text:
L'église de la Sainte-Famille est érigée dans la commune d'Annonay, département de l'Ardèche en région Auvergne-Rhône-Alpes. Son architecture de style contemporain est l'œuvre de l'architecte André Devys. L'édifice est situé dans le quartier des Perrières au nord du centre d'Annonay, rue Albert-Schweitzer.

Historique
Les documents cités dans la bibliographie de l'article permettent d'établir la chronologie suivante :
 1925 : Joseph Besset achète un terrain à « Ma Campagne » pour y installer les ateliers de sa carrosserie (fondée en 1913). Ils deviendront Iveco Bus.
 1949 : Vincent Auriol, président de la République, pose la première pierre d'un nouveau quartier d'Annonay : la cité des Perrières ().
 1957 : mille habitants aux Perrières, des écoles, des commerces, de plus en plus d'immeubles, des lieux de cultes éloignés. Premières réunions des catholiques du quartier. Réflexions sur la création d'une paroisse (janvier - février). Nomination d'un curé pour la « future paroisse des Perrières » (). Projet d'une église et d'un foyer paroissial : obtention du permis de construire, début du terrassement () et lancement d'une souscription publique (). Première messe aux Perrières et bénédiction d'une croix marquant l'emplacement de la future église (). Début du chantier de construction dans l'ancienne propriété Berger devenue propriété diocésaine d'une église provisoire devant devenir foyer paroissial en attendant la construction de l'église définitive ().
1958  : réception des entrepreneurs et partage du verre de l'amitié lors de la pose de la toiture (). Signature du décret d'érection de la paroisse par , évêque de Viviers (). Première kermesse paroissiale à proximité du chantier (7 et ). Ouverture et bénédiction d'un nouveau lieu de culte à Annonay (). Le programme de construction est revu, l'édifice deviendra lieu de culte définitif au fil des décennies. La croix de 1957 demeure au Sud du site paroissial.
 1960 : consécration de l'autel renfermant des reliques de saint Étienne et de saint Agrève ().
 1965-1970 : réaménagement du chœur pour pouvoir « célébrer face au peuple » à l'issue du concile Vatican-II
 Années 1990 :	campagne de travaux à l'intérieur de l'église.
 2002 : profanation de l'église (). L'ensemble du site est réaménagé.
 2003 : création de la paroisse « Sainte-Claire » d'Annonay, de Roiffieux et de La Vocance par fusion des paroisses catholiques existantes ( janvier).
 2009 : spécialisation cultuelle des églises catholiques d'Annonay. La Sainte-Famille : une église et un site intéressant pour des célébrations avec des enfants, des malades ou en plein air. L'église accueille aussi des messes dominicales, diverses célébrations comme des « célébrations pénitentielles »,  et une partie des funérailles des Annonéens.
 2012 : campagne de travaux à l'intérieur de l'église.
 2013 - 2015 : le projet de rénovation urbaine du quartier du Zodiaque mise en œuvre depuis 2010 entraine une reconfiguration cadastrale du site de la Sainte-Famille avec des acquisitions et des transferts de propriété entre l'association diocésaine de Viviers et la ville d'Annonay. La ville d'Annonay crée un parking devant l'entrée du site et une nouvelle liaison vers le parc de Déomas. Suivant l'engagement pris par le conseil municipal du , la croix sur socle de pierre bénite le  est déplacée durant la première quinzaine de  d'une dizaine de mètres plus au Nord. La paroisse Sainte-Claire en profite pour la rénover. Ce petit monument garde le souvenir de la première messe « aux Perrières » et du projet initial d'aménagement du site.
 2021 : création de la paroisse « Bienheureux Gabriel Longueville » du Bassin d'Annonay par fusion des paroisses « Sainte-Claire » d'Annonay, de Roiffieux et de La Vocance et « Saint-Christophe lès Annonay » ( mai) .
 2022 : rénovation des toitures (novembre)

Description générale
L'église a été bâtie en partie en pierres du Pilat par l'entreprise Parriche. Ses plans et son style s'inspire des églises rurales d'autrefois : une nef unique sans transept.
Le bâtiment mesure 36 m de long pour 11,20 m de large. Il comporte un préau de 2 m de large sur toute la longueur de la façade principale. La surface de l'ensemble est d'environ 403 m² (475 m² en tenant compte du préau).
Pas de clocher ici, mais sur le pignon, se trouve un campanile œuvré par le ferronnier annonéen Montélimard. .

Vocable

« La Sainte Famille » a été choisi comme patronne de l'église par les fidèles du quartier car, en 1957, il était essentiellement composé de familles avec des enfants en bas âge. D'autres vocables avaient été cependant proposés : « saint Pie X » ou « Le Saint Esprit ».

Visite de l'édifice

Le sanctuaire

Dans le chœur de l'édifice se trouvent plusieurs éléments aux fonctions liturgiques précises :
 la croix,
 l'ambon et le pupitre d'animation en chêne. Leur unique pied est travaillé,
 l'autel, œuvre du marbrier annonéen Gay. La table en granite bleu des Pyrénées repose sur un pied massif taillé dans un seul bloc du même granite,
 le tabernacle est ici un coffre recouvert de cuivre martelé posé sur un petit autel dont le pied en chêne rappelle celui de l'ambon et du pupitre d'animation.
Dans le chœur, se trouve aussi le siège de présidence : un fauteuil à l'assise rouge.

La menuiserie et le mobilier en bois (excepté la charpente) a été confectionné par Maurice Palayer, jeune artisan et ancien enfant de chœur -turiféraire exactement- du curé Poinas résidant en 1957 à la paroisse St François et qui gérait les travaux de l'église en construction. Les bancs de 3 m de long sont fait d'un seul tenant avec du chêne de Bourgogne.

Vitraux
Des vitraux éclairent le chœur de l'église. À gauche de l'autel, trois verrières forment un triptyque représentant la Trinité.

À droite les trois vitraux figurent :
L'alpha et l'oméga.
La salutation mariale.
L'Eglise.

Sculptures

Dans le jardin d'agrément autour de l'église se trouve une statue : Notre-Dame de la Famille. Cette sculpture, bénite le , a été restaurée dans les années 1990.

Dans l'église se trouve un Christ en croix.

Tableau
Dans le chœur à gauche, un tableau : La Sainte Famille.

Autre élément
La balustrade du préau en fer forgé est constituée par la date de bénédiction de l'église : .

Cloche
Une seule cloche est suspendue dans le campanile. Ses inscriptions : « - PARRAIN Sr JEAN DACHIS - MARRAINE MARIE ????ON - - ? MICHALON FONDU EN DECEMBRE 1821 - - BURDIN FONDEUR LYON - » Elle porte en effigie : un Christ en croix et une Vierge à l'Enfant.

Chronologie des curés

1958 – 1971
Un curé et des vicaires ont la charge de la paroisse dont le territoire correspond au quartier.

1971 – 2003
Une équipe presbytérale dont les membres sont « curés in solidum » (responsables solidairement) a la charge de l'ensemble des paroisses catholiques d'Annonay, de Roiffieux et de Vidalon.

2003 – 2021
Avec la création de la paroisse Sainte-Claire dont le territoire comprend Annonay, Roiffieux et la vallée de La Vocance, une Equipe d'Animation Pastorale (E.A.P.) composée de laïcs en mission et de prêtres nommés « curés in solidum » à la charge de la paroisse nouvelle.

Depuis 2021
Avec la création de la paroisse « Bienheureux Gabriel Longueville » dont le territoire correspond au bassin de vie d'Annonay, une Équipe d'Animation Pastorale (E.A.P.) présidée par un prêtre nommé « curé » à la charge de la paroisse nouvelle.

Notes et références

Voir aussi

Article connexe
 Annonay

Bibliographie
 Églises en Ardèche.- Document du Service Diocésain de la Pastorale des Réalités du Tourisme et des Loisirs du diocèse de l'Ardèche et de la Commission d'Art Sacré.- 2010.
 En Communauté.- bulletin de la paroisse catholique Sainte-Claire d'Annonay - Vocance paraissant depuis 1966.- bimensuel.
 Paroisse Sainte-Claire d'Annonay-Vocance.- Livre d'Or de la paroisse de la Sainte-Famille.- Album photos, coupures de presse et souvenirs divers parcourant diverses époques de 1957 à nos jours.
 Perrier Jacques ().- Visiter une église.- Centurion, Paris.-1993.- 143 p.

Lien externe

Annonay Sainte-Famille
Annonay
Annonay


McHale & Co. Solicitors :: McHale & Co Secures Personal Injury Settlement for Injured Mechanic in Lincoln.
Home > Solicitors for You > Personal Injury Claims > Case Studies > McHale & Co Secures Personal Injury Settlement for Injured Mechanic in Lincoln.
We were approached in 2008 by a client in Lincoln who had been involved in a nasty accident at work. The client was a vehicle mechanic and he had fallen down an inspection pit in the garage he worked. He landed on his right leg causing a severe fracture to the talus bone in his foot/ankle.
Typically the client was taken to hospital and underwent an operation for internal fixation with screws, pins and plates. He was discharged home to recover but unfortunately he was left with a permanent disability in his ankle. He now permanently walks with a walking stick and on a particularly bad day he requires two walking sticks. The client was 34 years old at the time of the accident. As you can imagine, his permanent disability has put a kibosh on returning to his job as a mechanic or any other job that require any manual handling and/or manual labour.
Having been a mechanic for nearly 10 years, there were real issues in the case as to what career or work the client could undertaken. It was agreed with the Defendant Solicitors' that they would fund some occupational rehabilitation and as a result the client found the confidence to return to college to obtain an NVQ Level in Teaching Assistance. The client even arranged work experience at a local school to assist him with his course and to enable him to start moving on with his life.
Despite some considerable areas of disagreement in relation to future treatment, care and career prospects, we managed to secure a hefty settlement for the client in excess of £380,000.00.
The client is thrilled with the settlement and is looking forward to moving into his new one storey property with his wife and 4 children.


text:
<?php

namespace console\models;

use Yii;

/**
 * This is the model class for table "field_data_field_google".
 *
 * @property string $entity_type
 * @property string $bundle
 * @property integer $deleted
 * @property string $entity_id
 * @property string $revision_id
 * @property string $language
 * @property string $delta
 * @property integer $field_google_value
 */
class FieldDataFieldGoogle extends \yii\db\ActiveRecord
{
    /**
     * @inheritdoc
     */
    public static function tableName()
    {
        return 'field_data_field_google';
    }

    /**
     * @return \yii\db\Connection the database connection used by this AR class.
     */
    public static function getDb()
    {
        return Yii::$app->get('db_tvod1');
    }

    /**
     * @inheritdoc
     */
    public function rules()
    {
        return [
            [['entity_type', 'deleted', 'entity_id', 'language', 'delta'], 'required'],
            [['deleted', 'entity_id', 'revision_id', 'delta', 'field_google_value'], 'integer'],
            [['entity_type', 'bundle'], 'string', 'max' => 128],
            [['language'], 'string', 'max' => 32]
        ];
    }

    /**
     * @inheritdoc
     */
    public function attributeLabels()
    {
        return [
            'entity_type' => 'Entity Type',
            'bundle' => 'Bundle',
            'deleted' => 'Deleted',
            'entity_id' => 'Entity ID',
            'revision_id' => 'Revision ID',
            'language' => 'Language',
            'delta' => 'Delta',
            'field_google_value' => 'Field Google Value',
        ];
    }
}

redpajama_set_name: RedPajamaGithub


A takeoff on Viennese crunch, this rich and delicious brittle bark is sure to satisfy your sweet tooth. It's also quick and easy to make; a great treat to have on hand!
Line a baking sheet with parchment paper. Place sugars in a small saucepan over medium flame, and heat without stirring until completely melted (about 8 minutes). Remove from heat and stir in margarine until melted. Stir in cashews and quickly pour onto parchment paper, spreading and flattening with the back of a spoon. Allow brittle to cool for 10 minutes.
Meanwhile, in a separate small pot, heat chocolate over low flame until melted and smooth. Spread chocolate over brittle and sprinkle with chopped almonds. Allow brittle bark to harden, then break into pieces. Store in a cool, dry place.


Welcome to KentJobs.co.uk, which is brought to you by KM Media Group, the independent, family-owned media company serving the people and local communities of Kent and Medway.
We're dedicated to helping you find a brilliant new job, make a career change or return to the workplace. Our online jobs board, KentJobs.co.uk, is here to help you find great employment opportunities in your local area.
You can also find details about current vacancies in the KentJobs section of your KM Media Group newspaper or hear about them on your local kmfm radio station.
We also post jobs on Twitter and Facebook - so why not stay in touch ?


What Do Falling Corporate Profits Mean With Stocks Near Their Highs?
If you've followed our commentary for awhile, you may have noticed that we don't cover fundamental or economic data too often. That is for a good reason: we don't use it, at all. Occasionally, however, a data point will cross the radar that piques our interest for whatever reason. So it is with the current state of U.S. Corporate Profits. The U.S. Bureau of Economic Analysis released the latest data today revealing that Corporate Profits (after Tax with Inventory Valuation Adjustment and Capital Consumption Adjustment) were down 9% for the 1st quarter and are now down 16% from their peak in the 3rd quarter of 2013.
the S&P 500 made a 2-year high at some point within the same quarter.
As it turns out, there have been 21 quarters meeting that criteria since 1960.
Many of the occurrences came in clusters in 1980, 1986-1987 and 1998-2000. There were also single occurrences in 1961, 2007, 2011 and the 1st quarter of last year. Without going into great depth of analysis, one can tell by the inauspicious dates that these circumstances have not worked out well in the past. The stock market may not have rolled over immediately in every occasion (e.g., 1986, 1998, 2014), but it usually ended up paying the piper.
Specifically, the average drawdown over the 2 years following these quarters was -18.6%. This compares with an average 2-year drawdown of -7.3% following all quarters since 1960.
We don't follow economic and fundamental data too often since we've never found it very helpful in our investment decision-making process. At times, however, a certain data series will garner our attention. Often times, as is the case with Corporate Profits presently, it grabs our attention because it is receiving very little attention elsewhere. From just a cursory look at the current trend of falling Corporate Profits, however, it would appear to be a potential negative influence on the stock market that is trading near its all-time highs – if not immediately, then eventually.


