Uma das músicas q mais gosto da banda Europe!
Actuación de Europe en el programa Sábado Noche presentado por Carlos Herrera y Bibi Andersen.
From the southern part of Portugal called Algarve we go on a roadtrip along the atlantic coast until we reach Porto, an ancient city known for its wine. Come along and discover the beaches and old buildings of Portugal!!!
Live at Germany on February 11 ,1989 .This is the audience recording of the Concert . This Show was Broadcasted by RTL MOSH ( Only 3 songs and Interview ) Uploaded By Stein-Vidar .
Europe plays "Coast To Coast" (a song from their new album at that time "Out Of This World") in Gothenburg, Sweden in 1989 during their "Out Of This World Tour"
We are a group of travellers exploring every country in the world, graduating with our Global Degree. Portugal is country 7/45 in our Europe Season. From exploring Lisbon to getting sketchy on the coast of Lagos. Portugal was an amazing time. Here's how you do it properly! Subscribe to follow our journey. Your support helps us travel the world! SPECIAL THANKS Good Morning Hostel Lisbon - http://goodmorninghostel.com/ Around Lisbon Tours - http://aroundlisbon.pt/ Tag Hostel Lagos - http://taghostel.com/ Kayak Tours - http://www.kayaktours.com.pt/ Algarve Sup - http://www.algarvesup.com/ ------------------------------------------------ POWERED BY: http://www.worldnomads.com http://www.flightcentre.ca/ https://www.monstercat.com/ FOLLOW WHAT WE'RE UP TO FACEBOOK http://www.facebook.com/globaldegreetv INSTAGRAM http://www.instagram.com/globaldegree http://www.globaldegree.tv All filmed on a Sony A7s, Canon 5D Mark iii, and Gopro Hero 4 If you're still reading this comment: "Those waffles look good!"
Coast To Coast is the fourth studio album by English rock band The Dave Clark Five. It's notable for containing two U.S. and U.K. songs "Anyway You Want It" and "Everybody Knows (I Still Love You)". Other songs from this album were used in the Dave Clark Five film "Having A Wild Weekend (Catch Us If You Can in the U.K.)" such as "I Can't Stand It" and "When".
All tracks written by Dave Clark and Mike Smith, except where noted.
This World is a BBC television current affairs documentary programme which broadcast by BBC Two in United Kingdom, and it also airs worldwide throughout occasionally at BBC World News on digital services, satellite and cable in many countries. The series is mainly focus on social issues and current affairs stories around the world.
This World was announced in December 2003 on BBC online and introduce to launch in early January 2004 which replace from its previous programme - Correspondent.
The website includes additional features and a discussion facility for public comment on the programmes. The BBC streams episodes in RealVideo format via its website for a limited period after they have been shown, and sells them on DVD and VHS by mail order.
Starting from 2009 onwards, the series was available and streaming at BBC iPlayer after the programme broadcast, with a limited period (Replacing with RealVideo and available in United Kingdom only).
The division between seasons of This World is based on the UK version of each episode, international episode are based on its airing on BBC World (Currently BBC World News). Subsequent airings of the international version random follows the original UK order (Including changes from the original title on some episode).
Out of This World is a 1976 library recordings release of atmospheric sounds and effects from the BBC Radiophonic Workshop. The album was divided into four sections (two on each side), each representing a different theme: "Outer Space", "Magic and Fantasy", "Suspense and the Supernatural" and "The Elements".
The album was produced by Glynis Jones and features sounds created by the various members of the Workshop. While many of the tracks are basic sound effects, such as those from the TV series Doctor Who, others are like ambient music, designed to create soundscapes rather than fulfilling a specific music or sound effects requirement.
The album was re-released on CD in 1991 in simulated stereo as Essential Science Fiction Sound Effects Vol. 2. It was released on vinyl LP by AudioGo and Discovery Records on 21 April 2012 as part of Record Store Day.
Track 61, "Passing clouds", by Roger Limb, was used on "Eye No", the opening song on Lovesexy, the album by Prince, released in 1988. It was also released as "Clouds", but wrongly credited to Delia Derbyshire, on a 2-CD bootleg called "Wow!", released by "Eye" Records in 2008.
Out of This World is a card trick created by magician Paul Curry in 1942. Many performers have devised their own variations of this trick. It is often billed as "the trick that fooled Winston Churchill" due to a story describing how it was performed for him during World War II. The method behind the trick is simple and essentially self-working, and can be enhanced by the presentation of the performer and the use of other principles of magic.
The performer takes a deck of cards, and places on the table two face-up "marker" cards, one black and one red; the black on the left and the red on the right. The performer tells the subject that he or she is going to deal cards face-down from the deck and the object of the exercise is for the subject to use the power of their mind to identify whether each card in the deck is black or red.
The performer takes one card at a time from the deck, face down, and asks the subject to attempt to divine whether it is black or red. The subject states their choice, and the performer then places the card in line with the appropriately coloured marker card, overlapping it at the bottom.


text:
CD - unabridged
Audio (5 discs)
Product Number: BN6296
Narrator/s: Tanya Eby
Publisher: Brilliance Audio
"WHO DEFINES FAMILY? To Tina Harrington, the definition was simple: anyone she loved was a part of her family. Including all the people-and animals-she'd invited into her home when they'd had nowhere else to go. She was their safe harbor, and they were hers. And she would protect them from whoever challenged their right to be a family-namely her new neighbor, the handsome and high-powered Drew Landry. Drew was determined to ensure that Tina wasn't taking advantage of her vulnerable houseguests. But as soon as he met Tina and her unconventional family, he realized he'd made a big mistake. Because he was drawn to her like a ship to a sheltered shore. And he'd set in motion a series of events that could destroy everything she held dear.. BONUS BOOK INCLUDED IN THIS VOLUME! A Cold Creek Homecoming by New York Times bestselling author RaeAnne Thayne Quinn Southerland still hadn't forgiven Tess Claybourne for the way she'd treated him years ago. But the woman tending his ailing mother was not the same girl he'd known. Could this be their second chance?"
Author(s): Sherryl Woods
The Calamity Janes
by Sherryl Woods
The Backup Plan
Do You Take This Reb...
Waking Up in Charles...
Flirting with Disast...

redpajama_set_name: RedPajamaCommonCrawl


text:
Home » articles » St. Vincent: Daddy's Home – Review
St. Vincent: Daddy's Home – Review
May 13, 2021 by Jamie Parmenter
St. Vincent creates beautiful moments of ingenuity in the 70s funk-inspired album, Daddy's Home.
We all know that Annie Clark, aka, St. Vincent, has always been one for innovation, and new record Daddy's Home is no different. With the title reflecting the release of her father after ten years imprisonment for stock fraud, it doesn't actually focus too much on the subject, instead delving into Dad's collection of 70s funk vinyl for inspiration, creating a mesmerising blend of the old and the new.
Opener Pay Your Way In Pain sets up the record nicely, giving us a classy piano intro before being interrupted by heavy, reverb-laden synths and settled drums. Focussing on a 70s era funky flow, Clark updates the sound found on those old records with a new, modern twist, Prince-esque vocal cues and a penchant for slinky, sweaty, sexuality. She still has her way with words as she cheekily side-steps into a dig about people looking down on her but also that she doesn't give a shit: "I went to the park just to watch the little children, the mothers saw the heels and said I wasn't welcome." Down and Out Downtown then follows with the lesser-used fade-in perfectly suiting the track's slow and sultry warmth. Clark is having fun here, using her vocals to guide the music forward in a tantalising dance of passion and process.
Although the album is built around those old, funky 70s sounds and her father's imprisonment, there's only one track that directly brings it up, namely the album titled, Daddy's Home. Here, St Vincent slows things downs and strips them back even more, letting the impressive musicianship slink forward like a smooth criminal before she gently caresses each syllable and vocal shred with enough emotion and seediness to make it almost seem like a playful parody.
It's a mesmerising listen that twists and turns in a way that only St. Vincent can do. The Melting Of The Sun then goes for a trip to funky town as a walking bassline encases melodies that Bowie would be proud of against settled sitar and mentions of past songwriters in their proudest moments.
What's interesting about the record is Annie Clark's ability to consistently innovate, even if she's sticking to a specific sound at the core. Many lesser artists would run out of steam on a 70s funk-influenced record, but Annie manages to keep things fresh throughout. Down is a beautifully simple yet playful piece of pop ingenuity that saunters along with confidence and panache before hitting you with a delightfully camp chorus.
My Baby Wants A Baby then settles into soulful ballad territory sounding like something John Lennon would enjoy calling his own. Candy Darling is another well-tailored and lovingly arranged piece that floats along on a dreamy soundscape of perfectly encapsulated pop ingenuity.
Daddy's Home from St. Vincent is the perfect amalgamation of warm, 70s funk layered beneath unique and ever-inventive pop production. Harsh realities may have inspired the record but supreme musicianship and cleverly written tracks are what hold it all together.
About Jamie Parmenter
Editor and Founder of Vinyl Chapters.
Always bothering people to tell a story about a record in their vinyl collection.


The only requirement these Epic exclusives have are a VPN and a torrent client.
If Epic want exclusives to tempt people to their store, they better get developing more games as simply snatching third party games is rubbing up the consumer the wrong way.
People need to accommodate change, as much as people don't like having multiple launchers to manage. It's not like it takes much of a thought process. I have all my launchers disabled at startup and if i want to play a game, I'll do a limited bandwidth download whilst playing a local game or an esports title. What are the real downsides other than having everything in somewhere that isn't one place. The Ability to have a variety of platforms provides external and external pressure/competativeness. Both of which benefit the consumer by in the case of epic games we are able to see developers grow their game engines and provide us with better and more emersive experiances.
Are the same games cheaper on Epic, or the publishers are keeping all savings to them?
Cause if they are not cheaper what is the point going "Epic way"?
You do get limited competition with steam through key sellers (Humble bundle, fanatical etc) so I don't think the competition argument is that strong. I've got 1000s of steam games and most of what I want is there. I gave up on EA when they went separate. I'll do the same here.
I got the closed beta running on medium @1080p pretty well between 40-55fps on the Ryzen 5 2400G(Vega 11 graphics) but didn't run very stable after that crashing a few times. It'll probably be sorted by the time it goes gold though.
I've hated steam over the years.Its just basically so they can farm info directly from the player base and maximise ingame revenues.
Your basically saying "get with the times" but the times are crap. I wanna click and go, not login, sort this out or that..
If you want it on Steam then you can pre-purchase it up to April 16th. Otherwise, it's off to the Epic Store with you.
No discount though, it's a full £50!
Nope...Epic need to stop buying up exclusives and screwing over the gaming community. They can provide competition and help themselves plus the gaming community by selling games on their store...as well as steam and other platforms. Exclusives harm everyone except Epic, especially gamers.
Sadly it is unlikely to change as people still give in and buy them. Much like when micro transactions and free to play started to invade our games, the only way to stop that madness would have been to just not buy the darn games...but guess what - people did anyway and now we have micro transactions in almost everything In a way, we only have ourselves to blame.
This time around i'm holding fast and flat out refusing to buy anything that Epic steal away as an exclusive - with the only exception being Metro Exodus since I pre ordered that on steam well before this mess came out. As sad I will be to miss out on games like Borderlands 3, its better than supporting hugely anti competiive and anti gamer practices.


It's a little noisy here so my voice might sound like a drill sergeant, don't mistake that for "these outcomes matter"
If you finish dancing before the music is done, let each couple in turn just sashay down until the music is ready to end.
Linda Leslie notes: timing here is not essential!
/+This gives a gentlespoon-gentlespoon-ladle-ladle line with heads at the ends on the opposite side of the square from their home position.
++Two ladles (in each line) roll away with a half sashay; two gentlespoons roll away (right to left); center two roll away (gentlespoon rolls ladle); everybody roll away.
Intended to teach roll away to beginners.
All swings are short swings in the middle of the set, ending with dancers facing the music. When waiting while others swing, face down.
Reel 4 x 40 bars. Four-couple set so sashays aren't numbingly long.
ladles allemande left 1½ around while the gentlespoons orbit clockwise ½ around. Ladles star if more than two.
In this mixer dancers form scattered groups with any number of couples. That makes for fun chaos, less transition panic (compared with 2-couple scatter mixers), and more flavors to enjoy as circle sizes vary.
The A2 allemande ends with ladles in the center facing counterclockwise and gentlespoons on the outside facing clockwise. If there are two ladles they allemande left for 8 beats; if there are more than two they star left for 8 beats. Either way, the gentlespoons orbit clockwise around the ladles. Then all balance and swing a new partner and promenade to a new group.
I tell the dancers it's a zero-stress dance. "If a couple wants to join your circle, let them in!" And no problem if you happen to keep the same partner now and then.
Three face three in lines facing up/down the hall. Teach "home position".
neighbors do si do once "ends do-si-do your opposite"
neighbors do si do once "middles do-si-do your opposite"
neighbors swing "ends swing your opposite"
neighbors swing "middles swing your opposite"
Linda Leslie notes: Although I don't have the exact time that I wrote this dance, it must predate the first date I called it: 10/22/93 for the Appalachian Mountain Club, Joy Street, Boston. Especially useful for one night stands! Any combination of three. All dancers CAN do this dance, and it really mixes up the dancers. Great for weddings, etc. For these type evenings, I don't talk about #1 or #2 lines of three, or progression: rather I describe "home position". Once we have walked through the dance once, I then point out the couples who have come out at the top and/or bottom, asking them to wait out one time through the dance, turn around, and that this is when "home" changes. Works like a charm.....and the dancers are very happy finding new opposites for each cycle.
Repeat for side couples, then all four ladles and all four gentlespoons. The transition from star to grand right and left is a challenge for beginning square dancers and provides an opportunity for the "team" to be successful. Introduction and ending are caller's choice.
Concentric circles, gentlespoons in inside circle facing partners in outer circle.


