ku Œp f k kf k k k m. We shall see that a scattering transform shares similarities with the Fourier transform modulus, where the path p plays the role of a frequency variable. However, as opposed to a Fourier modulus, a scattering transform is stable under the action of diffeomorphisms, because it is computed by iterating on wavelet transforms and modulus operators, which are stable. For complexvalued functions, xsf is also defined on negative paths, and xsf. p/ D xsf.p/ if f is real. If p then xsf.p/ is nonlinear but it preserves amplitude factors: (2.7) 8 2 R xs.f /.p/ D jj xsf.p/: A scattering has similar scaling and rotation covariance properties as a Fourier transform. If f is scaled and rotated, 2 l g ı f.x/ D f.2 l gx/, then (2.9) implies that U Œ.2 l g ı f / D 2 l g ı U Œ2 l g f, and cascading this result shows that (2.8) 8p 2 P U Œp.2 l g ı f / D 2 l g ı U Œ2 l gp f: Inserting this result in the definition (2.6) proves that (2.9) xs.2 l g ı f /.p/ D 2 dl xsf.2 l gp/: Rotating f thus rotates identically its scattering, whereas if f is scaled by 2 l, then the frequency paths p are scaled by 2 l. The extension of the scattering transform in L 2.R d / is done as a limit of windowed scattering transforms, which we now introduce. DEFINITION 2.4. Let 2 and P be a set of finite paths p D. ; 2 ; : : : ; m / with k 2 ƒ and hence j k j D 2 j k > 2. A windowed scattering transform is
10 34 STÉPHANE MALLAT defined for all p 2 P by (2.2) S Œp f.x/ D U Œp f? 2.x/ D U Œp f.u/ 2.x u/du: The convolution with 2.x/ D 2 d.2 x/ localizes the scattering transform over spatial domains of size proportional to 2 : S Œp f.x/ D ˇˇjf? j? 2ˇˇ j? m j? 2.x/: It defines an infinite family of functions indexed by P, denoted by S ŒP f WD fs Œp f g p2p : For complex-valued functions, negative paths are also included in P, and if f is real, S Œ p f D S Œp f. Section 2.3 proves that for appropriate wavelets, kf k 2 D P p2p ks Œp f k 2. However, the signal energy is mostly concentrated on a much smaller set of frequency-decreasing paths p D. k / km for which j kc j j k j. Indeed, the propagator U Œ progressively pushes the energy towards lower frequencies. The main theorem of Section 2.5 proves that a windowed scattering is Lipschitz-continuous under the action of diffeomorphisms. Since.x/ is continuous at, if f 2 L.R d / then its windowed scattering transform converges pointwise to its scattering transform when the scale 2 goes to : 8x 2 R d lim! 2d S Œp f.x/ D./ U Œp f.u/du (2.2) D./ p xs.p/: However, when increases, the path set P also increases. Section 3 shows that fp g 2 defines a multiresolution path approximation of a much larger set xp including paths of infinite length. This path set is not countable as opposed to each P, and Section 3 introduces a measure and a metric on xp. Section 3.2 extends the scattering transform xsf.p/ to all f 2 L 2.R d / and to all p 2 xp, and proves that xsf 2 L 2. xp ; d/. A sufficient condition is given to guarantee a strong convergence of S f to xsf, and it is conjectured that it is valid on L 2.R d /. Numerical examples illustrate this convergence and show that a scattering transform has strong similarities to a Fourier transforms modulus when mapping the path p to a frequency variable! 2 R d. 2.3 Scattering Propagation and Norm Preservation We prove that a windowed scattering S is nonexpansive and preserves the L 2.R d / norm. We denote by S Œ WD fs Œp g p2 and U Œ WD fu Œp g p2 a family of operators indexed by a path set. A windowed scattering can be computed by iterating on the one-step propagator defined by U f D fa f;.u Œ f / 2ƒ g;
11 GROUP INVARIANT SCATTERING 34 FIGURE 2.. A scattering propagator U applied to f computes each U Œ f D jf? j and outputs S Œ f D f? 2. Applying U to each U Œ f computes all U Œ ; 2 f and outputs S Œ D U Œ? 2. Applying iteratively U to each U Œp f outputs S Œp f D U Œp f? 2 and computes the next path layer. with A f D f? 2 and U Œ f D jf? j. After calculating U f, applying again U to each U Œ f yields a larger infinite family of functions. The decomposition is further iterated by recursively applying U to each U Œp f. Since U Œ U Œp D U Œp C and A U Œp D S Œp, it holds that (2.22) U U Œp f D fs Œp f;.u Œp C f / 2ƒ g: Let ƒ m be the set of paths of length m with ƒ D f g. It is propagated into (2.23) U U Œƒ m f D fs Œƒ m f; U ŒƒmC f g: Since P D S m2n ƒm, one can compute S ŒP f from f D U Œ f by iteratively computing U U Œƒ m f for m going from to, as illustrated in Figure 2.. Scattering calculations follow the general architecture of convolution neural networks introduced by LeCun []. Convolution networks cascade convolutions and a pooling nonlinearity, which is here the modulus of a complex number. Convolution networks typically use kernels that are not predefined functions such as wavelets but which are learned with back-propagation algorithms. Convolution network architectures have been successfully applied to a number of recognition tasks [] and are studied as models for visual perception [2, 7]. Relations between scattering operators and path formulations of quantum field physics are also studied in [9]. The propagator U f D fa f;.jw Œ f j/ 2ƒ g is nonexpansive because the wavelet transform W is unitary and a modulus is nonexpansive in the sense that jjaj jbjj ja bj for any.a; b/ 2 C 2. This is valid whether f is real or complex.