Dr. Phattar Burarak is keen in communication technology especially in media production, broadcasting, television broadcasting, small local broadcast media; Communication for Development; Marketing Communication, Journalism, citizen journalism; Communication and New Media. She used to be a newspaper reporter of the Manager Newspaper; also used to be a Visiting Professor at the University of Economics and Finance, Yunnan, Kunming city, Yunnan Province, China. She is also a reader of scholarly articles and research papers. She also works on the development of local media to develop the community such as the development and implementation of experimental community television services in the community; the community participation in television services to serve their own development, etc.


marble floor design » Italian marble medallion to enhance your home in CA, FL.
No matter the kind of luxury marble floor that you need, either for your living room or even the dining room, there are various marble medallions from which you can choose from. However, in order for you to enhance the luxury that is provided by the marble medallion in the house, there is a need for you to be keen while choosing. This is due to the fact that there are various designs, patterns and styles and this may leave you confused on the best choice.
One of the best designs which is popular from Orange County California to Manhattan New York and which you can opt for is the round marble medallion. These are available in various color schemes and sizes. Even where you may opt for an Italian Medallion, it is still possible for you various color schemes and sizes. Other than your preferred design, it is necessary to go for the right size that suits the space you intend to have the marbles installed. Where you intend to have a marble border in your living room, you should ensure that you know the place properly in advance. This is in order to avoid wasting any tiles for your house in Montana. On top of this, it will be possible for you to determine the tiles that you will require. This ensures that you have marble medallions that are enough.
The marble inlaid color is also an aspect which you will require to consider carefully as they are available in various color schemes. You should opt for those colors that are able to complement the color of your dining room, living room or any other place that you may be interested in placing them. This will help to bring out your sense of style and taste in regard to the marble carpet.


Thank you for a very enjoyable game created by Munro in 2015 is presented by the Colorado Springs Fine Arts Centre in July 2017.
The artwork, shown for the first time outside of the UK, creates a visual abstraction of the chess game between the spaceship supercomputer HAL 9000 and the astronaut Dr. Frank Poole. The work is a rare step away from Munro's preferred medium, Light.


text:
Biomarker Testing Is Key to Personalized, At-Home Fertility Care
For decades, women trying to conceive have been striving to meet an arbitrary "normal" menstrual cycle. Now, as healthcare rapidly expands beyond the hospital, personalized biomarker testing is crucial to understanding each woman's unique fertility profile, writes OOVA founder Dr. Amy Divaraniya.
By Dr. Amy Divaraniya
The menstrual cycle is a core component of a woman's health. Some even go as far as coining it a woman's fifth vital sign because various aspects, like the length of period flow, the color of blood, and the intensity of bleeding, can confirm if everything is okay—or highlight issues that may be arising inside a woman's body.
Many focus solely on the period to evaluate the menstrual cycle, however, it is far more complicated. It is also important to know how long the four phases of a woman's cycle are, confirm if she ovulates, learn how her hormones behave, and determine if she has an elevated hormone baseline.
For decades, women have been striving to meet the standards of an arbitrary "normal" 28-32-day cycle, with a period that lasts for 5 days and ovulation that occurs on cycle day 14. I say it's arbitrary because our lives have changed drastically since this standard was defined. Looking back even one generation, most will notice that their parents conceived their first child at a significantly younger age than women today. More women are striving to acquire higher education, hit career milestones, travel, find the right partner, and enjoy life together before growing their families. Unfortunately, while doing all these things, our bodies are aging and it becomes harder to conceive naturally.
It is critical to learn each woman's unique hormone profile, detect fluctuations, then determine what is happening in her cycle.
So what happens if you do not have a 28-32-day cycle? What happens if you don't have a regular period? Do you still ovulate? Can you still get pregnant naturally? These are questions that flood the minds of women who do not fit the mold of that "normal" cycle. These women make up over 35 percent of the female population, including those with irregular periods, imbalanced hormones, or reproductive disorders such as polycystic ovary syndrome (PCOS) or endometriosis.
If a woman has reproductive disorders, her hormone profile will be very different from someone who does not. For example, in order for ovulation to happen, the luteinizing hormone (or LH levels) must elevate to signal the ovary to release the egg. However, how is that spike detected? Does it have to reach some predetermined threshold? If so, what happens if a woman has systemic elevated LH levels because she has PCOS? She's not always ovulating. This is why personalized healthcare is so important. It is critical to learn each woman's unique hormone profile, detect fluctuations, then determine what is happening in her cycle. Without this information, medical advice and care is being based on standards that are irrelevant for individual woman.
Oova's urine test monitors quantitative levels of fertility hormones in the privacy of a woman's home with the same accuracy as a blood test.
With the advancement of technology (and a daily blood draw at a fertility clinic), it has become a lot more accessible to get the personalized data we need to truly understand each woman's unique fertility profile. This enabled clinicians to capture and monitor daily hormone levels throughout a cycle, but at the height of the COVID-19 pandemic, fertility clinics were being asked to close their doors because fertility care was considered an elective procedure. Couples and clinics were left in a horrible state of limbo where they weren't sure how to proceed.
This gave our technology at Oova an amazing opportunity to shine. We created a urine test to monitor quantitative levels of fertility hormones in the privacy of a woman's home with the same accuracy as a blood test. The only link missing was for a clinician to be able to view this data seamlessly. We solved this problem by creating a HIPAA compliant dashboard: when a woman gets her hormone results in the Oova app, her clinician sees them in their dashboard, all in real-time. This has become a true game changer for the healthcare industry. As healthcare rapidly expands beyond the hospital—with COVID-19 fueling an even greater demand for at-home monitoring—patients no longer want to get poked and prodded unnecessarily. Patients also don't want to go to the clinic for routine work that can be done in their home. Now clinics can continue to monitor their patients remotely, saving them time, money and resources, while patients are testing in the privacy of their homes.
But it is not enough to just monitor hormone levels. The critical element is learning and embracing every woman's unique hormone profile. This is vital for women who have irregular cycles or reproductive disorders. These women may have trends, baselines, and hormone behavior that is completely consistent for them each cycle, but off from the "normal" curve. By learning every woman's unique fertility profile, we can determine if her curve is normal for her body.
Let Oova decode your fertility by ordering your kit today
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The past year has been an unprecedented time for society. Every industry was challenged and put to the test. However, no industry was challenged as heavily as the healthcare system, from patient care, clinician demand, medical access, to what is considered an elective procedure. It highlighted how flawed our system was and how desperately change was needed. We can start addressing this change by applying technology to enable personalized care, and getting to the root of health issues.
Oova is on a mission to revolutionize healthcare and we started with an area that was in dire need of innovation. There has been such little innovation, especially when it comes to fertility, as most at-home urine testing (think regular ovulation predictor kits) has been limited to a binary positive or negative results. With the incorporation of technological advancements, we can now get accurate quantitative values for critical biomarkers, providing clinical data in the privacy of your home. This is not only easing the health journey for women but also increasing accessibility for fertility care for those who cannot afford to go to the clinic every day. We have just unveiled the tip of the iceberg when it comes to what our technology can do. We plan to disrupt as many areas of healthcare that we can. This is just the beginning of Oova's journey.
Dr. Amy Divaraniya, a PhD in Biomedical Sciences with a focus on Genetics and Genomics from Mount Sinai School of Medicine, is the founder and CEO of Oova. While trying to conceive for 18 months, Amy was shocked by the lack of precise, science-based fertility options. This drove her to develop Oova's lab-grade fertility translator to track multiple hormones and create a personalized fertility profile to indicate specific moments in a woman's cycle.
11 Essential Books About Women's Health, Written by Women
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21 Innovative Period Products for Your Menstrual Cycle
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We Need to Talk About Menopause
100 percent of people born with a uterus will go through menopause if they live long enough—but only 7 percent of them will get the help they need for it, reveals Gennev founder Jill Angelo.
Transforming How America Treats Chronic Disease
Through her growing tele-health platform, Parsley Health founder Robin Berzin is making holistic healthcare more accessible across the country.
By Lauren Fisher
Womanhood Is Not a Pre-Existing Condition
Across every age group, women are saddled with higher health care costs than men. What will it take for "being female" to be covered by insurance?
By Leslie Goldman
How Female VCs Are Thinking About Women's Healthcare
Research shows women raise less than half the capital of male founders on average, which doesn't bode well for an industry where 80 percent of founders identify as women.
By Ellen Sheng

redpajama_set_name: RedPajamaCommonCrawl


text:
Q: Laravel Spatie media library getFullUrl method not working inside the loop I am developing API in laravel. When I try to fetch an uploaded image url inside a loop that gives me this error,
Spatie\MediaLibrary\Exceptions\UrlCannotBeDetermined: Storage path myAppPath\storage\app is not part of public path myAppPath\public in file myAppPath\vendor\spatie\laravel-medialibrary\src\Exceptions\UrlCannotBeDetermined.php on line 11
But when I try to get url directly it works.
like this ==>
$log = Log::find($id)->getMedia('images');
return $log[0]->getFullUrl();

this is my code. below code is not working
$logs = Log::where(function ($query) {
$query->where('owner', '=', Auth::user()->id)
->orWhere('owner', '=', Auth::user()->partner->id);
})
->orderBy('date', 'desc')
->get();

    $resArr= array();

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        for ($j=0;$j<count($mediaArray);$j++){
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            );
            array_push($m,$med);
        }

        $a = array(
            'id'=>$logs[$i]->id,
            'title'=>$logs[$i]->title,
            'owner'=>$logs[$i]->owner,
            'body'=>$logs[$i]->body,
            'date'=>$logs[$i]->date,
            'mood'=>$logs[$i]->mood,
            'created_at'=>$logs[$i]->created_at,
            'partner_log_id'=>$logs[$i]->partner_log_id,
            'media'=>$m,
            'partner_log'=>$logs[$i]->partnerLog,

        );

        array_push($resArr,$a);
    }
    return \response()->json(['status'=>'success','data'=>$resArr]);

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return [

    /*
     * The disk on which to store added files and derived images by default. Choose
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         */
        'extra_headers' => [
            'CacheControl' => 'max-age=604800',
        ],
    ],

    'responsive_images' => [

        /*
         * This class is responsible for calculating the target widths of the responsive
         * images. By default we optimize for filesize and create variations that each are 20%
         * smaller than the previous one. More info in the documentation.
         *
         * https://docs.spatie.be/laravel-medialibrary/v7/advanced-usage/generating-responsive-images
         */
        'width_calculator' => Spatie\MediaLibrary\ResponsiveImages\WidthCalculator\FileSizeOptimizedWidthCalculator::class,

        /*
         * By default rendering media to a responsive image will add some javascript and a tiny placeholder.
         * This ensures that the browser can already determine the correct layout.
         */
        'use_tiny_placeholders' => true,

        /*
         * This class will generate the tiny placeholder used for progressive image loading. By default
         * the medialibrary will use a tiny blurred jpg image.
         */
        'tiny_placeholder_generator' => Spatie\MediaLibrary\ResponsiveImages\TinyPlaceholderGenerator\Blurred::class,
    ],

    /*
     * When urls to files get generated, this class will be called. Leave empty
     * if your files are stored locally above the site root or on s3.
     */
    'url_generator' => null,

    /*
     * Whether to activate versioning when urls to files get generated.
     * When activated, this attaches a ?v=xx query string to the URL.
     */
    'version_urls' => false,

    /*
     * The class that contains the strategy for determining a media file's path.
     */
    'path_generator' => null,

    /*
     * Medialibrary will try to optimize all converted images by removing
     * metadata and applying a little bit of compression. These are
     * the optimizers that will be used by default.
     */
    'image_optimizers' => [
        Spatie\ImageOptimizer\Optimizers\Jpegoptim::class => [
            '--strip-all', // this strips out all text information such as comments and EXIF data
            '--all-progressive', // this will make sure the resulting image is a progressive one
        ],
        Spatie\ImageOptimizer\Optimizers\Pngquant::class => [
            '--force', // required parameter for this package
        ],
        Spatie\ImageOptimizer\Optimizers\Optipng::class => [
            '-i0', // this will result in a non-interlaced, progressive scanned image
            '-o2', // this set the optimization level to two (multiple IDAT compression trials)
            '-quiet', // required parameter for this package
        ],
        Spatie\ImageOptimizer\Optimizers\Svgo::class => [
            '--disable=cleanupIDs', // disabling because it is known to cause troubles
        ],
        Spatie\ImageOptimizer\Optimizers\Gifsicle::class => [
            '-b', // required parameter for this package
            '-O3', // this produces the slowest but best results
        ],
    ],

    /*
     * These generators will be used to create an image of media files.
     */
    'image_generators' => [
        Spatie\MediaLibrary\ImageGenerators\FileTypes\Image::class,
        Spatie\MediaLibrary\ImageGenerators\FileTypes\Webp::class,
        Spatie\MediaLibrary\ImageGenerators\FileTypes\Pdf::class,
        Spatie\MediaLibrary\ImageGenerators\FileTypes\Svg::class,
        Spatie\MediaLibrary\ImageGenerators\FileTypes\Video::class,
    ],

