text:
Education in One World: Perspectives from Different Nations. BCES Conference Books, Volume 11
Popov, Nikolay, Ed.; Wolhuter, Charl, Ed.; Almeida, Patrícia Albergaria, Ed.; Hilton, Gillian, Ed.; Ogunleye, James, Ed.; Chigisheva, Oksana, Ed.
Bulgarian Comparative Education Society
This volume contains papers submitted to the 11th Annual International Conference of the Bulgarian Comparative Education Society (BCES), held in Plovdiv, Bulgaria, 14-17 May 2013, and papers submitted to the 1st International Distance Partner Conference, organized by the International Research Centre "Scientific Cooperation," Rostov-on-Don, Russia. The 11th BCES Conference theme is "Education in One World: Perspectives from Different Nations." The Distance Partner Conference theme is "Contemporary Science and Education in a Globally Competitive Environment." The book consists of 92 papers, written by 141 authors, and grouped into 7 parts. Parts 1-4 comprise papers submitted to the 11th BCES Conference, and Parts 5-7 comprise papers submitted to the Distance Partner Conference. Studies presented in the book cover all levels of the educational system--preschool, primary, secondary, postsecondary, and higher education. Topics in the field of general, special, and vocational education are examined. Methodologies used in the studies represent a multiplicity of research methods, models, strategies, styles, and approaches. Various types of studies can be seen--national and international, case and comparative, descriptive and analytical, theoretical and empirical, historical and contemporary, scientific and essayistic, and critical and indifferent. The following papers are included in this volume: (1) Editorial Preface (Nikolay Popov, Charl Wolhuter, Patrícia Albergaria Almeida, Gillian Hilton, James Ogunleye, and Oksana Chigisheva); and (2) Introduction: Globalization in the One World--Impacts on Education in Different Nations (Nicholas Sun-Keung Pang). Part 1: Comparative Education & History of Education--(3) William Russell on Schools in Bulgaria (Nikolay Popov and Amra Sabic-El-Rayess); (4) Prolegomena to an International-Comparative Education Research Project on Religion in Education (Charl Wolhuter); (5) Perspectives on Tolerance in Education Flowing from a Comparison of Religion Education in Estonia and South Africa (Johannes L. van der Walt); (6) Perspectives on Tolerance in Education Flowing from a Comparison of Religion Education in Mexico and Thailand (Ferdinand J. Potgieter); (7) Do Teachers Receive Proper In-Service Training to Implement Changing Policies: Perspective from the South African Case? (Elize du Plessis); (8) Towards understanding different faces of school violence in different "worlds" of one country (Lynette Jacobs); (9) Transforming Life Skills Education into a Life-Changing Event: The Case of the Musical "The Green Crystal" (Amanda S. Potgieter); (10) Accessing Social Grants to Meet Orphan Children School Needs: Namibia and South Africa Perspective (Simon Taukeni and Taole Matshidiso); (11) Educational achievement as defining factor in social stratification in contemporary Spain (Manuel Jacinto Roblizo Colmenero); and (12) From Times of Transition to Adaptation: Background and Theoretical Approach to the Curriculum Reform in Estonia 1987-1996 (Vadim Rouk). Part 2: Pre-Service and In-Service Teacher Training & Learning and Teaching Styles--(13) What lessons to take from educational reforms in Asia-Pacific region? Factors that may influence the restructuring of secondary education in East Timor (Ana Capelo, Maria Arminda Pedrosa, and Patrícia Albergaria Almeida); (14) The Culture of Experiential Community Based Learning: Developing Cultural Awareness in Pre-Service Teachers (Alida J. Droppert); (15) Theory in Educational Research and Practice in Teacher Education (Leonie G. Higgs); (16) Comparative study of learning styles in higher education students from the Hidalgo State Autonomous University, in Mexico (Emma Leticia Canales Rodríguez and Octaviano Garcia Robelo); (17) Equity and Competitiveness: Contradictions between the Identification of Educational Skills and Educational Achievements (Amelia Molina García); (18) Adult Reading in a Foreign Language: A Necessary Competence for Knowledge Society (Marta Elena Guerra-Treviño); (19) The teaching profession as seen by pre-service teachers: A comparison study of Israel and Turkey (Zvia Markovits and Sadik Kartal); (20) Teaching/learning theories--How they are perceived in contemporary educational landscape (Sandra Ozola and Maris Purvins); (21) Learning Paths in Academic Setting: Research Synthesis (Snežana Mirkov); (22) Innovation Can Be Learned (Stanka Setnikar Cankar and Franc Cankar); (23) Rethinking Pedagogy: English Language Teaching Approaches (Gertrude Shotte); (24) Repercussions of Teaching Training in the Sociology of Work in Mexico (Claudio-Rafael Vasquez-Martinez, Graciela Giron, Magali Zapata-Landeros, Antonio Ayòn- Bañuelos, and Maria Morfin-Otero); (25) Listening to the Voices of Pre-Service Student Teachers from Teaching Practice: The Challenges of Implementing the English as a Second Language Curriculum (Cathrine Ngwaru); (26) In-Service Training and Professional Development of Teachers in Nigeria: Through Open and Distance Education (Martha Nkechinyere Amadi); (27) Symbols of Hyphenated Identity Drawing Maps (IDM) for Arab and Jewish Students at the University of Haifa (Rachel Hertz-Lazarowitz, Abeer Farah, and Tamar Zelniker); (28) The contemporary transdisciplinary approach as a methodology to aid students of humanities and social sciences (Petia Todorova); (29) Instructional Objectives: Selecting and Devising Tasks (Milo Mileff); and (30) Problem Orientated Education on the Basis of Hyper-Coded Texts (Play and Heuristic) (Valeri Lichev). Part 3: Education Policy, Reforms and School Leadership--(31) Using e-learning to enhance the learning of additional languages--A pilot comparative study (Gillian L. S. Hilton); (32) Challenges of Democratisation: Development of Inclusive Education in Serbia (Vera Spasenovic and Slavica Maksic); (33) Nurturing child imagination in the contemporary world: Perspectives from different nations (Slavica Maksic and Zoran Pavlovic); (34) The abusive school principal: A South African case study (Corene de Wet); (35) Thinking Styles of Primary School Teachers in Beijing, China (Ying Wang and Nicholas Sun-Keung Pang); (36) Breaking the cycle of poverty through early literacy support and teacher empowerment in Early Childhood Education (J. Marriote Ngwaru); (37) Designing Cooperative Learning in the Science Classroom: Integrating the Peer Tutoring Small Investigation Group (PTSIG) within the Model of the Six Mirrors of the Classroom Model (Reuven Lazarowitz, Rachel Hertz-Lazarowitz, Mahmood Khalil, and Salit Ron); and (38) The Effects of Educational Reform (Claudio-Rafael Vasquez-Martinez, Graciela Giron, Ivan De-La-Luz-Arellano, and Antonio Ayon-Bañuelos). Part 4: Higher Education, Lifelong Learning and Social Inclusion--(39) Interactions between vocational education and training and the labour market in Europe: A case study of Ireland's formalised feedback mechanisms (James Ogunleye); (40) At the Intersections of Resistance: Turkish Immigrant Women in German Schools (Katie Gaebel); (41) Intellectual capital import for the benefit of higher education (Airita Brenca and Aija Gravite); (42) Lessons from the training programme for women with domestic violence experience (Marta Anczewska, Joanna Roszczynska-Michta, Justyna Waszkiewicz, Katarzyna Charzynska, and Czeslaw Czabala); (43) Loneliness and depression among Polish university students: Preliminary findings from a longitudinal study (Pawel Grygiel, Piotr Switaj, Marta Anczewska, Grzegorz Humenny, Slawomir Rebisz, and Justyna Sikorska); (44) Psychosocial difficulties experienced by people diagnosed with schizophrenia--Barriers to social inclusion (Marta Anczewska, Piotr Switaj, Joanna Roszczynska-Michta, Anna Chrostek, and Katarzyna Charzynska); (45) Lifelong Learning from Ethical Perspective (Krystyna Najder-Stefaniak); (46) Contemporary perspectives in adult education and lifelong learning--Andragogical model of learning (Iwona Blaszczak); (47) Examining the reasons black male youths give for committing crime with reference to inner city areas of London (Elizabeth Achinewhu-Nworgu, Chioma Nworgu, Steve Azaiki, and Helen Nworgu); (48) Restructuring Nigerian Tertiary (University) Education for Better Performance (Stephen Adebanjo Oyebade and Chika Dike); (49) Keeping abreast of continuous change and contradictory discourses (Marie J. Myers); (50) Process Management in Universities--Recent Perspectives in the Context of Quality Management Oriented towards Excellence (Veronica Adriana Popescu, Gheorghe N. Popescu, and Cristina Raluca Popescu); (51) Greek Primary Education in the Context of the European Life Long Learning Area (George Stamelos, Andreas Vassilopoulos, and Marianna Bartzakli); (52) Bologna Process Principles Integrated into Education System of Kazakhstan (Olga Nessipbayeva); (53) Methodology of poetic works teaching by means of innovative technologies (Bayan Kerimbekova) [title provided in English and Bulgarian, paper is in Bulgarian]; (54) About the use of innovations in the process of official Kazakh language teaching in level on the basis of the European standards (Kuralay Mukhamadi) [title provided in English and Bulgarian, paper is in Bulgarian]; and (55) A Study of Para-Verbal Characteristics in Education Discourse (Youri Ianakiev) [title provided in English and Bulgarian, abstract in English, and paper in Bulgarian]. Part 5: Educational Development Strategies in Different Countries and Regions of the World: National, Regional and Global Levels [title is in English and Bulgarian]--(56) Establishing sustainable higher education partnerships in a globally competitive environment (Oksana Chigisheva); (57) Modernising education: International dialogue and cooperation (Elena Orekhova and Liudmila Polunina); (58) The communication between speech therapist and parents as a way of correction work improvement with children having poor speech (Elena Popova) [title is in English and Bulgarian, paper in Bulgarian]; (59) ESP teaching at the institutions of higher education in modern Russia: Problems and perspectives (Nadezhda Prudnikova); (60) Competency-based approach to education in international documents and theoretical researches of educators in Great Britain (Olga Voloshina-Pala); (61) EU strategies of integrating ICT into initial teacher training (Vitaliya Garapko); (62) Socialisation channels of the personality at the present development stage of the Russian society (Evgenii Alisov) [title and abstract in English and Bulgarian, paper in Bulgarian]; (63) Perspectives of competence approach introduction into the system of philological training of language and literature teachers (Elena Zhindeeva and Elena Isaeva) [title and abstract in English and Bulgarian, paper in Bulgarian]; (64) Organization of special education in the primary school of the European Union (Yelena Yarovaya) [title and abstract in English and Bulgarian, paper in Bulgarian]; (65) Formation of professionally-innovative creative sphere of future Master degree students in the Kazakhstan system of musical education (Gulzada Khussainova) [title and abstract in English and Bulgarian, paper in Bulgarian]; (66) Ethnocultural component in the contemporary musical education of the Republic of Kazakhstan (Gulnar Alpeisova) [title in English and Bulgarian, abstract in English, and paper in Bulgarian]; (67) The main tendencies of scientific research within doctoral studies of PhD (Yermek Kamshibayev) [title and abstract in English and Bulgarian, paper in Bulgarian]; (68) Organizational and pedagogical conditions of education quality improvement in the professional college (Igor Artemyev and Alexander Zyryanov) [title and abstract in English and Bulgarian, paper in Bulgarian]; (69) The imperative of responsibility in a global society as a determinant of educational strategy development (Irina Rebeschenkova) [title and abstract in English and Bulgarian, paper in Bulgarian]; (70) Pedagogical understanding of diversification of mathematical education as a strategy of development of vocational training at the university (Irina Allagulova) [title and abstract in English and Bulgarian, paper in Bulgarian]; (71) Prerequisites of the establishment and evolution of concepts and categories on the problem of ethnic and art competence formation (Leonora Bachurina and Elena Bystray) [title and abstract in English and Bulgarian, paper in Bulgarian]; (72) Education institutionalization as a stratification manipulator (Oksana Strikhar) [title and abstract in English and Bulgarian, paper in Bulgarian]; (73) The Concept of Teaching Musical Art on the Basis of Using Interscientific Connections at the Lessons (Oksana Strikhar) [title and abstract in English and Bulgarian, paper in Bulgarian]; and (74) The key strategic priorities of the development of the additional professional education at the Economic University. Regional aspect (Evelina Pecherskaya) [title and abstract in English and Bulgarian, paper in Bulgarian]. Part 6: Key Directions and Characteristics of Research Organization in Contemporary World [title in English and Bulgarian]--(75) Metaphors in the press: The effectiveness of working with newspaper tropes to improve foreign language competence (Galina Zashchitina); (76) Legal portion in Russian inheritance law (Roza Inshina and Lyudmila Murzalimova); (77) Formation of healthy (sanogenic) educational environment in innovative conditions (Anatoly Madzhuga and Elvira Ilyasova) [title and abstract in English and Bulgarian, paper in Bulgarian]; (78) "The Sacred Truth" (T. Bondarev's teaching as an element of L. N. Tolstoy's philosophy) (Valentina Litvinova) [title in English and Bulgarian, abstract in English, paper in Bulgarian]; (79) The destiny of man (Vasiliy Shlepin) [title and abstract in English and Bulgarian, paper in Bulgarian]; (80) Diversity of the world in the culture of the city Astana (Gulnar Alpeisova) [title in English and Bulgarian, abstract in English, paper in Bulgarian]; (81) The study of self-expression and culture of self-expression in pedagogy and psychology in the context of the problems of tolerant pedagogical communication (Elizaveta Omelchenko and Lubov Nemchinova) [title and abstract in English and Bulgarian, paper in Bulgarian]; (82) Infrastructural support of innovative entrepreneurship development in Ukraine (Iryna Prylutskaya) [title and abstract in English and Bulgarian, paper in Bulgarian]; (83) Guidelines and peculiarities of network mechanisms of an organization running (Natalia Fomenko) [title and abstract in English and Bulgarian, paper in Bulgarian]; (84) The influence of information technologies on medical activity and the basic lines of medical services (on the example of the portal of the state services) (Nataliya Muravyeva) [title and abstract in English and Bulgarian, paper in Bulgarian]; (85) Economic expediency of the integration cooperation between pharmaceutical complex of Russia and the CIS (Natalia Klunko) [title and abstract in English and Bulgarian, paper in Bulgarian]; (86) Research of prospects of the Russian tourism (Tatyana Sidorina, Marina Artamonova, Olga Likhtanskaya, and Ekaterina Efremova) [title and abstract in English and Bulgarian, paper in Bulgarian]; and (87) The influence of globalization on contemporary costume changes (Julia Muzalevskaya) [title and abstract in English and Bulgarian, paper in Bulgarian]. Part 7: International Scientific and Educational Cooperation for the Solution of Contemporary Global Issues: From Global Competition to World Integration [title in English and Bulgarian]--(88) An overview on Gender problem in Modern English (Daria Tuyakaeva); (89) Focus-group as a qualitative method for study of compliance in cardiovascular disease patients (Olga Semenova, Elizaveta Naumova, and Yury Shwartz); (90) The development of the social and initiative personality of children in the system of additional education (Andrei Matveev) [title and abstract in English and Bulgarian, paper in Bulgarian]; (91) Proceedings in criminal cases in respect of juveniles in the Criminal Procedure Code of Russia and Ukraine: Comparative and legal aspect (Vitaliy Dudarev) [title and abstract in English and Bulgarian, paper in Bulgarian]; (92) Some implementation issues of the UN Convention against transnational organized crime in the criminal legislation (A case of the Republic of Kazakhstan and the Russian Federation) (Gulnur Yensebayeva and Gulnur Tuleubayeva) [title and abstract in English and Bulgarian, paper in Bulgarian]; and (93) Hepatitis B immunization in children with hematological malignancies (Umida Salieva, Lubov Lokteva, Malika Daminova, and Naira Alieva) [title and abstract in English and Bulgarian, paper in Bulgarian]. A list of contributors is included. (Individual papers contain references.) [For Volume 10 (2012), see ED567040.]
Descriptors: Conferences (Gatherings), Conference Papers, Comparative Education, Global Approach, Educational History, Religious Education, Inservice Teacher Education, Faculty Development, Teacher Competencies, Violence, School Safety, Daily Living Skills, Grants, Social Services, Social Stratification, Academic Achievement, Curriculum Development, Educational Change, Preservice Teacher Education, Performance Factors, Experiential Learning, Multicultural Education, Educational Research, Educational Theories, Educational Practices, Cognitive Style, Equal Education, Competition, Ability Identification, Adult Reading Programs, Student Teacher Attitudes, Work Attitudes, Teaching (Occupation), Learning Theories, English Instruction, Teaching Methods, Educational Innovation, Educational Sociology, English (Second Language), Distance Education, Interdisciplinary Approach, Behavioral Objectives, Educational Policy, Electronic Learning, Inclusion, Child Development, Case Studies, Principals, Educational Malpractice, Elementary School Teachers, Teacher Attitudes, Teacher Empowerment, Emergent Literacy, Early Childhood Education, Cooperative Learning, Peer Teaching, Vocational Education, Labor Market, Resistance (Psychology), Immigrants, Higher Education, Womens Education, Depression (Psychology), Psychological Patterns, Individual Development, Schizophrenia, Social Integration, Barriers, Lifelong Learning, Ethics, Adult Education, Andragogy, Black Studies, Males, Criminals, Quality Control, Educational Principles, Technology Uses in Education, Poetry, Academic Standards, Second Language Instruction, Immunization Programs, Child Health, Delinquency, Court Litigation, Human Body, Focus Groups, Gender Issues, Tourism, Pharmaceutical Education, Organizational Change, Medical Services, Influence of Technology, Entrepreneurship, Self Expression, Cultural Differences, Educational Environment, Music Education, Mathematics Education, Educational Strategies, Educational Development, Educational Quality, Educational Improvement, Scientific Research, Masters Programs, Doctoral Programs, Special Education, Personality Development, Technology Integration, Competency Based Teacher Education, Speech Language Pathology, Parent Teacher Cooperation, Sustainability, Partnerships in Education, Foreign Countries
Bulgarian Comparative Education Society. Blvd Shipchenski prohod 69 A, 1574 Sofia, Bulgaria. e-mail: info@bces-conference.org; Web site: http://www.bces-conference.org
Publication Type: Books; Collected Works - Proceedings
Education Level: Higher Education; Postsecondary Education; Elementary Education; Early Childhood Education; Adult Education
Language: EnglishBulgarian
Authoring Institution: Bulgarian Comparative Education Society (BCES)
Identifiers - Location: Bulgaria; China (Beijing); Estonia; European Union; Germany; Ireland; Israel; Kazakhstan; Mexico; Namibia; Nigeria; Poland; Russia; Serbia; South Africa; Spain; Thailand; Timor-Leste; Turkey; Ukraine; United Kingdom (Great Britain); United Kingdom (London)