12 342 STÉPHANE MALLAT As a consequence, (2.24) ku f U hk 2 D ka f A hk 2 C kw f W hk 2 kf hk 2 : 2ƒ kjw Œ f j jw Œ hjk 2 Since W is unitary, setting h D also proves that ku f k D kf k, so U preserves the norm. For any path set the norms of S Œ f and U Œ f are ks Œ f k 2 D p2 ks Œp f k 2 and ku Œ f k 2 D p2 ku Œp f k 2 : Since S ŒP iterates on U, which is nonexpansive, the following proposition proves that S ŒP is also nonexpansive [2]. PROPOSITION 2.5. The scattering transform is nonexpansive: (2.25) 8.f; h/ 2 L 2.R d / 2 ks ŒP f S ŒP hk kf hk: PROOF. Since U is nonexpansive, it results from (2.23) that ku Œƒ m f U Œƒm hk2 ku U Œƒ m f D ks Œƒ m f U U Œƒ m hk2 S Œƒ m hk2 C ku Œƒ mc f U Œƒ mc hk 2 : Summing these equations for m going from to proves that (2.26) ks ŒP f S ŒP hk 2 D ks Œƒ m f S Œƒ m hk2 md kf hk 2 : Section 2.2 explains that each U Œ f D jf? j captures the frequency energy of f over a frequency band covered by y and propagates this energy towards lower frequencies. The following theorem proves this result by showing that the whole scattering energy ultimately reaches the minimum frequency 2 and is trapped by the low-pass filter 2. The propagated scattering energy thus goes to as the path length increases, and the theorem implies that ks ŒP f k D kf k. This result also applies to complex-valued functions by incorporating negative paths. ; 2 ; : : : ; m / in P. THEOREM 2.6. A scattering wavelet is said to be admissible if there exists 2 R d and, with jy.!/j jy.2!/j and y./ D, such that the function C (2.27) y.!/ D jy.! /j 2 k. jy.2 k.! //j 2 / kd
13 satisfies (2.28) D inf GROUP INVARIANT SCATTERING 343 C j!j2 j D r2g y.2 j r!/j y.2 j r!/j 2 > : If the wavelet is admissible, then for all f 2 L 2.R d / (2.29) lim ku m! Œƒm f k2 D lim ks Œƒ n m! f k2 D and ndm (2.3) ks ŒP f k D kf k: PROOF. We first prove that lim m! ku Œƒ m f k D is equivalent to having P lim m! ndm ks Œƒ n f k2 D and to ks ŒP f k D kf k. Since ku hk D khk for any h 2 L 2.R d / and U U Œƒ n f D fs Œƒ n f; U ŒƒnC g, (2.3) ku Œƒ n f k2 D ku U Œƒ n f k2 D ks Œƒ n f k2 C ku Œƒ nc f k 2 : Summing for m n < proves that lim m! ku Œƒ m f k D is equivalent P to lim m! ndm ks Œƒ n f k2 D. Since f D U Œƒ f, summing (2.3) for n < m also proves that (2.32) kf k 2 D so m nd ks ŒP f k 2 D ks Œƒ n f k2 C ku Œƒ m f k2 ; ks Œƒ n f k2 D kf k 2 nd if and only if lim m! ku Œƒ m k D. We now prove that condition (2.27) implies that lim m! ku Œƒ m f k2 D. It relies on the following lemma, which gives a lower bound of jf? j convolved with a positive function. LEMMA 2.7. If h then for any f 2 L 2.R d / (2.33) jf? j? h sup 2R d jf?? h j with h.x/ D h.x/e ix : The lemma is proved by computing ˇˇˇˇ jf? j? h.x/ D f.v/.u v/dvˇ ˇh.x u/du ˇˇˇˇ D f.v/.u v/e i.x u/ h.x u/dvˇ ˇdu ˇ f.v/.u v/h.x u/e i.x u/ dv du ˇ D
14 344 STÉPHANE MALLAT D ˇ D ˇ f.v/.x v u /h.u /e iu du dv ˇ f.v/? h.x v/dv ˇ D jf?? h j; which finishes the lemma s proof. Appendix A uses this lemma to show that the scattering energy propagates progressively towards lower frequencies and proves the following lemma: LEMMA 2.8. If (2.28) is satisfied and (2.34) kf k 2 w D then (2.35) j D r2g C j kw Œ2 j r f k 2 < 2 ku ŒP f k 2 max. C ; /kf k 2 C kf k 2 w : PROOF. The class of functions for which kf k w < is a logarithmic Sobolev class corresponding to functions that have an average modulus of continuity in L 2.R d /. Since ku ŒP f k 2 D C md ku Œƒ m f k2 ; if kf k w < then (2.35) implies that lim m! ku Œƒ m f k D. This result is extended in L 2.R d / by density. Since 2 L.R d / and y./ D, any f 2 L 2.R d / satisfies lim n! kf f n k D, where f n D f? 2 n and 2 n.x/ D 2 nd.2 n x/. We prove that lim m! ku Œƒ m f nk 2 D by showing that kf n k w <. Indeed, kw Œ2 j r f n k 2 D jf y.!/j 2 jy.2 n!/j 2 j y.2 j r!/j 2 d! C 2 2n 2j jf y.!/j 2 d!; because has a vanishing moment so j y.!/j D O.j!j/, and the derivatives of are in L.R d / so j!j jy.!/j is bounded. Thus we have that kf n k w <. Since U Œƒ m is nonexpansive, ku Œƒ m f U Œƒm f nk kf f n k, so ku Œƒ m f k kf f nk C ku Œƒ m f nk: Since lim n! kf f n k D and lim m! ku Œƒ m f nk D, we have lim ku m! Œƒm f k2 D for any f 2 L 2.R d /.
Mathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which areMore information
MA651 Topology. Lecture 6. Separation Axioms.