    /*
     * The engine that should perform the image conversions.
     * Should be either `gd` or `imagick`.
     */
    'image_driver' => env('IMAGE_DRIVER', 'gd'),

    /*
     * FFMPEG & FFProbe binaries paths, only used if you try to generate video
     * thumbnails and have installed the php-ffmpeg/php-ffmpeg composer
     * dependency.
     */
    'ffmpeg_path' => env('FFMPEG_PATH', '/usr/bin/ffmpeg'),
    'ffprobe_path' => env('FFPROBE_PATH', '/usr/bin/ffprobe'),

    /*
     * The path where to store temporary files while performing image conversions.
     * If set to null, storage_path('medialibrary/temp') will be used.
     */
    'temporary_directory_path' => null,

    /*
     * Here you can override the class names of the jobs used by this package. Make sure
     * your custom jobs extend the ones provided by the package.
     */
    'jobs' => [
        'perform_conversions' => Spatie\MediaLibrary\Jobs\PerformConversions::class,
        'generate_responsive_images' => Spatie\MediaLibrary\Jobs\GenerateResponsiveImages::class,
    ],
];

This is filesystems.php file.
<?php

return [

    /*
    |--------------------------------------------------------------------------
    | Default Filesystem Disk
    |--------------------------------------------------------------------------
    |
    | Here you may specify the default filesystem disk that should be used
    | by the framework. The "local" disk, as well as a variety of cloud
    | based disks are available to your application. Just store away!
    |
    */

    'default' => env('FILESYSTEM_DRIVER', 'local'),

    /*
    |--------------------------------------------------------------------------
    | Default Cloud Filesystem Disk
    |--------------------------------------------------------------------------
    |
    | Many applications store files both locally and in the cloud. For this
    | reason, you may specify a default "cloud" driver here. This driver
    | will be bound as the Cloud disk implementation in the container.
    |
    */

    'cloud' => env('FILESYSTEM_CLOUD', 's3'),

    /*
    |--------------------------------------------------------------------------
    | Filesystem Disks
    |--------------------------------------------------------------------------
    |
    | Here you may configure as many filesystem "disks" as you wish, and you
    | may even configure multiple disks of the same driver. Defaults have
    | been setup for each driver as an example of the required options.
    |
    | Supported Drivers: "local", "ftp", "sftp", "s3"
    |
    */

    'disks' => [

        'local' => [
            'driver' => 'local',
            'root' => storage_path('app'),
        ],

        'public' => [
            'driver' => 'local',
            'root' => storage_path('app/public'),
            'url' => env('APP_URL').'/storage',
            'visibility' => 'public',
        ],

        's3' => [
            'driver' => 's3',
            'key' => env('AWS_ACCESS_KEY_ID'),
            'secret' => env('AWS_SECRET_ACCESS_KEY'),
            'region' => env('AWS_DEFAULT_REGION'),
            'bucket' => env('AWS_BUCKET'),
            'url' => env('AWS_URL'),
        ],

    ],

    /*
    |--------------------------------------------------------------------------
    | Symbolic Links
    |--------------------------------------------------------------------------
    |
    | Here you may configure the symbolic links that will be created when the
    | `storage:link` Artisan command is executed. The array keys should be
    | the locations of the links and the values should be their targets.
    |
    */

    'links' => [
        public_path('storage') => storage_path('app/public'),
    ],

];

Hope this helps.
Any suggestion, please????

redpajama_set_name: RedPajamaStackExchange


The same KIT KAT wafers you love, draped in delicious dark chocolate. Learn more about product, nutrition and allergy information.
Give yourself a break and enjoy layers of crispy wafers covered in smooth milk chocolate.
Each bar is individually wrapped for snacking on the go or sharing with friends.
These candy bars are perfect for filling candy bowls or giving out as holiday favors.
Contains only 210 calories per serving size. KIT KAT Bars are OU Certified as a Kosher food.
Item Description: Kit Kat 1.5 oz.


The Soul Keeper download is an RPG action production created by the American team of HELM Systems. The action of the title takes place in a fantastic land, which looks a bit like The Elder Scrolls V: Skyrim. The project is based on Unreal Engine 4 technology by Epic Games. This engine powers games such as Daylight, Dead Island 2, Fable Legends or Street Fighter V.
Download "The Soul Keeper download PC" Installer (Supports Resumable Downloads).
If you feel any problem in running The Soul Keeper download PC, then feel free to comment down below.


Man at Arms TD is a top-down tower defense game for iPad from developer Inert Soap.
The game begins with the player setting up a new profile and visiting the World Map where you can see the stages currently available to you. Just tap on a level to jump right in – you'll see a brief description of the stage as well as the ability to alter the difficulty level before getting started.
The layout of the game is similar to other Tower Defense titles and fans of the genre will be able to quickly get in to the action. The map has a number of designated zones where you can lay down defenses – simply tap on one to construct buildings such as barracks, artillery and archer outposts. Once you're finished with your defensive placements, the game begins and successive waves of enemy troops will enter from one side of the map and attempt to make it across to the other side. You gain coins through defeating enemies that enable you to upgrade your defences as you play.
Each stage ends either after all of your lives are exhausted, or you manage to defeat the enemy. The spoils of victory include special cards which can be used later in the game. Each card confers special abilities or upgrades which can be compiled in a 'defensive deck' or combined in Mike's Mansion of Magic to create new and more powerful cards.
The are many game stages in Man at Arms TD and the levels cover a wide range of beautifully drawn environments including deserts, forests and swamps.
Man at Arms TD features high-quality cartoon-style graphics, with bright colors and an excellent level of visual detail. There are a huge number of unique enemies that you encounter throughout the game so you have to keep modifying your strategy to keep up. There are fast enemies that zip past your troops, flying enemies as well as ranged monsters. The interface is very intuitive though, and allows you to manage your defenses with ease during the heat of battle.
In addition to all of the pre-scripted gameplay, Man at Arms TD also comes with a free level editor that lets you create your own stages to play or to send to friends.
Man at Arms TD is available now on the App Store for $4.99.
Man at Arms TD requires iOS 5.1 or later. Compatible with iPad. Follow Tapscape for reviews of all the best iPhone tower defense games.


text:
Q: Find Each line of a textfile in another text file using python I have 2 text files, 1 which contain keywords in a each line around 75 lines ex. __key25words_
Second file contains thousand of lines I want to find a keyword in secondfile if it is present anywhere in second file I want to print that line similar way it should check for all keywords in second file and print if any matches
Note: am able to print a keyword but only when second file starts a line with that key word my script not printing even if keyword exist in middle of line in second file

A: Assuming that file1 contains  keywords and file2 contains the lines to be compared ,
then this code may help you
with  open('file1.txt' , 'r+') as file1 :
    for keyword in file1.readlines() :
        keyword = keyword.replace('\n' , '')
        print ("Looking for " , keyword , " in file" )
        with  open('file2.txt' , 'r+') as file2:
            for line in file2.readlines():
                if keyword in line:
                    print(line)


text:
Factcheck.org: Obama's False Medicare Accusation
By Brooks Jackson On 10/19/08 at 8:00 PM EDT
In a TV ad and in speeches, Obama is making bogus claims that McCain plans to cut $880 billion from Medicare spending and to reduce benefits.
A TV spot says McCain's plan requires "cuts in benefits, eligibility or both."
Obama said in a speech that McCain plans "cuts" that would force seniors to "pay more for your drugs, receive fewer services, and get lower quality care."
These claims are false, and based on a single newspaper report that says no such thing. McCain's policy director states unequivocally that no benefit cuts are envisioned. McCain does propose substantial "savings" through such means as cutting fraud, increased use of information technology in medicine and better handling of expensive chronic diseases. Obama himself proposes some of the same cost-saving measures. We're skeptical that either candidate can deliver the savings they promise, but that's no basis for Obama to accuse McCain of planning huge benefit cuts.
Update, Oct. 21: A second Obama ad claims that McCain's plan would bring about a 22 percent cut in benefits, "higher premiums and co-pays," and more expensive prescription drugs.
The Obama campaign began the Medicare assault with a 30-second TV ad released Oct. 17, which it said would run "across the country in key states."
Obama: I'm Barack Obama and I approved this message.
The ad quotes the Wall Street Journal as saying McCain would pay for his health care plan with "major reductions to Medicare and Medicaid," which the ad says would total $882 billion from Medicare alone, "requiring cuts in benefits, eligibility, or both."
Obama elaborated on the theme Oct. 18 in a stump speech in St. Louis, Mo., claiming flatly that seniors would face major medical hardships under McCain:
Update, Oct. 21: A second and even more misleading Obama ad begins: "How will your golden years turn out?" It states flatly that McCain's plan would mean a 22 percent cut in benefits, higher premiums, higher co-pays, and more expensive prescription drugs, and claims that both nursing home care and a patient's choice of doctor could be affected.
As the narrator says that McCain's plan "means a 22 percent cut in benefits," the ad displays a footnote citing an Oct. 6 Wall Street Journal story as its authority.
But, in fact, the Journal story makes no mention of any 22 percent reduction, or any reduction at all. To the contrary, the story's only mention of what might happen to benefits is a quote from McCain adviser Douglas Holtz-Eakin promising to maintain "the benefit package that has been promised." The story quotes him as saying "savings" would come from eliminating Medicare fraud and by reforming payment policies to lower the overall cost of care.
But in fact, McCain has never proposed to cut Medicare benefits, or Medicaid benefits either. Obama's claim is based on a false reading of a single Wall Street Journal story, amplified by a one-sided, partisan analysis that piles speculation atop misinterpretation. The Journal story in turn was based on an interview with McCain economic adviser Douglas Holtz-Eakin. He said flatly in a conference call with reporters after the ad was released, "No service is being reduced. Every beneficiary will in the future receive exactly the benefits that they have been promised from the beginning."
Twisting Facts to Scare Seniors
Here's how Democrats cooked up their bogus $882 billion claim.
On Oct. 6, the Journal ran a story saying that McCain planned to pay for his health care plan "in part" through reduced Medicare and Medicaid spending, quoting Holtz-Eakin as its authority. The Journal characterizes these reductions as both "cuts" and "savings." Importantly, Holtz-Eakin did not say that any benefits would be cut, and the one direct quote from him in the article makes clear that he's talking about economies:
Wall Street Journal, Oct. 6: Mr. Holtz-Eakin said the Medicare and Medicaid changes would improve the programs and eliminate fraud, but he didn't detail where the cuts would come from. "It's about giving them the benefit package that has been promised to them by law at lower cost," he said.
Holtz-Eakin complains that the Journal story was "a terrible characterization" of McCain's intentions, but even so it clearly quoted him as saying McCain planned on "giving [Medicare and Medicaid beneficiaries] the benefit package that has been promised."
Nevertheless, a Democratic-leaning group quickly twisted his quotes into a report with a headline stating that the McCain plan "requires deep benefit and eligibility cuts in Medicare and Medicaid" – the opposite of what the Journal quoted Holtz-Eakin as saying. The report was issued by the Center for American Progress Action Fund, headed by John D. Podesta, former chief of staff to Democratic President Bill Clinton. The report's authors are a former Clinton administration official, a former aid to Democratic Sen. Bob Kerrey and a former aid to Democratic Sen. Barbara Mikulski.
The first sentence said – quite incorrectly – that McCain "disclosed this week that he would cut $1.3 trillion from Medicare and Medicaid to pay for his health care plan." McCain said no such thing, and neither did Holtz-Eakin. The Journal reporter cited a $1.3 trillion estimate of the amount McCain would need to produce, over 10 years, to make his health care plan "budget neutral," as he promises to do. The estimate comes not from McCain, but from the Urban-Brookings Tax Policy Center. McCain and Holtz-Eakin haven't disputed that figure, but they haven't endorsed it either.
Nevertheless, the report assumes McCain would divide $1.3 trillion in "cuts" proportionately between the two programs, and comes up with this: "The McCain plan will cut $882 billion from the Medicare program, roughly 13 percent of Medicare's projected spending over a 10-year period." And with such a cut, the report concludes, Medicare spending "will not keep pace with inflation and enrollment growth—thereby requiring cuts in benefits, eligibility, or both."
"Savings" vs. "Cuts"
For the record, Holtz-Eakin said in a telephone conference call with reporters Oct. 17, after the ad was released, that any shortfall in McCain's health care plan could be covered, without cutting benefits, by such measures as reducing "Medicare fraud and abuse," employing "a new generation of treatment models" for expensive chronic diseases, speeding adoption of low-cost generic drugs, and expanding the use of information technology in medicine.
Interestingly, Obama proposes to pay for his own health care plan in part through some of the same measures, particularly expanded use of I.T. and better handling of chronic disease. Whether either candidate can achieve the huge savings they are promising is dubious at best. As regular readers of FactCheck.org are aware, we're skeptical of Obama's claim that he can achieve his promised $2,500 reduction in average health insurance premiums, for example.
But achievable or not, "savings" are what McCain is proposing. It's a rank distortion for Obama's ad to twist that into a plan for "cuts in benefits, eligibility or both," and for Obama to claim in a speech that seniors will "receive fewer services, and get lower quality care."
Update, Oct. 21: The Center for American Progress Action Fund issued a rebuttal to this article <>, claiming our analysis is "flawed," that this article "relies solely on the denials of McCain senior policy adviser Douglas Holtz-Eakin" and that we failed to conduct a "thorough analysis of the implications" of McCain's health care proposals.
We disagree. Our criticism of both Obama and American Progress is that they themselves misinterpret and misrepresent what Holtz-Eakin said to the /Wall Street Journal/ in the first place. He was quoted in the /Journal/, and stated again to reporters in a conference call, that what McCain is proposing is to reduce the costs borne by Medicare and Medicaid, and that benefits will not be reduced. American Progress simply ignores that clear statement in its analysis, and the Obama ads take the extra step of telling seniors that McCain plans to cut benefits, when McCain says the opposite.
The American Progress argument rests on the idea that because McCain has also promised to make his health care plan budget neutral – neither raising nor cutting total federal spending – and that because American Progress' analysis concludes that he cannot achieve the savings that he claims, that McCain therefore must be forced to break his promise not to cut benefits.
We are also skeptical that McCain can achieve such savings, and we said so at the outset of our article. And we've twice called into questionthe campaign's claim that its plan is budget neutral.
But it is false logic to conclude that Medicare benefit cuts would be McCain's only option should his promised savings fail to materialize. McCain could simply run up the deficit. Or he could choose to water down his health care plan to make it less expensive.
It is certainly possible that McCain will break his promise not to cut benefits, just as it is possible that Obama will break his promise to raise taxes only on families making over $250,000 a year. We have no crystal ball, and we don't pretend we can predict the future. But for Obama or American Progress to state as a matter of fact that McCain will be forced to cut benefits, or that he is proposing any such thing, is simply a falsehood designed to frighten elderly voters
Republished with permission from factcheck.org.
Meckler, Laura "McCain Plans Federal Health Cuts: Medicare, Medicaid Spending Would Be Reduced to Offset Proposed Tax Credit" Wall Street Journal 6 Oct 2008.
Factcheck.org: Obama's False Medicare Accusation | U.S.