text:
Pop/Bollywood Music
RM's Indigo is a raw, profound insight into the real Kim Nam-joon and the struggle to maintain his identity
RM's Indigo is a raw, profound insight into the real Kim Nam-joon and the struggle to maintain his identity
Indigo is RM's canvas where he has bared the colours of his soul---it sees many things, the loneliness of an artist, the cost of fulfilling dreams, the struggle to keep one's identity in the face of monumental fame.
Written by Lakshana N Palat
BTS RM's Indigo is out (Photo: Instagram/ RM)
Listen to this article Your browser does not support the audio element.
If J-Hope wanted to explore the darker side of himself with Jack in The Box, RM desires to understand his real and raw self in Indigo. The young rapper, an art enthusiast, has brought out his shades of emotions on a personal canvas, a reflection to his own soul.
While all the members have much burdens on the shoulders, RM carries an unusual weight. At a young age, he became the leader of a band, found himself engulfed in fame and stardom, as he became part of the band that later revolutionised the face of K-Pop. From being the songwriter and rapper, he evolved into being the representative of the band owing to his English fluency. They never envisioned that they would become an instrument in South Korean diplomacy, years down the line.
RM has given powerful speeches at the UN, visited the White House and while all these milestones fuelled the band's fame, RM has always expressed his worry that he lost himself in the last few years. He became mechanical, like he said during the emotional 2022 Festa and music had become something that everyone else enjoyed, except them. "We've lost our direction," he had said tearfully.
After the band announced solo activities in June, RM went back to basics and found his soul again—and hence, the release of his solo album, Indigo. This album is a place where he could finally express his rawest and truest self, almost like a promise that he owed to himself and no one else. With this pensive and philosophical album, he takes his fans into his actual world that is untouched by the shackles of success. It's the 'last archive of his twenties' where he delves into the melancholy transition into adulthood.
In Indigo, he isn't any of these things; a leader, a fellow member or the face of South Korea—-he's just himself. Finally.
The search for identity
'I want to be human before I do some art', is RM's refrain in the song, Yun, the first in the album. It's a question that he has been confronting for a while now—his own identity behind the stoic answers in interviews and the flashing lights on stage. Yun includes comforting beats and vocals from Erykah Badu and it sets the tone for the rest of the songs of the album. He begins with saying that he doesn't want to be the trendsetter anymore; he just wishes to turn back time. Yet, while the beats are soothing, RM's contained frustration reverberates through the track.
Of course, no track in the album can be without a cleverly veiled reference to RM's passion for art—something that consumes his soul as much as music does, or maybe more. Still Life, a Korean-English collaboration with Anderson Paak, is upbeat and is a fun banter between Paak and RM. "You can't lock me in the frame, I'm moving, " the song reiterates RM's determination to avoid being slotted into one perception. He's moving forward without being held back from the constraints of the past and the expectations from the future. With All Day, RM collaborates with Tablo for another upbeat track where both of them extrapolate on the difficulties of crafting something original and following the herd. In Forg_tful, he attempts something new with his vocals and uses his a rather raspy lower register.
Five years ago, RM had released the cheery and positive Change. However, in Change Pt 2, the rapper is rather resigned and weary as evident in the opening lines 'Things change, people change, everything changes. Love change, friends change…' Everything has fallen apart, and he's just wondering how to reconstruct himself with the pieces. While it seems like heartbreak at the outset, RM looks into himself and realises that he isn't the person that he used to be.
The angst of youth
Change Pt 2 is followed by Lonely (RM had clearly stated that the songs should be listened to in order). In this track—with again a seeming heartbreak at the center—RM turns to poignant angst and resentment, with a hint of aggression. He reiterates the line, "I tried a million times to let you go, so many memories on the floor…"
Hectic, featuring Colde, is a disco-themed pulsing track, which begins with RM's words, "Yesterday was a hard day, there was nothing romantic…" It reflects his feelings on a regular work day from dawn till dusk—-the hard gruel and toil of his days should not be romanticised and idealised, as one can interpret from the lyrics of the song. Yet, with No.2 featuring Park Jiyoon, he strives to look forward to the morning.
The consuming power of dreams
It's in Wild Flower, where RM shows his true, raw and vulnerable self completely, without any inhibitions. He doesn't mince words and lays bare his honesty for the fans to see. His desperation echoes in the song as he almost begs the world to let him be himself, as the fame has turned into 'shackles'. It's a plaintive and painful plea as he says, 'I can't go to the stars again, I can't…'
Throughout the years, RM has verbalised his struggle with fame as he once emphasised, "We are just seven normal boys from Korea." Fame has had its moments of unbridled joy and pride, but it has also come at a price—compromising on the core aspects of himself. In this track, He shows the Kim Nam-joon, who just wanted to be a poet at one point. He just wants to escape to the fields of flowers when times get cruel, as he explains, 'When your dreams devour you, when your own heart underestimates you…'
Indigo is RM's canvas where he has bared the colours of his soul—it sees many things, the loneliness of an artist, the cost of fulfilling dreams, the struggle to keep one's identity in the face of monumental fame.
Struggling with lack of raw material, CAI urges Centre to withdraw import duty on cotton