MA651 Topology. Lecture 6. Separation Axioms. This text is based on the following books: Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology by Nicolas Bourbaki CounterexamplesMore information
1 if 1 x 0 1 if 0 x 1
Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is orMore information
Metric Spaces. Chapter 1
Chapter 1 Metric Spaces Many of the arguments you have seen in several variable calculus are almost identical to the corresponding arguments in one variable calculus, especially arguments concerning convergenceMore information
The Dirichlet Unit Theorem
Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same ifMore information
The Ideal Class Group
Chapter 5 The Ideal Class Group We will use Minkowski theory, which belongs to the general area of geometry of numbers, to gain insight into the ideal class group of a number field. We have already mentionedMore information
BANACH AND HILBET SPACE EVIEW CHISTOPHE HEIL These notes will briefly review some basic concepts related to the theory of Banach and Hilbert spaces. We are not trying to give a complete development, butMore information
Factorization Theorems
Chapter 7 Factorization Theorems This chapter highlights a few of the many factorization theorems for matrices While some factorization results are relatively direct, others are iterative While some factorizationMore information
Chapter 1. Metric Spaces. Metric Spaces. Examples. Normed linear spaces
Chapter 1. Metric Spaces Metric Spaces MA222 David Preiss email@example.com Warwick University, Spring 2008/2009 Definitions. A metric on a set M is a function d : M M R such that for all x, y, zMore information
Math 5311 Gateaux differentials and Frechet derivatives
Math 5311 Gateaux differentials and Frechet derivatives Kevin Long January 26, 2009 1 Differentiation in vector spaces Thus far, we ve developed the theory of minimization without reference to derivatives.More information
Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011
Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011 A. Miller 1. Introduction. The definitions of metric space and topological space were developed in the early 1900 s, largelyMore information
Wavelet analysis. Wavelet requirements. Example signals. Stationary signal 2 Hz + 10 Hz + 20Hz. Zero mean, oscillatory (wave) Fast decay (let)
Wavelet analysis In the case of Fourier series, the orthonormal basis is generated by integral dilation of a single function e jx Every 2π-periodic square-integrable function is generated by a superpositionMore information
INDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS STEVEN P. LALLEY AND ANDREW NOBEL Abstract. It is shown that there are no consistent decision rules for the hypothesis testing problemMore information
Mathematical Methods of Engineering Analysis
Mathematical Methods of Engineering Analysis Erhan Çinlar Robert J. Vanderbei February 2, 2000 Contents Sets and Functions 1 1 Sets................................... 1 Subsets.............................More information
2.3 Convex Constrained Optimization Problems
42 CHAPTER 2. FUNDAMENTAL CONCEPTS IN CONVEX OPTIMIZATION Theorem 15 Let f : R n R and h : R R. Consider g(x) = h(f(x)) for all x R n. The function g is convex if either of the following two conditionsMore information
THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS KEITH CONRAD 1. Introduction The Fundamental Theorem of Algebra says every nonconstant polynomial with complex coefficients can be factored into linearMore information
PUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5.
PUTNAM TRAINING POLYNOMIALS (Last updated: November 17, 2015) Remark. This is a list of exercises on polynomials. Miguel A. Lerma Exercises 1. Find a polynomial with integral coefficients whose zeros includeMore information
M.Sc. program in mathematics INTRODUCTORY SET THEORY Katalin Károlyi Department of Applied Analysis, Eötvös Loránd University H-1088 Budapest, Múzeum krt. 6-8. CONTENTS 1. SETS Set, equal sets, subset,More information
FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES CHRISTOPHER HEIL 1. Cosets and the Quotient Space Any vector space is an abelian group under the operation of vector addition. So, if you are have studiedMore information
1. Prove that the empty set is a subset of every set.
1. Prove that the empty set is a subset of every set. Basic Topology Written by Men-Gen Tsai email: email@example.com Proof: For any element x of the empty set, x is also an element of every set sinceMore information
Metric Spaces. Chapter 7. 7.1. Metrics
Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give someMore information
Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm.
Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm. We begin by defining the ring of polynomials with coefficients in a ring R. After some preliminary results, we specializeMore information
9 More on differentiation
Tel Aviv University, 2013 Measure and category 75 9 More on differentiation 9a Finite Taylor expansion............... 75 9b Continuous and nowhere differentiable..... 78 9c Differentiable and nowhere monotone......More information
CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e.
CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e. This chapter contains the beginnings of the most important, and probably the most subtle, notion in mathematical analysis, i.e.,More information
Metric Spaces Joseph Muscat 2003 (Last revised May 2009)
1 Distance J Muscat 1 Metric Spaces Joseph Muscat 2003 (Last revised May 2009) (A revised and expanded version of these notes are now published by Springer.) 1 Distance A metric space can be thought ofMore information
Continued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents IntroductionMore information
Mathematics for Econometrics, Fourth Edition
Mathematics for Econometrics, Fourth Edition Phoebus J. Dhrymes 1 July 2012 1 c Phoebus J. Dhrymes, 2012. Preliminary material; not to be cited or disseminated without the author s permission. 2 ContentsMore information
PSEUDOARCS, PSEUDOCIRCLES, LAKES OF WADA AND GENERIC MAPS ON S 2 Abstract. We prove a Bruckner-Garg type theorem for the fiber structure of a generic map from a continuum X into the unit interval I. WeMore information
SOLUTIONS TO EXERCISES FOR. MATHEMATICS 205A Part 3. Spaces with special properties
SOLUTIONS TO EXERCISES FOR MATHEMATICS 205A Part 3 Fall 2008 III. Spaces with special properties III.1 : Compact spaces I Problems from Munkres, 26, pp. 170 172 3. Show that a finite union of compact subspacesMore information
Mathematics for Computer Science/Software Engineering. Notes for the course MSM1F3 Dr. R. A. Wilson
Mathematics for Computer Science/Software Engineering Notes for the course MSM1F3 Dr. R. A. Wilson October 1996 Chapter 1 Logic Lecture no. 1. We introduce the concept of a proposition, which is a statementMore information
A QUIK GUIDE TO THE FOMULAS OF MULTIVAIABLE ALULUS ontents 1. Analytic Geometry 2 1.1. Definition of a Vector 2 1.2. Scalar Product 2 1.3. Properties of the Scalar Product 2 1.4. Length and Unit VectorsMore information
Random graphs with a given degree sequence
Sourav Chatterjee (NYU) Persi Diaconis (Stanford) Allan Sly (Microsoft) Let G be an undirected simple graph on n vertices. Let d 1,..., d n be the degrees of the vertices of G arranged in descending order.More information
GROUPS ACTING ON A SET MATH 435 SPRING 2012 NOTES FROM FEBRUARY 27TH, 2012 1. Left group actions Definition 1.1. Suppose that G is a group and S is a set. A left (group) action of G on S is a rule forMore information
x if x 0, x if x < 0.