On Thursday, September 15, 2005 at 6:24 AM (PDT) we recorded a quake (preliminary ml 3.9) located in Baja California.
Initial source parameters for the recent event are shown below. The epicentral distance (distance separating the epicenter and the closest station in the recording network) for Anza is 114 km (1.02 degrees).
We operate a broadband station on Mt. Soledad. At 06:24 (PDT) station SOL recorded the P wave from this event. Here is a view of that event as picked by our analyst. A red P indicates a P wave arrival, a red S indicates an S wave arrival.


text:
Arrondissement Châteauroux (fr. Arrondissement de Châteauroux) je správní územní jednotka ležící v departementu Indre a regionu Centre-Val de Loire ve Francii. Člení se dále na 11 kantonů a 82 obce.

Kantony
 Ardentes
 Argenton-sur-Creuse
 Buzançais
 Châteauroux-Centre
 Châteauroux-Est
 Châteauroux-Ouest
 Châteauroux-Sud
 Châtillon-sur-Indre
 Écueillé
 Levroux
 Valençay

Externí odkazy

Chateauroux
Indre

redpajama_set_name: RedPajamaWikipedia


text:
John Henry McDowell, född 7 mars 1942 i Boksburg i Sydafrika, är en sydafrikansk filosof. Han är sedan 1986 professor vid University of Pittsburgh. Bland hans influenser finns Kant, Hegel och Marx. McDowell är ledamot av American Academy of Arts and Sciences och British Academy.

Biografi
John McDowell föddes i Boksburg 1942. Han studerade vid Zimbabwes universitet och senare vid New College i Oxford.

McDowell är en av företrädarna för moralisk partikularism, vilket innebär att det inom moralens område inte existerar någon övergripande, generell princip. Istället bedöms de enskilda handlingarna utifrån sin unika kontext.

Bibliografi (urval)
 1973 – Plato, Theaetetus
 1982 – The Varieties of Reference (redaktör: Gareth Evans)
 1994 – Mind and World
 1998 – Mind, Value, and Reality
 1998 – Meaning, Knowledge, and Reality
 2009 – Having the World in View: Essays on Kant, Hegel, and Sellars
 2009 – The Engaged Intellect: Philosophical Essays

Källor

Noter

Externa länkar

Födda 1954
Levande personer
Män
Sydafrikanska filosofer
Sydafrikanska professorer
Alumner från University of London
Personer verksamma vid University of Pittsburgh
Hegelianer
Hegelforskare
Kantforskare
Wittgensteinforskare
Ledamöter av British Academy
Kunskapsteoretiker
Metafysiker
Etiker
Sydafrikanska forskare under 1900-talet
Sydafrikanska forskare under 2000-talet
Personer från Boksburg


text:
\section{Introduction}}

We denote by $\mathbb{Z}, \mathbb{Z}_+, \mathbb{N}$ and $\mathbb{C}$
the set of all integers, nonnegative integers, positive integers and
complex numbers, respectively.

The Schr\"{o}dinger group is the symmetry group of the free particle
Schr\"{o}dinger equation. The Lie algebra $\mathfrak{S}$ of this
group in the case of $(1+1)-$dimensional space-time is called the
Schr\"{o}dinger algebra which plays an important role in physics
applications (see [1-4]). Concretely, the Schr\"{o}dinger algebra
has a basis $\{e, h, f,p, q, z\},$ subject to the Lie brackets
\begin{eqnarray}
\begin{split}
&[h,e]=2e, &[&h,f]=-2f, &[&e,f]=h,\\
&[h,p]=p, &[&h,q]=-q, &[&p,q]=z,\\
&[e,q]=p, &[&p,f]=-q, &[&f,q]=0,\\
&[e,p]=0, &[&z,\mathfrak{S}]=0. &
\end{split}
\end{eqnarray}
 From the definition, we see that the
Schr\"{o}dinger algebra $\mathfrak{S}$ can be viewed as a semidirect
product $ \mathfrak{S}=\mathfrak{H}\rtimes sl_2 $ of two
subalgebras: a Heisenberg subalgebra
$\mathfrak{H}=\mathrm{span}\{p,q,z\}$ and
$sl_2=\mathrm{span}\{e,h,f\}.$ We also see that
$\mathfrak{S}=\bigoplus_{i=-2}^2\mathfrak{S}_i$ is
$\mathbb{Z}-$graded, where
$$\mathfrak{S}_{-2}=\mathbb{C}f,\mathfrak{S}_{-1}=\mathbb{C}q,
\mathfrak{S}_{0}=\mathbb{C}h+\mathbb{C}z,
\mathfrak{S}_{1}=\mathbb{C}p, \mathfrak{S}_{2}=\mathbb{C}e.$$
Obviously,
\begin{eqnarray}
\mathfrak{S}=\mathfrak{n}^{+}\oplus\mathfrak{h}\oplus\mathfrak{n}^{-}
\end{eqnarray}
is the Cartan decomposition according to the Cartan subalgebra
$\mathfrak{h}=\mathbb{C}h+\mathbb{C}z,$ where
$\mathfrak{n}^{-}=\mathrm{span}\{f,q\},
\mathfrak{n}^{+}=\mathrm{span}\{p,e\}.$

A module $V$ for Schr\"{o}dinger algebra $\mathfrak{S}$ is called a
weight module if it is the sum of all its weight spaces
$V_{\lambda}=\{v\in V|hv=\lambda v\}$ for some $\lambda\in
\mathbb{C}$. Vectors in $V_{\lambda}$ are called weight vectors. A
weight $\mathfrak{S}-$module $V$ is called a Harish-Chandra module
if all weight spaces are finite-dimensional.  A nonzero weight
vector $v\in V$ is called a highest weight vector if
 $\mathfrak{n}^{+}v=0$. $V$ is called a highest
weight module if it is generated by a highest weight vector $v$.

 Recently, the weight modules for
the Schr\"{o}dinger algebra are deeply studied.  In [5], the
irreducible lowest weight modules are classified by using the
technique of singular vectors. In [6], it is proved that all the
weight spaces of a simple weight module  which is neither a highest
weight module nor a lowest weight module have the same dimension. By
using the results in [5] and [6] and $\mathrm{Mathieu}^{,}$s
twisting functors (see [7]),  the author proved that the
Harish-Chandra modules can be twisted from a class of irreducible
highest weight modules in [8]. Thus the classification of simple
weight modules with finite-dimensional weight spaces are completely
obtained: such module is either a dense $sl_2-$module (see [9]) or a
highest (lowest) weight module or a module ``twisted" from a highest
weight module. Moreover, The classification of simple weight modules
over the Schr\"odinger algebra are completed in [10].

The first series of nonweight simple modules for complex semisimple
Lie algebras were constructed by B. Kostant in [11]. These modules
were called Whittaker modules because of their connection to
Whittaker equation in number theory. D. Arnal and G. Pinczon studied
in [12] a generalization of Whittaker modules.
  Since then the study of property
 of  Whittaker modules over various
algebras are open. The prominent role played by Whittaker modules is
illustrated by the main result in [13] about the
 classification of all simple modules for the Lie algebra
$sl_2:$  the simple $sl_2-$modules fall into three families: highest
(lowest) weight modules, Whittaker modules and a third family
obtained by localization.  Whittaker modules have many important
properties and are studied for many algebras, such as Virasoro
algebra, Heisenberg algebras, affine Lie algebras,  generalized Weyl
algebras, quantum enveloping algebra $U(sl_2)$, Schr\"{o}dinger-witt
algebra, generalized Virasoro algebra, twisted Heisenberg-Virasoro
algebra as well as a class of algebras similar to $U(sl_2)$ etc.
(see Refs. [14-23]). Moreover, quantum deformation of Whittaker
modules and modules induced from Whittaker modules are studied(see
Refs.[24],[25]).

In this paper, we classify the simple Whittaker modules over the
Schr\"{o}dinger algebra.

In section 2, we give some notations, identities and some lemmas
which will be used in the following sections. If $V$ is a simple
module over the Schr\"{o}dinger algebra then $z$ acts as a scalar
$\dot{z}$, called the level.

In section 3, we study the Whittaker modules of zero level. For
simplicity, we first  study  the Whittaker modules over a subalgebra
generated by $e,p,h,q,z$  in subsection 3.1. In subsection 3.2,
the classification of simple Whittaker modules of zero level is
given. A quasi-central element plays an important role in this part.

In section 4, we use the Casimir operator to study the simplicity of
Whittaker modules for semisimple Lie algebra $sl_2$ first. Then by
using the simple Whittaker modules for Heisenberg algebra
$\mathfrak{H}$ and $sl_2,$ we give the classification of simple
Whittaker modules of nonzero level for Schr\"{o}dinger algebra.

\noindent{\section{Definitions, identities and some lemmas}}

In this section, we first give some fundamental definitions and some
identities which will be used frequently in this paper.

\begin{defi} Let $\phi:\mathfrak{n}^{+}\rightarrow\mathbb{C}$ be any
nonzero Lie algebra homomorphism. Let $V$ be a
$\mathfrak{S}$-module.

 $\mathrm{(i)}$ A nonzero vector $v\in V$ is
called a Whittaker vector of type $\phi$ if $xv=\phi(x)v$ for all
$x\in\mathfrak{n}^{+}.$

$\mathrm{(ii)}$ $V$ is called a Whittaker module for $\mathfrak{S}$
of type $\phi$ if $V$ contains a cyclic Whittaker vector $v$ of type
$\phi$.
\end{defi}

\begin{defi}  Let
$\phi:\mathfrak{n}^{+}\rightarrow\mathbb{C}$ be any nonzero Lie
algebra homomorphism. Define a one-dimensional
$\mathfrak{n}^{+}$-module $\mathbb{C}_{\phi}=\mathbb{C}w$ by
$ew=\phi(e)w, pw=\phi(p)w.$ The induced module
$$W_{\phi}=U(\mathfrak{S})\otimes_{U(\mathfrak{n}^{+})}\mathbb{C}_{\phi}$$
is called the universal Whittaker module of type $\phi$ in the sense
that for any Whittaker module $V$ with whittaker vector $v$ of type
$\phi,$ there is a unique surjective homomorphism
$f:W_{\phi}\rightarrow V$ such that $uw\mapsto uv, \forall u\in
U(\mathfrak{b}^{-}).$
\end{defi}

For simplicity,  we always denote $\phi(x)$ by $\dot{x}$ for $x=e,p$
in the following.