In the early 1990, the community of Marcus saw a need for nursing care in the senior living community. From 1990-1995, over six hundred supporters donated in excess of one million dollars for the construction and operation of Heartland Care Center. The sole purpose of the facility is to serve the residents and their families with outstanding nursing care, quality staff, and a home-like atmosphere for everyone. The supporters of this facility gave their time, effort, money, and dedication to make a successful building. The facility opened its doors in August of 1994.
We want to Thank the community of Marcus for recognizing the need and pursued the dream until it became a reality. That dream/passion for commitment continues today in the form of volunteerism that enhances our residents lives.
Heartland Care Center is a community non profit facility that has a Board of Directors that meet monthly to oversee the management of the facility.


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First Chamorro Bishop
Chose life of religion
Education after WWII
Becomes bishop in 1970
From bishop to archbishop
Archbishop Felixberto C. Flores
Bishop Olano and Archbishop Flores
Bishop Apollinaris Baumgartner
Mass Offering
Archbishop Flores at Work
Oka Point Loop Statue of Archbishop Flores
Archbishop Felixberto C. Flores was the first Chamorro bishop. He was born Felixberto Camacho Flores on 13 January 1921, in Hagåtña, the son of León Flores of Laoag, Ilocos Norte, Philippines, and Ana Martínez Camacho of Hagåtña. He attended the Guam Institute, a private school in Hagåtña run by Nieves M. Flores, and graduated as valedictorian of his class in 1938.
The following year, Flores embarked on the USS Gold Star bound for Manila, where he enrolled at the Ateneo de Manila, a Jesuit university, intending to become an engineer. He finished his first year with high honors.
Despite what seemed to be the potential for a successful secular career, Flores left the Ateneo and chose, instead, to pursue the priesthood. The Jesuits were an attractive option, but Flores wanted to serve the church on Guam, and so he became a candidate for the diocesan priesthood. Flores entered San Jose Seminary in Manila in 1940. The seminary, run by the Jesuits, was the same institution where three earlier Chamorro priests attended: Fathers Jesus Duenas, Oscar Calvo and Jose Manibusan.
When World War II broke out in December 1941, classes at the Seminary were temporarily stopped. Classes eventually resumed until 1945 when the American advance in the Pacific came close to the Philippines. The Japanese interned all Americans, and others, at the Santo Tomas University concentration camp, and Flores was among them.
After the war, Flores changed seminaries and entered Saint John's Seminary in Brighton, Massachusetts, the diocesan seminary for the Archdiocese of Boston. Two years before he finished at Brighton, Flores started taking educational administration classes at Fordham University in New York. Flores was one of an outstanding vanguard of young Chamorro men and women who attended educational institutions of higher learning in the US mainland immediately following the WWII. Many later became leaders on Guam.
On 30 April 1949, Flores was ordained a priest in the Agana Cathedral by Bishop Apollinaris W. Baumgartner, OFM Cap. He was the first Chamorro priest to have been partly educated in the U.S. mainland. Baumgartner quickly put Flores' talent and education to work. For the twenty years of his priesthood before becoming bishop, Flores was intimately tied to the central workings of the church on Guam.
He was made priest of Asan in 1949 but was then transferred as rector of the cathedral in Hagåtña in 1955. He was superintendent of Catholic Schools from 1951 to 1955 and Chancellor or the Vicariate (and then the diocese) from 1952 to 1969. He taught at the Academy of Our Lady of Guam (1949-1969), was associate editor of the diocesan paper Umatuna si Yuus (1949-1969), headed various Catholic organizations (Holy Name Society, Society for the Propagation of the Faith, and Apostleship of Prayer), and was a frequent host of Catholic television and radio programs.
Flores' prominent role in church affairs lead him to be named a papal chamberlain a few days before the tenth anniversary of his ordination in 1959. Papal chamberlain is one of the highest honors bestowed by the Church. Flores was now called Monsignor Felixberto C. Flores. In 1963, he was given the title of domestic prelate which means he was an honorary member of the papal household.
By the year 1968, Bishop Baumgartner's health had declined seriously enough that Archbishop George Hamilton Pearce of Suva, Fiji, was made apostolic administrator of the Diocese of Hagåtña. An apostolic administrator runs the daily affairs of a diocese when the bishop is incapacitated. In 1969, Flores took his place as apostolic administrator.
On 19 March 1970, the news was announced that Rome had nominated Flores to the episcopacy (in line to be the next Bishop). He would be given a titular see (leader of the Catholic Church in title only) but remain apostolic administrator of the Hagåtña diocese, as Bishop Baumgartner was still suffering bad health. Flores was consecrated bishop on 17 May 1970, by the apostolic delegate (the Pope's diplomatic representative) to the United States, Archbishop Luigi Raimondi.
On 18 December 1970, Bishop Baumgartner passed away. Flores was named his successor as Bishop of the Diocese of Agana on 24 April 1971, taking effect on May 15 of that year.
Flores was to lead the Catholic Church on Guam for the next fourteen years. During this time, the diocese underwent much positive development. The earlier religious orders on Guam stabilized and became semi-autonomous regions of their mainland provinces. New religious orders came to work in the diocese (the Augustinian Recollects, Good Shepherd Sisters, Dominican Sisters, Marist Brothers). Flores ordained a dozen or so Chamorro priests, both diocesan and Capuchin.
Diocesan activities, especially in religious education, received much attention and were expanded. In 1979, Flores signed the incorporation papers establishing Catholic Social Services.
One diocesan venture which became a challenge to sustain was the Medical Center of the Marianas, a Catholic hospital which Flores built in 1977 at a cost of $14 million. Within a year it became clear that the promised financial backing he was depending on was not coming through. Flores lobbied with the government to purchase the hospital, which it did in 1979.
Flores was also very vocal on social issues, including a 1977 referendum on casino gambling. The diocese opposed casino gambling saying that it would increase crime and addiction to gambling and the voters echoed this position, defeating the proposal by a vote of three to one.
One of Flores' largest achievements was securing the historic visit of Pope John Paul II to Guam on 22-23 February 1981. Following this event, the diocese enjoyed unprecedented attention and affirmation from Rome. On 8 December 1983, Father Anthony S. Apuron, OFM Cap, was appointed an Auxiliary Bishop to assist Flores.
On 8 March 1984, Rome announced that Flores was to be raised to the rank of Archbishop and the diocese as well to become a metropolitan archdiocese. The cathedral in Hagåtña was raised to the rank of Minor Basilica. In the same year, on 8 November, the Vatican announced that the Northern Marianas, which had been a part of the Archdiocese, was to be made its own diocese with its own bishop.
On 6 October 1985, Diego Luís de San Vitores was beatified in Rome. For years, Flores and others worked hard to obtain this recognition of what many recognize as the heroic virtue of an evangelist and martyr. Flores led the Guam pilgrimage to the beatification ceremonies in Rome. It was a high point in the church's life for Flores and many Guam Catholics.
After the beatification, Flores went to San Francisco, California, to take care of his health. For years he had struggled with renal and cardiac issues. Two hours after an apparently successful open-heart surgery, he died of cardiac arrest in San Francisco on 25 October. Many remarked that at least he had seen many of his dreams for the church on Guam realized in the months and years before his death. He was buried in the sanctuary of the Dulce Nombre de María Cathedral-Basilica.
By Eric Forbes, OFM Cap.
Archdiocese of Agana "The Roman Catholic Archdiocese of Agana: History." (accessed August 9, 2010).
Capuchin Franciscan Friars of the Pacific "Our Legacy." (accessed August 9, 2010).
Catholic Social Services "This is My Story." (accessed August 9, 2010).


This formula brings together a number of herbal plants that are recognized for their beneficial application in addressing difficulty with sleep. This compound works to reduce the anxiety, irritability, restlessness, physical tension, and worry that one will often see associated with occasional sleeplessness. This compound may also be used at times when nervous excitement and anxiety affect normal and healthy sleep function.


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\section{INTRODUCTION}
The problem of maximization of a symmetric form
$$\left\{\begin{array}{l} F(\vec\sigma)=\sum_{i,j=1}^n J_{ij}\sigma_i\sigma_j
\to {\rm max},\ \sigma_i=\{\pm 1\},\\
\vec\sigma=(\sigma_1,\ldots,\sigma_n),\ J_{ij}=J_{ji},\ i,j=1,2,\ldots,n,
\end{array}\right.\eqno(1)$$
which is quadratic in {\it spin} variables $\sigma_i$, is under investigation.
This problem arises in the Ising model, in the
surface physics, in the theory of optimal coding, in factor analysis, in the
theory of neural networks and in the optimization problems
 \cite{Hopf,Dom,Surf1,Sour,Par,Brav2,Lit1}. The aim is to obtain an
effective method for the search of the global maximum of the functional and
a constructive description of the set of its local maxima.

The solution of the problem (1) is sought among the $n$-dimensional vectors
$\vec\sigma$. These vectors define $2^n$ configurations; in what follows they
are called {\it the configuration vectors}. The configuration vector
which gives the solution of the problem (1) is called {\it the ground state}.

Let's settle with the choice of the diagonal elements of the matrix
${\bf J}=(J_{ij})_{i,j=1}^n$. Since for any configuration vector the diagonal
gives the same contribution to the value of the functional, the local maxima
of
the functional do not depend on $J_{ii}$. Consequently, $J_{ii}$ can be chosen
in an arbitrary suitable form. In the same time, the set of the local maxima
of
the functional $F$ is the same as the set of the fixed points of a neural
network whose connection matrix differs from $\bf J$ only by zero diagonal
elements\footnote{For details see \cite{Lit3}.}. In what follows, we call
$\bf J$ {\it the connection  matrix} irrespective of its diagonal. And with
regard to neural networks we have in mind that its diagonal elements are equal
zero.

Without the loss of generality $\bf J$ can be considered as a positive
semidefinite matrix. (It always can be achieved by a suitable choice of its
diagonal elements.) Then the matrix $\bf J$ can be presented as a product
 $${\bf J} ={\bf S}^{\rm T}\cdot\bf S,\eqno(2)$$
where $\bf S$ is a $(p\times n)$-matrix with {\it real} matrix elements. The
number of its rows is equal to the rank of the matrix $\bf J$ and it does not
exceed $n-1$:
$$p=\mbox{rank }{\bf J}\le n-1.$$
The representation (2) for the connection matrix plays an important role in
the
theory of neural networks. If all the elements of the matrix $\bf S$ are equal
$\{\pm 1\}$, the connection matrix is called the Hebb matrix and the related
neural network is called the Hopfield network. Of course, in general case
when according to Eq. (2) the matrix $\bf S$ is obtained from an {\it
arbitrary}
symmetric matrix $\bf J$, not necessarily all the elements of $\bf S$
are equal $\{\pm 1\}$. By analogy with the Hopfield model, in this case Eq.(2)
is called {\it the Hebb type} representation and the related neural network is
called {\it the Hopfield type} network. Summarizing, we can say that the local
maxima of the problem (1) are the fixed points of a Hopfield type network.

We investigate the problem (1) in the case of the
connection matrix constructed with regard to the Hebb-like rule (2) from
the $(p\times n)$-matrix $\bf S$  of the form
$${\bf S}=\left(\begin{array}{ccccccc}
1-x&1&\ldots&1&1&\ldots&1\\
1&1-x&\ldots&1&1&\ldots&1\\
\vdots&\vdots&\ddots&\vdots&\vdots&\ldots&\vdots\\
1&1&\ldots&1-x&1&\ldots&1\end{array}\right), \eqno(3.a)$$
where $x$ is an arbitrary real number. The connection matrix $\bf J$ has the
form:
$${\bf J}={\bf S}^{\rm T}{\bf\cdot S}=\left(\begin{array}{ccccccc}
p-2x+x^2&p-2x&\ldots&p-2x&p-x&\ldots&p-x\\
p-2x&p-2x+x^2&\ldots&p-2x&p-x&\ldots&p-x\\
\vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\vdots\\
p-2x&p-2x&\ldots&p-2x+x^2&p-x&\ldots&p-x\\
p-x&p-x&\ldots&p-x&p&\ldots&p\\
\vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\
p-x&p-x&\ldots&p-x&p&\ldots&p\end{array}\right).\eqno (3.b) $$
According to the conventional neural network tradition, we treat the
$n$-dimensional vectors $\vec s^{(l)}$, which are the rows of the matrix
$\bf S$, as $p$ {\it generalized memorized patterns} embedded in the network
memory. Then the following meaningful interpretation of the problem can be
suggested:
{\it the network had to be learned by p-time showing of the standard
$$\vec\varepsilon (n)= (1,1,\ldots,1),$$
but an error crept into the learning process and in fact the network was
learned with the help of $p$ distorted copies $\vec s^{(l)}$ of the standard
$\vec\varepsilon (n)$:
$$\vec s^{(l)}=(1,\ldots,1,\underbrace{1-x}_l,1,\ldots,1),\quad l=1,2,\ldots,
p.\eqno(4)$$}
The problem (1)-(3) will be called the {\it basic model}. In what follows we
present its generalization.