Chapter 3 Sequences In this chapter, we discuss sequences. We say what it means for a sequence to converge, and define the limit of a convergent sequence. We begin with some preliminary results about theMore information
F. ABTAHI and M. ZARRIN. (Communicated by J. Goldstein)
Journal of Algerian Mathematical Society Vol. 1, pp. 1 6 1 CONCERNING THE l p -CONJECTURE FOR DISCRETE SEMIGROUPS F. ABTAHI and M. ZARRIN (Communicated by J. Goldstein) Abstract. For 2 < pMore information
Undergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics
Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics An Introductory Single Variable Real Analysis: A Learning Approach through Problem Solving Marcel B. Finan c All RightsMore information
MOP 2007 Black Group Integer Polynomials Yufei Zhao. Integer Polynomials. June 29, 2007 Yufei Zhao firstname.lastname@example.com
Integer Polynomials June 9, 007 Yufei Zhao email@example.com We will use Z[x] to denote the ring of polynomials with integer coefficients. We begin by summarizing some of the common approaches used in dealingMore information
IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL. 1. Introduction
IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL R. DRNOVŠEK, T. KOŠIR Dedicated to Prof. Heydar Radjavi on the occasion of his seventieth birthday. Abstract. Let S be an irreducibleMore information
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +More information
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b aMore information
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES Contents 1. Random variables and measurable functions 2. Cumulative distribution functions 3. DiscreteMore information
Properties of BMO functions whose reciprocals are also BMO
Properties of BMO functions whose reciprocals are also BMO R. L. Johnson and C. J. Neugebauer The main result says that a non-negative BMO-function w, whose reciprocal is also in BMO, belongs to p> A p,andMore information
Point Set Topology. A. Topological Spaces and Continuous Maps
Point Set Topology A. Topological Spaces and Continuous Maps Definition 1.1 A topology on a set X is a collection T of subsets of X satisfying the following axioms: T 1.,X T. T2. {O α α I} T = α IO α T.More information
The Goldberg Rao Algorithm for the Maximum Flow Problem
The Goldberg Rao Algorithm for the Maximum Flow Problem COS 528 class notes October 18, 2006 Scribe: Dávid Papp Main idea: use of the blocking flow paradigm to achieve essentially O(min{m 2/3, n 1/2 }More information
An example of a computable
An example of a computable absolutely normal number Verónica Becher Santiago Figueira Abstract The first example of an absolutely normal number was given by Sierpinski in 96, twenty years before the conceptMore information
and s n (x) f(x) for all x and s.t. s n is measurable if f is. REAL ANALYSIS Measures. A (positive) measure on a measurable space
RAL ANALYSIS A survey of MA 641-643, UAB 1999-2000 M. Griesemer Throughout these notes m denotes Lebesgue measure. 1. Abstract Integration σ-algebras. A σ-algebra in X is a non-empty collection of subsetsMore information
REAL ANALYSIS LECTURE NOTES: 1.4 OUTER MEASURE CHRISTOPHER HEIL 1.4.1 Introduction We will expand on Section 1.4 of Folland s text, which covers abstract outer measures also called exterior measures).More information
CARDINALITY, COUNTABLE AND UNCOUNTABLE SETS PART ONE With the notion of bijection at hand, it is easy to formalize the idea that two finite sets have the same number of elements: we just need to verifyMore information
Ideal Class Group and Units
Chapter 4 Ideal Class Group and Units We are now interested in understanding two aspects of ring of integers of number fields: how principal they are (that is, what is the proportion of principal idealsMore information
LINEAR ALGEBRA W W L CHEN c W W L Chen, 1997, 2008 This chapter is available free to all individuals, on understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied,More information
RESULTANT AND DISCRIMINANT OF POLYNOMIALS SVANTE JANSON Abstract. This is a collection of classical results about resultants and discriminants for polynomials, compiled mainly for my own use. All resultsMore information
Stochastic Inventory Control
Chapter 3 Stochastic Inventory Control 1 In this chapter, we consider in much greater details certain dynamic inventory control problems of the type already encountered in section 1.3. In addition to theMore information
Rotation Rate of a Trajectory of an Algebraic Vector Field Around an Algebraic Curve
QUALITATIVE THEORY OF DYAMICAL SYSTEMS 2, 61 66 (2001) ARTICLE O. 11 Rotation Rate of a Trajectory of an Algebraic Vector Field Around an Algebraic Curve Alexei Grigoriev Department of Mathematics, TheMore information
MATH 4330/5330, Fourier Analysis Section 11, The Discrete Fourier Transform
MATH 433/533, Fourier Analysis Section 11, The Discrete Fourier Transform Now, instead of considering functions defined on a continuous domain, like the interval [, 1) or the whole real line R, we wishMore information
Some Polynomial Theorems. John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 firstname.lastname@example.com.
Some Polynomial Theorems by John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 email@example.com This paper contains a collection of 31 theorems, lemmas,More information
Gambling Systems and Multiplication-Invariant Measures
Gambling Systems and Multiplication-Invariant Measures by Jeffrey S. Rosenthal* and Peter O. Schwartz** (May 28, 997.. Introduction. This short paper describes a surprising connection between two previouslyMore information
Extension of measure
1 Extension of measure Sayan Mukherjee Dynkin s π λ theorem We will soon need to define probability measures on infinite and possible uncountable sets, like the power set of the naturals. This is hard.More information
. INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.More information
Revision of ring theory
CHAPTER 1 Revision of ring theory 1.1. Basic definitions and examples In this chapter we will revise and extend some of the results on rings that you have studied on previous courses. A ring is an algebraicMore information
Taylor and Maclaurin Series
Taylor and Maclaurin Series In the preceding section we were able to find power series representations for a certain restricted class of functions. Here we investigate more general problems: Which functionsMore information
P (A) = lim P (A) = N(A)/N,
1.1 Probability, Relative Frequency and Classical Definition. Probability is the study of random or non-deterministic experiments. Suppose an experiment can be repeated any number of times, so that weMore information
Partitioning edge-coloured complete graphs into monochromatic cycles and paths
arxiv:1205.5492v1 [math.co] 24 May 2012 Partitioning edge-coloured complete graphs into monochromatic cycles and paths Alexey Pokrovskiy Departement of Mathematics, London School of Economics and PoliticalMore information
CHAPTER 5. Product Measures
54 CHAPTER 5 Product Measures Given two measure spaces, we may construct a natural measure on their Cartesian product; the prototype is the construction of Lebesgue measure on R 2 as the product of LebesgueMore information
Ri and. i=1. S i N. and. R R i
The subset R of R n is a closed rectangle if there are n non-empty closed intervals {[a 1, b 1 ], [a 2, b 2 ],..., [a n, b n ]} so that R = [a 1, b 1 ] [a 2, b 2 ] [a n, b n ]. The subset R of R n is anMore information
Vector Spaces II: Finite Dimensional Linear Algebra 1
John Nachbar September 2, 2014 Vector Spaces II: Finite Dimensional Linear Algebra 1 1 Definitions and Basic Theorems. For basic properties and notation for R N, see the notes Vector Spaces I. DefinitionMore information
Notes on metric spaces
Notes on metric spaces 1 Introduction The purpose of these notes is to quickly review some of the basic concepts from Real Analysis, Metric Spaces and some related results that will be used in this course.More information
1 VECTOR SPACES AND SUBSPACES What is a vector? Many are familiar with the concept of a vector as: Something which has magnitude and direction. an ordered pair or triple. a description for quantities suchMore information
Chapter 13: Basic ring theory
Chapter 3: Basic ring theory Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 42, Spring 24 M. Macauley (Clemson) Chapter 3: Basic ringMore information
TOPIC 3: CONTINUITY OF FUNCTIONS. Absolute value We work in the field of real numbers, R. For the study of the properties of functions we need the concept of absolute value of a number. Definition.. LetMore information
Finite Sets. Theorem 5.1. Two non-empty finite sets have the same cardinality if and only if they are equivalent.