We give some useful identities in $U(\mathfrak{S})$:
\begin{lemm}In $U(\mathfrak{S}),$ the following identities hold:
\begin{align}
&[p,f^n]=-nqf^{n-1},\\
&[p,h^n]=\sum_{i=1}^{n}(-1)^i\binom{n}{i}
h^{n-i}p,\\
&[q,h^n]=\sum_{i=1}^{n}\binom{n}{i}
h^{n-i}q,\\
&[e,q^n]=nq^{n-1}p+\frac{n(n-1)}{2}q^{n-2}z,\\
&[e,f^n]=nf^{n-1}h-n(n-1)f^{n-1},\\
&[e,h^n]=\sum_{i=1}^{n}(-2)^i\binom{n}{i} h^{n-i}e.
\end{align}
\end{lemm}
\begin{proof} We only prove (2.6), the other identities are obvious or can be
proved similarly. Let $R_h$ (resp. $L_h$) be the right (resp. left)
multiplication operator by $h.$ Then
\begin{align*}
& [e,h^n]=R_h^n(e)+h^ne=(L_h-\mathrm{ad}h)^ne+h^ne.
\end{align*}
Using the binomial formula since $L_h$ and $\mathrm{ad}h$ are
commuting, we obtain (2.6).
\end{proof}

\begin{lemm} For an arbitrary homomorphism of Lie algebra $\phi:\mathfrak{n}^+\rightarrow\mathbb{C},$
if  $W_{\phi}$ is the universal Whittaker module type $\phi$, then
any Whittaker vectors of $W_{\phi}$ are all of type $\phi.$
\end{lemm}
\begin{proof}
Suppose $\phi^{'}:\mathfrak{n}^{+}\rightarrow \mathbb{C}$ is a Lie
algebra homomorphism which is different from $\phi$ and $v^{'}=\sum
a_{ijk}f^iq^jh^kw$ is a Whittaker vector of type $\phi^{'}.$ Define
$A:=\mathrm{max}\{i+j+k|a_{ijk}\neq0\}$ and let $\succ$ be the
lexicographic order on $\mathbb{N}\times\mathbb{N}\times\mathbb{N}.$
Suppose $(\bar{i},\bar{j},\bar{k})$ is the maximal element of
$\{(i,j,k)|i+j+k=A\}$ and $a_{\bar{i}\bar{j}\bar{k}}=1.$ Then
\begin{align}
v=f^{\bar{i}}q^{\bar{j}}h^{\bar{k}}w+\sum_{i+j+k<A\ or\
(i,j,k)\prec(\bar{i},\bar{j},\bar{k})}a_{ijk}f^iq^jh^kw.
\end{align}
If $\phi^{'}(x)\neq \phi(x),$ $x=e$ or $p,$ on one hand, we have
\begin{align}
xv=\phi^{'}(x)v.
\end{align}
On the other hand, we have
\begin{align}
xv=\phi(x)v+([x,f^{\bar{i}}q^{\bar{j}}h^{\bar{k}}]+\sum_{i+j+k<A\
or\ (i,j,k)\prec(\bar{i},\bar{j},\bar{k})}a_{ijk}[x,f^iq^jh^k])w.
\end{align}
By Lemma 2.3 and identity (2.9), we see that
\begin{align}
xv=\phi(x)f^{\bar{i}}q^{\bar{j}}h^{\bar{k}}w+\sum_{i+j+k<A\ or\
(i,j,k)\prec(\bar{i},\bar{j},\bar{k})}b_{ijk}f^iq^jh^kw,
\end{align} where  $b_{ijk}\in\mathbb{C}.$
Then the lemma follows from (2.7), (2.8) and (2.10).
\end{proof}

\begin{lemm} Suppose $\mathfrak{L}$ be a
finite-dimensional Lie algebra with the triangular decomposition
$\mathfrak{L}=\mathfrak{L}^{+}\oplus \mathfrak{L}_0\oplus
\mathfrak{L}^{-}$ according to the Cartan subalgebra
$\mathfrak{L}_0$, $\phi:\mathfrak{L}^{+}\rightarrow \mathbb{C}$ is a
Lie algebra homomorphism.

$\mathrm{(i)}$ If $V$ is a Whittaker $\mathfrak{L}-$module with
cyclic Whittaker vector of type $\phi$,  then every submodule $W$ of
$\mathfrak{L}-$module $V$ contains a Whittaker vector of type
$\phi.$

$\mathrm{(ii)}$ If the vector space of Whittaker vectors of type
$\phi$ is one dimensional, then $V$ is irreducible.
\end{lemm}
\begin{proof} We see that $\{x-\phi(x)|x\in\mathfrak{L}^+\}$ act locally nilpotent on $V$ since $\mathrm{ad}x$
acts locally nilpotent on $U(\mathfrak{L})$ for any
$x\in\mathfrak{L}^+.$ Thus  for any nonzero submodule $W$ of $V$ and
any nozero vector $v\in W,$ $U(\mathfrak{L}^+)v$ is a
finite-dimensional $\mathfrak{L}^+$-submodule. By Lie Theorem, (i)
holds. (ii) is a direct corollary of (i).
\end{proof}

The following result is a direct corollary of Lemma 2.5.
\begin{coro}
Let $V$ be any Whittaker module of type $\phi$ for Schr\"{o}dinger
algebra $\mathfrak{S}.$ Then every submodule $W$ of $V$ contains a
Whittaker vector of type $\phi.$
\end{coro}

\vskip 3mm\noindent{\section{Whittaker modules of zero level}}

In this section, we classify the simple Whittaker modules of zero
level for the Schr\"{o}dinger algebra $\mathfrak{S}.$ For
simplicity, we first study the Whittaker vectors and Whittaker
modules for the subalgebra $\mathfrak{S}_1$ generated by $\langle
p,q,e,h\rangle$ in subsection 3.1. Then in subsection 3.2, we study
the Whittaker $\mathfrak{S}$-modules induced from the simple
Whittaker $S_1-$modules, and determine the classification of simple
Whittaker $\mathfrak{S}$-modules of zero level. In our study, a
quasi-central element (see Lemma 3.5) of $U(\mathfrak{S})$ given in
[26] plays an important role.

\vskip 3mm\noindent{\subsection{Whittaker modules of zero  level for
subalgebra $\mathfrak{S_1}$ }}

We denote the subalgebra of Schr\"{o}dinger algebra $\mathfrak{S}$
generated by $\{e,p,h,q\}$ by $\mathfrak{S}_1$. The universal
Whittaker module of type $\phi$ for $\mathfrak{S}_1$ is denoted by $
W_{\phi}.$ Then for any $v=\in W_{\phi},$ $v$ can be written in the
form:
\begin{equation} \label{eq:1}
v=\sum_{i=0}^na_ih^i\psi_i(q)w,
\end{equation}
where $\psi_i(x)\in\mathbb{C}[x],\psi_n(x)\neq0$ and $a_n\neq0$.

\begin{prop}
$\mathrm{(i)}$  Suppose $\dot{e}\neq0,\dot{p}=0$ and $W_{\phi}$ is
the universal Whittaker module for $\mathfrak{S}_1$ with cyclic
Whittaker vector $w$. Then $v\in W_{\phi}$ is a Whittaker vector if
and only if $v=uw$ for some $u\in\mathbb{C}[q].$

$\mathrm{(ii)}$ If $\dot{p}\neq0,$ the set of Whittaker vectors in
$W_{\phi}$ is $\mathbb{C}w$.
\end{prop}
\begin{proof}
$\mathrm{(i)}$  By (2.4) and $[p,q]w=0,$ it is obvious that $uw$ is
a Whittaker vector if $u\in\mathbb{C}[q]$. For any Whittaker vector
$v\in W_{\phi}$ with form (3.1), if $n\neq0,$ by (2.6), we have
\begin{equation} \label{eq:1}
(e-\dot{e})^{n}v=a_{n}(-2)^{\frac{n(n+1)}{2}}n!\dot{e}\psi_{n}(q)w\neq0,
\end{equation}
a contradiction.

$\mathrm{(ii)}$  For any nonzero vector $v\in
W_{\phi}\setminus\mathbb{C}w,$ we can assume $
v=\sum_{i=0}^na_ih^i\psi_i(q)w, $ where $a_i\in\mathbb{C}, a_n\neq0,
\psi_i(x)\in\mathbb{C}[x]$ and $n+\mathrm{deg}(\psi_i(x))>0.$

If $n=0, \mathrm{deg}(\psi_i(x))>0,$ we see that
$v=\sum_{j=0}^mb_jq^jw,$ where $m>0,b_j\in\mathbb{C},b_m\neq0.$ By
(2.4), we can easily get that
$$(e-\dot{e})v=\dot{p}\sum_{j=1}^mjb_jq^{j-1}w\neq0,$$ which means
$v$ is not a Whittaker vector.

If  $n\neq0,$ by (2.2) we see that
$$(p-\dot{p})v=-n\dot{p}a_nh^{n-1}\psi_n(q)w+\sum_{k<n-1}c_kh^k\psi_k^{'}(q)w\neq0,$$where
$c_k\in\mathbb{C},\psi_k^{'}(x)\in \mathbb{C}[x].$ Thus $v$ is not a
Whittaker vector.
\end{proof}

\begin{lemm}  If $\dot{e}\neq0,\dot{p}=0$, $M$ is a
$\mathfrak{S}_1$-submodule of $W_{\phi},$
 then
$M$ is a maximal submodule of $W_{\phi}$ if and only if there exists
an a scalar $\xi$ such that $M$ is generated by $(q-\xi)w.$
\end{lemm}
\begin{proof}
Suppose there exists a scalar $\xi$ such that $M$ is generated by
$(q-\xi)w.$ For arbitrary vector $v\in M$, $v$ is of the form
$$v=\sum_{i,j}a_{i,j}h^iq^j(q-\xi)w.$$ By
$(e-\dot{e})(q-\xi)w=0=p(q-\xi)w$ and identities (2.2), (2.3) and
(2.6) we see that $w\notin M.$ Thus $M$ is a proper submodule of
$W_{\phi}.$ For any $v^{'}\notin M,$ we have $v^{'}\equiv
\sum_{i}a_ih^iw(\mathrm{mod}M).$ Thus we can assume that
$$v^{'}=h^kw+\sum_{j<k}a_jh^jw\in (U(\mathfrak{S}_1)w+M).$$
Then by using (2.6) we can easily deduce that $w\in
(U(\mathfrak{S}_1)w+M),$ which means
$U(\mathfrak{S}_1)w+M=W_{\phi}.$ Thus $M$ is maximal.

Now suppose $M$ is a maximal $\mathfrak{S}_1-$submodule of
$W_{\phi}$ and $v\in M$ is an arbitrary nonzero vector. Then $v$
must be of the form (3.1).  We choose $v\in M$ such that $n$ be the
minimal. If $n\neq0,$ by (3.2), we see that $\psi_{n}(q)w\in M$,
which is contrary to the minimal of $n.$ Thus there exists
polynomial $0\neq\psi_0(x)\in\mathbb{C}[x]$ such that $\psi_0(q)w\in
M.$ Set
$$\underline{k}=\mathrm{min}\{\mathrm{deg}(\psi(x))|\psi(q)w\in M\}$$ and
choose monic polynomial $\underline{\psi}(x)\in\mathbb{C}[x]$ such
that $\mathrm{deg}(\underline{\psi}(x))=\underline{k}$ and
$\underline{\psi}(q)w\in M$. We see that $\underline{k}>0$ since
$w\notin M.$ Now we claim that $M$ is generated by
$\underline{\psi}(q)w.$ In fact, for any $0\neq v\in M$ with form
(3.1), similar as we have done in (3.2), we find that $\psi_i(q)w\in
M$ for all $i$ with $a_i\neq0.$ Thus $\underline{\psi}(x)|\psi_i(x)$
by the definition of $\underline{\psi}(x)$ and division algorithm in
$\mathbb{C}[x].$ So $M=U(\mathfrak{S}_1)\underline{\psi}(q)w.$
Furthermore, we see that $\underline{\psi}(x)$ is irreducible since
$M$ is maximal, which means there exists a $\xi\in\mathbb{C}$ such
that $\underline{\psi}(x)=x-\xi$ since
$\underline{\psi}(x)\in\mathbb{C}$ is monic.
\end{proof}

By Lemma 3.2 we see that if
$\phi(e)=\dot{e}\neq0,\phi(p)=\dot{p}=0,$ $M_{\xi}$ is a maximal
$\mathfrak{S}_1-$submodule of $W_{\phi}$ generated by $(q-\xi)w,$
then the corresponding simple quotient module
$W_{\phi}/M_{\xi}=\mathbb{C}[h]\bar{w}$ as vector space. We will
denote the simple quotient module $W_{\phi}/M_{\xi}$ by
$\mathbb{C}[h]_{\phi}^{\xi}\bar{w},$ where $\bar{w}=w+M_{\xi}.$

\begin{prop}$\mathrm{(i)}$  If $\dot{e}\neq0,\dot{p}=0,$
  any simple Whittaker $\mathfrak{S}_1-$module of
type $\phi$ is isomorphic to $\mathbb{C}[h]_{\phi}^{\xi}\bar{w}$ for
some $\xi\in\mathbb{C}.$

$\mathrm{(ii)}$ If $\dot{p}\neq0,$ then $W_{\phi}$ is simple
Whittaker $\mathfrak{S}_1-$module.
\end{prop}
\begin{proof}
$\mathrm{(i)}$ For any simple Whittaker $\mathfrak{S}_1-$module $V$,
there exists a maximal submodule $M$ of $W_{\phi}$ such that $V\cong
W_{\phi}/M.$ By Lemma 3.2, there exists some $\xi\in\mathbb{C}$ such
that $M=M_{\xi},$ so we have $V\cong
\mathbb{C}[h]_{\phi}^{\xi}\bar{w}$ for some $\xi\in\mathbb{C}.$

$\mathrm{(ii)}$ This is the direct corollary of Lemma 2.5 and
Proposition 3.1 (ii).
\end{proof}

\vskip 3mm\noindent{\subsection{Simple Whittaker modules for
$\mathfrak{S}$ of zero level }}

In subsection 3.1, we obtain all the simple $\mathfrak{S}_1-$modules
$\mathbb{C}[h]_{\phi}^{\xi}$ (when $\dot{e}\neq0,\dot{p}=0$) and
$W_{\phi}$ (when $\dot{p}\neq0)$. For $\dot{e}\neq0,\dot{p}=0,\xi\in
\mathbb{C},$ in the following, we set
 $$L_{\phi}^{\xi}:=U(\mathfrak{S})\otimes_{U(\mathfrak{S}_1)}\mathbb{C}[h]_{\phi}^{\xi}.$$
For $\dot{p}\neq0,$ we set
$$\widetilde{W}_{\phi}=U(\mathfrak{S})\otimes_{U(\mathfrak{S}_1)}V.$$

In this subsection, we first study the Whittaker vectors and
simplicity of the above induced Whittaker modules. Then we classify
the simple Whittaker modules of zero level over $\mathfrak{S}$.