When $x$ is equal zero, the network is found to be learned by $p$ copies of
the standard $\vec\varepsilon (n)$. It is well-known that in this case the
vector $\vec\varepsilon (n)$ itself is the ground state and the functional has
no other local maxima. For continuity reasons, it is clear that the same is
true for a sufficiently small distortion $x$. But when $x$ increases the
ground
state changes. For the problem (1)-(3) we succeeded in obtaining the
analytical
description of the dependence of the ground state on the value of the
distortion parameter.

In \ref{2} the main mathematical results are presented; in \ref{3} we give
the interpretation of the obtained results in terms of the neural network
and the Ising model. Some proofs which are not significant for the
understanding of the structure of the solution enter Appendix.

We would like to mention that our approach is very close to the problem of
generalization in neural networks \cite{Fon,Kre,Sil}. Under the classical
setting of the problem, one is investigating the network ability to
reconstruct the set of memorized patterns if during the learning their
distorted copies have been used. In contrast to the classical setting, in our
case the distortions are deterministic ones. Moreover, the value of the
distortion $x$ is the same for all the memorized patterns and only one
coordinate is distorted every time. In \ref{2} it is shown that some of
these
restrictions can be omitted. However, the most important restriction of our
approach is investigation of the generalization in the case
of one memorized pattern only, but not the
set of memorized patterns. Nevertheless, the meaningful interpretation of the
obtained results (see \ref{3}) is of interest for the more general case.

{\bf Notations.}
We denote by $\vec \varepsilon (k)$ the configuration vector
which is collinear to the bisectrix of the principle orthant of the space
$\rm R^k$. The vector, which after $p$ distortions
generates the set of the memorized patterns $\vec s^{(l)}$, is called
{\it the standard-vector} or {\it the standard}. Next, $n$ is the number of
the spin variables, $p$
is the number of the memorized patterns and $q=n-p$ is the number of the
nondistorted coordinates of the standard. Configuration vectors are
denoted by small Greek letters. We use small Latin letters to denote vectors
whose coordinates are real. The n-dimensional vectors are numerated by
superscripts enclosed in brackets and their coordinates are numerated by
subscripts. Matrices are denoted by half-bold capital Latin letters.

\section{Mathematical Results}
\label{2}
Let us look for the local maxima among configuration  vectors
whose last coordinate is positive.

\subsection{Basic Model}
\label{2.1}
Since $q$ last columns of the matrix $\bf S$ are the same, the configuration
vector which is "under the suspicion" to provide an extremum is of the
form\footnote{For details see \cite{Lit2}.}
$$\vec\sigma^*=(\underbrace{\sigma_1,\sigma_2,\ldots,\sigma_p}_
{\vec\sigma^\prime},\underbrace{1,\ldots,1}_q),\eqno(5)$$
where {\it we denote by $\vec\sigma^\prime$ the $p$-dimensional part of the
vector $\vec{\sigma}^{*}$, which is formed by its first $p$ coordinates.}
The direct calculations (or see \cite{Lit5}) show that
$$F(\vec \sigma^*)\propto x^2 - 2x(q+p\cos w)\cos w +(q+p\cos w)^2,\eqno(6)$$
where
$$\cos w=\frac{\sum_{i=1}^p \sigma_i}p=\frac{(\vec\sigma^\prime,
\vec\varepsilon (p))}{\parallel\vec\sigma^\prime\parallel\cdot\parallel
\vec\varepsilon (p)\parallel}\eqno(7)$$
is the cosine of the angle between $p$-dimensional vectors $\vec\sigma^\prime$
and $\vec\varepsilon (p)$. Depending on the number of the negative coordinates
of the  vector $\vec\sigma^\prime$, $\cos w$ takes the values
$$\cos w_k=1-\frac{2k}{p},\quad k=0,1,\ldots,p.$$
Consequently, $2^p$ "suspicious-looking" vectors $\vec\sigma^*$ are
grouped into the $p+1$ {\it classes } $\Sigma_k$ and
the functional $F(\vec\sigma^*)$ has the same value $F_k$ for all the
vectors from the same class:
$$F(\vec\sigma^*)\equiv F_k\quad \forall \vec\sigma^*\in\Sigma_k.$$
The classes $\Sigma_k$ are numerated by the number $k$ of the negative
coordinates which have the relevant vectors $\vec\sigma^*$. The
number of the vectors in the class $\Sigma_k$ is equal to $(^p_k)$.

To find the ground state under a given value of $x$, it is necessary to
determine the greatest of the values $F_0(x),F_1(x),\ldots,F_p(x)$. Under the
comparison the common term $x^2$ can be omitted. Therefore, to find out how
the ground state depends on the parameter $x$, it is necessary to examine the
family of the straight lines
$$L_k(x)=(q+p\cos w_k)^2 - 2x(q+p\cos w_k)\cos w_k. \eqno(8)$$
In the region where the $L_k(x)$ majorizes all the other straight lines, the
ground state belongs to the class $\Sigma_k$ and is $(^p_k)$-times degenerate.
The analysis of the relative position of the straight lines $L_k(x)$ is given
in Appendix and leads to the following

\noindent
{\bf Theorem.} {\it As $x$ varies from $-\infty$ to $\infty$ the ground state
in consecutive order belongs to the classes
$$\Sigma_0,\Sigma_1,\ldots,\Sigma_{k_{max}}.$$
The rebuilding of the ground state from the class $\Sigma_{k-1}$ into the class
$\Sigma_k$ occurs at the point $x_k$ of intersection of the straight lines
$L_{k-1}(x)$ and $L_k(x)$:
$$x_k=\frac{p}{2}\left(1+\frac{q}{q+p(\cos w_{k-1}+\cos w_k)}\right)=
p\cdot\frac{n-(2k-1)}{n+p-2(2k-1)},\quad k=1,2,\ldots,k_{max}.\eqno(9)$$
If $\frac{p-1}{n-1}<\frac13$, one after another all the $p$ rebuildings
of the ground state take place according to the
above scheme: $k_{max}=p$. And if $\frac{p-1}{n-1}>\frac13 $, the last
rebuilding is the one with the number $k_{max}=\left[\frac{n+p+2}4\right]$.
The functional has no other local maxima.}

The Theorem allows to solve a lot of practical problems. For example, let $n$
and $p$ be fixed. Then, for any preassigned $x$ it is sufficient to know
between which $x_k$ and $x_{k+1}$ it finds itself to determine which of the
classes $\Sigma_k$ provide the global maximum of the functional
$F(\vec\sigma)$, what is its value and the degeneration of the ground state.
On
the contrary, for the fixed $x$, it is possible to find out such $n$ and $p$
for which the ground state belongs to the preassigned class $\Sigma_k$,
{\it etc.}

\begin{figure}[htb]
\begin{center}
\leavevmode
\epsfxsize = 16.2truecm
\epsfysize = 7.5truecm
\epsffile{figure1.ps}
\caption{The typical behavior of the straight lines
$L_k(x),\ k=0,1,\ldots,p$. The rebuildings of the ground state occurs at the
points $x_k$ of the intersection of the straight lines $L_{k-1}$ and $L_k$.
Inside the interval $(x_k,x_{k+1})$ the ground state belongs to the class
$\Sigma_k$. When $x$ increases: a). all the rebuildings of the ground state
occur
($k_{max}=5$), because $\frac{p-1}{n-1}<\frac13$; b). only
$k_{max}=\left[\frac{n+p+2}4\right]=3$ rebuildings of
the ground state occur, because
$\frac{p-1}{n-1}>\frac13$.}
\end{center}
\end{figure}

The points $x_k$ (9), where the rebuildings of the ground state take place will
be called the {\it rebuilding points}.

In Fig.1 the typical examples of the relative position of the straight lines
$L_k(x)$ are presented for the cases $\frac{p-1}{n-1}<\frac13$ ({\it a})
and $\frac{p-1}{n-1}>\frac13$ ({\it b}). When $x$ changes from $-\infty$ to
$x_1$, the ground state is the standard-vector $\vec\varepsilon (n)$ (it
exhausts the class $\Sigma_0$). In the point $x_1$ the ground state rebuilds
from
the class $\Sigma_0$ to the class $\Sigma_1$ and becomes $p$-times degenerate.
When $x$ achieves the value $x_2$, the ground state rebuilds from the class
$\Sigma_1$ to the class $\Sigma_2$ and becomes $p\choose 2$-times degenerate,
and so on. As $x$ increases, at first the value of the functional
{\it for the ground state} monotonically decreases and then, after reaching
the minimum value, increases monotonically. For what follows let us note that
if $p<n-1$, then $k_{max}\ge [\frac{p+1}2]$ and $x_{k_{max}}\ge p$.

The case $p=n$ worth to be specially mentioned. Here all the rebuilding points
$x_k$ stick to one point
$$x^\prime\equiv x_k =\frac{n}2,\quad k=1,2,3,\ldots,\left[\frac{n+1}2\right].
\eqno(10) $$
For any $x$ from the left of $x^\prime$ the ground state is the
standard-vector
$\vec\varepsilon (n)$, and for $x$ from the right of $x^\prime$ the ground
state belongs to the class $\Sigma_{[\frac{n+1}2]}$ and is
$n\choose [\frac{n+1}2]$-times degenerate.

The interval $x_1 < x < x_{k_{max}}$ will be called {\it
the rebuilding region} of the ground state. This region is examined in details
in Appendix. Here we would like to mention only, that its left boundary
$x_1\ge\frac{p}2$ and it is the monotonically increasing function of $p$ as
well as of
$n$. And also, when $p=const$ and $n\to\infty$ the rebuilding region contracts
to the point
$$x^{\prime\prime}=p.\eqno(11)$$
In this case for  $x<x^{\prime\prime}$ the ground state is the standard-vector
and for $x>x^{\prime\prime}$ the ground state belongs to the class $\Sigma_p$;
again it is a nondegenerate one.

\subsection{Generalization of the Basic Model}
\label{2.2}
Let's examine some generalizations of the basic model, which seem to be the
most important. From our point of view they significantly widen the class of
possible connection matrices for which the
effective solution of the problem (1) can be obtained.

\subsubsection{An arbitrary cofiguration standard-vector}
\label{2.2.1}
An arbitrary configuration vector
$$\vec{\alpha}=(\alpha_1,\ldots,\alpha_p,\alpha_{p+1},\ldots,\alpha_n)$$
can be used in place of the standard $\vec\varepsilon (n)$, and its $p$
distorted copies of the type (4) can be used as memorized patterns. These
changes affect the elements of the connection matrix:
$$J_{ij}^{(\alpha)}=J_{ij}\alpha_i\alpha_j,\ i,j=1,2,\ldots,n.\eqno(12)$$
If we introduce a diagonal $(n\times n)$-matrix
$${\bf A}=\mbox{diag
}(\alpha_1,\ldots,\alpha_p,\alpha_{p+1}\ldots,\alpha_n),$$
then ${\bf J}^{(\alpha)}={\bf A}\cdot{\bf J}\cdot{\bf A}$ and the functional
we
have to maximize, takes the form:
$$F^{(\alpha)}=({\bf J^{(\alpha)}}\vec\sigma,\vec\sigma)=({\bf J}\cdot
{\bf A}\vec\sigma,{\bf A}\vec\sigma)=({\bf J}\vec\xi,\vec\xi)=F(\vec\xi).$$
Here $\vec\xi={\bf A}\vec\sigma$, and the functional $F(\vec\xi)$ has been
examined above. Consequently, all the results of the basic model remain
correct, though here they are relative to configuration vectors
$$\vec\sigma^*=(\alpha_1\sigma_1,\alpha_2\sigma_2,\ldots,\alpha_p\sigma_p,
\alpha_{p+1},\ldots,\alpha_n).$$
Thus, $\vec\varepsilon (n)$ can be used as the standard-vector without the
loss
of generality.