MATH 337 Cardinality Dr. Neal, WKU We now shall prove that the rational numbers are a countable set while R is uncountable. This result shows that there are two different magnitudes of infinity. But weMore information
Fixed Point Theorems in Topology and Geometry
Fixed Point Theorems in Topology and Geometry A Senior Thesis Submitted to the Department of Mathematics In Partial Fulfillment of the Requirements for the Departmental Honors Baccalaureate By Morgan SchrefflerMore information
Exercises with solutions (1)
Exercises with solutions (). Investigate the relationship between independence and correlation. (a) Two random variables X and Y are said to be correlated if and only if their covariance C XY is not equalMore information
The Henstock-Kurzweil-Stieltjes type integral for real functions on a fractal subset of the real line
The Henstock-Kurzweil-Stieltjes type integral for real functions on a fractal subset of the real line D. Bongiorno, G. Corrao Dipartimento di Ingegneria lettrica, lettronica e delle Telecomunicazioni,More information
Separation Properties for Locally Convex Cones
Journal of Convex Analysis Volume 9 (2002), No. 1, 301 307 Separation Properties for Locally Convex Cones Walter Roth Department of Mathematics, Universiti Brunei Darussalam, Gadong BE1410, Brunei DarussalamMore information
Chapter 5. Banach Spaces
9 Chapter 5 Banach Spaces Many linear equations may be formulated in terms of a suitable linear operator acting on a Banach space. In this chapter, we study Banach spaces and linear operators acting onMore information
S(A) X α for all α Λ. Consequently, S(A) X, by the definition of intersection. Therefore, X is inductive.
MA 274: Exam 2 Study Guide (1) Know the precise definitions of the terms requested for your journal. (2) Review proofs by induction. (3) Be able to prove that something is or isn t an equivalence relation.More information
Numerical Analysis Lecture Notes
Numerical Analysis Lecture Notes Peter J. Olver 5. Inner Products and Norms The norm of a vector is a measure of its size. Besides the familiar Euclidean norm based on the dot product, there are a numberMore information
ON FIBER DIAMETERS OF CONTINUOUS MAPS PETER S. LANDWEBER, EMANUEL A. LAZAR, AND NEEL PATEL Abstract. We present a surprisingly short proof that for any continuous map f : R n R m, if n > m, then thereMore information
10. Graph Matrices Incidence Matrix
10 Graph Matrices Since a graph is completely determined by specifying either its adjacency structure or its incidence structure, these specifications provide far more efficient ways of representing aMore information
8.1 Examples, definitions, and basic properties
8 De Rham cohomology Last updated: May 21, 211. 8.1 Examples, definitions, and basic properties A k-form ω Ω k (M) is closed if dω =. It is exact if there is a (k 1)-form σ Ω k 1 (M) such that dσ = ω.More information
Convex analysis and profit/cost/support functions
CALIFORNIA INSTITUTE OF TECHNOLOGY Division of the Humanities and Social Sciences Convex analysis and profit/cost/support functions KC Border October 2004 Revised January 2009 Let A be a subset of R mMore information
No: 10 04. Bilkent University. Monotonic Extension. Farhad Husseinov. Discussion Papers. Department of Economics
No: 10 04 Bilkent University Monotonic Extension Farhad Husseinov Discussion Papers Department of Economics The Discussion Papers of the Department of Economics are intended to make the initial resultsMore information
Inner Product Spaces and Orthogonality
Inner Product Spaces and Orthogonality week 3-4 Fall 2006 Dot product of R n The inner product or dot product of R n is a function, defined by u, v a b + a 2 b 2 + + a n b n for u a, a 2,, a n T, v b,More information
Math212a1010 Lebesgue measure.