redpajama_set_name: RedPajamaCommonCrawl


Which roads are closed for the European Championship cycling in Glasgow tomorrow?
Tomorrow marks the end of the games in Glasgow as the European Championships enter their final day.
The event is being hailed as a huge success with millions tuning in to watch events across the continent and over 360,000 spectators attending them in person in Glasgow.
The final day will include with a huge Celtic Connections shindig on George Square, with revellers also gathering for one last big party in the hidden Tontine Lane on Sunday night. If you are counting your pennies at the mo', both of those events are completely free.
But before then we have the small matter of the men's cycling road race which will again see a big part of the city centre on lockdown. There won't be quite as many closures as there were on Wednesday for the time trial, the poor people of Torrance will be able to get in and out of their homes, but there are key closures that anyone thinking of heading into town should be aware of.
For more info, see the Glasgow City Council website .


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The NetworkWorld Asia (NWA) Information Management Awards, inaugurated in 2012, recognizes Asia's leading providers of Data Management, Security and Storage solutions for their huge advances made over the last few years.
The 7th annual Information Management Awards this year will bear testimony to the innovation and responsiveness of these solution and service providers in meeting the growing needs of end-user organizations as a result of the burgeoning adoption and application of big data, IoT, cybersecurity, the hybrid cloud and AI.
As the only regional editors' and CIOs' choice awards program for this fast-growing field, the prestige and honor in winning an award in NWA's Information Management Awards cannot be overstated.
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Q: ParseFile if empty from server using Android I am working on an Android application for parse.com in which I am taking image file from parse.com. I want to make a condition in which if image file is not available on server then it will not crash the application it should give a toast message "Image File Not Available" or it will show default image file.
My code is given below:
public static ParseUser user = getCurrentUser();

progressDialog = ProgressDialog.show(UserInfoActivity.this, "","Downloading Image...", true);

ParseFile fileObject = (ParseFile) user.get("profileImage");
fileObject.getDataInBackground(new GetDataCallback() {

    public void done(byte[] data, ParseException e) {

        if (e == null) {

            Log.d("test", "We've got data in data.");
            // Decode the Byte[] into Bitmap
            Bitmap bmp = BitmapFactory.decodeByteArray(data, 0,data.length);
            Bitmap circleBitmap = Bitmap.createBitmap(bmp.getWidth(), bmp.getHeight(), Bitmap.Config.ARGB_8888);
            BitmapShader shader = new BitmapShader(bmp, TileMode.CLAMP, TileMode.CLAMP);
            Paint paint = new Paint();
            paint.setShader(shader);
            Canvas c = new Canvas(circleBitmap);
            c.drawCircle(bmp.getWidth() / 2, bmp.getHeight() / 2,
            bmp.getWidth() / 2, paint);

            // Get the ImageView from main.xml
            ImageView image = (ImageView) findViewById(R.id.image);

            // Set the Bitmap into the ImageView
            image.setImageBitmap(circleBitmap);

            // Close progress dialog
            progressDialog.dismiss();
        } else {

            Toast.makeText(getApplicationContext(), "asfdsasa", Toast.LENGTH_LONG).show();
            Log.d("test","There was a problem downloading the data.");
        }
    }
});

A: From the Android API for ParseObject https://parse.com/docs/android/api/com/parse/ParseObject.html

has(String key)
            Whether this object has a particular key.

So, you should simply be able to:
if (getCurrentUser().has("profileImage")) {
    // fetch and show
} else {
    // use default image
}

You might have to fetch the data for the user before the check using user.fetchIfNeededInBackground()
After that you can get the data by calling getCurrentUser().get("profileImage"). No need to do any additional calls, as the user object has already been populated with the data from the profile image.
Bonus:
Looks as if you want to draw the image on a circular canvas.
The Android Bootstrap library has methods to do just that, which might be useful: https://github.com/Bearded-Hen/Android-Bootstrap

redpajama_set_name: RedPajamaStackExchange


text:
Ford vs. Chevy spat leads to stabbing
CALHOUN - Sheriff's officials say a northwest Georgia man was stabbed after an argument with another man over which vehicles are better: Ford or Chevrolet.
The Calhoun Times and Rome News-Tribune reports that several people were driving off-road vehicles in a field when an argument escalated into a fistfight around 1 a.m. Saturday.
Gordon County sheriff's officials said in a statement the fight ended when one man pulled a knife and stabbed the other one.
Sheriff's detectives said witnesses told them the assault stemmed from a disagreement over the superiority of Ford vehicles versus Chevrolet vehicles.
Authorities said 21-year-old Zachary Woody of Calhoun has been charged with aggravated assault.
Authorities say the man who was stabbed was hospitalized with serious injuries.


text:
Jepson ist der Familienname folgender Personen:
 Edgar Alfred Jepson (1863–1938), englischer Schriftsteller
 Helen Jepson (1906–1997), US-amerikanische Opernsängerin (Sopran)
 Paul Newton Jepson (1893–1949), US-amerikanischer Arzt und Chirurg
 Selwyn Jepson (1899–1989), englischer Schriftsteller
 Willis Linn Jepson (1867–1946), US-amerikanischer Botaniker

Siehe auch:
 Jepsen

redpajama_set_name: RedPajamaWikipedia


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Summer 2010 - There is nothing quite like diving into a refreshingly icy pool on a hot and humid day, or nothing quite as relaxing as spending time in a hot tub. Hollandia Pools & Spas (Hollandia) can provide these experiences, because the company has experienced decades of success in the swimming pool, water play and water park facilities construction industry.
Though much has changed in all avenues of society since Hollandia was founded in 1950 (and incorporated in 1953), one thing remains constant for the company: its dedication to building the best products available for each and every client. Thanks to its hard work, Hollandia (also doing business as Hollandia Gardens Ltd.) is now the second largest company of its kind in Canada.
From its headquarters in London, Ontario, Hollandia employs 50 people. With its in-house staff, the company takes a design-build approach, and performs the majority of its trades, including structural concrete, mechanical, installation, finishes, service and concrete flat work, among others. Just as impressively, the company works throughout all of Canada from this one location.
On the rare occasion that Hollandia does work with subcontractors, the company has the "attitude that building relationships with suppliers and monitoring what they do" is crucial to success, explains Walter Schmoll, Hollandia's president and CEO.
After graduating with a degree in architecture, Schmoll went to work for an architectural firm in Toronto. He got a call one day in 1980 inquiring whether he would be interested in trying his hand at building pools. "I thought I might try it for a year, but I am still here," Schmoll says with a chuckle.
Hollandia's status is due, in part, to its highly trained employees. The company's well-known name attracts new employees who swiftly enter into a rigorous training program. Each position has its own training schedule, and once training is complete, the new recruit is out on his or her own.
Hollandia completes about 60 percent of its projects in the commercial market and the remaining 40 percent in the residential market. With each project, the company promises to build a pool that matches its environment. It offers award-winning designs, but can also complete a customized package to meet a client's requirements.
Most recently, the company completed a 50-meter pool for the University of Western Ontario and another 50-meter pool with a slide for the City of London. Another recent project was also one of the company's most interesting. For this project, Hollandia imported a pool from Italy to be installed locally in Portage la Prairie. The project, about $3 million in scope, involved a pre-engineered substructure made with stainless steel.
"It's very efficient and high-quality. The pool went in quickly and has an extensive life to it," Schmoll explains. "We can also offer a better warranty than if it was just concrete," he adds. Hollandia offers various regular maintenance, occasional check-up or emergency repair services.
For a company like Hollandia with only one location, working on projects far from its headquarters could theoretically prove to be a challenge. However, the company knows the ropes, and the traveling takes place glitch-free. For the Portage la Prairie project, for example, Hollandia rented two furnished homes for its employees to stay in during the completion of the project. "I hate having them live out of suitcases in a hotel," Schmoll says emphatically.
With decades of experience and a world-class team of professionals, Hollandia has other feathers in its cap, too. One of the most impressive is the ISO 9001: 2008 certification that the company received in March 2007. This certification formalizes all of Hollandia's processes and work instructions. It establishes a system of continuous improvement that involves all of the company's employees. Customer satisfaction is also monitored within this certification, allowing Hollandia a true read of its impact on its clients.
The future for Hollandia promises to be a bright one. The company hopes and plans to expand into the United States within the near future, providing the same dedication to 100-percent conformance to the contract's schedule and budget, while exceeding customer expectations in terms of build quality and aesthetic value. "We have made some contact with designers to pave some inroads," Schmoll says. In the meantime, Hollandia Pools & Spas will keep itself busy amassing more acclaimed work in Canada. "There is a lot of potential for growth here," Schmoll predicts optimistically.