\subsubsection{Rotation of the memorized patterns}
\label{2.2.2}
An interesting generalization of the basic model is obtained when the
memorized patterns (4) are subjected to rotation as a whole. Let the
nontrivial
part of the rotation matrix ${\bf U}=(u_{ij})$ has the dimension $p\times p$.
It acts on the first $p$ coordinates of $n$-dimensional vectors
transforming the distorted part of the standard into a $p$-dimensional
vector $\vec u$:
$$\vec u={\bf U}\cdot\vec\varepsilon (p)=\left(\begin{array}{c}u_1\\u_2\\
\vdots\\u_p\end{array}\right),\quad
u_l=\sum_{i=1}^p u_{li},\mbox{ }l=1,2,\ldots,p,\mbox{ }\parallel\vec
u\parallel^2=p.\eqno(13)$$
As a result of the rotation the $l$-th memorized pattern takes the form
$$(u_1-xu_{1l},u_2-xu_{2l},\ldots,u_p-xu_{pl},1,\ldots,1),\eqno(14)$$
and it is convenient to choose the elements of the relevant connection matrix
${\bf J}^{(U)}$ in the form
$$J_{ij}^{(U)}=J_{ij}u_iu_j,\ i,j=1,2,\ldots,n.\eqno(15)$$
From Eqs. (14), (15) it directly follows that if the standard-vector
$\vec\varepsilon (n)$ is not changed as a result of the rotation (in other
words, if $u_1=u_2=\ldots=u_p=1$) then the connection matrix ${\bf J}^{(U)}$
coincides with the matrix $\bf J$. Consequently, all the results of the
basic model are valid, though in this case the memorized patterns (14) are
obtained from $\vec\varepsilon (n)$ by simultaneous (!) distortion of its $p$
coordinates.

But if the rotation results in $u_l$ which are not all equal unity, the
standard-vector shifts. Then it is convenient to introduce a diagonal
$(n\times n)$-matrix
$${\bf D}=(u_1,u_2,\ldots,u_p,1,\ldots,1)$$
and present the connection matrix ${\bf J}^{(U)}$  as a triple product
${\bf J}^{(U)}={\bf D}\cdot{\bf J}\cdot{\bf D}$. By analogy with the above
reasonings it is easy to show that again only the configuration vectors
$\vec\sigma^*$ (5) are "under the suspicion" to provide an extremum of the
functional $F^{(U)}(\vec\sigma)=({\bf J}^{(U)}\vec\sigma,\vec\sigma)$. Again
the value of the functional $F^{(U)}(\vec \sigma^*)$ is given by Eq.(6), but
$\cos w$ has to be calculated with the help of other equation:
$$\cos  w=\frac{\sum_{i=1}^p \sigma_i\cdot u_i}p=
\frac{(\vec\sigma^\prime,\vec u)}{\parallel \vec \sigma^\prime
\parallel \cdot \parallel \vec u \parallel}\eqno(16)$$
(compare with Eq.(7)). Thus, the value of the functional
$F^{(U)}(\vec\sigma^*)$ is completely defined by the distortion $x$ and the
value of the cosine of the angle between $\vec\sigma^\prime$ and $\vec u$.
Consequently, again the vectors $\vec\sigma^*$ are grouped in the classes
$\Sigma^{(U)}_k$, inside which the value of the functional
$F^{(U)}(\vec\sigma^*)$ is constant. The vectors $\vec\sigma^*$ belong to the
same class if their $p$-dimensional parts are equidistant from the vector
$\vec u$. The number of the classes $\Sigma^{(U)}_k$ is equal to the number of
{\it the different} values of $\cos w$ (16).

Thus, it is necessary to put in order $2^p$ vectors $\vec\sigma^\prime$ with
regard to their proximity to the vector $\vec u$. This problem will be
discussed in another publication. But now let us assume that the desired
ordering has been done and a decreasing sequence of the values of $\cos w$
(16) has been obtained:
$$\cos w_0>\cos w_1>\ldots>\cos w_t.\eqno(17)$$
Here the number of the classes $\Sigma^{(U)}_k$ is equal to $t+1$. We
denote by $v_k$ the number of the vectors $\vec\sigma^*$ in the  $k$-th class:
$$v_0+v_1+\ldots+v_t=2^p.$$
Since the half of the vectors $\vec\sigma^\prime$ can be
obtained from the other half of the full set of these vectors by changing
signs of all their coordinates, for every $k$ we have obvious equalities:
$$\cos w_k=-\cos w_{t-k},\quad v_k=v_{t-k}.$$
Therefore, $\cos w_k$ is negative beginning from some number $k$.

We see that as above to determine the dependence of the ground
state on the parameter $x$, it is necessary to examine the family of
the straight lines $L_k(x)$ (8) where the cosines are calculated according to
Eq.(16). The analysis of the placement of the straight
lines gives the result which generalizes the Theorem (the proof is given in
Appendix):

\noindent
{\it As $x$ varies from $-\infty$ to $\infty$ the ground state
in consecutive order belongs to the classes
$$\Sigma^{(U)}_0,\Sigma^{(U)}_1,\ldots,\Sigma^{(U)}_{k_{max}}.$$
The rebuilding of the ground state from the class $\Sigma^{(U)}_{k-1}$ into the
class
$\Sigma^{(U)}_k$ occurs at the point $x_k$ of intersection of the straight
lines $L_{k-1}(x)$ and $L_k(x)$:
$$x_k=\frac{p}{2}\left(1+\frac{q}{q+p(\cos w_{k-1}+\cos w_k)}\right),
\quad k=1,2,\ldots,k_{max},\eqno(18)$$
where $\cos w_k$ belong to the ordered sequence
(17). If $x_1>\frac34p$, one after another all the $t$ rebuildings
of the ground state take place according to the
above scheme: $k_{max}=t$. And if $x_1<\frac34p$, the rebuildings of the
ground
state come to an end when the denominator in Eq.(18) becomes negative.}

{\bf Note 1.} When the distortion $x$ belongs to the interval $(x_k,x_{k+1})$,
all the configuration vectors from the class $\Sigma^{(U)}_k$ are
the local maxima of the functional $F^{(U)}(x)$. At the same time the set of
these vectors is the set of the fixed points of the Hopfield type network
whose connection matrix is ${\bf J}^{(U)}$ (we remind that its diagonal
elements are equal zero). But the composition of each of the classes
$\Sigma^{(U)}_k$ is
determined by the values of $\{u_l\}_{l=1}^p$ (13) only. And the choice  of
$\{u_l\}_{l=1}^p$ is completely in the researcher's hand. In other words,
selecting
$\{u_l\}_{l=1}^p$ and the distortion parameter $x$, the Hopfield type network
with a preassigned set of fixed points can be created.

Let's explain this statement with the help of a simple example. Let's create
the network with the only fixed points
$$\begin{array}{rrrrrrrrrrl}\vec\alpha=(&-1,&-1,&-1,&1,&1,&1,&1,&\ldots,&1&)\\
\vec\beta=(&1,&1,&1,&-1,&-1,&1,&1,&\ldots,&1&)\\
\vec\gamma=(&1,&1,&1,&1,&1,&-1,&1,&\ldots,&1&).\end{array}$$
Then it is necessary, firstly, to group these vectors into one class
$\Sigma^{(U)}_k$, and, secondly, to choose such distortion $x$ which provides
the belonging of the ground state to this class. The first step can be easily
done, if with the help of a rotation of the memorized patterns the standard
$\vec\varepsilon (n)$ transforms into the vector
$$\vec u=(u_1,u_1,u_1,\frac{3u_1}2,\frac{3u_1}2,3u_1,u_7,\ldots,u_p),\quad
0<u_1<\frac{3u_1}2<3u_1<u_7<\ldots<u_p.$$

Indeed, it is evident that in this case the first six values of the cosines
(16) and the relevant classes $\Sigma_k^{(U)}$ are:
$$\begin{array}{lll}\cos w_0=\frac{9u_1+\sum_{i=7}^p u_i}p,&
\Sigma_0^{(U)}=&\vec\varepsilon (n)=\begin{array}{rrrrrrrrrrl}(&1,&1,&1,&1,&1,
&1,&1,&\ldots,&1&)\end{array}\\
\cos w_1=\cos w_0-2\cdot\frac{u_1}p,&
\Sigma_1^{(U)}=&\left\{\begin{array}{rrrrrrrrrrl}(&-1,&1,&1,&1,&1,&1,&1,
&\ldots,&1&)\\
(&1,&-1,&1,&1,&1,&1,&1,&\ldots,&1&)\\
(&1,&1,&-1,&1,&1,&1,&1,&\ldots,&1&)\end{array}\right.\\
\cos w_2=\cos w_0-3\cdot\frac{u_1}p,&
\Sigma_2^{(U)}=&\left\{\begin{array}{rrrrrrrrrrl}(&1,&1,&1,&-1,&1,&1,&1,
&\ldots,&1&)\\
(&1,&1,&1,&1,&-1,&1,&1,&\ldots,&1&)\end{array}\right.\\
\cos w_3=\cos w_0-4\cdot\frac{u_1}p,&
\Sigma_3^{(U)}=&\left\{\begin{array}{rrrrrrrrrrl}(&-1,&-1,&1,&1,&1,&1,&1,&
\ldots,&1&)\\
(&1,&-1,&-1,&1,&1,&1,&1,&\ldots,&1&)\\
(&-1,&1,&-1,&1,&1,&1,&1,&\ldots,&1&)\end{array}\right.\\
\cos w_4=\cos w_0-5\cdot\frac{u_1}p,&
\Sigma_4^{(U)}=&\left\{\begin{array}{rrrrrrrrrrl}(&-1,&1,&1,&-1,&1,&1,&1,&
\ldots,&1&)\\
(&-1,&1,&1,&1,&-1,&1,&1,&\ldots,&1&)\\
(&1,&-1,&1,&-1,&1,&1,&1,&\ldots,&1&)\\
(&1,&-1,&1,&1,&-1,&1,&1,&\ldots,&1&)\\
(&1,&1,&-1,&-1,&1,&1,&1,&\ldots,&1&)\\
(&1,&1,&-1,&1,&-1,&1,&1,&\ldots,&1&)\\
\end{array}\right.\\
\cos w_5=\cos w_0-6\cdot\frac{u_1}p,&
\Sigma_5^{(U)}=&\left\{\begin{array}{rrrrrrrrrrl}(&-1,&-1,&-1,&1,&1,&1,&1,&
\ldots,&1&)
\\
(&1,&1,&1,&-1,&-1,&1,&1,&\ldots,&1&)\\
(&1,&1,&1,&1,&1,&-1,&1,&\ldots,&1&)\end{array}\right.
\end{array}$$
Thus, the class $\Sigma_5^{(U)}$ consists of the given configuration
vectors $\vec\alpha, \vec\beta\mbox{ and }\vec\gamma$ exactly. Cosequently, if
the distortion parameter $x$ is chosen inside the interval $(x_5,x_6)$, where
$x_k$
are defined by Eq.(18), the relevant network of the Hopfield type has the
preassigned set of the fixed points.

We would like to mention that an additional analysis is required to find out
the limits of our method for the construction of networks with a preassigned
set of the fixed points.

\subsubsection{Different distortions of the memorized patterns}
\label{2.2.3}
The distortions $x_l$ can differ for every of the memorized patterns (4):
$$\vec s^{(l)}=(1,\ldots,1,\underbrace{1-x_l}_l,1,\ldots,1),\quad
l=1,2,\ldots,p.$$
With the help of the reasoning the same as in \ref{2.1}, it is easy to
show that as above only the configuration
vectors $\vec\sigma^*$ (5) are important. And the value of the functional
$F(\vec\sigma^*)$ is
$$F(\vec\sigma^*)=\sum_{l=1}^px_l^2 - 2(q+p\cos w)
\sum_{l=1}^p\sigma_lx_l+p(q+p\cos w)^2,$$
where again $\cos w$ is given by Eq.(7). Under the comparison we have to
omit
the term $\sum_{l=1}^px_l^2$. Now we obtain that to determine the ground
state dependence
on {\it the set} of the distortions $\{x_l\}_1^p$, it is necessary to examine
the family of the linear functions
$$L(\vec\sigma^*,\{x_l\}_1^p)=(q+p\cos w)^2-\frac{2(q+p\cos w)}p
\sum_{l=1}^p\sigma_lx_l.\eqno(19)$$
For this analysis one can try to use a probabilistic approach, considering
$x_l$ as
independent realizations of a stochastic variable, which, let us say,
is Gaussianly distributed. This way brings us nearer to the classical problem
of the network ability to generalization (see Introduction). We have not
investigate this approach.