Math212a1010 Lebesgue measure. October 19, 2010 Today s lecture will be devoted to Lebesgue measure, a creation of Henri Lebesgue, in his thesis, one of the most famous theses in the history of mathematics.More information
RANDOM INTERVAL HOMEOMORPHISMS. MICHA L MISIUREWICZ Indiana University Purdue University Indianapolis
RANDOM INTERVAL HOMEOMORPHISMS MICHA L MISIUREWICZ Indiana University Purdue University Indianapolis This is a joint work with Lluís Alsedà Motivation: A talk by Yulij Ilyashenko. Two interval maps, appliedMore information
E3: PROBABILITY AND STATISTICS lecture notes
E3: PROBABILITY AND STATISTICS lecture notes 2 Contents 1 PROBABILITY THEORY 7 1.1 Experiments and random events............................ 7 1.2 Certain event. Impossible event............................More information
Example 4.1 (nonlinear pendulum dynamics with friction) Figure 4.1: Pendulum. asin. k, a, and b. We study stability of the origin x
Lecture 4. LaSalle s Invariance Principle We begin with a motivating eample. Eample 4.1 (nonlinear pendulum dynamics with friction) Figure 4.1: Pendulum Dynamics of a pendulum with friction can be writtenMore information
MEASURE AND INTEGRATION. Dietmar A. Salamon ETH Zürich
MEASURE AND INTEGRATION Dietmar A. Salamon ETH Zürich 12 May 2016 ii Preface This book is based on notes for the lecture course Measure and Integration held at ETH Zürich in the spring semester 2014. PrerequisitesMore information
Duality of linear conic problems
Duality of linear conic problems Alexander Shapiro and Arkadi Nemirovski Abstract It is well known that the optimal values of a linear programming problem and its dual are equal to each other if at leastMore information
Sets and Cardinality Notes for C. F. Miller
Sets and Cardinality Notes for 620-111 C. F. Miller Semester 1, 2000 Abstract These lecture notes were compiled in the Department of Mathematics and Statistics in the University of Melbourne for the useMore information
UNIT 2 MATRICES - I 2.0 INTRODUCTION. Structure
UNIT 2 MATRICES - I Matrices - I Structure 2.0 Introduction 2.1 Objectives 2.2 Matrices 2.3 Operation on Matrices 2.4 Invertible Matrices 2.5 Systems of Linear Equations 2.6 Answers to Check Your ProgressMore information
University of Ostrava. Fuzzy Transforms
University of Ostrava Institute for Research and Applications of Fuzzy Modeling Fuzzy Transforms Irina Perfilieva Research report No. 58 2004 Submitted/to appear: Fuzzy Sets and Systems Supported by: GrantMore information
The fundamental group of the Hawaiian earring is not free (International Journal of Algebra and Computation Vol. 2, No. 1 (1992), 33 37) Bart de Smit
The fundamental group of the Hawaiian earring is not free Bart de Smit The fundamental group of the Hawaiian earring is not free (International Journal of Algebra and Computation Vol. 2, No. 1 (1992),More information
Limit processes are the basis of calculus. For example, the derivative. f f (x + h) f (x)
SEC. 4.1 TAYLOR SERIES AND CALCULATION OF FUNCTIONS 187 Taylor Series 4.1 Taylor Series and Calculation of Functions Limit processes are the basis of calculus. For example, the derivative f f (x + h) fMore information
Classification of Cartan matrices
Chapter 7 Classification of Cartan matrices In this chapter we describe a classification of generalised Cartan matrices This classification can be compared as the rough classification of varieties in termsMore information
Information Theory and Coding Prof. S. N. Merchant Department of Electrical Engineering Indian Institute of Technology, Bombay
Information Theory and Coding Prof. S. N. Merchant Department of Electrical Engineering Indian Institute of Technology, Bombay Lecture - 17 Shannon-Fano-Elias Coding and Introduction to Arithmetic CodingMore information
Stationary random graphs on Z with prescribed iid degrees and finite mean connections
Stationary random graphs on Z with prescribed iid degrees and finite mean connections Maria Deijfen Johan Jonasson February 2006 Abstract Let F be a probability distribution with support on the non-negativeMore information
Week 5: Binary Relations
1 Binary Relations Week 5: Binary Relations The concept of relation is common in daily life and seems intuitively clear. For instance, let X be the set of all living human females and Y the set of allMore information
Math 241, Exam 1 Information.
Math 241, Exam 1 Information. 9/24/12, LC 310, 11:15-12:05. Exam 1 will be based on: Sections 12.1-12.5, 14.1-14.3. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/241fa12/241.html)More information
Lebesgue Measure on R n
8 CHAPTER 2 Lebesgue Measure on R n Our goal is to construct a notion of the volume, or Lebesgue measure, of rather general subsets of R n that reduces to the usual volume of elementary geometrical setsMore information


text: Negro Leagues Baseball Museum MLB And Players Union Donate $1 Million To Negro Leagues Museum In Kansas City By Laura Ziegler • Jun 22, 2017 Courtesy @MLB_Players Twitter There was a ceremonial presentation of a $1 million check to officials with The Negro Leagues Baseball Museum in Kansas City's historic 18th & Vine neighborhood Wednesday. It happened again before the Kansas City Royals game at Kauffman Stadium later that afternoon. KCUR's Band Of The Week: Julian Vaughn By Bill Brownlee • May 17, 2017 Julian Vaughn The smooth-jazz bassist and Kansas City native Julian Vaughn joins respected smooth-jazz guitarist — and retired New York Yankees slugger — Bernie Williams at the Negro Leagues Baseball Museum on May 20, in an event billed as Jazz & Jackie: A Monarchs Salute to Jackie Robinson. Don Motley, Force Behind Negro Leagues Baseball Museum, Dies At 89 By Kyle Palmer • Nov 21, 2016 NLBM Don Motley, who coached amateur baseball around Kansas City for decades and later in life became a driving force for the Negro Leagues Baseball Museum, died Sunday. He was 89. "Coach Motley, as he was affectionately known, gave nearly two decades of his illustrious life to help build the NLBM and guided it to unprecedented fiscal heights," Bob Kendrick, president of the museum, said in a statement. "His impact on the organization will be felt for generations to come." Motley served as executive director for the museum from 1991 until he retired in 2008. Kansas City Council Committee Deadlocks On $27M Jazz District Improvement Plans By Steve Bell • Jun 9, 2016 Andrea Tudhope / KCUR 89.3 A four hour hearing Wednesday left a Kansas City Council committee undecided on the fate of a $27.6 million plan for improvements to the 18th and Vine Jazz District. The request for city participation had virtually quadrupled from the $7 million version first proposed. The plan had grown in its 6-month life to include historic preservation, infrastructure improvements, a fountain, outdoor amphitheater, apartments, retail and a connecting corridor to the Crossroads Arts District. Incoming Director On Her Vision For The American Jazz Museum In Kansas City By Lisa Rodriguez • Jan 22, 2016 File Photo / Paul Andrews Listen Listening... / 22:12 Listen to Steve Kraske's full interview with Cheptoo Kositany-Buckner on 'Up To Date.' The challenges facing the American Jazz Museum will fall to new leadership. Cheptoo Kositany-Buckner takes the reins of the 20-year-old organization after spending the past 25 years with the Kansas City Public Library. State Awards