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I make this all the time .. I love it and Rubens are amazing either way ... I also just make plan sauerkraut with chunks of Brats in them omg it is heaven to me!
INGREDIENTS: Pork, water, corn syrup, and less than 2% of the following: pork broth with natural flavorings, salt, dextrose, natural flavors, BHA, propyl gallate, citric acid.


text:
Which Stereotypes Are Not True About Southern Society?
Sara Lewis answered
There are many stereotypes about the Southern States in the US and most are unfair. Here are a couple that are definitely inaccurate:
The South is all conservative and has outdated values
Sure, there are some places where this may be true, but there are also areas where Liberalism and progressive values are far more prevalent.
Carrborro, North Carolina: Carrboro is generally regarded as one of the most liberal towns in the South, and even featured in a recent list of the most liberal places in the US. It's population of 20,000 is made of many hippies, vegans and University of North Carolina alumni, and up until recently was governed by an openly gay mayor.
People are uneducated Rednecks
Whilst it is true that most academia is centred around the northern states, the south is not far behind. Duke University in Durham, NC is ranked 8th out of all Universities in the US (and 25th in the world), ahead of Dartmouth and Northwestern, with Rice and Vanderbilt also making the top 20.
What's Your Stereotype About Colombia?
The opinion I first had was invoked by the media and yes, I see it as fields of marajauna, cocoa with...
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Paranormal Beliefs
Werewolves are like well you and other humans us werewolves are just gifted or cursed it depends how...
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Middle colonies economy...
Which Of The Statements Is NOT True About Our Federalist System?
A. They called for a strict interpretation ofthe constitution b. George Washington Founded their party...
What's your stereotype about England?
Nice country full of uptight, repressed, and class-ridden people....
Which of the following is not true regarding social development?
Which of the following is not true regarding social development? A. Social developmental transitions...
Which statement is true about 4x-2(3+2x)=-6?
It is a tautology, meaning it is true for all values of x. It is a linear equation with an infinite number...
Is there room for debate on the pressures faced by our youth to conform to the norms of society and its stereotypical groups in order to make friends and be accepted? Will there be a time when we are not judged by appearance or social status?
I think there is always room to debate anything! That being said, here's my two cents: Unfortunately,...
which of the following is true about hiv?
I'd go for number 3....
Which of the following is a true statement about U.S. District Courts? a.The Judiciary Committee does not hold hearings. b.They are assigned other duties by the Senate. c.They have no appellate jurisdiction. d.They have no original jurisdiction.
I know the answer to your question but I'm not telling you because I think you need to do your own homework...
Is It Legal To Make An Employee Punch Out For Bathroom Breaks?
What Is The Religion In Southern Ireland?
What Is Southern Blotting?
Mikhail Gorbachev's Policies Changed The Soviet Union, Would You Consider Him A Patriot, Or In A Sense, A Subversive?
How Do You Know If You're A Vampire?
Can My Child Get Social Security When Father Is Incarcerated?
What Are The Most Addictive Drugs?
What Is The Salary Of An Assistant Manager In IDBI Bank?
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Is Adam Lambert Gay?
What Does It Mean When I Don't Know How I Feel About My Boyfriend?
Why does my son drink until he gets drunk?can't he tell when he has had enough


text:
by Web Aggregator | Sep 10, 2020 | Blog, Business Technology, Home Technology, New Products, Windows
If you're an Office 365 user, you'll be pleased to know that that Word for the web now has two cool new features: Transcribe and support for voice commands. Currently, the transcription capabilities are somewhat limited, but are still robust enough to be...
Google Play Music Changing To YouTube Music By December 2020
by Web Aggregator | Sep 10, 2020 | Blog, Business Technology, Google, Home Technology, New Products
Google has been planning to consolidate its holdings for a long while now, and part of that process includes doing away with Google Play Music and rolling it into YouTube Music. The company's original music service now has an official end of life date: December...
Feature Similar To iPhone Airdrop Was Added To Android
by Web Aggregator | Sep 10, 2020 | Apple, Blog, Google, New Products
Apple users are almost certainly familiar with the Airdrop feature, which allows content to be shared between two iPhones that are in close proximity to each other. Recently, Google announced the rollout of their new 'Nearby Share' feature, which seeks to...
SpaceX Starlink Satellite Getting Closer To Providing Service
by Web Aggregator | Aug 5, 2020 | Blog, Business Technology, Home Technology, New Products
How would you like to get your internet signal from outer space? You can! Elon Musk's SpaceX company has moved another step closer to offering internet connectivity via its Starlink service. However, at the moment, there's an important catch and caveat....
New Image Format Will Soon Get Browser Support
by Web Aggregator | Aug 5, 2020 | Blog, Business Technology, Google, Home Technology, New Products, Windows
In 2015, a new image format was born of a collaboration between Cisco, Google, and Xiph.org. Dubbed AVIF, the format is based on the Av1 video codec. It is royalty-free and widely regarded as the most highly optimized image format to be developed to date, superior to...
Mac OS Will Get Some Big Changes Soon
by Web Aggregator | Jul 13, 2020 | Apple, Blog, Business Technology, Home Technology, New Products
Big changes are in the works for Apple's venerable MacOS, as the company prepares for a future where the desktop OS shares increasing amounts in common with both iOS and iPadOS. A recent preview of the latest desktop OS reveals an interface that has been...


Apple is developing the new technology related to wireless charging for new iPhone in 2017, Bloomberg reports. That technology is the long range wireless charging that they develop together with the partners in the United States and Asia.
This technology will become a solution for Apple in fixing the disadvantage of the current wireless charging. At this moment, wireless charging has "the loss of power over distance" issue. This means that the further a device is away from the receiver, the longer the batteries would take to charge.
Apple has adopted the wireless charging for Apple Watch by using the charging inductive technic, meaning the Watch must be touching the charging pad in order to gain power. This is similar to many Android manufacturers have implemented wireless charging into their devices over the years. If Apple succeeds to develop the new long range wireless charging technology, it will become the best implementation of wireless charging technology for a mobile device. Moreover, we can make sure our iPhone keeps on charging, even though it's a bit further from the wireless charging adapter.
The previous rumor said that iPhone 7 will be released this year and adopt the wireless charging, but according to this latest report, Apple is targeting to adopt this technology in 2017. So, are you interested in this technology?


text:
\section{Introduction}

We explore algorithms that use dual spaces to compute local information about polynomial ideals over the complex numbers, as an alternative to working directly with polynomials.  This strategy often has advantages in the regime of approximate numerical computations and for hybrid symbolic-numerical algorithms.  Many computations are reduced to linear algebra, allowing numerical linear algebra techniques to be applied.

Given generators of a polynomial ideal $I$ and a point $p$ in its vanishing set, the dual space of $I$ at $p$ is the vector space dual of the extension of $I$ in the local ring at $p$, and it uniquely encodes the local properties of $I$ there.  Certain combinatorial information about the dual space, such as dimension, can be accurately computed even when $p$ is only known approximately but with high enough precision. Existing methods of numerical algebraic geometry (see e.g., \cite{SVW9} and~\cite{Sommese-Wampler-book-05}) offer efficient algorithms to approximately compute points on the vanishing set of an ideal.

The contributions of this paper are:
\begin{itemize}
 \item Establishing a correspondence between a local monomial order on a local ring (primal order) and the corresponding dual order on the monomials of the dual space.
 \item Characterizing the dual space of colon (quotient) ideals.
 \item Introducing the notion of {\em eliminating dual spaces} as a computational tool.
 \item Applying these ideas to construct an algorithm for detecting embedded primary components of a curve.
\end{itemize}
Additionally, we consolidate the necessary background on dual spaces, state their properties in a modern language, and provide proofs where references are unavailable.

The idea of studying systems of polynomials through dual spaces dates back to Macaulay~\cite{M16}.
Most of the recent work using Macaulay's machinery concerns zero-dimensional ideals or, geometrically speaking, isolated points. This includes algorithms for computing a basis of the dual space~\cite{Mourrain:inverse-systems,DZ05} and the local Hilbert function at an isolated point~\cite{griffin2011numerical}, as well as various deflation procedures~\cite{Lec-deflation-02,LVZ,Hauenstein-Wampler:isosingular}.
Several studies depart from the zero-dimensional setting: the local dimension test~\cite{Bates-et-al:local-dimension-test}, computations using dual spaces for homogeneous ideals~\cite{Hauenstein:preprint2011}, and an numerical algorithm for local Hilbert polynomial in the general case~\cite{krone2014numerical}.

As an application of eliminating dual spaces we present an algorithm for detecting embedded components of a curve. This fills in a key missing piece in the program to numerically compute a primary decomposition of a polynomial ideal as laid out in \cite{Leykin:NPD} in the case of one-dimensional ideals.  We remark that eliminating dual spaces may, in theory, be used to provide an embedded component test in dimension more than one. However, we did not pursue this direction since for practical computation this potential technique appears to be inferior to an alternative treatment we develop in~\cite{krone2014numerical}. Our algorithms for dual spaces and detecting embedded components are implemented in the {\em Macaulay2} computer algebra system~\cite{M2www} and the code is posted at~\cite{ECTwww}.

The rest of the paper is structured as follows.  Section~\ref{Sec:dualSpace} gives an expository background on dual spaces of ideals in a local ring, laying out the facts that will be used later in the paper and giving proofs where they were not readily available in the literature.  Additionally we characterize the dual space of quotient ideals.  Section~\ref{Sec:Hilbert} describes the correspondence between the initial terms of an ideal with respect to a (primal) local monomial order and the initial terms of the dual space in the dual order.  Such a relation has been observed before in special settings such as zero-dimensional ideals with a graded order, but we give a fully general result.  This section also describes the relation between truncated dual spaces and the Hilbert function.  Section~\ref{Sec:eliminating} introduces the eliminating dual space, which generalize the truncated dual space, and can be used to compute truncated dual spaces of a quotient ideal.  These ideas are applied in~\ref{Sec:embedded-test} to the problem of detecting the embedded components on a curve, and examples are provided.

\section{Macaulay dual spaces}\label{Sec:dualSpace}

For $\alpha \in \POStoN$ and $y\in{\mathbb C}^N$, define
\begin{itemize}
\item $x^\alpha = x_1^{\alpha_1}\cdots x_N^{\alpha_N}$,
\item $|\alpha| = \sum_{i=1}^N \alpha_i$,
\item $\alpha! = \alpha_1!\alpha_2!\dots\alpha_N!$,
\item ${\partial}^\alpha = \frac{1}{\alpha!}\frac{{\partial}^{|\alpha|}}{{\partial} x^\alpha}$, and
\item ${\partial}^\alpha[y]: R \rightarrow {\mathbb C}$~~defined by~
    ${\partial}^\alpha[y](g) = ({\partial}^\alpha g)(y)$.
\end{itemize}
The differential functional $\partial^\alpha[y]$ sometimes would be written $\partial^{x^\alpha}[y]$ (e.g. $\partial^1 - \partial^{y} + \partial^{x^2yz}$) and
when the point $y$ is implied $\partial^\alpha[y]$ would be written as~$\partial^\alpha$.
For $y \in {\mathbb C}^N$, let $D_y = \operatorname{span}_{{\mathbb C}}\left\{\partial^\alpha[y]~|~\alpha \in \POStoN \right\}$
be the vector space of differential functionals at $y$. This linear space is graded by {\em order}, for a finite sum $q = \sum c_\alpha{\partial}^\alpha$,
\[
\ord q = \max_{c_\alpha\neq 0} |\alpha|.
\]
The {\em homogeneous} part of order $i$ of $q\in D_y$ is referred to as $q_i$. This grading is the associated graded linear space of the filtration $D_y^*$:
\[D_y^0 \subset D_y^1 \subset D_y^2 \subset \ldots \text{, where }D_y^i = \{q\in D_y~|~\ord q\leq i\}\}.\]

\begin{definition}
The {\it Macaulay dual space}, or simply {\em dual space},
of differential functionals that vanish
at $y$ for an ideal $I\subset{\mathbb C}[x]={\mathbb C}[x_1,\dots,x_N]$~is
\begin{equation}\label{Eq:Dual}
D_y[I] = \{q\in D_y~|~q(g)=0\hbox{~for all~}g\in I\}.
\end{equation}
The dual space $D_y[I]$ is a linear subspace of $D_y$, a basis of $D_y[I]$ is called a {\it dual basis} for $I$.
\end{definition}