On the other hand, we can remain in the framework of the geometric
approach, which is rather fruitful for this problem. Let's show this.
Without loss of generality we assume that the distortions $x_l$ are arranged
in increasing order. And for simplicity we assume them to be positive. Then,
$$0<x_1<x_2<\ldots<x_p.\eqno(20)$$
As in \ref{2.1} we have
$$\cos w_k=1-\frac{2k}p\quad\Rightarrow\quad q+p\cos w_k=n-2k,\mbox{ }k=0,1,
\ldots,p,$$
and the family of the functions (19) decomposes into $p+1$ subfamilies
$L_k(\vec\sigma^*,\{x_l\}_1^p)$. Here the subscript $k$ indicates that the
relevant configuration vector $\vec\sigma^*$ belongs to the class $\Sigma_k$:
$$L_k(\vec\sigma^*,\{x_l\}_1^p)=(n-2k)^2-\frac{2(n-2k)}p
\sum_{l=1}^p\sigma_lx_l,
\quad\vec\sigma^*\in\Sigma_k,\quad k=0,1,\ldots,p.\eqno(21)$$

The decomposition of the family of the functions (19) into $p+1$ subfamilies
(21) allows to find easily the ground state for a given set of the distortions
$\{x_l\}_1^p$. Indeed, to determine the ground state we have, firstly, to find
the vector $\vec\sigma^*(k)$ inside every class $\Sigma_k$, which provides
maximum $L_k^*$ of the function $L_k(\vec\sigma^*,\{x_l\}_1^p)$:
$$L^*_k=L_k(\vec\sigma^*(k),\{x_l\}_1^p)\ge L_k(\vec\sigma^*,\{x_l\}_1^p)
\quad\forall\vec\sigma^*\in\Sigma_k,\quad k=0,1,\ldots,p.$$
And, secondly, we have to determine the greatest among $p+1$ values $L_k^*$.
The last can be done easily, and the determination of the
vectors $\vec\sigma^*(k)$ does not require additional calculations at all:
if $n-2k>0$, we have to minimize the factor $\sum_{l=1}^p\sigma_lx_l$ (see
Eq.(21)); if $n-2k<0$ the same factor has to be maximized.
From inequalities (20) it is evident that the factor is minimal at the vector
$$\vec\sigma^*(k)=(\overbrace{1,1\ldots,1,\underbrace{-1,-1,\ldots,-1}_{k}}^
{p},1,1,\ldots,1),$$
and it is maximal at the vector
$$\vec\sigma^*(k)=(\overbrace{\underbrace{-1,-1\ldots,-1}_{k},1,1,\ldots,1}^{p
},1,1,\ldots,1).$$
Thus, when the distortions $x_l$ are different, the geometric approach allows
to determine the ground state very quickly  and efficiently. It seems, that
the potentiality of the geometric approach  does not exhausted by the mentioned
examples.

{\bf Note 2.} We denote by $\bar x$ the average over all the distortions
$$\bar x = \frac{\sum_{l=1}^p x_l}p.$$
We write the distortions $x_l$ in the form:
$$x_l=\bar x+d_l,\quad l=1,\ldots,p,$$
where $d_l= x_l - \bar x$ is the deviation of the $l$-th distortion from the
mean value $\bar x$. Now Eq.(19) can be rewritten as
$$L(\vec\sigma^*,\{x_l\}_1^p)=(q+p\cos w)^2-2\bar x(q+p\cos w)\cos w
-\frac{2(q+p\cos w)}p\sum_{l=1}^p\sigma_ld_l.\eqno(22)$$
Let's discuss the case of slightly different distortions. That is, the
deviations of all the distortions from the mean value $\bar x$ are
sufficiently small:
$$\mid d_l\mid <<1,\quad l=1,\ldots,p.$$
Then it is clear
that the first two terms give the main contribution in Eq.(22). We would
like to remind that just these two terms were analyzed in \ref{2.1} (see
Eq.(8)). Consequently,
{\it in the zeroth approximation} we have, that if $\bar x$ is inside the
interval $[x_k,x_{k+1}]$ ($x_k$ are defined by Eqs.(9)), the ground state
belongs to the class $\Sigma_k$, and the global maximum of the functional is
${p\choose k}$-times degenerate. What will happen when the third term in
Eq.(22) is taken into account? It can be expected that for
sufficiently small $d_l$, this term does not lead the ground state
out of the class $\Sigma_k$, but only lifts of the ground
state degeneracy. Numerical experiments confirm this assumption: when all
$d_l$ are not very great and $\bar x\in (x_k,x_{k+1})$, {\it all} the local
maxima of the functional (1) belong to the class $\Sigma_k$, the values of
the functional at vectors $\vec\sigma^*\in\Sigma_k$ differ slightly and one
(or several) of these vectors provide(s) the global maximum of the functional.
Thus, we explain one of the possible reasons, why the functional (1) has a lot
of slightly different local maxima. One frequently faces such situation in the
neural network theory, Ising model and the optimization problems.

\subsubsection{Other generalizations of the basic model}
\label{2.2.4}
Without dwelling on the details, we would like to mention other possible
generalizations of the basic model\footnote{See also Note 3 of \ref{3.2}.}.
\vskip 2mm
\noindent
${\bf 1^\circ.}$ {\bf The normalized memorized patterns.}

\noindent
The memorized patterns (4) can be normalized to unit to prevent their length
being dependent on the varying parameter $x$. It is easy to see, that in this
case all the reasonings, which lead to the Theorem, are valid. It can be shown
that here the maximum value of the functional {\it for the ground state}
decreases monotonically as function of $x$.
\vskip 2mm
\noindent
${\bf 2^\circ.}$ {\bf The identical distortion of a group of coordinates of
the standard.}

\noindent
Let $m$ coordinates of the standard $\vec \varepsilon (n)$ be distorted
{\it simultaneously and identically}.
Then $(p\times n)$-matrix of the memorized patterns has the form
$${\bf S}^{(m)}=
\left(\begin{array}{ccccccccccccc}
1-x&\ldots&1-x&1&\ldots&1&\ldots&1&\ldots&1&1&\ldots&1\\
1&\ldots&1&1-x&\ldots&1-x&\ldots&1&\ldots&1&1&\ldots&1\\
\vdots&\stackrel{m}{\longleftrightarrow}&\vdots&\vdots&
\stackrel{m}{\longleftrightarrow}&\vdots&\cdots&\vdots&
\stackrel{m}{\longleftrightarrow}&\vdots&\vdots&
\stackrel{q}{\longleftrightarrow}&\vdots\\
1&\ldots&1&1&\ldots&1&\ldots&1-x&\ldots&1-x&1&\ldots&1\end{array}
\right).$$
Of course, now $n=p\times m+q$. Let's note that the nondiagonal elements of
the
relevant connection matrix ${\bf J}^{(m)}$ have the terms, which are quadratic
in distortions.

With regard to the arguments of \ref{2.1}, it is not difficult to show
that the maximum of the functional
$F^{(m)}=({\bf J}^{(m)}\cdot\vec\sigma,\vec\sigma)$ is achieved only at
"piecewise constant" configuration vectors
$$\vec\sigma^*=(\underbrace{\sigma_1,\ldots,\sigma_1}_m,\underbrace{
\sigma_2,\ldots,\sigma_2}_m,\ldots,
\underbrace{\sigma_p,\ldots,\sigma_p}_m,\underbrace{1,\ldots,1}_q).$$
The form of these vectors is dictated by the memorized patterns matrix
${\bf S}^{(m)}$. The value
of the functional $F^{(m)}$ at the vectors $\vec\sigma^*$ is
$$F^{(m)}(\vec\sigma^*)\propto x^2 - 2x\cos w\left(\frac{q}m+p\cos w\right)+
\left(\frac{q}m+p\cos w\right)^2, $$
where $w$ is the angle between the $p$-dimensional vectors
$\vec\sigma^\prime=(\sigma_1,\sigma_2,\ldots,\sigma_p)$ and
$\vec\varepsilon (p)$, and $\cos w$ is given by Eq.(7). As above, $2^p$
vectors $\vec\sigma^*$ are distributed among the classes $\Sigma_k^{(m)}$
at which the value of the functional $F^{(m)}$ is constant:
$$F^{(m)}(\vec\sigma^*)=F^{(m)}_k\quad\forall\vec\sigma^*\in\Sigma_k^{(m)},
\quad k=0,1,\ldots,p.$$
The structure of the classes $\Sigma_k^{(m)}$ is much alike the structure of
the classes $\Sigma_k$ from \ref{2.1}. The only difference is that here
the value $-1$ have not one of the coordinates $\sigma_l$ of the vector
$\vec\sigma^*$, but a whole group of $m$ such coordinates. Again, as above,
the
number of the vectors $\vec\sigma^*$ in the class $\Sigma_k^{(m)}$ is equal to
$p\choose k$. The same reasoning as in \ref{2.1} leads to the generalization of
the Theorem:

\noindent
{\it The rebuildings of the ground state from the class
$\Sigma_{k-1}^{(m)}$
into the class $\Sigma_k^{(m)}$ occur at the points $x_k^{(m)}$
$$x_k^{(m)}=\frac{p}{2}\left[1+\frac{q}{q
+mp(\cos w_{k-1}+\cos w_k)}\right]=
p\cdot\frac{\frac{n}m-(2k-1)}{\frac{n}m+p-2(2k-1)},\quad k=1,\ldots,k_{max}.$$
If $p<\frac{q}{2m}+1$, one after another all the $p$ rebuildings of the ground
state take place according to the above scheme: $k_{max}=p$. And if
$p>\frac{q}{2m}+1$, the last rebuilding is the one whose number is
$k_{max}=\left[\frac{\frac{n}{m}+p+2}4\right]$. The functional has no other
local maxima.}

For what follows we would like to note, that when $m$ increases the rebuilding
points
$x_k^{(m)}$ with $k<\frac{p+1}2$ move to the left, and the rebuilding points
$x_k^{(m)}$ with $k>\frac{p+1}2$ move to the right. If $p$ is an odd number,
necessarily there is the rebuilding point with the number
$k=\frac{p+1}2$:\quad $x_{\frac{p+1}2}=p$. When $m$ increases this point
does not move.

\section{Discussion and Interpretations}
\label{3}
Let us discuss the results which mainly refer to the basic model.

\subsection{Neural networks}
\label{3.1}
In connection with neural networks the Theorem has to be interpreted in the
framework of the meaningful setting of the problem, which has
been given in Introduction. Then the Theorem means: {\it The quality of
"the truth" (i.e., the standard $\vec\varepsilon (n)$) which is understood by
the network depends on the distortion value $x$ during the learning stage and
on the length $p$ of the learning sequence}. We would like to point out some
important details of the suggested interpretation.

${\bf 1^\circ.}$ In agreement with the common sense the error of the network
increases with the increase of the distortion $x$: when $x$ belongs to the
interval $(x_k,x_{k+1})$, "the truth" which is understood by the
network (the class $\Sigma_k$) differs from the standard $\vec\varepsilon (n)$
by $k$ coordinates.

Also it is quite reasonable that the left boundary of the rebuilding region
$x_1$ is the increasing function of $p$ and $n$. Indeed, when $n$ and $x$ are
fixed, merely due to increase of the number of the memorized patterns $p$ the
value of $x_1$ can be forced to exceed $x$ (of course, if $x$ is not too
large). As a result $x$ turns out to be on the left of $x_1$, i.e. in the
region where {\it the only} fixed point is the standard $\vec\varepsilon (n)$.
In other words, only
by an increase of the length $p$ of the learning sequence, one can attain that
the network understands correctly "the truth" it is tried to be learned.
This conclusion is in agreement with the practical experience according which
the greater the length of the learning sequence, the better the signal can be
read through noise. From this point of view the most reliable network of
the considered type is the one with $p=n$: for any distortions
$x<x^\prime=\frac{n}2$ it steadily reproduces the standard-vector (see the
comment to Eq.(10)).

In the same way when $p$ and $x$ are fixed, merely due to an increase of the
number $n$ the value of $x_1$ can be forced to exceed $x$. This result is
reasonable too: when $p$ is fixed and the number $n$ increases, the relative
number $p/n$ of the distorted coordinates of the standard-vector decreases.
Naturally, the less the relative weight of the distortion, the better must be
the result of the learning.

${\bf 2^\circ.}$ Under an increase of the distortion parameter not only the
error of the network increase, but also the number of "noncorrect truths"
which
are understood by the network: when $x\in(x_k,x_{k+1})$ the number of the
network
fixed points is $p\choose k$. Such multiple degeneracy of the "truth"
is in agreement with the common idea that in general
there is only one "real truth", but the number of the "noncorrect truths" is
large. And also, the greater is the distortions at the learning stage, the
more can be the number of "noncorrect truths".

However, when $x$ becomes greater than $x_{\frac{p+1}2}=p$, the number of
"noncorrect truths" decreases. And this number decreases monotonically under
the further
increase of the distortion parameter. Moreover, according to the Theorem, when
$\frac{p-1}{n-1}<\frac13$, for all $x$ whose values exceed the right boundary
of the rebuilding region, the ground state belongs to the class $\Sigma_p$.
And, in spite of the very great distortion, the ground state is a
nondegenerate
one! In other words, the number of "noncorrect truths" is a nonmonotone
function of the distortion $x$.

This result seems to be a paradox. This paradox can be explained when it is
related to the well-known feature of our perception: we interpret
deviations in the image of a standard {\it as permissible ones only till some
threshold.} If only this threshold is exceeded, the distorted patterns are
interpreted as {\it quite different} standard. The Hopfield network under
consideration demonstrate just the same behavior with $x_{\frac{p+1}2}=p$
being the boundary of the permissible distortions of the standard. Let's
prove the last statement.