\begin{lemm} Suppose $\dot{p}\neq0.$ For $\bar{\mathfrak{c}}\in U(\mathfrak{S})$ with form $\bar{\mathfrak{c}}=f+\bar{\mathfrak{c}},$ where
$\bar{\mathfrak{c}}_1\in U(\mathfrak{S}_1),$ such that
$\bar{\mathfrak{c}}w$ is a Whittaker vector, then
$$\bar{\mathfrak{c}}=f-\frac{1}{\dot{p}}q(1+h)-\frac{\dot{e}}{\dot{p}^2}q^2+c,$$
where $c\in\mathbb{C}.$
\end{lemm}
\begin{proof}
Suppose $\bar{\mathfrak{c}}=f+\sum_{i=0}^na_i\psi_i(q)h^i,$ where
$a_i\in\mathbb{C},a_{n}\neq0,\psi_i(x)\in\mathbb{C}[x].$ Then by
$(p-\dot{p})\bar{\mathfrak{c}}w=0,$ we get that
$$-qw+\dot{p}\sum_{i=1}^na_i\psi_i(q)\big(\sum_{k=1}^i(-1)^k\binom{i}{k}h^{i-k}\big)w=0.$$
So $n=1$  and $-qw-\dot{p}a_1\psi_1(q)w=0.$ Thus
$\psi_1(q)=-\frac{1}{a_1\dot{p}}q.$ If we assume that
$\psi_1(x)=b_0+b_1x+\cdots+b_sx^s,$ then we can rewrite
$\bar{\mathfrak{c}}$ as following
$$\bar{\mathfrak{c}}=f+a_0(b_0+b_1q+\cdots+b_sq^s)-\frac{1}{\dot{p}}qh.$$
By $(e-\dot{e})\bar{\mathfrak{c}}w=0,$ we get that $s=2$ and
$a_0b_1\dot{p}+2a_0b_2\dot{p}q+1+\frac{2\dot{e}}{\dot{p}}q=0.$ So
$$b_1=-\frac{1}{a_0\dot{p}}, b_2=-\frac{\dot{e}}{a_0\dot{p}^2},$$
which means that
$\bar{\mathfrak{c}}=(f-\frac{1}{\dot{p}}q(1+h)-\frac{\dot{e}}{\dot{p}^2}q^2)+a_0b_0.$
\end{proof}

From Lemma 3.4, we see that
$$(\bar{\mathfrak{c}}-a)w=(f-\frac{1}{\dot{p}}q(1+h)-\frac{\dot{e}}{\dot{p}^2}q^2)w=(fp^2-q(1+h)p-q^2e)(\frac{1}{\dot{p}^2}w).$$

The following key lemma is from [10], we will call $\mathfrak{c}$ a
quasi-central element of $U(\mathfrak{S}):$
\begin{lemm} Let $\mathfrak{c}=fp^2-q(1+h)p-q^2e.$ Then the center of
$U(\mathfrak{S}/\mathbb{C}z)=\mathbb{C}[\mathfrak{c}].$
\end{lemm}

We denote by $U(\langle q,h\rangle)$ the universal enveloping
algebra of the subalgebra generated by $q$ and $h.$ For
$u=\sum_{i=0}^na_i\psi_{i}(q,h)f^i\in U(\mathfrak{S}),$ where
$\psi_{i}(q,h)\in U(\langle q,h\rangle)$,  $a_i\in\mathbb{C},$
$a_n\neq0,$ $\psi_{n}(q,h)\neq0,$ we define $\mathrm{deg}_f(u)=n.$
Define
$$W_{\phi}^k:=\mathrm{span}\{uw|\mathrm{deg}_f(u)\leq k.\}$$

\begin{lemm}
If $\dot{p}\neq0,$ then for any $a\in \mathbb{C}, i\in\mathbb{N},$
there exists $\psi_k(q,h)\in U(\langle q,h\rangle),$ $
k=0,\cdots,i,$ such that
$$f^iw=\frac{1}{\dot{p}^{2i}}(\mathfrak{c}-a)^iw+\psi_1(q,h)(\mathfrak{c}-a)^{i-1}w+\cdots+\psi_{i-1}(q,h)(\mathfrak{c}-a)w+\psi_i(q,h)w.$$
\end{lemm}
\begin{proof}  Suppose Lemma
3.6 is proved for $i\leq k.$  By Lemma 3.4 and Lemma 3.5, we have
\begin{align*}
&f^{k+1}w=
f^{{k}}(f-\frac{1}{\dot{p}}q(1+h)-\frac{\dot{e}}{\dot{p}^2}q^2)w(\mathrm{mod}W_{\phi}^k)
\\
&\equiv f^{{k}}(fp^2-q(1+h)p-q^2e)\frac{1}{\dot{p}^2}w
(\mathrm{mod}W_{\phi}^k)
\\
&\equiv f^{{k-1}}(fp^2-q(1+h)p-q^2e)f\frac{1}{\dot{p}^2}w
(\mathrm{mod}W_{\phi}^k)\\
&\equiv f^{{k-2}}(fp^2-q(1+h)p-q^2e)^2\frac{1}{\dot{p}^4}w
(\mathrm{mod}W_{\phi}^k)\\
&\equiv \cdots\\
&\equiv (fp^2-q(1+h)p-q^2e)^{k+1}\frac{1}{\dot{p}^{2(k+1)}}w
(\mathrm{mod}W_{\phi}^k).
\end{align*}
By induction, Lemma 3.6 holds.
\end{proof}

\begin{prop} $\mathrm{(i)}$ If $\dot{e}\neq0,  \dot{p}=0, \xi\neq0,$  the set of  Whittaker vectors of
$L_{\phi}^{\xi}$ is $\mathbb{C}\bar{w}.$

$\mathrm{(ii)}$ If $\dot{p}\neq0, $  the set of  Whittaker vectors
of $\widetilde{W}_{\phi}$ is $\mathbb{C}[\mathfrak{c}]w,$ where
\begin{align*}
\mathfrak{c}=fp^2-q(1+h)p-q^2e.
\end{align*}
\end{prop}
\begin{proof}
$\mathrm{(i)}$ Suppose $\dot{e}\neq0,\dot{p}=0,\xi\neq0.$ For any
nonzero $v\in L_{\phi}^{\xi},$ we can assume that
$$v=\sum_{k=0}^{n}a_kf^k\psi_k(h)\bar{w},$$ where
$a_n\neq0,\psi_k(x)\in \mathbb{C}[x],\psi_n(x)\neq0.$ Then if $v$ is
a Whittaker vector, by identities (2.1) and (2.2),  we have
$$pv=\sum_{k=0}^{n}a_k(-k)f^{k-1}q\psi_k(h)\bar{w}=0.$$ By (2.3) and the fact
that $q\bar{w}=\xi\bar{w}\neq0,$ we have $n=0.$ So $v$ can be
written by
$$v=\psi_0(h)\bar{w}=h^k\bar{w}+a_{k-1}h^{k-1}\bar{w}+\cdots +a_0h_0,$$
where $k\in\mathbb{Z}_+$ and $a_i\in\mathbb{C}, i=0,i,\cdots,k-1.$
Then by using $(e-\dot{e})v=0$ and (2.6) we deduce that $k=0$, which
means $v\in \mathbb{C}\bar{w}.$

$\mathrm{(ii)}$ By Lemma 3.5, we see that every nonzero vector of
$\mathbb{C}[\mathfrak{c}]w$ is a Whittaker vector. For any Whittaker
vector $v\in \widetilde{W}_{\phi}$, there exists $0\neq
u=\sum_{i=0}^na_i\psi_{i}(q,h)f^i\in U(\mathfrak{S})$ with
$\mathrm{deg}_f(u)=n$ such that $v=uw.$ If $n=0,$ by Proposition
3.1(ii), we see that $\psi_{0}(q,h)\in\mathbb{C}^*,$ so $v\in
\mathbb{C}w\subseteq\mathbb{C}[\mathfrak{c}]w.$ Suppose for $n\leq
k$ we have proved that $v\in \mathbb{C}[\mathfrak{c}]w.$ Now we
begin to investigate the case of $n=k+1.$
 Using $(p-\dot{p})$ and
$(e-\dot{e})$ to act in turn on $v$, we can easily deduce that
$\psi_{k+1}(q,h)\in\mathbb{C}^{*}.$ For simplicity, we can assume
that $a_{k+1}=1=\psi_{k+1}(q,h).$ So by Lemma 3.6, we see that
\begin{align*}
v&=f^{k+1}w+\sum_{i=0}^ka_i\psi_{i}(q,h)f^iw\in\mathbb{C}[\mathfrak{c}]w.
\end{align*}Then, by
induction, Proposition 3.7 holds.
\end{proof}
\begin{theo}
$\mathrm{(i)}$ If $\dot{e}\neq0,\dot{p}=0,\xi\neq0,$ the Whittaker
module
 $L_{\phi}^{\xi}$
 is simple.

 $\mathrm{(ii)}$ If $\dot{p}\neq0,$  for any
 $a\in\mathbb{C},$ the Whittaker module $\widetilde{W}_{\phi}$ has the following filtration:
 $$\widetilde{W}_{\phi}=W^0\supset W^1\supset W^2\supset\cdots \supset W^n\supset
 \cdots,$$ where $W^i$ is a Whittaker submodule defined by $W^i=U(\mathfrak{S})(\mathfrak{c}-a)^iw.$ More precisely, $\widetilde{W}_{\phi}/W^1$
  is simple and $W^i/W^{i+1}\cong \widetilde{W}_{\phi}/W^1$ for all
 $i\in\mathbb{N}.$
\end{theo}
\begin{proof}
$\mathrm{(i)}$ follows from Corollary 2.6 and Proposition 3.7.

$\mathrm{(ii)}$ Suppose $\dot{p}\neq0.$ By Lemma 3.6, we see that
for any $0\neq v\in \widetilde{W}_{\phi}\setminus W_1,$
 there exists $\psi_i(q,h)\in
U(\langle q,h\rangle), i=0,\cdots,n, \psi_0(q,h)\neq0,
\psi_n(q,h)\neq0$ such that
$$v=\psi_n(q,h)(\mathfrak{c}-a)^nw+\psi_{n-1}(q,h)(\mathfrak{c}-a)^{n-1}w+\cdots+\psi_{1}(q,h)(\mathfrak{c}-a)w+\psi_0(q,h)w.$$
Thus $\psi_0(q,h)w\in (U(\mathfrak{S})v+W_1).$ Then by Proposition
3.3 (ii), we have $w\in(U(\mathfrak{S})v+W_1)$ and
$U(\mathfrak{S})v+W_1=\widetilde{W}_{\phi},$ which means $W_1$ is a
maximal submodule of $\widetilde{W}_{\phi}.$

By Lemma 3.5 and Lemma 3.6, for any element $v\in W^i,$ $v$ is of
the form
$$v=\sum_{k=i}^n\psi_k(q,h)(\mathfrak{c}-a)^kw,$$ where $n\geq i, \psi_n(q,h)\neq0.$
Define linear map:
\begin{align*}
g:\ \ \ W^i\ \ \ &\rightarrow\ \ \ \widetilde{W}_{\phi}/W^1\\
\psi(q,h)(\mathfrak{c}-a)^{k+i}w & \mapsto
\psi(q,h)(\mathfrak{c}-a)^{k}\bar{w}.
\end{align*}
It is easy to see that $g$ is a module homomorphism and
$\mathrm{ker}(g)=W^{i+1}$, which means that $W^{i}/W^{i+1}\cong
\widetilde{W}_{\phi}/W^1.$
\end{proof}

In the following, we denote $\widetilde{W}_{\phi}/W^1$ by
$M_{\phi}^{a}.$ So $M_{\phi}^{a}$ is a simple  Whittaker module with
cyclic Whittaker vector $\bar{w}:=w+W^1.$ By Proposition 3.7 (ii)
and the definition of $M_{\phi}^{a}$, we have the following lemma
immediately:

\begin{lemm}
The set of Whittaker vectors of $M_{\phi}^{a}$ is
$\mathbb{C}\bar{w}.$
\end{lemm}

\begin{theo} Suppose $V$ is any simple Whittaker module of zero level  of type $\phi$ for
$\mathfrak{S}$.

$\mathrm{(i)}$ If $\dot{e}\neq0,\dot{p}=0,$ then $V$ is either a
simple $sl_2-$module of type $\phi$ or  isomorphic to
 $L_{\phi}^{\xi}$
 for some $\xi\neq0.$

 $\mathrm{(ii)}$ If $\dot{p}\neq0,$ then  $V$   is
isomorphic to $M_{\phi}^a$ for some $a\in\mathbb{C}.$
\end{theo}
\begin{proof}
If $V$ is a simple Whittaker module of with cyclic Whittaker vector
$w_v$ of type $\phi$, by the universal property of
$\widetilde{W}_{\phi},$ there exists a surjective module
homomorphism $g:\widetilde{W}_{\phi}\rightarrow V$ with
$uw\rightarrow uw_v.$

$\mathrm{(i)}$ If $\phi$ satisfies
$\phi(e)=\dot{e}\neq0,\phi(p)=\dot{p}=0,$ let $V_1\subseteq V$ be
the $\mathfrak{S}_1$-module generated by $w_v.$ Then as
$\mathfrak{S}_1$-module, $V$ has the following direct sum
decomposition:
$$V=V_1\oplus fV_1/V_1\oplus f^2V_1/fV_1\oplus\cdots\oplus f^nV_1/f^{n-1}V_1\oplus\cdots.$$
Since $V$ is simple as $\mathfrak{S}-$module, we see that $V_1$ is
simple as $\mathfrak{S}_1$-module. Then by Proposition 3.3, there
exists $\xi\in \mathbb{C}$ such that $V_1\cong
\mathbb{C}[h]_{\phi}^{\xi}\bar{w}.$

\noindent{\bf{Case (1)}} If $\xi=0,$ then $qw_v=0,$ combing with
$ew_v=zw_v=0,$ we see that $V$ is just a simple Whittaker
$sl_2-$module.