The following theorem of Macaulay describes the dimension of the dual space at an isolated solution $y$. The following statement appears in the classical text of Macaulay~\cite{M16}.
\begin{theorem}\label{Thm:dimDual}
A solution $y\in\Var(I)$ is isolated with multiplicity $m$ if and only if $\dim_{\mathbb C} D_y[I] = m$.
\end{theorem}

\begin{definition} A subspace $S \subset D_y$ is {\em homogeneous} if it is spanned by homogeneous functionals $q\in D_y^{\ord q}\setminus D_y^{\ord q-1}$. If, in addition, $S$ is spanned by ${\partial}^\alpha[y],\ \alpha\in A$ for some subset $A \subset ({\mathbb Z}_{\geq 0})^N$, then $S$ is called {\em monomial}.
\end{definition}

\subsection{Local ring vs. Dual space}\label{Sec:Local-vs-Dual}

For the purpose of this section, without a loss of generality, we may assume $y=0\in{\mathbb C}^N$.
Consider the local ring $R_0 = R_{\mathfrak m}$ where ${\mathfrak m} = (x_1,\ldots,x_N)$. Let the space of dual functionals be defined as above replacing $R$ (polynomial) with $R_0$ (rational functions with denominators not vanishing at 0).

\begin{remark}\label{Rem:OneToOne} Ideals in $R$ with no primary components away
from the origin are in one-to-one correspondence with ideals in the local ring $R_0$:
\begin{itemize}
\item an ideal $I\subset R$ defines the extension $IR_0\subset R_0$;
\item an ideal $I\subset R_0$ corresponds to the ideal $I\cap R \subset R$
with no primary components away from the origin.
\end{itemize}
\end{remark}

\begin{proposition}
For ideal $I \subset R$, the dual space $D_0[I]$ is identical to the dual space of its extension in $R_0$, $D_0[IR_0]$.
\end{proposition}
\begin{proof}
Any rational function $g \in R_0$ can be expressed as a power series $g = \sum_\alpha c_\alpha x^\alpha$.  If $q \in D_0[I]$, then for any $f \in I$, $q(x^\alpha f) = 0$ for all monomials $x^\alpha$.  Then
\[ q(gf) = \sum_{\alpha \in ({\mathbb Z}_{\geq 0})^N} c_\alpha q(x^\alpha f) = 0 \]
so $q \in D_0[IR_0]$.  For $q$ not in $D_0[I]$ there is some $f\in I$ with $q(f) \neq 0$, and $f$ is also in $IR_0$.
\end{proof}
As a result we will speak interchangeably about the dual space of an ideal $I$ at the point $0$ and the dual space of its extension in the localization of $R$ at $0$, $IR_0$.

The following lemma provides another characterization of the extension of an ideal $I$ in the local ring, which will help describe the close connection between $IR_0$ and the Macaulay dual space.

\begin{lemma}\label{Lem:ExtensionAsIntersection}
 For any ideal $I \subset R$,
 \[ IR_0 \cap R = \bigcap_{k=1}^\infty (I + {\mathfrak m}^k). \]
\end{lemma}
\begin{proof}
 Let $\hat{R}$ denote the completion of $R$ with respect to the maximal ideal ${\mathfrak m}$ (the formal power series ring $\hat{R} = {\mathbb C} [[x_1,\ldots,x_N]]$).  The kernel of the map of $R$-modules $R/I \to \widehat{R/I}$ is $\bigcap_k {\mathfrak m}^k(R/I)$, and by the exactness of completion $\widehat{R/I} \cong \hat{R}/I\hat{R}$ (see \cite{Atiyah-Macdonald} Chapter 10).  Composing the quotient map $R \to R/I$ with the above, we see that $I\hat{R} \cap R$, which is the kernel of natural map $R \to \hat{R}/I\hat{R}$, is $\bigcap_k (I + {\mathfrak m}^k)$.

 For any $f \in I\hat{R} \cap R$, there is $h \in I$, $g \in \hat{R}$ such that $f = hg$, so $g = h/f$ is a rational function in $R_0$.  Therefore $f \in IR_0 \cap R$, and so $IR_0 \cap R = I\hat{R} \cap R$.
\end{proof}

\begin{proposition}\label{Prop:IdealFromDual}
For ideal $I \subset R$, $f \in IR_0 \cap R$ if and only if $q(f) = 0$ for all $q \in D_0[I]$.
\end{proposition}
\begin{proof}
It follows from the definition that $f \in I$ implies $q(f) = 0$ for all $q \in D_0[I]$.

 Let $R^k$ be the space of polynomials with degree $\leq k$, let $f^k$ denote the truncation of $f$ to degree $k$ and let $I^k \subset R^k$ be the set $\{ f^k : f \in I\}$.  Since $R^k$ is a finite dimensional vector space,
 \[ (I^k)^\perp = D_0^k[I] = D_0[I + {\mathfrak m}^{k+1}]. \]
 Suppose for some polynomial $f$ that $q(f) = 0$ for all $q \in D_0[I] = \bigcup_k D_0^k[I]$.  Because $(I^k)^{\perp\perp} = I^k$, we have $f^k \in I^k$ which implies $f \in I + {\mathfrak m}^{k+1}$.  By Lemma~\ref{Lem:ExtensionAsIntersection}, $f \in IR_0 \cap R$.
\end{proof}

\begin{corollary}\label{Coro:DualIsInjective}
For ideals $J_1,J_2 \subset R_0$, $J_1 \subset J_2$ if and only if $D_0[J_1] \supset D_0[J_2]$.
\end{corollary}
\begin{proof}
It's clear that $J_1 \subset J_2$ implies $D_0[J_1] \supset D_0[J_2]$.  Suppose $J_1 \not\subset J_2$, so there is polynomial $f \in J_1$ with $f \notin J_2$.  By Proposition~\ref{Prop:IdealFromDual} there is $q \in D_0[J_2]$ with $q(f) \neq 0$, so $D_0[J_2] \not\subset D_0[J_1]$.
\end{proof}
An immediate consequence of this corollary is that an ideal $J \subset R_0$ is uniquely determined by its dual space $D_0[J]$.

\begin{corollary}\label{Coro:HomoMono}
The dual space $D_0[J]$ is homogeneous (respectively, monomial) iff the ideal $J\subset R_0$ is homogeneous (respectively, monomial), i.e., generated by homogeneous elements with respect to filtration
$\{{\mathfrak m}^k\}_{k\geq 0}$ (respectively, by monomials).
\end{corollary}
\begin{proof}
Given a homogeneous (respectively, monomial) dual space $L=D_0[J]$ of an ideal $J \subset R_0$ it is straightforward to write down homogeneous (respectively, monomial) $I\subset R$ such that $D_0[I] = L$. Namely, its homogeneous part of order $k$ is the set of polynomials orthogonal to $L^k/L^{k-1}$; for the monomial case, it is particularly explicit: a monomial $x^\alpha$ belongs to $I$ iff ${\partial}^\alpha\notin L$. The extension $IR_0$ is determined by $L$ uniquely according to Proposition~\ref{Coro:DualIsInjective}, hence, $IR_0 = J$.
\end{proof}

\begin{remark}
  One could easily extend the definition of homogeneous and monomial ideals to the local ring $R_y$ for an arbitrary point $y\in{\mathbb C}^N$: in particular, an ideal is called monomial if it is generated by elements of the form $(x-y)^\alpha$, $\alpha\in \POStoN$.
\end{remark}

Macaulay dual bases allow for testing ideal membership at a solution \cite{MMM96}
as stated in the following proposition. This can be readily generalized for homogeneous ideals using the following corollary, Remark~\ref{Rem:OneToOne}, and Proposition~\ref{Coro:DualIsInjective}.

\begin{corollary}[Lemma~11 of \cite{Hauenstein:preprint2011}]\label{Coro:HomoMembership}
A polynomial $f \in R$ of degree $d$ is a member of a homogeneous ideal $I\subset R$ iff $f$ is annihilated by $D_0^{d}[I]$.
\end{corollary}
\begin{proof}
It follows from the proof of Corollary~\ref{Coro:HomoMono} that $D_0^d[I]$ determines $J = I R/{\mathfrak m}^{d+1}$.
Now, $f\in I$ iff its image $\bar f \in J$ iff $f$ is annihilated by $D_0^{d}[I]$.
\end{proof}

The statement of Corollary~\ref{Coro:HomoMembership} corrects that of Theorem~$4.6$ of~\cite{Leykin:NPD} where the assumption of homogeneity was missed as shown in \cite{DualExample}. The local membership test without the assumption of homogeneity is a much harder task, addressed in \cite{krone2014numerical}.

\subsection{Action of differentiation on the dual space}
An alternative characterization of the dual space can be given via Proposition~\ref{Prop:closedness}. There is a natural action of $R_0$ on $D_0$ by pre-multiplication.  Specifically for $q \in D_0$ and $g \in R_0$ let $g\cdot q \in D_0$ denote the functional defined by $(g\cdot q)(f) = q(gf)$.  It can be checked that this gives $D_0$ an $R_0$-module structure. The action of each variable $x_i$ can also be considered as {\em differentiating} functionals in $D_0$ by ${\partial}_i$ (up to normalization).  Let $\der_{x_i}: D_0 \to D_0$ denote the map defined by the action of $x_i$.
\begin{eqnarray*}
  \der_{x_i}:  D_0 &\to& D_0\\
        {\partial}^\alpha &\mapsto& {\partial}^{\alpha - e_i}, \ \ \ \ (i=1,\ldots,N),
\end{eqnarray*}
where ${\partial}^\beta$ is taken to be $0$ when any entry of $\beta$ is less than zero.