Here the notation for the standard $\vec\varepsilon (n)$ differs from the one
used above, and it is
$$\vec\varepsilon^{(+)}=(\underbrace{1,1,\ldots,1}_p,1,\ldots,1).\eqno(23)$$
We also introduce another standard:
$$\vec\varepsilon^{(-)}=(\underbrace{-1,-1,\ldots,-1}_p,1,\ldots,1).\eqno(24)$$
In their not coincident parts the
standards $\vec\varepsilon^{(+)}$ and $\vec\varepsilon^{(-)}$ are opposed with
each other, i.e. they are two opposite statements. Any of the
network fixed points $\vec\sigma^*$ (5) is an intermediate statement between
$\vec\varepsilon^{(+)}$ and $\vec\varepsilon^{(-)}$, which is drawn towards
either one edge of the scale ($\vec\varepsilon^{(+)}$), or the other
($\vec\varepsilon^{(-)}$).

Which distortions of the standard $\vec\varepsilon^{(+)}$ have to be assumed
as
permissible? Evidently, those for which the result of the learning
resembles the standard $\vec\varepsilon^{(+)}$ more, than its opposition
$\vec\varepsilon^{(-)}$. This is, the fixed points $\vec\sigma^*$  have to
differ from the standard $\vec\varepsilon^{(+)}$ not more than by $\frac{p}2$
first coordinates: $\vec\sigma^*\in \Sigma_k,\mbox{ }k\le\frac{p}2.$ And this
takes place only if $x< x_{\frac{p+1}2}=p$. But if $x>p$, the fixed points
$\sigma^*$ belong to the class $\Sigma_k$ with $k>\frac{p}2$, and they more
resemble the standard $\vec\varepsilon^{(-)}$, than $\vec\varepsilon^{(+)}$.

Let's point out the following curious circumstance. Let not the one, but a
whole group $m$ of the coordinates of the standard $\vec\varepsilon^{(+)}$
be distorted. Then the rebuilding points $x_k^{(m)}$ with the numbers
$k<\frac{p+1}2$
move to the left when $m$ increases (see the last paragraph of item $2^\circ$
in \ref{2.2.4}). And this is clear: now the total distortion of the
standard is $m$-times enlarged and the network earlier ceases to extract the
standard from the memorized patterns.
It is somewhat unexpected that the boundary $x_{\frac{p+1}2}^{(m)}=p$ of the
permissible distortions of the standard does not depend on $m$. It might
appear
that when the total distortion is $m$-times enlarged, the boundary of
the permissible distortions is reached $m$-times quicker. However, this is not
the case. Apparently, the reason is nonadditivity of the influence of the
distortion $x$ on the result of the learning. Out of the simultaneous
distortion of
$m$ coordinates of the standard, it does not follow at all that $x_k^{(m)}$
becomes $m$-times less than $x_k$. When $m$ increases, the rebuilding point
$x_1^{(m)}$ undergoes the most shift to the left. The shift of $x_2^{(m)}$ is
somewhat smaller, the shift of $x_3^{(m)}$ is still smaller, and so on. The
rebuilding points $x_k^{(m)}$ with the numbers $k \sim p/2$ shift only
negligible. It might be assumed that when the
distortion $x$ is so big that the result  of the learning differs from the
standard significantly (it is almost impossible to understand the standard),
it is not so important one or several parameters are distorted. Although
formally
just $x_{\frac{p+1}2}=p$ is the boundary of the permissible distortions.

In our opinion, the most interesting conclusion here is that the Hopfield type
network understands the distortions {\it as permissible} only up to some
threshold. That is, a nontrivial feature of the human perception is inherent
in
the artificial network of the examined type.

${\bf 3^\circ.}$ When the distortion $x$ is not very large the memorized
patterns $\vec s^{(l)}$ are interpreted by the network as the distorted
copies of the standard $\vec\varepsilon^{(+)}$.
But if during the learning stage the distortion $x$ is very large (it exceeds
$p$), the network ceases to understand the memorized patterns (4) as
distortions
of the standard $\vec\varepsilon^{(+)}$. It does not mean that in this
case the memorized patterns (4) are understood as distortions of the
opposite standard $\vec\varepsilon^{(-)}$: the permissible distortions of
$\vec\varepsilon^{(-)}$ are the vectors
$$(\overbrace{-1,\ldots,-1,\underbrace{-1+y}_l,-1,\ldots,-1}^p,1,\ldots,1),
\quad l=1,2,\ldots,p\eqno(25)$$
where $y$ is not so large. The question is, what do the memorized patterns (4)
with large distortions $x$ mean? We think, that the increase of $x$ above the
the threshold $p$ can be understood as
the more and more {\it negation} of the standard $\vec\varepsilon^{(+)}$.  As
if the network is learned by the memorized patterns, which {\it denies} the
standard $\vec\varepsilon^{(+)}$. In other words, the network is
{\it relearned} by presentation of negative examples.

There is big and clear to everybody difference between \underline{the
relearning} with the help of negative examples and \underline{the learning} of
the opposite truth. In the framework of our model in the last case slightly
distorted $n$-dimensional vectors of the type (25)
ought to be used as memorized patterns. Then, with no problem the network
understands the standard $\vec\varepsilon^{(-)}$. The relearning is entirely
other case. It is well known: the better the
incorrect truth has been understood, the more difficult (and sometimes even
impossible) to correct it; it is comparatively easy to correct the result
slightly, but it is much more difficult to revise it in the main, {\it etc.}

We think, that the dependence of $k_{max}$ on $p$ (see the Theorem) is the
reflection of just these problems. When the number $p$ of the parameters which
have to be corrected is not very great ($\frac{p-1}{n-1}<\frac13$), the
network can be relearned by simple presentation of negative examples. In this
case $k_{max}=p$ and, when "the denial" of the standard
$\vec\varepsilon^{(+)}$
is rather strong ($x>x_p$), as  "a new" truth the network understands
the opposite standard $\vec\varepsilon^{(-)}$. But if the number of the
corrected parameters is great ($\frac{p-1}{n-1}>\frac13$), to relearn the
network it is not sufficient to present the negative examples. In this case
$\left[\frac{p+1}2\right]<k_{max}<p$
and whatever large $x$ is $(x_{k_{max}}<x<\infty)$, as a new truth the network
understands not the opposite standard $\vec\varepsilon^{(-)}$, but one of the
statements intermediate between $\vec\varepsilon^{(+)}$ and
$\vec\varepsilon^{(-)}$. Though the understood truth is drawn towards
$\vec\varepsilon^{(-)}$, since $k_{max}>\frac{p}2$.

Of course, our interpretation is open for discussion. But it seems that in
real life there are a lot of examples, which confirm our
conception\footnote{In
this connection we are tempted to compare the development of the reforms ("the
relearning") in Russia and in China. In Russia very great number of the
parameters of the social order were simultaneously changed on their
opposition.
They are a
very wide group of economic regulations, a group of parameters related to home
and foreign policy, the parameters related to human rights and to the
destruction of
"the iron curtain" and so on. On the contrary, according to massmedia, in China
only a restricted group of economic regulations has been changed up to now.

And what are the results? Of course, now Russia is not a semi-fascist Soviet
State, but it does not reach (until now?) the stable State system of the
modern
democratic type. Already for a long time the situation in Russia is
characterized as a state, which is intermediate between these two oppositions.
The permanent instability and variations of the internal situation in Russia
are
associated with "the degeneracy of its present ground state". In the same time
(again, if the massmedia information is trustworthy), China demonstrates a
steady increase of well-being of the people. And this stipulates the direction
of its development.

Being not a specialist, I have no right to intrude into such a special field
of
knowledge as politology. But in accordance with our results, it appears that,
may be, "a step by step" strategy of reforms is more prospective. That is, the
result will be better if at first one group of the parameters is corrected
reliably, then the second group, then the third and so on.
According to the Theorem, taken separately each
group of the parameters can be corrected reliably (of course, if the Theorem
is applicable in this case). On the other hand, the final result of "the
relearning" of these two colossi is unknown yet...}.

\subsection{The Ising model}
\label{3.2}
The interpretation of the Ising model in
terms of the matrix $\bf S$ is not known yet. Therefore here the obtained
results are interpreted starting from the form of the Hamiltonian $\bf J$
(3.b). Let's write it in the form
$$ \bf J\propto\left(\begin{array}{cc}
\bf A&\bf B\\
\bf B^{\rm T}&\bf C\end{array}\right),$$
where the diagonal elements of $(p\times p)$-matrix $\bf A$ and
$(q\times q)$-matrix $\bf C$ are equal zero, and
$$\left\{\begin{array}{ll}a_{ij}=1-2y,&i,j=1,2,\ldots,p,\quad i\ne j;\\
b_{ik}=1-y,&i=1,2,\ldots,p,\ k=1,2,\ldots,q;\\
c_{kl}=1,&k,l=1,2,\ldots,q,\quad k\ne l;\\
y=\frac{x}p.&\ \end{array}\right.\eqno(26)$$
The matrix $\bf J$ corresponds to a spin system with the infinitely large
interaction radius. The system consists of two subsystems, which are
homogeneous with
respect to the spin interaction. The interaction between the $p$ spins of the
first subsystem is equal to $1-2y$; the interaction between the $q$ spins of
the second subsystem is equal to $1$; the crossinteraction between the spins
of
the subsystems is equal to $1-y$. When $p=n$ all the spins are interacting
with each other in the same way and we have the usual mean-field
approximation for the Ising model at zero temperature.

While $y<\frac12$, all the spins are interacting in the ferromagnetic way;
when
$\frac12<y$, the interaction between the spins of the first subsystem becomes
of antiferromagnetic type, and when $1<y$ the crossinteraction is of
antiferromagnetic type too. The Theorem allows to trace how the ground state
depends on the variation of the parameter $y$.

Let $p<n$. For $y\in (-\infty,\frac12)$ the ground state is the ferromagnetic
one since $\frac12<y_1=\frac{x_1}p$, and for $x<x_1$ the ground state is the
standard-vector $\vec\varepsilon^{(+)}$ (23). It is interesting that the
ground state remains ferromagnetic even if $\frac12<y<y_1$, i.e. when the
antiferromagnetic interactions already are shown up in the system. In other
words, when $p<n$, there is {\it "a gap"} between the value of $y$
corresponding to the destruction of the ferromagnetic
interaction, and the value of $y$ corresponding to the
destruction of the ferromagnetic ground state. Only after a "sufficient
amount" of the antiferromagnetic interaction is accumulated, the first
rebuilding
occurs and the ground state ceases to be the ferromagnetic one. Then,
when another critical "portion" of the antiferromagnetic interaction is
accumulated, the next rebuilding of the ground state occurs (it happens when
$y$
exceeds $y_2=\frac{x_2}p$), and so on. After the parameter $y$ reaches the
value $y^{\prime\prime}=1=\frac{x^{\prime\prime}}p$, the crossinteraction
becomes antiferromagnetic too. But the rebuildings of the ground state go on,
since usually $x_{k_{max}}> p$ (see the comment to the Theorem in \ref{2.1}).

The energy $E=-\ F$ {\it of the ground state} as a function of the parameter
$y$ has breaks at the points $y_k=\frac{x_k}p$. It increases till
$y\le 1$ and decreases when $y>1$. However, if
the memorized patterns are normalized to unit, the energy of the ground state
is a monotonically increasing function of $y$ (see item $1^\circ$ in
\ref{2.2.4}).

It is natural to treat $p=const,\mbox{ }n\to\infty$ as an infinitely large
sample with a few number of impurities. In this case all
$y_k$ stick to the point $y^{\prime\prime}=1$ (see the comment to Eq.(11)).
Depending {\it only on the type of the crossinteraction} between the
impurities and the rest of the sample, the ground state is either the
ferromagnetic one, or the spins of the impurities are opposite with respect to
all other spins.

Finally, let's discuss the case $p=n$. Then all $y_k$ stick to the
point  $y^\prime=\frac12$ (see the comment to Eq.(10)). Here the
destruction of the ferromagnetic interaction occurs simultaneously with the
change of the ground state ( "the gap" disappears). As long as the
interaction of the spins is ferromagnetic ($y<\frac12$), the ground state is
ferromagnetic too. But when the interaction of the spins becomes
antiferromagnetic ($y>\frac12$), the ground state turns out to be
$n\choose \frac{n+1}2$-times degenerate. From the right of $\frac12$ it
is natural to associate the state of the system with the spin glass phase.

{\bf Note 3.} We would like to remind that the Hamiltonian can be
generalized (see Eqs.(12) and (15)). Nevertheless, at heart of all these
generalizations is the strict relation between the elements of the
submatrices $\bf A$ and $\bf B$ of the Hamiltonian:
$$1+a=2b,$$
see Eqs.(26). This restriction on the matrix elements is
justified by nothing. It would be interesting to examine the case when the
matrix elements $a$ and $b$ do not related in any way. We have succeeded in
the
generalization of our consideration on this case as well as on the case when
the linear term $h\sum_{i=1}^n\sigma_i$ has been added to the functional
$F(\vec\sigma)$ (1). In the Ising model due to such a term the external
magnetic field can be taken into account. Now we prepare these results for
publication.