Tax Credits For MLB Urban Youth Academy By Elle Moxley • Dec 22, 2015 Elle Moxley / KCUR The Missouri Development Finance Board has awarded $4 million in tax credits to the Kansas City MLB Urban Youth Academy. The baseball park could open as early as next fall near 18th and Vine near the Negro Leagues Baseball Museum. It's expected the city will be able to leverage $8 million in donations with the credits, which will go to donors who contribute at least $5,000 to the Academy. Kansas City Council Committee Endorses Tax Credits For MLB Youth Academy By Steve Bell • Oct 8, 2015 Populous With Kansas City revved on Royals post-season play, a city council committee laid the groundwork for phase two of the development of the planned Major League Baseball Urban Youth Academy in Parade Park near 18th and Vine. Mayor Sly James announced the $14 million baseball project in late September, explaining that funding for phase one was in place. The Royals and, Major League Baseball had committed $2 million and the MLB Players Association another $1 million. The state of Missouri and the city were to combine resources to match those contributions. With Major League Baseball's Help, Kansas City Will Get New Youth Sports Complex By Elle Moxley • Sep 25, 2015 Elle Moxley / KCUR Parade Park in Kansas City's 18th and Vine District will soon be the home of Major League Baseball's newest Urban Youth Academy. Kansas City Mayor Sly James announced Friday fundraising for the first phase of the project – which will include two full-size baseball diamonds, two youth fields, a walking trail, a playground, park upgrades and, ultimately, an indoor training facility – is complete. The outdoor fields are slated to open next fall. Revitalizing the historic jazz district and former site of Municipal Stadium has long been one of James' goals. Get Dapper: Menswear Tips For Kansas City Guys By Gina Kaufmann • Jun 16, 2015 Alyson Raletz / KCUR Tom Paolini makes custom suits for a living, so it stands to reason that he goes out and about in Kansas City wearing spiffy, fashion-forward suits. "I'm looking forward to the day where I'm wearing a suit and I don't get a compliment," Paolini said on Central Standard. "Right now, you're more the exception." Bob Motley, Last Surviving Negro League Ump, Recalls Baseball History By NPR News & NPR Staff • Aug 25, 2014 Bob Motley, a 91-year-old from Kansas City, Mo., has lived through remarkable times in our history. His story is one of a black man in love with baseball. Racial integration didn't come to the major leagues until 1947, when Jackie Robinson broke the color line at first base for the Brooklyn Dodgers. But it was another 19 years before a black man, Emmett Ashford, appeared behind home plate. In the interim, black umpires called balls and strikes in the Negro League. Paseo YMCA Takes One Step Closer To Reopening By Sylvia Maria Gross • Jul 31, 2013 Sylvia Maria Gross / KCUR For years, the old YMCA building was an eyesore at the entrance of the 18th and Vine District. But over the last few years, the façade has been restored and the windows replaced. And, on the south wall, a giant mural has gone up of the legendary baseball player, coach and keeper of the Negro Leagues flame Buck O'Neil. The Negro Leagues got their start in Kansas City, when eight independent black baseball teams met at the Paseo YMCA in 1920. Buck O'Neil played for the Kansas City Monarchs, and had a major role in opening the Negro Leagues Baseball Museum. 'Kansas City Swing' By Bill Anderson, Matthew Long-Middleton & Claire McInerny • Apr 23, 2013 Kansas City's role in the integration of baseball is a popular subject these days. The recent release of 42, the movie about Jackie Robinson's integration into Major League Baseball premiered in Kansas City, home of the Negro Leagues Baseball Museum, this month. UMKC Theatre is also tackling the subject with the debut of its play Kansas City Swing. [VIDEO] In This Scene...A Storm Is Brewing In Baseball By Julie Denesha • Apr 17, 2013 Julie Denesha / KCUR The play Kansas City Swing is set in Kansas City in 1947 when the nation was on the cusp of great change. Red Carpet Conversations About '42' By Greg Echlin • Apr 15, 2013 courtesy Kansas City Sports Commission The movie "42" tells a story of baseball Hall of Famer Jackie Robinson, who once played for the Kansas City Monarchs. He broke the color barrier with his entry into Major League Baseball. A recent sneak preview of the movie in Kansas City featured an appearance by iconic actor Harrison Ford and a handful of his co-stars. '42' Screened In Kansas City By Greg Echlin • Apr 12, 2013 Greg Echlin / KCUR As a benefit for the Negro Leagues Baseball Museum and the Kansas City Sports Commission, the movie "42" was shown on Thursday night at the AMC theaters on Barry Road. The chilly temperatures didn't stop those who gathered near the red carpet to get a glimpse of an A-list actor. Actor Harrison Ford And '42' In Kansas City By Greg Echlin • Mar 21, 2013 D. Stephens / Courtesy Legendary Pictures Productions LLC Actor Harrison Ford plays Branch Rickey in "42," the movie on baseball Hall of Famer Jackie Robinson to be released next month. Rickey is the former general manager who signed Robinson to break the color line in Major League Baseball. Ford will be in Kansas City next month for a special screening. Project Fly-Threads Inspired By History By Matthew Long-Middleton & Jabulani Leffall • Dec 5, 2012 Project Fly - The Wilkinson Dress Shirt The Negro Leagues Baseball Museum has teamed up with the KC clothing boutique Method to provide young men with a smart "fly" dress shirt as they embark on interviews that can transform their lives. Listen Listening... / 25:48 House Rejects Senate Budget Bills By Marshall Griffin • May 1, 2012 The Missouri House rejected most of the budget bills passed last week by the Senate. The move was part of the normal procedure for preparing for final budget negotiations. Bob Kendrick Leaves The Negro Leagues Baseball Museum By Susan B. Wilson • Feb 22, 2010 Kansas City, Mo. – For the past twelve years, Bob Kendrick has been a passionate voice on behalf of the Negro Leagues Baseball Museum at 18th and Vine. As marketing director, Kendrick worked closely with the late Buck O'Neil to promote the legacy of black baseball. But last year, in a controversial move by the museum's board, he was passed over when they selected a new president.


text:
|David Little
|Date of birth
21 March 1937
|Place of birth
Wadesboro, North Carolina, USA
|IMDb profile
David Little is an actor who played seven different characters in Law & Order, Law & Order: Criminal Intent.
|Matt Luck
|Jack Haviland
|Fred Dillon
|L&O: "Happily Ever After"
|L&O: "Forgiveness"
|L&O: "Sweeps"
|L&O: "Scoundrels"
|Mitch Weiss
|Judge Michael Schneider
|Mr. Hewlitt
|L&O: "Girlfriends" • "Sundown"
|L&O: "Can I Get a Witness?" • "Vendetta" • "House of Cards" • "Invaders"
|CI: "Last Rites"


text:
The hotel is located close to the main gate to Petra, so it is a good choice for staying some nights when visiting the historic site. The hotel has wifi in the lobby. There is a restaurant on the top floor, where a buffet dinner is served in the evening and breakfast in the morning. Our double room was a bit small but considering the price during the local high season (April), it was a good deal.