The following statements (from Proposition~\ref{Prop:closed} to Corollary~\ref{Coro:truncated-for-homogeneous}) appear, perhaps in alternative phrasing, in many works addressing the duality at hand (see, for example,~\cite{Mourrain:inverse-systems}). We collect the essential pieces, stated in our language, and complete with our own short proofs to guide reader's intuition for this paper.

\begin{proposition}\label{Prop:closed}
For a subspace $L \subset D_0$ the following are equivalent:
\begin{itemize}
 \item $L$ is the dual space of some ideal \mbox{$J_L \subset R_0$}.
 \item $L$ is closed under differentiation by each variable: $x_i\cdot L \subset L$ for all \mbox{$1 \leq i \leq N$}.
 \item $L$ is an $R_0$-submodule of $D_0$.
\end{itemize}
\end{proposition}
\begin{proof}
For any $L \subset D_0$ define

\[ J_L = \{ f\in R_0: q(f) = 0 \text{ for all } q \in L\}. \]

If $L$ is closed under differentiation, then $J_L$ is closed under multiplication by each $x_i$, and therefore under multiplication by all monomials in $R_0$.  Express any $g \in R_0$ as $g = \sum_\alpha c_\alpha x^\alpha$.  Then if $f \in J_L$ and $q \in L$, $q(gf) = \sum_\alpha c_\alpha q(x^\alpha f)$ and each term is zero, so $gf \in J_L$.  Therefore $J_L$ is an ideal and $D_0[J_L] = L$.  Conversely if $L = D_0[J_L]$ and $q \in L$ then $(x_i\cdot q)(f) = q(x_i f) = 0$ for all $f \in J_L$ so $x_i\cdot q \in L$.
\end{proof}

Consider the map $$\Dual : \{\text{ideals of } R_0\}  \to \{R_0\text{-submodules of }D_0\}$$ defined by $\Dual(J) = D_0[J]$.  By Corollary~\ref{Coro:DualIsInjective} and Proposition~\ref{Prop:closed}, this map is a bijection.  This provides another way to characterize the dual space.

\begin{proposition}\label{Prop:closedness}
For ideal $J=\langle f_1,\dots,f_n\rangle\subset R_0$, let $L$ be the maximal $R_0$-submodule of $D_0$ that satisfies $q(f_i) = 0$ for all $q \in L$ and all $0 \leq i \leq n$.  Then $L = D_0[J]$.
\end{proposition}
\begin{proof}
$D_0[J]$ is closed under differentiation and satisfies $q(f_i) = 0$ for all $q \in D_0[J]$ and $0 \leq i \leq n$, so $D_0[J] \subseteq L$.  The ideal $J_L$ contains $\{f_1,\ldots,f_n\}$, so $J \subseteq J_L$ which implies $L \subseteq D_0[J]$.
\end{proof}

\begin{remark}
 For an ideal $J \subset R_0$, the dual space $D_0[J]$ is finitely-generated as an $R_0$-module only when it is a finite dimensional vector space.  If $D_0[J]$ is generated by a single functional $p$, then $J$ is exactly the {\em apolar ideal} of $p$~(see, for instance, \cite{iarrobino1999power} for the definition).
\end{remark}

A result of Proposition~\ref{Prop:closedness} is that for $I = \ideal{f_1,\dots,f_n}$, a dual element $q$ is in $D_0[I]$ if and only if $q(f_i) = 0$ and $x_j\cdot q \in D_0[J]$ for each $0 \leq i \leq n$ and $0 \leq j \leq N$.  Note that this leads to a completion scheme for computing~$D_y^k[I]$ proposed in \cite{Mourrain:inverse-systems}, assuming $y$ is in the vanishing set of $I$:
\begin{algorithmic}
\STATE $D_y^0[I] \leftarrow \operatorname{span}_{\mathbb C}({\partial}^0)$
\FOR{$i = 1 \to k$}
\STATE $D_y^i[I] \leftarrow \{ q\in D_y \mid x_j\cdot q \in D_y^{i-1}[I] \mbox{ for all } j=1,\ldots,N \mbox{ and } q(f_i) = 0 \mbox{ for all } i=1,\ldots,n \}$
\ENDFOR
\end{algorithmic}
Moreover, the above algorithm makes apparent that if $D_y^i[I] = D_y^{i+1}[I]$ for some $i \geq 0$ then $D_y^i[I]$ is equal to all higher truncations, and so is equal to $D_y[I]$.  This gives an effective stopping criterion for computing $D_y[I]$ when it is finite dimensional.

\begin{proposition}\label{Prop:intersection}
For ideals $J_1,J_2 \subset R_0$,
\begin{itemize}
 \item $D_0[J_1 + J_2] = D_0[J_1] \cap D_0[J_2].$
 \item $D_0[J_1 \cap J_2] = D_0[J_1] + D_0[J_2].$
\end{itemize}
\end{proposition}
\begin{proof}
The first statement follows from the definition of the dual space, as does $D_0[J_1] + D_0[J_2] \subset D_0[J_1 \cap J_2].$

Let $L = D_0[J_1] + D_0[J_2]$.  It's clear that $L$ is an $R_0$-submodule, so it is the dual space of an ideal $J_L$.  The fact that $D_0[J_1] \subset L$ implies $J_L \subset J_1$ and similarly $J_L \subset J_2$.  Therefore $D_0[J_1 \cap J_2] \subset L$.
\end{proof}

\begin{corollary}\label{Coro:truncated-for-homogeneous} If $J_1$ and $J_2$ are homogeneous ideals of $R_0$, then the equality holds for the truncated dual spaces:
$$D_0^d[J_1\cap J_2] = D_0^d[J_1]+D_0^d[J_2],\ \text{for all }d\in {\mathbb N}.$$
\end{corollary}
\begin{proof}
  This follows from the fact that if $q\in D_0[J]$ for a homogeneous $J$, then $q_d$, the part of $q$ of order $d$, is also in $D_0[J]$.
\end{proof}

\begin{remark}
  For truncated dual space, in general, only one inclusion holds:
  $$D_0^k[J_1\cap J_2] \supset D_0^k[J_1]+D_0^k[J_2].$$
  However, because $D_0^k[J_1\cap J_2]$ is finite dimensional, it follows that
  $$D_0^k[J_1\cap J_2] \subset D_0^l[J_1]+D_0^l[J_2]$$
  for $l$ large enough.
\end{remark}

\begin{example}
Let $I_1 = \ideal{x_1}$ and $I_2 = \ideal{x_1-x_2^2}$ in $R = {\mathbb C} [x_1,x_2]$.  Then
\[ D_0^1[I_1] + D_0^1[I_2] = \operatorname{span}\{1,\partial_2\}, \]
\[ D_0^1[I_1\cap I_2] = \operatorname{span}\{1,\partial_1,\partial_2\}, \]
\[ D_0^2[I_1] + D_0^2[I_2] = \operatorname{span}\{1,\partial_2,\partial_2^2,\partial_2^2+\partial_1\}. \]
There are strict inclusions
\[ D_0^1[I_1] + D_0^1[I_2] \subsetneq D_0^1[I_1\cap I_2] \subsetneq D_0^2[I_1] + D_0^2[I_2]. \]
\end{example}

\subsection{Dual space of quotient ideals}\label{Subsec:quotient-dual}
Recall that for $g \in R_0$, the map $\der_{g}:D_0 \to D_0$ denotes the action of $g$ on $D_0$ by pre-multiplication, or equivalently by ``differentiation'' with respect to $g$.

\begin{proposition}
 For all non-zero $g\in R_0$, the map $\der_g:D_0 \to D_0$ is surjective and $\ker \der_g = D_0[\ideal{g}]$.
\end{proposition}
\begin{proof}
 Note $g\cdot D_0$ is closed under differentiation.  If $\der_g$ is not surjective, then $g\cdot D_0$ is the dual space of some non-trivial ideal $I \subset R_0$ by Proposition~\ref{Prop:closed}.  Choose some non-zero $f \in I$.  Since $gf \neq 0$, there exists some functional $q$ with $q(gf) \neq 0$.  Then $g\cdot q(f) = q(gf) \neq 0$, which is a contradiction since $g\cdot q$ should annihilate $f$.  To show $\ker \der_g = D_0[\ideal{g}]$, if $q \in D_0[\ideal{g}]$ then $g\cdot q(f) = q(gf) = 0$ for all $f \in R_0$.  The only functional that is zero on all elements of $R_0$ is the zero functional so $g\cdot q = 0$.  Conversely if $q \notin D_0[\ideal{g}]$ then $g\cdot q(f) = q(gf) \neq 0$ for some $f \in R$, so $g\cdot q \neq 0$.
\end{proof}

\begin{theorem}\label{thm:dualOfColonIdeal}
	$D_0[I:\ideal{g}] = g\cdot D_0[I]$.
\end{theorem}

\begin{proof}
For $I$ homogeneous, the statement is shown in~\cite[Theorem 22]{Hauenstein:preprint2011}. Here we consider the general case.

If $p \in D_0[I]$, then $g\cdot p(f) = p(gf) = 0$ for all $f$ such that $gf \in I$.  These are precisely the polynomials $f$ in $I:\ideal{g}$, and so $g\cdot p \in D_0[I:\ideal{g}]$.

For any $q \in D_0[I:\ideal{g}]$, because $\der_g$ is surjective we can choose some $p \in D_0$ such that $g\cdot p = q$.  Then for all $f \in I:\ideal{g}$, we have $q(f) = p(gf) = 0$, so \[p \in D_0[g(I:\ideal{g})] = D_0[I \cap \ideal{g}] = D_0[I] + D_0[\ideal{g}]. \]
Therefore $p = p' + u$ for some $p' \in D_0[I]$ and $u \in D_0[\ideal{g}]$.  Then $q = g\cdot p = g\cdot p' + g\cdot u$ but $g\cdot u = 0$ so $q \in g\cdot D_0[I]$.
\end{proof}

\section{Local Hilbert function and its regularity}\label{Sec:Hilbert}

The Hilbert function of an ideal $I \subset R_0$ provides combinatorial information about $I$ that can be computed numerically using truncated dual spaces.

\subsection{Primal and dual monomial order}
Let $\geq$ be a local monomial order ($1$ is the largest monomial), which we shall refer to as a {\em primal order}. For
$g = \sum_{\alpha} a_\alpha x^\alpha$, a nonzero polynomial,
the {\em initial term} with respect to $\geq$ is
the largest monomial with respect to $\geq$ that has a nonzero coefficient, namely
$$\initial_{\geq}(g) = \max_\geq\{x^\alpha~|~a_\alpha\neq 0\}.$$
For an ideal $I$, the {\it initial terms} of $I$ with respect to $\geq$
is the set of initial terms with respect to $\geq$ of all the elements of $I$, namely
$$\initial_{\geq}(I) = \{\initial_{\geq}(f)~|~f \in I\}.$$
A monomial is called a {\it standard monomial} of $I$
with respect to $\geq$ if it is not a member of $\initial_\geq(I)$.

We shall order the monomial differential functionals via the {\em dual order}:
$$
\partial^\alpha \succeq \partial^\beta\ \Leftrightarrow\ x^\alpha \leq x^\beta,
$$
the order opposite to $\geq$.

The {\em initial term} $\initial_\succeq(q)$ of $q$
is the largest monomial differential functional that has a nonzero coefficient.
The {\em initial support} of a dual space with respect to $\succeq$
is the set of initial terms with respect to $\succeq$ of all the elements in the dual space (which can be considered as a subset of $\POStoN$).

A dual basis that has distinct initial terms is called a {\em reduced dual basis}.
Using a (possibly infinite dimensional) Gaussian elimination procedure,
it is easy to see that any dual basis can be transformed into a reduced dual basis.

\begin{theorem}[Theorem~3.1 of \cite{LVZ}]\label{Thm:LVZ}
Let $I_0$ be a 0-dimensional ideal of $R_0$.
The initial support of the dual space $D_0[I_0]$ is the set of standard monomials for $I=I_0\cap R$, i.e.,
\begin{equation}\label{Eq:InitStdMon}
\initial_\succeq(D_0[I_0]) = \initial_\succeq(D_0[I]) = \{\partial^\alpha~|~x^\alpha \notin \initial_\geq(I)\}.
\end{equation}
\end{theorem}
\begin{proof}
 Note that $D_0[I]$ is finite dimensional.  Choose a monic reduced basis $B$ for $D_0[I]$ such that the lead term of each element does not occur in any other element (using Gaussian elimination).

 Suppose ${\partial}^\alpha\in \operatorname{in}_\succeq(D_0[I])$ so some $p \in B$ has $\operatorname{in}_\succeq(p) = {\partial}^\alpha$.  For any monic polynomial $f$ with $\operatorname{in}_\geq(f) = x^\alpha$, $f$ and $p$ have no terms with the same exponent except their respective lead terms, so $p(f) = 1$ and $f \notin I$.