\subsection{Factor analysis}
\label{3.3}
In the framework of factor analysis a $(p\times n)$-matrix $\bf S$
with {\it real} matrix elements (see Eq.(2)) plays an important role.
Here the matrix $\bf S$
is an empirical matrix of the "objects-parameters" which includes the
exhausting quantitative description of the phenomenon under
consideration \cite{Brav2}. Its rows are $p$ objects defined by $n$ parameters
which are available for the measurement:
$$\begin{array}{cll}
\vec s^{(l)}=(s^{(l)}_1,s^{(l)}_2,\ldots,s^{(l)}_n) \in\rm R^n,&
l=1,2,\ldots,p&\mbox{ -- the vector-objects,}\\
\vec s_i=\left(\begin{array}{c}s_i^{(1)}\\s_i^{(2)}\\\vdots\\ s_i^{(p)}
\end{array}\right)\in\rm R^p,&i=1,2,\ldots,n&\mbox{ -- the vector-parameters.}
\end{array}$$
The main idea of the factor analysis is as follows: by proceeding from $n$
$p$-dimensional vector-parameters we have to introduce such $t$ artificial
characteristics, the so called {\it factors}, that their number is {\it much
less} than $n$ ($t\ll n$), but the description of the objects in terms of
these
factors does not lead to the loss of an important information about the
objects. The factors $\vec f_j$ are constructed as linear combinations of
the vector-parameters $\vec s_i$,
$$\vec f_j=\sum_{i=1}^n a_i^{(j)}\vec s_i,$$
which in the sense of a measure of proximity minimize the total deviation of
each factor from all the vector-parameters. It has been shown \cite{Lit1},
that for the so called {\it centroid method}, which is widespread in
factor analysis, the problem of the construction of the first factor is
equivalent to the maximization of the functional (1) with the connection
matrix
of the Hebb type. The interpretation of the obtained results in terms of
factor analysis will be presented elsewhere.

\section{Acknowledgments}
The author is grateful to Prof. Alexander Ezhov for helpful advices on the
substance of the work and to Dr. Inna Kaganova for preparation of this
manuscript. Wonderful atmosphere and personal contacts with the members of
workshop "Statistical Physics of Neural Networks" which took place at
Max-Plank-Institute of Physics of Complex Systems (March, 1999, Dresden,
Germany) helped the author in the interpretation of the obtained results.

\section{Appendix}
\subsection{The family of the straight lines $L_k(x)$}
\label{4.1}
The family of the straight lines (8),
$$L_k(x)= (n-2k)^2-2x\frac{p-2k}{p}(n-2k),\quad k=0,1,\ldots,p,$$
is under investigation. When $x=0$ we have $L_k(0)= (n-2k)^2$. Then if
$p \le n/2$ the values $L_k(0)$ decrease monotonically when $k$ increases:
$$L_0(0)>L_1(0)>L_2(0)>\ldots>L_p(0).$$
And if $p > n/2$, beginning from the number $n-p$ the values $L_k(0)$
become twice degenerate:
$$L_0(0)>L_1(0)>\ldots>L_{n-p}(0)=L_p(0)>L_{n-p+1}(0)=L_{p-1}(0)>\ldots $$
Depending on the number $k$ the straight lines $L_k(x)$
behave themselves in different ways: for $k<\frac{p}{2}$ and $k>\frac{n}{2}$
the slope of $L_k(x)$ is negative, and for $\frac{p}{2}<k<\frac{n}{2}$ it is
positive. The straight lines with the numbers $k=\frac{p}{2}$
and $k^\prime=\frac{n}{2}$ (if such $k$ and $k^\prime$ are found) are parallel
to the axis $OX$.

We denote by $x_{(k,k^\prime)}$ the abscissa of the point of intersection of
the straight lines $L_k(x)$ and $L_{k^\prime}(x)$:
$$x_{(k,k^\prime)}=\frac{p}{2}\left[1+\frac{q}{q+p(\cos w_k+\cos w_
{k^\prime})}\right]=p\frac{n-(k^\prime+k)}{n+p-2(k^\prime+k)},\quad 0\le k<
k^\prime\le p.\eqno(27)$$
We would like to point out that the abscissa
$x_{(k,k^\prime)}$ depends on the sum $k + k^\prime$ only. When $n+p$ is odd,
the number of {\it different} abscissae
$x_{(k,k^\prime)}$ is exactly $2p-1$. When $n+p$ is even, there can be such
numbers
$k^\prime$ and $k$, that $n+p-2(k^\prime+k)=0$. The relevant straight lines
$L_k(x)$ and $L_{k^\prime}(x)$ are parallel (they intersect at the infinity).

Generally, the abscissa $x_{(k,k^\prime)}$ increases when the sum
$k+k^\prime$ increases
$$x_{(0,1)}<x_{(0,2)}<x_{(0,3)}=x_{(1,2)}<x_{(0,4)}=x_{(1,3)}<\ldots$$
But the increase is strictly monotonic only for
$$\frac{p-1}{n-1}\le\frac13.$$
In this case the denominator of the expression for $x_{(k,k^\prime)}$ is
nonnegative for all $k$ and $k^\prime$, and all the intersections of the
straight lines take place in the region $x>0$. But if
$$\frac{p-1}{n-1} > \frac13,$$
the abscissae $x_{(k,k^\prime)}$ increases monotonically up to the value
$k+k^\prime = \frac{n+p}2$ where it has a discontinuity and becomes negative,
and
after that $x_{(k,k^\prime)}$ again increases monotonically with increase of
$k+k^\prime$.
It is easy to
verify that for $\frac13<\frac{p-1}{n-1}\le\frac12$ all the abscissae
$x_{(k,k^\prime)}$ with $k+k^\prime>\frac{n+p}2$ remain negative. But if
$\frac{p-1}{n-1}>\frac12$, at least $x_{(p-1,p)}$ appears in the region $x>0$.
Nevertheless, in this case also $x_{(p-1,p)}$ remains less than
$x_{(0,1)}$. In Fig.2 the dependence of $x_{(k,k^\prime)}$ on $k+k^\prime$ is
shown both for $\frac{p-1}{n-1}$ less and greater than $\frac13$.

\begin{figure}[htb]
\begin{center}
\leavevmode
\epsfxsize = 16.2truecm
\epsfysize = 7.5truecm
\epsffile{figure2.ps}
\caption{The dependence of the abscissa $x_{(k,k^\prime)}$ of the intersection
of the straight lines $L_k(x)$ and
$L_{k^\prime}(x)$ on $k+k^\prime$. Its character is
defined by the value of the parameter $\frac{p-1}{n-1}$:
a). for $\frac{p-1}{n-1}\le\frac13$ the curve $x_{(k,k^\prime)}$ increases
monotonically, remaining all the time in the region $x>0$;
b). for $\frac13<\frac{p-1}{n-1}$ the monotonically increasing curve
$x_{(k,k^\prime)}$
has discontinuity when $k+k^\prime=\frac{n+p}2$, after that again it
increases monotonically.}
\end{center}
\end{figure}

\subsection{The proof of the Theorem}
We divide the proof into two steps.

${\bf 1^\circ.}$ Let's observe how the ground state changes as $x$ increases.
For $x=0$ the ground state is the one with $k=0$  (see above
the inequalities for $L_k(0)$), and this state is nondegenerate. As $x$
increases, the state with $k=0$ remains the ground state, till the straight
line $L_0(x)$ does not intersect any other straight line $L_{k^\prime}$. From
Eq.(27) it is
evident, that the first intersection happens with the straight line $L_1(x)$
at the point
$$x_1\equiv x_{(0,1)}= p\frac{n-1}{n+p-2}.$$
After this the ground state is the state with $k=1$ from the class $\Sigma_1$.
This state is $p$-times degenerate.

As $x$ increases further, the state with $k=1$ remains the ground state till
the straight line $L_1(x)$ does not intersect any other straight line
$L_{k^\prime}(x)$. From Eq.(27) it is evident, that the first
intersection takes place with the straight line $L_2(x)$ at the point
$$x_2\equiv x_{(1,2)}= p\frac{n-3}{n+p-6}.$$
After this the ground state is the state with $k=2$ from the class $\Sigma_2$.
And this state is $p\choose 2$-times degenerate.

By analogy with the above reasoning and taking into account the behavior of
the
family of the straight lines $L_k(x)$ (see \ref{4.1}), we obtain that
when $x$ increases the rebuildings of the ground state happen at the points
$$x_k\equiv x_{(k-1,k)}=\frac{p}{2}\left[1+\frac{q}{q+p(\cos w_{k-1}+
\cos w_k)}\right]= p\frac{n-(2k-1)}{n+p-2(2k-1)}\quad k=1,2,3,\ldots.$$
At the point of the $k$-th rebuilding the $p\choose {k-1}$-times degenerate
ground state from the class $\Sigma_{k-1}$ is replaced by the
$p\choose k$-times degenerate ground state from the class $\Sigma_k$.

The rebuildings of the ground state take place till the increase of $x$ is
accompanied by the intersections of the straight lines $L_{k-1}(x)$ and
$L_k(x)$. In other words, till $2k-1$ is less than $\frac{n+p}2$. Since for
$\frac{p-1}{n-1}\le\frac13$ even $2p-1 < \frac{n+p}2$, in this case as $x$
increases one after the other all the $p$ possible rebuildings of the ground
state take place. Consequently, for all $x > x_p \equiv x_{(p-1,p)}$, the
ground state remains in the class
$\Sigma_p$. (Strictly speaking, if $\frac{p-1}{n-1}=\frac13$, the last
rebuilding of the ground state happens at the infinity.)

But if $\frac{p-1}{n-1}>\frac13$, the rebuildings of the ground state come to
an end when $2k-1$ becomes more than $\frac{n+p}2$. For these $k$ the abscissa
$x_{(k-1,k)}$ of the intersection of the straight lines $L_{k-1}$ and $L_k$
is negative, and under the further increase of $x$ there is no new
intersection
of the straight line $L_{k-1}$ with any other straight line $L_{k^\prime}$. In
other words, the state with the number $k-1$ remains the ground state under
the
unlimited increase of $x$. In this case the number $k_{max}$ of the last
rebuilding is defined from the equation
$$2k_{max}-1=\frac{n+p}2\Longrightarrow k_{max}=\left[\frac{n+p+2}4\right].
\eqno(28)$$
Thus, in the region $x > 0$ the last rebuilding of the ground state takes
place
when $x=x_{k_{max}}$. (If $\frac{n+p+2}4=\left[\frac{n+p+2}4\right]$, the
last rebuilding takes place at the infinity.) For negative $x$ there are no
rebuildings of the ground state, since in this region the straight line
$L_0(x)$ majorizes all other straight lines.

${\bf 2^\circ.}$ To complete the proof, let's show that the functional has no
other local maxima. The configuration vector, which provides a local maximum,
has the form (5) necessarily. Let for some $x$ this vector belong to the class
$\Sigma_k$. Then the inequalities
$$L_k(x)\ge L_{k-1}(x),L_{k+1}(x)$$
must be fulfilled. Removing the brackets and collecting the similar terms we
obtain:
$$x[2\pm(n+p-4k)]\ge p[1\pm(n-2k)].\eqno(29)$$

A). Let $n+p-4k\ge 2$. It means that $k+1\le\frac{n+p+2}4$, or, in other
words,
that $k\in [1,k_{max}-1]$. The sign "+" in Eq.(29) results in the inequality
$$x\ge p\frac{n-(2k-1)}{n+p-2(2k-1)}=x_k,$$
and the sign "--" results in the inequality
$$x\le p\frac{n-(2k+1)}{n+p-2(2k+1)}=x_{k+1}.$$
Thus, for $k\in [1,k_{max}-1]$ the state with the number $k$ is the local
maximum only if $x$ belongs to the interval $[x_k,x_{k+1}]$, where this state
is the global maximum.

B). Let now $-2\le n+p-4k<2$. It means that $k>k_{max}-1$ and $k\le k_{max}$;
in other words, that $k=k_{max}$.
The sign "+" in Eq.(29) results in the inequality $x\ge x_{k_{max}}$, and the
sign "--" results in the inequality $x\ge x_{k_{max}+1}$. But we know that
the
value of $x_{k_{max}+1}$ is already negative (see above \ref{4.1}).
Consequently, the simultaneous fulfillment of the last two inequalities takes
place in the region $x\ge x_{k_{max}}$. And this means that the state with
the number $k_{max}$ is the local maximum of the functional only if it is its
global maximum.

C). And, finally, let $n+p-4k<-2$; in other words, let $k>k_{max}$.
The sign "+" in Eq.(29) results in the inequality $x\le x_k$, and the sign
"--" results in the inequality $x\ge x_{k+1}$. But as it has been shown in
\ref{4.1}, even for $k>k_{max}$ we have the ordering $x_k < x_{k+1}$.
Consequently, for $k>k_{max}$ the simultaneous fulfillment of
inequalities (29)
is impossible. This completes the proof.