- Official Description (provided by the hotel):
- A three-star hotel of charm Ideally located in Wadi Musa a beautifully restored historic city. With the atmosphere of freedom, clearness and calmness. The hotel is situated in 300 meters from the main entrance of the ancient Lost City of Petra the new 7 wonders of the world.The hotel's location, will make it easy to conduct your activities, whether for tourism or pleasure. You are just a few minutes away, stroll from Market Circle, the local Centre. The hotel has an exceptional location at the hill foot (facing one of the most picturesque views). ... more less
- Reservation Options:
- TripAdvisor is proud to partner with Booking.com, Priceline, Agoda, Hotwire and TripOnline SA so you can book your Candles Hotel reservations with confidence. We help millions of travelers each month to find the perfect hotel for both vacation and business trips, always with the best discounts and special offers.
- Also Known As:
- Candles Hotel Jordan/Petra - Wadi Musa


text:
Bumper Grrrls 2Favorites [+]
Pay-Per-Minute Viewing
This premium movie consumes minutes at 2.0 times the normal rate. Why?
In an effort to encourage studios to send us their newest and best movies as close to their street release dates as possible, we must offer certain titles at a premium rate. Other content is placed at a premium rate because studios consider those videos to be rare, hard to find or special in some other way. While many premium titles revert to the regular price over time, some studios with rare content, require us to keep their content at premium prices for even longer. Movies at less than 1.0x rate are a limited time special discount streaming rate.
If you have any questions about pricing, please contact Customer Support.
If you have any questions about pricing, please contact Customer Support.
You can now enjoy this movie on your iPhone®, iPad®, or compatible Smart TV & Android devices.
Purchase Options
Pay-Per-MinutePurchase Minutes
Streaming Rentals
Available MP4 sizes:SD 480p (1.56 GB)
DRM-free downloads offer content that is not limited by standard Digital Rights Management (DRM) access technologies. These technologies allow hardware manufacturers, publishers, copyright holders and individuals to limit the usage of digital content.
When you purchase DRM-free downloadable content, that content is available for you to download on any computer and save to a disk whenever you like. You can also download to multiple computers. There is no timeframe associated with your purchase and the content will not expire.
Running Time: 102 minutes
Released: 06/2010
Studio Name: Club Jenna
Series: Bumper Grrrls
Director: Jim Powers
Categories: Gonzo Reality Based
Description: Club Jenna pops the clutch and tells the world to "eat my dust"! Ahryan Astyn, Ashli Orion and Sindee Jennings are on a whore ride of their life! Follow them through a fuck-fest across America - blown gaskets and cocks left in their path, no gearshift is safe! The Bumper Grrrls are reckless, loose, hot and sweaty. No one takes this ride for free!
5 Scenes from Bumper Grrrls 2
Bumper Grrrls 2: Scene 1 Free Preview
Sex Acts: Breast Play, Rim Job, Orgy, Voyeurism, Masturbation, Vaginal Sex, Deep Throating, Oral Sex
Positions: Reverse Missionary, Standing Doggy Style, Side Saddle, Missionary, Laying, Cowgirl
Bumper Grrrls 2: Scene 2 Free Preview
Sex Acts: 3-Way, Vaginal Sex, Voyeurism, Clubbing, Modeling, Deep Throating, Oral Sex, Blowjob
Positions: Side Saddle, Sitting, Missionary, Laying, Reverse Cowgirl, Cowgirl
Settings: Night Club
Bumper Grrrls 2: Scene 3 Free Preview
Sex Acts: 3-Way, Vaginal Sex, Voyeurism, Cum Swap, Spanking/Paddling, Deep Throating, Oral Sex, Blowjob
Positions: Standing Doggy Style, Standing, Missionary, Laying, Reverse Cowgirl
Bumper Grrrls 2: Scene 4 Free Preview
Sex Acts: Vaginal Sex, Voyeurism, Modeling, Deep Throating, Oral Sex
Positions: Standing Side Saddle, Standing Doggy Style, Standing, Standing Missionary, Laying, Reverse Cowgirl, Cowgirl
Bumper Grrrls 2: Scene 5 Free Preview
Sex Acts: Vaginal Sex, Orgy, Voyeurism, Hair Pulling, Handjob, Modeling, Deep Throating, Oral Sex, Blowjob
Recommended Movies
AEBN Cams


text:
Chelsea Marie Goff Model
GraceModel Female Orlando, Florida, US
My MM URL: http://www.modelmayhem.com/gracecornelison
Mayhem # 8486
About Me
I grew up in Des Moines, & Iowa isn't exactly a fashion metropolis. However, Ive recently move to Orlando, & am ready to take the fashion world on full force! Because I am new to the area, I am trying to build my portfolio & meet new photographers, stylists etc...So please, contact me if you are interested in collaborating! Limited TFP, but if the shots are worth it, Im in!
I was discovered through ProScout, and and taken under the wing of the president of PS himself which led to my being called back by agencies including NEXT, Karin, Traque & Elite. I'm just a classically trained dancer who just happens to love the camera!
Have you worked with Grace? Add Credits for Grace! >>
Chelsea Marie Goff Model
Kelly Sedivec-Ealy Photographer
AndyS Photographer
Anna Victoria Model
Natashia Rose Model
Cydnie Owens Model
Loading Content


text:
Here is my latest Ask Alice video that has been posted on YouTube. To receive notification all of my new Ask Alice videos, subscribe to the SaleHoo YouTube Channel.
If you have a question for me, send an email to firstname.lastname@example.com, contact me via Twitter, or leave a comment below.
This week, Sharon asks: How can I tell if my supplier has sent me fake items? I help Sharon out by revealing the top 3 ways to spot a fake.
Discover four of the most profitable niches on eBay and get details for the best suppliers for these niches. Enter your details below and we will send them instantly.
Your list of profitable markets and supplier details are on their way to your email inbox.
Be sure to add firstname.lastname@example.com to your safelist to ensure we can deliver the free suppliers details to you.
If you haven't received an email from us in 5 minutes, please check your spam folder or email us at firstname.lastname@example.com.