 Suppose ${\partial}^\alpha \notin \operatorname{in}_\succeq(D_0[I])$.  Let $\{p_1,\ldots,p_s\} \subset B$ be the basis elements with ${\partial}^\alpha$ in their monomial support.  For each $p_i$ let $c_i$ be the coefficient of ${\partial}^\alpha$ and let ${\partial}^{\beta_i} = \operatorname{in}_\succeq(p_i)$.  The following polynomial
  \[ f = x^\alpha + \sum_{i = 1}^s \frac{x^{\beta_i}}{c_i} \]
 has $p(f) = 0$ for all $p \in B$, and $\operatorname{in}_\geq(f) = x^\alpha$.  By Proposition~\ref{Prop:IdealFromDual}, $f \in I_0$ so $x^\alpha \in \operatorname{in}_\geq(I_0) = \operatorname{in}_\geq(I)$.
\end{proof}

\begin{corollary}\label{Coro:dim-local-dual}
For an ideal $I\subset R_0$, $\dim_{\mathbb C} \left(R_0/(I + {\mathfrak m}^{k+1})\right) = \dim_{\mathbb C} D_0^{k}[I]$ where ${\mathfrak m} = \ideal{x_1,\ldots,x_N}$.
\end{corollary}
\begin{proof}
Choosing a graded primal order $\geq$, a vector space basis for the quotient $R_0/(I + {\mathfrak m}^{k+1})$ is the set of monomials
\[ \{x^\alpha~|~x^\alpha \notin \initial_\geq(I), \text{ and } |\alpha| \leq k\}. \]
By Theorem~\ref{Thm:LVZ} this is corresponds to a basis for $\initial_\succeq(D_0^{k}[I])$ which has the same dimension as $D_0^{k}[I]$.
\end{proof}

We can extend Theorem~\ref{Thm:LVZ} to ideals of arbitrary dimensions.

\begin{theorem}\label{thm:complementary-staircases}
For an ideal $I \subset R$ the monomial lattice ${\mathbb N}^N$ is a disjoint union of $\operatorname{in}_\succeq D_0[I]$ and $\operatorname{in}_\geq I$.
\end{theorem}
\begin{proof}
By Theorem~\ref{Thm:LVZ}, ${\mathbb N}^N \setminus \operatorname{in}_\succeq D_0[I + {\mathfrak m}^{k+1}] = \operatorname{in}_\geq (I+{\mathfrak m}^{k+1})$, so then
 \[ {\mathbb N}^N \setminus \bigcup_k  \operatorname{in}_\succeq D_0[I + {\mathfrak m}^{k+1}] = \bigcap_k \operatorname{in}_\geq (I+{\mathfrak m}^{k+1}). \]
  By definition $\bigcup_k  \operatorname{in}_\succeq D_0[I + {\mathfrak m}^{k+1}] = \operatorname{in}_\succeq D_0[I]$, while by Lemma~\ref{Lem:ExtensionAsIntersection}
\[\bigcap_k \operatorname{in}_\geq (I+{\mathfrak m}^{k+1}) = \operatorname{in}_\geq (IR_0 \cap R) = \operatorname{in}_\geq(I)\,.\]
\end{proof}

\subsection{Hilbert function and regularity index}
\begin{definition}
For an ideal $I\subset R_0$ define the {\em Hilbert function} as
\begin{align*}
H_I(k) &= \dim_{\mathbb C} ({\mathfrak{gr}}(R_0/I)_k) = \dim_{\mathbb C}\left(\frac{I+{\mathfrak m}^{k}}{I+{\mathfrak m}^{k+1}}\right) \\
&= \dim_{\mathbb C}\left(R_0/(I+{\mathfrak m}^{k+1})\right) - \dim_{\mathbb C}\left(R_0/(I+{\mathfrak m}^{k})\right).
\end{align*}
\end{definition}
This is the same as $HS_{R_0/I, {\mathfrak m}}$, the Hilbert-Samuel function of the $R_0$-module $R_0/I$ where $R_0$ is filtered by $\{{\mathfrak m}^k\}$.

The Hilbert function is determined by the initial ideal with respect to the primal monomial order (that respects the degree).
\begin{proposition} For an ideal $I\subset R_0$
  $$H_I(k) = H_{I,0}(k)= H_{\operatorname{in}_\geq(I\cap R)}(k),\ \text{for all }k\in{\mathbb N}.$$
\end{proposition}

Alternatively, truncated dual spaces determine the Hilbert function.  By Corollary~\ref{Coro:dim-local-dual} it can be seen that
$$H_I(k) = \dim_{\mathbb C} D_0^k[I] - \dim_{\mathbb C} D_0^{k-1}[I],\ \text{for }k\geq 0, $$
where $\dim_{\mathbb C} D_0^{-1}[I]$ is taken to be 0.

For some $m\geq 0$ the Hilbert function is a polynomial in $k$ for all $k \geq m$ (see, e.g., \cite[Lemma 5.5.1]{Singular-book-02}), the {\em Hilbert polynomial} $\HP_I(k)$.  If the dimension of $I \subset R_0$ is $d$, then $\HP_I(k)$ is a polynomial of degree $d-1$.  In particular if $I$ is 0-dimensional then $\HP_I(k) = 0$ since $R_0/I$ is finite dimensional.

\begin{definition} The {\em regularity index} of the Hilbert function is
$$\rho_0(I) = \min\{\,m \,: \,H_I(k)=\HP_I(k)\text{ for all } k \geq m\,\}.$$
\end{definition}

The regularity index of an ideal is used as a stopping criterion for many algorithms which work iteratively by degree.  In particular we will make use of it in Algorithm~\ref{alg:saturation-test-for-a-curve}.

\begin{comment}
The Hilbert polynomial is closely tied to the notion of multiplicity.
\begin{definition}
For a 0-dimensional ideal $I$, the {\em multiplicity} $\mu(I)$ is defined as $\dim_{\mathbb C} (R_0/I)$.  For $I$ of dimension $d > 0$ with Hilbert polynomial $\HP_I(k) = a_{d-1}k^{d-1} + \cdots + a_0$ the multiplicity is defined as
 \[ \mu(I) = a_{d-1}(d-1)!. \]
\end{definition}
The multiplicity of $I$ can be interpreted geometrically as follows.  For $I \subset R_0$ with dimension $d$, let $L \subset R$ be a system of affine hyperplanes with codimension $d$.  If $L$ is chosen generically, then $J = (I \cap R) + L$ is a 0-dimensional ideal and the points of ${\mathbb V}(J)$ are smooth points of ${\mathbb V}(I \cap R)$.  The multiplicity $\mu(I)$ is equal to $\dim_{\mathbb C} (R/J)$.  It follows that if $Q_1,\ldots,Q_s$ are the primary components of $I$ that have maximal dimension, then $\mu(I) = \sum_{i=1}^s \mu(Q_i)$. For detailed discussion of multiplicity see \cite[Sec 5.5]{Singular-book-02}.

\begin{definition} The {\em regularity index} of the Hilbert function is
$$\rho_0(I) = \min\{\,m \,: \,H_I(k)=\HP_I(k)\text{ for all } k \geq m\,\}.$$
\end{definition}

Let us refer to the minimal monomial generators of a monomial ideal $M$ as g-corner{}s. We call a monomial $x^\alpha$ an {\em s-corner} of $M$ when $x_i x^\alpha \in M$ for all $i=1,\ldots,n$.  For a general ideal $I$, the g-corners and s-corners of $I$ will refer to the g-corners and s-corners of the monomial ideal $\operatorname{in}_\geq I$, respectively.\footnote{{\bf g-} and {\bf s-} stand for {\bf generators} of $\operatorname{in}_\geq I$ and monomials spanning the {\bf socle} of the quotient $R_0/\operatorname{in}_\geq I$, respectively.}

\begin{figure}[ht]\label{fig:stair}
  \centering
  \includegraphics[width=.7\columnwidth]{fig_staircase.ps}
  \caption{The "staircase" of monomial ideal $I = \ideal{x_1^3,x_1^2x_2^2,x_2^4}$ in the lattice of monomials.  The regularity index of the Hilbert function is $\rho_0(I)=5$.}
\end{figure}

The Hilbert function of $I$ can be computed in terms of the set $C$ of g-corner{}s of $\operatorname{in}_\geq(I)$.  The following formula is derived from a simple counting argument
\begin{equation}\label{Eq:HilbertFormula}
H_I(k) = \sum_{S \subseteq C} (-1)^{|S|} \binom{k - \deg \lcm(S) + N - 1}{N-1}
\end{equation}
where $\binom{k - \deg \lcm(S) + N - 1}{N-1}$ is taken to be 0 for all $k < \deg \lcm(S)$.  For $k - \deg \lcm(S)+N-1 \geq 0$ this binomial coefficient is a polynomial in $k$ of degree $N-1$.  The formula provides a way to explicitly produce the Hilbert polynomial from the set of g-corner{}s.  It also provides a bound on the regularity index,
\begin{equation*}
 \rho_0(I) \leq \deg \lcm(C)-N+1.
\end{equation*}

\begin{remark}
For a 0-dimensional ideal $I$,  The Hilbert regularity index $$\rho_0(I) = \max \{ |\alpha| \,:\, x^\alpha \text{ is an s-corner of }\operatorname{in}_\geq I\} + 1.$$
\end{remark}

\subsection{Computing the regularity index of an ideal}\label{Sec:LocalReg}

Let $F\subset R$ be a finite set of generators of $I$ and let $F^h\subset R[h]$ denote the homogenization of $F$.  Then it is possible to compute $\rho_0(I)$ using the relationship between the truncated dual spaces $D^k_0[I]$  and $D^l_0[\ideal{F^h}]$ together with a stop criterion for the latter that recovers all g-corner{}s (and, therefore, all corners) of $\ideal{F^h}$.  Here we summarize the algorithm presented in \cite{Krone:dual-bases-for-pos-dim} that accomplishes this.

Let $\varphi:R[h] \to R$ denote the dehomogenization map sending $h$ to 1.  We equip $R[h]$ with the unique graded local order $\geq$ such that for monomials $a,b \in R[h]$ with the same total degree, $a \geq b$ if and only if $\varphi(a) \geq \varphi(b)$.  By calculating a reduced dual basis of $D_0^k[\ideal{F^h}]$ for a given $k$, we find the monomials not represented in $\operatorname{in}_\succeq D_0^k[\ideal{F^h}]$, which by Theorem~\ref{Thm:LVZ} correspond to the monomials of $\operatorname{in}_\geq\ideal{F^h}$ of degree $\leq k$.  Examining these monomials, we deduce all g-corner{}s of $\ideal{F^h}$ which have degree $\leq k$.  The calculation is run for successively higher values of $k$ until all of the g-corner{}s of $\ideal{F^h}$ are found.  A stopping criterion is provided by the following proposition about homogeneous ideals.

\begin{proposition}\label{Prop:StopCrit}
Let $J$ be a homogeneous ideal given by homogeneous generating set $F$, let $G$ be a homogeneous minimal standard basis for $J$, and let $G^k = \{g \in G : \, \deg g \leq k\}$.  For any $k \geq \max_{f \in F}(\deg f)$, one of the following must be true:
\begin{itemize}
 \item $G^k = G$,
 \item or there is $f \in G \setminus G^k$ with
	\[ \deg f \leq \max_{g,h \in G^k}(\deg \lcm\{\operatorname{in}_\geq g,\operatorname{in}_\geq h\}) \leq 2k. \]
\end{itemize}
\end{proposition}
\begin{proof}
 This statement follows from Buchberger's criterion for a standard basis.  Since $k$ is chosen larger than the degrees of the generators, $G^k$ generates $J$.  If $G^k$ is not a standard basis, then there must be some $g,h \in G^k$ with S-polynomial $S(g,h)$ that does not reduce to 0.  Since $g$ and $h$ are homogeneous, $\deg S(g,h) = \deg \lcm\{\operatorname{in}_\geq g,\operatorname{in}_\geq h\}$, and the normal form of $S(g,h)$ has the same degree.