\noindent{\bf{Case (2)}} If $\xi\neq0,$ then as
$\mathfrak{S}-$module, $V=\mathbb{C}[f]V_1$ is a quotient module of
$L_{\phi}^{\xi}=U(\mathfrak{S})\otimes_{U(\mathfrak{S}_1)}\mathbb{C}[h]_{\phi}^{\xi}\bar{w}.$
Thus $V\cong L_{\phi}^{\xi}$ since $L_{\phi}^{\xi}$ is simple by
Theorem 3.8 (i).

$\mathrm{(ii)}$ If $\phi$ satisfies $\phi(p)=\dot{p}\neq0,$  by
Proposition 3.7 (ii),  for $n\in\mathbb{Z}_+$ and $x=e$ or $p$ we
have
$$x\mathfrak{c}^nw_v=x(\mathfrak{c}^ng(w))=g(x\mathfrak{c}^nw)=\dot{x}g(\mathfrak{c}^nw)=\dot{x}\mathfrak{c}^ng(w),$$
which means $\mathbb{C}[\mathfrak{c}]w_v$ is a subset of Whittaker
vector of $V$. Since $V$ is simple, the Whittaker vectors are of the
form $cw_v$ for some $c\in\mathbb{C},$ thus $\mathfrak{c} w_v=aw_v$
for some $a\in \mathbb{C}.$ So
$U(\mathfrak{S})(\mathfrak{c}-a)w_v=0,$ which means
$U(\mathfrak{S})(\mathfrak{c}-a)w\subseteq \mathrm{ker}(g).$ Thus
$V\cong \widetilde{W}_{\phi}/\mathrm{ker}(g)$ is a nonzero submodule
of $M_{\phi}^{a},$ thus $V\cong M_{\phi}^{a}$ since $ M_{\phi}^{a}$
is simple by Theorem 3.8.
\end{proof}

\begin{coro}
Let $V$ be a Whittaker $\mathfrak{S}-$module of level zero of type
$\phi:e \rightarrow \dot{e}, p \rightarrow \dot{p}$.

$\mathrm{(i)}$ If $\dot{e}\neq0,\dot{p}=0,$ then $V$ is simple if
and only if $q$ acts on cyclic Whittaker vector as a scalar.

 $\mathrm{(ii)}$ If $\dot{p}\neq0,$ then  $V$   is simple if and
 only if
$\mathfrak{c}$ acts as a zero.
\end{coro}

\begin{theo} $\mathrm{(i)}$ If
$\dot{e}\neq0,\dot{p}=0, \xi, \xi^{'}\neq0,$ then
$L_{\phi^{'}}^{\xi^{'}}\cong L_{\phi}^{\xi}$ if and only if
$\phi^{'}=\phi,$ and $\xi^{'}=\xi.$

$\mathrm{(ii)}$  If $\dot{p}\neq0,$ then $M_{\phi^{'}}^{a^{'}}\cong
M_{\phi}^{a}$ if and only if $\phi^{'}=\phi,$ and $a^{'}=a.$
\end{theo}
\begin{proof}
 $\mathrm{(i)}$ The sufficiency is trivial. By Proposition 3.7 (i) and Lemma 2.4, we see that
 the necessity holds.

 $\mathrm{(ii)}$ We need only to prove the necessity. By Lemma 3.9,
 we get
 $\phi^{'}=\phi$ immediately. Suppose $\bar{w}$ (resp.
 $\bar{w}^{'}$)
 is a cyclic Whittaker vector of $M_{\phi}^a$ (resp.
 $M_{\phi^{'}}^{a^{'}}$) and $g: M_{\phi}^a\rightarrow M_{\phi^{'}}^{a^{'}}$ is the module isomorphism such that $g(\bar{w})=g(w)$.
 Then we have
 $$0=(\mathfrak{c}-a^{'})\bar{w}^{'}=(\mathfrak{c}-a^{'})g(\bar{w})=g((\mathfrak{c}-a^{'})(\bar{w}))=(a-a^{'})\bar{w}^{'},$$
 that is $a^{'}=a.$
\end{proof}

\begin{rema}
By Theorem 3.10 and Theorem 3.11, the Whittaker modules of zero
level for Schr\"{o}dinger algebra are classified.
\end{rema}

\noindent{\section{Whittaker modules of nonzero level}}

In this section, we classify the simple Whittaker modules of nonzero
level. Our arguments use the  Casimir operator of semisimple Lie
algebra $sl_2$ and the description of simple modules over conformal
Galilei algebras in [24].

\begin{defi}
Let $\mathbb{C}_{\dot{p},\dot{z}}=\mathbb{C}w$ be a one-dimensional
vector space. By the action of
$$pw=\dot{p}w, zw=\dot{z}w,$$ $\mathbb{C}_{\dot{p},\dot{z}}$ can be viewed as a
$\mathbb{C}p\oplus\mathbb{C}z-$module. The induced module
$$M_{\mathfrak{H}}(\dot{p},\dot{z})=U(\mathfrak{H})\otimes_{U(\mathbb{C}p\oplus\mathbb{C}z)}\mathbb{C}_{\dot{p},\dot{z}}$$
is called a Whittaker module for Heisenberg Lie algebra
$\mathfrak{H}.$
\end{defi}

The following proposition is from [15], it can also be checked
easily:

\begin{prop}
For $\dot{z}\neq0,$ the following hold:

$\mathrm{(i)}$ $M_{\mathfrak{H}}(\dot{p},\dot{z})$ is the unique (up
to isomorphism) irreducible Whittaker $\mathfrak{H}-$module on which
$z$ acts by $\dot{z}\neq0$.

$\mathrm{(ii)}$ $M_{\mathfrak{H}}(\dot{p},\dot{z})\cong
M_{\mathfrak{H}}(\dot{p}^{'},\dot{z}^{'})$ if and only if
$\dot{p}=\dot{p}^{'}$ and $\dot{z}=\dot{z}^{'}$.
\end{prop}

\begin{defi}
Suppose $\mathbb{C}_{\dot{e}}=\mathbb{C}w$ be a $\mathbb{C}e-$module
by the action of $ew=\dot{e}w.$ The induced module
$$W_{sl_2}(\dot{e})=U(\mathfrak{H})\otimes_{U(\mathbb{C}e)}\mathbb{C}_{\dot{e}}$$
is called a Whittaker module for complex semisimple Lie algebra
$sl_2.$
\end{defi}

Let $v\in W_{sl_2}(\dot{e})$ be any nonzero vector, then $v$ is of
the form $v=\sum_{i,j}a_{i,j}f^ih^jw.$ Define
$$\mathrm{deg}_f(v)=\mathrm{max}\{i|a_{i,j}\neq0\}$$ and
$$W_{sl_2}^{k}(\dot{e})=\mathrm{span}\{v|v\in W_{sl_2}(\dot{e}),\mathrm{deg}_f(v)\leq k.\}$$

\begin{lemm}
Suppose $\dot{e}\neq0.$

$\mathrm{(i)}$ If $\bar{\Omega}\in U(sl_2)$ is of the form
 $\bar{\Omega}=f+\bar{\Omega}_1,$ where $\bar{\Omega}_1\in \mathbb{C}[h],$
such that $\bar{\Omega}w$ is a Whittaker vector, then
$\bar{\Omega}=f+\frac{h}{2\dot{e}}+\frac{h^2}{4\dot{e}}+c$ for some
$c\in\mathbb{C}.$

$\mathrm{(ii)}$ For any $a\in\mathbb{C},$ $i\in\mathbb{N},$ there
exists $\psi_0(x),\psi_1(x),\cdots,\psi_i(x)\in\mathbb{C}[x]$ such
that
$$f^iw=\frac{1}{(4\dot{e})^i}(\Omega-a)^iw+\psi_1(h)(\Omega-a)^{i-1}w+\cdots+\psi_{i-1}(h)(\Omega-a)w+\psi_i(h)w,$$
where $\Omega=4fe+2h+h^2$ is the Casimir element of $sl_2.$
\end{lemm}
\begin{proof}$\mathrm{(i)}$ Assume $\bar{\Omega}=f+\sum_{i=0}^na_ih^i, a_n\neq0$
and $\bar{\Omega}w$ is a Whittaker vector. By
$(e-\dot{e})\bar{\Omega}w=0$ and (2.6), we can easily get that
$\bar{\Omega}=f+\frac{h}{2\dot{e}}+\frac{h^2}{4\dot{e}}+c$ for some
$c\in\mathbb{C}.$

$\mathrm{(ii)}$ By $\mathrm{(i)}$ and the fact that $\Omega\in
Z(sl_2),$ the center of $U(sl_2),$ we have
\begin{align*} f^{i+1}w&=
f^{{i}}(f+\frac{h}{2\dot{e}}+\frac{h^2}{4\dot{e}})w(\mathrm{mod}W_{sl_2}^{i}(\dot{e}))
\\
&\equiv f^{{i}}(4fe+2h+h^2)\frac{1}{4\dot{e}}w
(\mathrm{mod}W_{sl_2}^{i}(\dot{e}))
\\
&\equiv f^{{i-1}}(4fe+2h+h^2)f\frac{1}{4\dot{e}}w
(\mathrm{mod}W_{sl_2}^{i}(\dot{e}))\\
&\equiv f^{{i-2}}(4fe+2h+h^2)^2\frac{1}{(4\dot{e})^2}w
(\mathrm{mod}W_{sl_2}^{i}(\dot{e}))\\
&\equiv \cdots\\
&\equiv (4fe+2h+h^2)^{i+1}\frac{1}{(4\dot{e})^{i+1}}w
(\mathrm{mod}W_{sl_2}^{i}(\dot{e})).
\end{align*}
By induction, $\mathrm{(ii)}$ holds.
\end{proof}

\begin{lemm}
For Whittaker $sl_2-$module $W_{sl_2}(\dot{e}),$ the following hold:

$\mathrm{(i)}$ If $\dot{e}=0,$  the set of Whittaker vectors of
$W_{sl_2}(0)$ is $\mathbb{C}[h]w.$

$\mathrm{(ii)}$ If $\dot{e}\neq0,$  the set of Whittaker vectors of
$W_{sl_2}(\dot{e})$ is $\mathbb{C}[\Omega]w.$
\end{lemm}
\begin{proof}
(i) Suppose $\dot{e}=0.$ By (2.6), any nonzero element of
$\mathbb{C}[h]w$ is a Whittaker vector. By (2.5) and (2.6),  we see
that $ev\neq0$ if $\mathrm{deg}_f(v)\neq0,$ which means $v$ is not a
Whittaker vector.

 (ii) Suppose $\dot{e}\neq0.$ On one hand, every nonzero vector of
$\mathbb{C}[\Omega]w$ is a Whittaker vector since $\Omega\in
Z(sl_2)$.

On the other hand, for any Whittaker vector
$v=\sum_{i=0}^na_i\psi_{i}(h)f^iw\in W_{sl_2}(\dot{e})$ with
$\mathrm{deg}_f(v)=n,$ by Lemma 4.4 (ii), we see that $v$ is of the
form
$$v=\sum_{i=0}^nb_i\varphi_i(h)(\Omega-a)^iw$$
for $\varphi_{x}\in\mathbb{C}[x],
\varphi_n(x)\neq0,b_i\in\mathbb{C},b_n\neq0.$ By (2.6), it is easy
to deduce that either $\varphi_i(x)=0$ or
$\mathrm{deg}(\varphi_i(x))=0.$
\end{proof}

\begin{prop}
For Whittaker $sl_2-$module $W_{sl_2}(\dot{e}),$ the following hold:

$\mathrm{(i)}$ If $\dot{e}=0,$ $V$ is a $sl_2-$submodule of
Whittaker module $W_{sl_2}(0),$  then $V$ is a maximal submodule if
and only if there exists a scalar $\alpha$ such that $V$ is
generated by $(h-\alpha)w.$

$\mathrm{(ii)}$ If $\dot{e}\neq0,$  for any
 $\dot{\Omega}\in\mathbb{C},$ the Whittaker module $W_{sl_2}(\dot{e})$ has the following filtration:
 $$W_{sl_2}(\dot{e})=W^0\supset W^1\supset W^2\supset\cdots \supset W^n\supset
 \cdots,$$ where $W^i$ is a Whittaker submodule defined by $W^i=U(sl_2)(\Omega-\dot{\Omega})^iw.$ More precisely,
  $M_{sl_2}(\dot{e},\dot{\Omega}):=W_{sl_2}(\dot{e})/W^1$
  is simple and $W^i/W^{i+1}\cong W_{sl_2}(\dot{e})/W^1$ for all
 $i\in\mathbb{N}.$
\end{prop}
\begin{proof} We can use the same arguments as in Lemma 3.2 and Theorem 3.8 (ii) to
complete the proof. We omit the details.
\end{proof}

\begin{rema}
It is follows from Proposition 4.6 (i) that if $\dot{e}=0,$ $V$ is a
maximal submodule of $W_{sl_2}(0)$ generated by $(h-\alpha)w,$ then
$W_{sl_2}(0)/V$ is a simple highest weight $sl_2$-module with
highest weight $\alpha$.
\end{rema}

\begin{theo}
Let $V$ be a simple Whittaker module of type $\phi:e \rightarrow
\dot{e}$.