\subsection{The region of the ground state rebuilding}
The region where the parameter $x$ changes from $x_1$ to $x_{k_{max}}$, will
be called {\it the rebuilding region} of the ground state. Let's examine this
region.

The left boundary of the rebuilding region is
$$x_1=p\cdot\frac{n-1}{n+p-2}\ge\frac{p}2,$$
where the equality takes place only for $p=n$. It is easy to verify that
$x_1$
is a monotone increasing function of both $p$ and $n$. These statements are
important for the meaningful interpretation of the obtained results.

The expression for the right boundary of the rebuilding region depends on the
relation between $p$ and $n$. If $\frac{p-1}{n-1}<\frac13$, then $k_{max}=p$
and
$$x_{k_{max}}=x_p=p\cdot\frac{n+1-2p}{n+2-3p}.$$
As function of $n$ the value of $x_p$  decreases. Consequently, for
$n\to\infty$ and fixed $p$ the rebuilding region contracts to the point
$x^{\prime\prime}=p$. And if $\frac{p-1}{n-1}>\frac13$, than
$$x_{k_{max}}=p\cdot\frac{n-p+4\kappa}{8\kappa},$$
where $4\kappa=1,2\mbox{ or }3$, and $k_{max}$ is given by Eq.(28).
When $n$ is fixed and $p\to n$, the rebuilding region also contracts to the
point. It is not difficult to see that this point is $x^\prime=\frac{n}{2}$.

In many respects the point $x^{\prime\prime}=p$ is remarkable (see \ref{3}).
It is easy to see that before $x$ becomes equal to
$x^{\prime\prime}=p$, exactly $\frac{p+1}2$ rebuildings of the ground state
take place. This result does not depend on $n$ (if only $p$ is less than
$n-1$).

\subsection{The rotation of the memorized patterns and $k_{max}$}

Let's prove the generalization of the Theorem for the case, when all the
memorized patterns (4) are subjected to the rotation as a whole (see
\ref{2.2.2}). For simplicity we assume that the number $t+1$ of different
values of the cosines (17) is even:
$$t+1=2r.$$
Let's agree the subscript $k$ to take the values $1, 2,\ldots, r=\frac{t+1}2$
only.

Since for every $k$ we have $\cos w_k=-\cos w_{t+1-k}$, then
$\cos w_{r-1}=-\cos w_r$ and the ordered sequence of the cosines has the form:
$$\cos w_0>\cos w_1>\ldots>\cos w_{r-1}>0>\cos w_r>\cos w_{r+1}>
\ldots>\cos w_{2r-1}.$$
It is not difficult to prove that the rebuilding of the ground state from the
class $\Sigma_{j-1}^{(U)}$ into the class $\Sigma_j^{(U)}$ happens at the point
$x_j$ (18). In what follows we'll prove that
{\it if $x_1>\frac34p$, one after
another all the $t$ rebuildings
of the ground state take place according to the above scheme: $k_{max}=t$. And
if $x_1<\frac34p$, the rebuildings of the ground state come to an end when the
denominator in Eq.(18) becomes negative.}

The numbers $k$ and $t+1-k$ are symmetric with regard to the ends of the
sequence of the rebuilding points. For every rebuilding point
$$x_k=\frac{p}2\cdot \left[1+\frac{q}{q+p(\cos w_{k-1}+\cos w_k)}\right],$$
the point $x_{t+1-k}$ is called {\it the dual} rebuilding point. The
expression for $x_{t+1-k}$,
$$x_{t+1-k}=\frac{p}2\cdot \left(1+\frac{q}{q-p(\cos w_{k-1}+
\cos w_k)}\right),$$
differs from the expression for $x_k$ by the sign in the denominator only.
The rebuilding point
$$x_r=\frac{p}2\cdot \left[1+\frac{q}{q+p(\cos w_{r-1}+\cos w_r)}\right]=
\frac{p}2\cdot \left[1+\frac{q}q\right]=p$$
is dual to itself. We begin the proof of the general statement with the proof
of the following

\noindent
{\bf Lemma.}

\noindent
{\it 1. The inequality $x_k>\frac{3}4p$ is equivalent to the inequality
$q-p(\cos w_{k-1}+\cos w_k)>0.$}

\noindent
{\it 2. For the value of $x_{t+1-k}$ to be negative, it is necessary and
sufficient to have $\frac{2}3p<x_k<\frac{3}4p.$}

The proof of the first statement is obtained from the convertible sequence of
the inequalities
$$x_k>\frac{3}4p\Longleftrightarrow\frac{q}{q+p(\cos w_{k-1}+\cos w_k)}>
\frac{1}2\Longleftrightarrow q-p(\cos w_{k-1}+\cos w_k)>0.$$
Let's prove the second statement. Let
$$x_{t+1-k}=\frac{p}2\cdot \left[1+\frac{q}{q-p(\cos w_{k-1}+
\cos w_k)}\right]<0.$$
Then, firstly, $q-p(\cos w_{k-1}+\cos w_k)<0$, and it is equivalent to the
inequality $x_k<\frac{3}4p$. And, secondly:
$$\frac{q}{q-p(\cos w_{k-1}+\cos w_k)}< -1,$$
from where we directly obtain $q+p(\cos w_{k-1}+\cos w_k)<3q$, and,
consequently,
$$x_k=\frac{p}2\cdot \left[1+\frac{q}{q+p(\cos w_{k-1}+\cos w_k)}\right]>
\frac{p}2\cdot \left[1+\frac{q}{3q}\right]=\frac{2}3p.$$
Assume the converse. Let $\frac{2}3p<x_k<\frac{3}4p$. This means that
$$-q<q+p(\cos w_{t-k}+\cos w_{t-k+1})<0,$$
from where it follows that
$$x_{t+1-k}=\frac{p}2\cdot \left[1-\frac{q}{\mid q+p(\cos w_{t-k}+
\cos w_{t-k+1})\mid}\right]<\frac{p}2\cdot \left[1-\frac{q}q\right]=0,$$
i.e. that the abscissa $x_{t+1-k}$ is negative. This concludes the proof of
the Lemma.

Generally, the first rebuilding point $x_1$ always exceeds $\frac{p}2$.
The proof of the main statement is in three steps.

\noindent
{\bf Step 1.} Let's examine the case $x_1>\frac{3}4p$.
Then according to the first conclusion of the Lemma
$q-p(\cos w_0+\cos w_1)>0$ and therefore a similar inequality is true for any
other $k$: $q-p(\cos w_{k-1}+\cos w_k)>0$. Consequently, all the rebuilding
points $x_j$ (18) are positive. And this is just the case
when in consecutive order one after another all $t$ possible
rebuildings of the ground state take place. It is interesting that this
case corresponds to very simple requirement to the first rebuilding point:
$x_1>\frac{3}4p$.

\noindent
{\bf Step 2.} Now we investigate the case $\frac{2}3p<x_1<\frac{3}4p$.
According to
the second statement of the Lemma, here the rebuilding point $x_t$,
which is dual to $x_1$, is negative: $x_t<0$. Then we pass from $x_1$ to $x_2$:
$x_1 < x_2$. If $x_2$ is still less than $\frac{3}4p$, its dual rebuilding
point $x_{t-1}$ is negative too. Then we must to pass from $x_2$ to
$x_3$:
if $x_3$ is also less than $\frac{3}4p$, its dual
rebuilding point $x_{t-2}$ is also negative, and so on. Since at least
$x_r=p>\frac{3}4p$,
sooner or later we get the number $k$, for which $x_k$ exceeds
$\frac{3}4p$: $x_k>\frac{3}4p$. Then according to the Lemma its dual rebuilding
point $x_{t+1-k}$ is already positive:
$$x_{t+1-k}=\frac{p}2\cdot
\left[1+\frac{q}{q-p(\cos w_{k-1}+\cos w_k)}\right]>0.$$
It is evident that $x_{t+1-k}$ is not only positive, but exceeds $x_1$:
$$x_{t+1-k}-x_1\sim \cos w_0+\cos w_1+\cos w_{k-1}+\cos w_k>0.$$
Thus, the last rebuilding of the ground state happens at the point with
the number $t+1-k$. The next rebuilding point, $x_{t-k+2}$, is already
negative. In other words, $k_{max}$ is exactly equal to $j\equiv t+1-k$, where
$$q+p(\cos w_{j-1}+\cos w_j)>0,\mbox{ but }q+p(\cos w_j+\cos w_{j+1})<0.$$
And just these inequalities give the statement we have to prove.

\noindent
{\bf Step 3.} Finally, we have to examine the case $x_1<\frac{2}3p$. This
inequality
is equivalent to $q-p(\cos w_0+\cos w_1)<-q$, from which it
follows that $x_t>0$:
$$x_t=\frac{p}2\cdot \left[1-\frac{q}{\mid q+p(\cos w_0+\cos w_1)
\mid}\right]>\frac{p}2\cdot \left[1-\frac{q}q\right]=0.$$
In the same time $x_t<x_1$:
$$x_1-x_t\sim
\frac{q}{q+p(\cos w_0+\cos w_1)}-\frac{q}{q-p(\cos w_0+\cos w_1)}>0.$$
We pass from $x_1$ to $x_2$. If $x_2$ is also less than $\frac{2}3p$, its dual
rebuilding point $x_{t-1}$ is also positive (but this time
it does not exceed $x_t$). Increasing the number $k$ we sooner or later get
the rebuilding point $x_k$ which exceeds $\frac{2}3p$. And then we
find ourselves under the conditions discussed in Step 2:
$\frac{2}3p<x_k<\frac{3}4p$. Here
the point $x_{t+1-k}$, which is dual to $x_k$, is negative. At the next steps
we get such a number $k$, that
$x_k$ exceeds $\frac{3}4p$. According to the above reasoning this means that
$k_{max}=t+1-k$. In other words, in the case $x_1<\frac{2}3p$, at first
$x_j$ increase up to the maximal positive value, then become negative, and
after that they increase. Beginning with a certain
number $j$, the values of $x_j$ are positive again, but less than $x_1$. This
concludes the proof.

Thus, the rebuilding points $x_j,\mbox{ }j=1,2,\ldots,t$ behave themselves in
different ways depending on the interval to which the first point $x_1$
belongs:
$$x_1>\frac{3}4p,\quad\frac{2}3p<x_1<\frac{3}4p,\quad x_1<\frac{2}3p.
\eqno(30)$$
The analogous result has been also obtained above, when in \ref{4.1}
the family of the straight lines $L_k(x)$ was investigated. There the behavior
of the rebuilding points were depended on the belonging of the ratio
$\frac{p-1}{n-1}$ to one of the three intervals
$$\frac{p-1}{n-1}<\frac{1}3,\quad\frac{1}3<\frac{p-1}{n-1}<\frac{1}2,
\quad\frac{1}2<\frac{p-1}{n-1}.\eqno(31)$$
It is easy to verify that for $u_1=u_2=\ldots=u_p=1$ the inequalities (30)
turn into inequalities (31). The case of an even $t=2r$ is examined in the
same way.

redpajama_set_name: RedPajamaArXiv


text:
Bartle hits the trail on election night
State Capitol Bureau
Intro: The state's leading opponent to stem cell research will start election night by slipping into his running shoes. Hillari Duthoo has more from the state Capitol. RunTime:
One of the hottest issues on the Missouri ballot is the stem cell initiative. Its origin and future in the Missouri Senate hinges on one man, Senator Matt Bartle.
While election night watch parties are getting underway, the Kansas City area republican will continue his own tradition of running ten miles alone.
He says he started doing this in his first run for office in 1998 and remembers every one since.
Actuality: BART01.WAV
Run Time: 00:14
Description: "Campaigning is so public. And this is kind of a quiet beginning to a very public night. But it's just been great. I just spend a lot of time praying out on these runs."
He says if the measure fails, he plans to bring it before the Senate in January.
From the state Capitol, I'm Hillari Duthoo.


FALLS CHURCH, Va. — The National Association of State EMS Officials released a guidebook on fatigue risk management.
NASEMSO partnered with the University of Pittsburgh School of Medicine to create "Evidence-Based Guidelines (EBG) for Fatigue Risk Management in Emergency Medical Services," which is available here.
The guidebook is meant to help mitigate the effects of fatigue, "with recommendations based on a comprehensive evaluation of the best available evidence related to numerous fatigue mitigation strategies," according to a press release from the organization.


Billy Casey currently plays drums and sings in a Fort Worth band called "Legends of the Mind". This project of 5 musicians is a psychedelic/symphonic rock band focusing on the music of four bands: * Pink Floyd * Moody Blues * Beatles * Jethro Tull * and others.
Members of Legends of the Mind are all veteran musicians with extensive studio and live experience. Scott Hawk (keyboard, flute, vocals), Larry Hughes (guitar, vocals), John D. Harris (bass guitar), Scott Douglas (guitar, vocals), Bill Casey (drums, vocals).


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