$\mathrm{(i)}$ If $\dot{e}=0,$ then there exists
$\alpha\in\mathbb{C}$ such that $V$ is isomorphic to a simple
highest weight $sl_2$-module with highest weight $\alpha$.

$\mathrm{(ii)}$ If $\dot{e}\neq0,$ then $V\cong
M_{sl_2}(\dot{e},\dot{\Omega})$ for some $\dot{\Omega}\in
\mathbb{C}.$
\end{theo}
\begin{proof}
$\mathrm{(i)}$ follows from Proposition 4.6 (i) and Remark 4.7. The
proof of $\mathrm{(ii)}$ is similar as that of Theorem 3.8 (ii). We
omit the details.
\end{proof}

\begin{coro}
Let $V$ be a Whittaker $sl_2-$module of type $\phi:e \rightarrow
\dot{e}$.

$\mathrm{(i)}$ If $\dot{e}=0,$ $V$ is simple if and only if $V$ is a
highest weight module.

$\mathrm{(ii)}$ If $\dot{e}\neq0,$ then $V$ is simple if and only if
 the Casimir element of $sl_2$ acts as a scalar.
\end{coro}

\begin{theo}
Let $W(\dot{e},\dot{p},\dot{z})$ be a Whittaker
$\mathfrak{S}-$module of type $\phi: e \rightarrow \dot{e}, p
\rightarrow \dot{p}$ of nonzero level $\dot{z}.$ Then
$W(\dot{e},\dot{p},\dot{z})$ is simple if and only if there exists a
simple Whittaker $\mathfrak{H}-$module
$M_{\mathfrak{H}}(\dot{p},\dot{z})$ and a simple Whittaker
$\mathfrak{sl_2}-$module
$M_{\mathfrak{sl_2}}(\dot{e}-\frac{1}{2\dot{z}}\dot{p}^2,\dot{\Omega})$
such that $W(\dot{e},\dot{p},\dot{z})\cong
M_{\mathfrak{H}}^{\mathfrak{S}}(\dot{p},\dot{z})\otimes
M_{sl_2}^{\mathfrak{S}}(\dot{e}-\frac{1}{2\dot{z}}\dot{p}^2,\dot{\Omega}).$
\end{theo}
\begin{proof} Suppose first that $W(\dot{e},\dot{p},\dot{z})$ is
simple Whittaker module of nonzero level.
 Denote by
$U(\mathfrak{H})_{(z)}$ the localization of $U(\mathfrak{H})$ with
respect to the multiplicative subset $\{z^{i}|i\in\mathbb{Z}_+\}$.
By Theorem 1 (i) in [26] (here $z$ is opposite to that of in [26]),
 we see that there is a unique algebra
homomorphism $\Phi:U(\mathfrak{S})\rightarrow U(\mathfrak{H})_{(z)}$
which is the identity on $U(\mathfrak{H})$ such that
$$\Phi(e)=\frac{1}{2z}p^2,\ \Phi(f)=-\frac{1}{2z}q^2,\
\Phi(h)=-\frac{1}{z}qp-\frac{1}{2}.$$ Then
$\mathrm{Res}_{\mathfrak{H}}^{\mathfrak{S}}W(\dot{e},\dot{p},\dot{z}),$
the restrictive module on $\mathfrak{H}$, contains a simple
Whittaker submodule $M_{\mathfrak{H}}(\dot{p},\dot{z}),$ which can
be viewed as a $\mathfrak{S}-$Whittaker module such that
$$ev=\Phi(e)v,fv=\Phi(f)v,hv=\Phi(h)v.$$
 We denote this new $\mathfrak{S}$-module by $W_{\mathfrak{H}}^{\mathfrak{S}}(\dot{p},\dot{z}).$
Then, by Theorem 3 (ii) in [26], we have a simple $sl_2-$Whittaker
module $M_{sl_2}(\dot{e}-\frac{1}{2\dot{z}}\dot{p}^2,\dot{\Omega})$
such that $W\cong
M_{\mathfrak{H}}^{\mathfrak{S}}(\dot{p},\dot{z})\otimes
M_{sl_2}^{\mathfrak{S}}(\dot{e}-\frac{1}{2\dot{z}}\dot{p}^2,\dot{\Omega}),$
where the simple $\mathfrak{S}-$module
$M_{sl_2}^{\mathfrak{S}}(\dot{e}-\frac{1}{2\dot{z}}\dot{p}^2,\dot{\Omega})$
is from simple $sl_2-$module
$M_{sl_2}(\dot{e}-\frac{1}{2\dot{z}}\dot{p}^2,\dot{\Omega})$ by
setting $\mathfrak{H}v=0.$

Conversely, if $M_{\mathfrak{H}}(\dot{p},\dot{z})$ is a simple
Whittaker $\mathfrak{H}$-module of nonzero level $\dot{z}$ and
$M_{sl_2}(\dot{e}-\frac{1}{2\dot{z}}\dot{p}^2,\dot{\Omega})$ is a
simple Whittaker $sl_2-$module. Then we can view
$M_{\mathfrak{H}}(\dot{p},\dot{z})$ as a $\mathfrak{S}-$ module,
denote by $M_{\mathfrak{H}}^{\mathfrak{S}}(\dot{p},\dot{z}),$ by
defining $ev=\Phi(e)v,fv=\Phi(f)v,hv=\Phi(h)v.$ We also can view
$M_{sl_2}(\dot{e}-\frac{1}{2\dot{z}}\dot{p}^2,\dot{\Omega})$  as a
$\mathfrak{S}-$module by defining
$\mathfrak{H}.M_{sl_2}(\dot{e}-\frac{1}{2\dot{z}}\dot{p}^2,\dot{\Omega})=0.$
then $M_{\mathfrak{H}}^{\mathfrak{S}}(\dot{p},\dot{z})\otimes
M_{sl_2}^{\mathfrak{S}}(\dot{e}-\frac{1}{2\dot{z}}\dot{p}^2)$ is a
Whittaker $\mathfrak{S}-$module of type $\phi: e\rightarrow\dot{e},
p\rightarrow\dot{p}$ of nonzero level $\dot{z}.$  By Theorem 3 (i)
in [26], we see that
$M_{\mathfrak{H}}^{\mathfrak{S}}(\dot{p},\dot{z})\otimes
M_{sl_2}^{\mathfrak{S}}(\dot{e}-\frac{1}{2\dot{z}}\dot{p}^2)$ is
simple. This completes the proof of Theorem 3.9.
\end{proof}

\begin{rema}
By Proposition 4.2, Theorem 4.8 and Theorem 4.10, the Whittaker
modules of nonzero level for Schr\"{o}dinger algebra are classified.
\end{rema}

\vskip 5mm

redpajama_set_name: RedPajamaArXiv


Architectural education shapes not only what a professional architect does but also how the profession of architecture conducts itself. It is important for it to be in sync with current times. The need of the hour is to rework the role of the architect from that of a solitary creator to being a collaborator or a facilitator in improving the built environment. The architectural process then, has to lay more emphasis on the relevance and usefulness of design rather than on its uniqueness and originality. This approach to architectural design and practice means that the studio will have to provide a platform where students learn to co-create design options. Architectural education must move away from a pedagogic premise ofidentifying design 'problems' and teaching to create design solutions, towards teaching to negotiatedesign options and working in a multidisciplinary team. If the merit of design lies in its social relevance rather than its signature style, then it will have to be taught and assessed accordingly in the studio.
The social relevance of a program hinges on whether it is true to its cultural context. After decades of experimenting with European and American designs and styles and by mostly failing to deliver relevant solutions it is perhaps time to look more closely at what our context really is and then ask what it needs. The Indian context and its many regional sub-contexts are very different from its European and American counterparts. The layers of our social and political histories, our cultural and architectural traditions, and the sheer variety of our communities and their needs mean that the architectural curriculum needs to address all these determinants of our built environment in the design studio. The program will also have to take into account the 'feminisation' of the profession. This is in a philosophical term to indicate how architecture is taught and practised in a less aggressive and more nurturing manner, and also points to the growing number of women joining the profession. Seventy percent of our students and more than fifty percent of our faculty in SSAA are women. This ought to have some bearing on how the program is taught and learnt. While not suggesting that it must cater to the majority, the program is sensitive to women's needs as students and as architects. The architectural profession has to help women practice as successful architects by providing them a compassionate environment to flourish in, and this begins in the Schools.
The design studio forms the core of the curriculum in the first three years. Each semester adopts a project that is developed in the studio aided by pedagogical inputs from two packs. The two packs of subjects that feed into the design studio can be broadly categorised as 'Thinking' and 'Doing' subjects. 'Thinking' pack includes: History, Critical Thinking, Research Method, and Multidisciplinary Approaches to Negotiating Design. 'Doing' pack includes: Building Construction, Structures and Services. The design project progresses in complexity over the three years from addressing individual needs (room, house, office) to a collective requirement (institution, housing and the urban and historical context). The project adopted can be 'actual' or 'hypothetical' but will include the widened ambit of the scope of the profession. Architectural inputs into critical, curatorial and conceptual practices of design negotiations that intersect with fields of art and visual culture, and with conservation and heritage practices.
In addition to these subjects, a supplement of 'skills modules' will form a pool of electives that the student can choose from over the years depending upon individual interests and aspirations. These can include, depending on demand from the design studio: Digital Fabrication, Parametric Design, latest computer software, traditional material techniques, preservation and restoration techniques, statistical data managing, and so on.
If the method of design is negotiation, then it cannot be assessed in terms of a solution to a problem, but evaluated as an embodiment of an ideology, a stance, an approach and an attitude. So the prevalent mathematical way of judging a design problem in terms of how it is solved poses two problems: one, identifying the 'problem' and two, considering design as a problem-solving exercise. In the redefined role of the architect as a collaborator, design is what can be negotiated, given the relevance and context. So good and bad design must negotiate its status in terms of what is viable, relevant, and sustainable.


text:
"Sound check" By MARTY HUGHLEY, The Oregonian, February 4, 2005
WHERE BASS INSTINCTS MEET MELODIC JOY -- The image is of something natural and lovely, a sign of growth and a symbol of optimism. In Belinda Underwood's song "Uncurling," "the simple joy of a delicate fern uncurling" serves as a metaphor for the emotional openness that new love can bring. But the phrase might also be used as a description of "Underwood Uncurling," the Portland singer/bassist's beguiling debut album.
A collection split evenly between well-chosen jazz standards and Underwood's own compositions, "Underwood Uncurling" is softly lyrical and intimate, sophisticated yet unassuming, contemplative but never ponderous. It's a very promising introduction to a young talent. And for Underwood herself, she sees it as a representation of her growth as a musician, songwriter and person.
The 28-year-old California native has been uncurling in several directions of late. In addition to jazz gigs, she performs in Beliss (an eclectic, ukulele-based folk duo with her younger sister Melissa), the Latin-jazz band Pachamanca and a Middle Eastern group called Wazn al Sharq.
She grew up in a musical household, her mother a jazz pianist (classically trained at University of Oregon, alongside the likes of bassist Glen Moore), her father a horn player and inventor of a popular pickup microphone for the acoustic bass. That background gave her the advantage of violin and piano lessons as a youngster, and a close-up look at jam sessions when her parents' friends dropped by. Family connections have continued to be a plus. For one thing, she can supplement her nightclub gigs with work for her father's company, Underwood Pickups. For another, she wound up with the internationally renowned Portland bassist David Friesen as a teacher and mentor.
Underwood had taken up the bass during high school, then while attending the University of California at Berkeley (initially to study astrophysics) began to take music more seriously. Friesen, an old friend of her parents, came to teach a master class.
"I remember one of my teachers calling me into her office and saying, 'You have to take this class,' " Underwood, a willowy beauty, recalled recently, sitting with drummer Martin Zarzar over steaming bowls of pho at a downtown Portland restaurant. "I showed up for the class -- late. And afterward he said that when I'd walked in he recognized me from having met me when I was a baby."
That brief class was a turning point. Feeling she'd learned so much in that single session, she asked Friesen to give her lessons. He said no. But she kept asking and months later, convinced of her seriousness, he relented on the condition that she move to Portland.
Arriving about two years ago, she found inspiration not only in Friesen, who she credits with pressing her to become a songwriter, but also in a jazz community she says is much more inviting than that in the Bay Area.
"Underwood Uncurling" takes advantage of that community in the talents of Zarzar (of Pink Martini and Pachamanca), guitarist Dan Balmer, saxophonist John Gross, and pianist Clay Giberson. And though Underwood handles the big upright herself on a few tracks, she leaves the heavy lifting for Friesen and another Portland stalwart, Phil Baker.
Friesen's involvement also helped secure another guest star, the famed Brazilian percussionist Airto Moreira. And at times Underwood's pillowy vocal tone and unhurried melodies recall the original incarnation of Chick Corea's Return to Forever, which included Moreira and his wife, singer Flora Purim. There's even a cover of Corea's "You're Everything" from that period.
This is a jazz singer, though, who counts Bobby Gentry and Bjork among her favorites, along with No. 1 influence Nancy King. As she continues to fashion those disparate tastes into a style of her own, chances are she'll uncurl more and more into, simply, a joy. BELINDA UNDERWOOD
Website by Belinda ~ 2012 Copyright Belinda Underwood


"The Scriptures tell us that Jesus came leaping! He came with such love--that He came leaping! When a man goes leaping--you may know that it was with a great eagerness.
Jesus came leaping! How so? He came leaping . . .
***from thence to the throne again!