Note that the assumption of homogeneity of $J$ is necessary.  With a local order, an S-polynomial may in general have higher degree than the LCM of the lead terms of it's constituent polynomials.
\end{proof}

Each time we find a new g-corner{} of $\ideal{F^h}$, we revise our bound on what degree to stop at according to Proposition~\ref{Prop:StopCrit}.  If we ever reach the bound without finding any new g-corner{}s then all g-corner{}s are guaranteed to be found and the algorithm stops.  However our stated goal is to find the g-corner{}s of $I = \ideal{F}$ rather than of $\ideal{F^h}$.  We recover the g-corner{}s of $I$ by the following fact:

\begin{theorem}[Theorem~14 of \cite{Krone:dual-bases-for-pos-dim}]
 If $C$ is a set of monomial generators of $\operatorname{in}_\geq \ideal{F^h}$ then $\varphi(C)$ generates $\operatorname{in}_\geq \ideal{F}$.
\end{theorem}
\end{comment}

\section{Quotient ideals and eliminating dual spaces}\label{Sec:eliminating}
Section~\ref{Subsec:quotient-dual} described the relationship between the dual space of an ideal $D_0[I]$, and the dual space of the quotient ideal by a principal ideal $D_0[I:\ideal{g}]$.  For applications (such as in Section~\ref{Sec:embedded-test}) it is useful to compute information even about the simplest case, where $g = x_1$.  We would like to find bases for the truncated dual spaces $D_0^d[I:\ideal{x_1}]$ but this proves difficult.

Let $>$ be a graded primal order on the monomials of the local ring $R_0$, and $\succ$ be the dual order for the dual monomials of $D_0$. For any $p \in D_0$, we must have $\ord \operatorname{in}_\succeq(x_1\cdot p) \leq \ord \operatorname{in}_\succeq(p) - 1$, since differentiation reduces the degree of each monomial by 1, but may also annihilate the lead term.  Therefore taking the derivative of the dual space truncated at degree $d+1$ we have $x_1\cdot D_0^{d+1}[I] \subset D_0^{d}[I:\ideal{x_1}]$.  Equality may not hold since there may be some functionals $q \in D_0^{d}[I:\ideal{x_1}]$ with $q = x_1\cdot p$ for some $p \in D_0[I]$ with lead term having degree higher than $d+1$ and is annihilated by $x_1$.  In general, finding $D_0^{d}[I:\ideal{x_1}]$ from the truncated dual space of $I$ may require calculating $D_0^{c}[I]$ up to a high degree~$c$.

\medskip
To overcome the difficulty of computing truncated dual spaces of colon ideals, we consider other filtrations of $D_0$ corresponding to gradings on $R_0$ other than the total degree grading.  For $A \subset \{x_1,\ldots,x_n\}$ define $\ord_A {\partial}^\alpha = \sum_{x_i\in A}\alpha_i$, the total order of all ${\partial}_i$ with $x_i \in A$.  For general $q \in D_0$ define $\ord_A q$ to be the maximum order of the terms of $q$.

\begin{definition}
 Fixing $A \subset \{x_1,\ldots,x_N\}$, the {\em eliminating truncated dual spaces} of $I$ are
$$E^d_0[I,A] = \{ q\in D_0[I] : \ord_A q \leq d\}$$
for all $d\in {\mathbb N}$.
\end{definition}
We often drop the word {\em truncated} when talking about eliminating dual spaces.

The truncated dual spaces $D_0^d[I]$ give a filtration of $D_0[I]$ corresponding to the maximal ideal ${\mathfrak m}$ of $R_0$
\[ D_0^d[I] = D_0[I + {\mathfrak m}^{d+1}]. \]
Similarly, the eliminating truncated dual spaces for $A$ correspond to the ideal $\ideal{A}$ in that
\[ E^d_0[I,A] = D_0[I + \ideal{A}^{d+1}]. \]

To see this, note that $E^d_0[I,A]$ is the intersection of $D_0[I]$ with $E^d_0[0,A] = D_0[\ideal{A}^{d+1}]$.  By Propositon \ref{Prop:intersection}, the intersection of these two dual spaces is $D_0[I + \ideal{A}^{d+1}]$.

For which ever grading of $R_0$ (and corresponding filtration of $D_0$) is chosen, it is useful to pick a local order $\geq$ (and corresponding dual order $\succeq$) that is compatible with the grading.  An order is compatible if for $x^\alpha \in (R_0)_i$ and $x^\beta \in (R_0)_j$ with $i < j$ then $x^\alpha > x^\beta$.  In the case of the total degree grading, such an order is a graded order.  For the grading given by $\ideal{A}$ a compatible local order is an {\em elimination order}, eliminating the variables in $A$.  In particular this is a block order in which the most significant block is a degree order on the variables in $A$ and the second block is an arbitrary order on the variables not in $A$.  Such an order ensures that $p \in E_0^d[0,A]$ if and only if $\operatorname{in}_\succeq p \in E_0^d[0,A]$.

\begin{remark}
 Dual spaces offer analogs to many operations in elimination theory.  The dual space of $I \cap {\mathbb C}[x_{m+1},\ldots,x_N]$ is equal to $D_0[I]|_{{\partial}_1 = 0,\ldots,{\partial}_m = 0}$.  The eliminating dual $E^0_0[I,A]$ is the dual space of $I + \ideal{A}$, the variety of which is the intersection of ${\mathbb V}(I)$ with the coordinate subspace in which the variables in $A$ are zero.  For a more detailed discussion in the case if homogeneous ideals see \cite{Hauenstein:preprint2011}.

 Let $\geq$ be an local elimination order for $x_1,\ldots,x_m$ with dual order $\succeq$ and consider ring extension $R' := {\mathbb C}(x_{m+1},\ldots,x_N)[x_1,\ldots,x_m] \supset R$.  In this extension with $\succeq'$ the corresponding dual order, the monomials in $\operatorname{in}_{\succeq'} D_0[IR']$ are the monomials of $\operatorname{in}_\succeq D_0[I]$ considering only the $x_1,\ldots,x_m$ parts.  These can be computed from $\operatorname{in}_\succeq E_0^d[I,\{x_1,\ldots,x_m\}]$ for sufficiently large $d$.
\end{remark}

Note that the eliminating dual space generalizes the usual truncated dual space since $E^d_0[I,\{x_1,\ldots,x_N\}] = D^d_0[I]$.  For general $A$, we have $E^d_0[I,A]\supset D^d_0[I]$.  Unlike $D_0^d[I]$, the eliminating truncated dual space can be infinite-dimensional.
\begin{proposition}
 If $I$ is an $m$-dimensional ideal that is in general position with respect to $x_1,\ldots,x_m$ then $\dim_{\mathbb C} E^d_0[I,\{x_1,\ldots,x_m\}] < \infty$.
\end{proposition}
\begin{proof}
 For $I$ satisfying these hypotheses the intersection of $\Var(I)$ with the space $\Var(x_1,\ldots,x_m)$ is 0-dimensional.  By Theorem \ref{Thm:dimDual}, $I + \ideal{x_1,\ldots,x_m}^{d+1}$ has dual space of finite dimension.
\end{proof}
In particular, if $I$ is a curve, after a generic change of coordinates one can finitely compute its eliminating dual spaces for $A = \{x_1\}$.  The following proposition provides a method to compute eliminating dual spaces of the quotient ideal $I:\ideal{x_1}$ as well.

\begin{proposition}~\label{prop:E_0 of colon ideal}
$E_0^{d}[I:\ideal{x_1}, \{x_1\}] = x_1 \cdot E_0^{d+1}[I,\{x_1\}]$
for all $d\in{\mathbb N}$.
\end{proposition}
\begin{proof}
Let $\succeq$ be a dual order on $D_0$ eliminating $x_1$.
	For any functional $p \in D_0$, either $\operatorname{in}_\succeq(p)$ is divisible by ${\partial}_1$, or $p$ has no terms divisible by ${\partial}_1$.  In the first case, $\operatorname{in}_\succeq(x_1\cdot p) = \operatorname{in}_\succeq(p)/{\partial}_1$.  In the second case $x_1\cdot p = 0$.  Therefore, in the view of Theorem~\ref{thm:dualOfColonIdeal}, any non-zero $q \in E_0^{d}[I:\ideal{x_1},\{x_1\}]$ must be the derivative of some $p \in E_0^{d+1}[I,\{x_1\}]$.
\end{proof}

This proposition is used in Algorithm~\ref{alg:saturation-test-for-a-curve}; see Example~\ref{example:cusp}.

Proposition~\ref{prop:E_0 of colon ideal} for curves does not hold in general (only a weaker Proposition~\ref{prop:E_0 of colon ideal_dim_m} does) and we are unable to use the eliminating dual spaces outside the specialized Algorithm~\ref{alg:saturation-test-for-a-curve}.

\begin{proposition}~\label{prop:E_0 of colon ideal_dim_m}
\begin{align}
  \label{eq: E^d supset... E^{d+1}} E_0^{d}[I:\ideal{x_1,\ldots,x_m}, \{x_1,\ldots,x_m\}] & \supset
  \sum_{i=1}^m x_i\cdot E_0^{d+1}[I,\{x_1,\ldots,x_m\}]
\end{align}
for all $d\in{\mathbb N}$.
\end{proposition}
\begin{proof} The inclusion (\ref{eq: E^d supset... E^{d+1}}) holds, since $I:\ideal{x_1,\ldots,x_m} = \bigcap_{i=1}^m I:\ideal{x_i}$ and, by Theorem~\ref{thm:dualOfColonIdeal},
\begin{align*}
  D_0[I:\ideal{x_1,\ldots,x_m}] =
  \sum_{i=1}^m D_0[I:\ideal{x_i}] =
  \sum_{i=1}^m x_i \cdot D_0[I].
\end{align*}

\end{proof}

\begin{remark} \label{rem:E_0}
Assuming it is finite, a basis for $E^d_0[I,\{x_1,\ldots,x_m\}]$ can be computed by finding a basis of the dual space of $I + \ideal{x_1,\ldots,x_m}^{d+1}$.  The dual space of a 0-dimensional ideal can be efficiently computed for example with the algorithm of \cite{Mourrain:inverse-systems} or others.
\end{remark}

\section{Detecting embedded points on curves}\label{Sec:embedded-test}

The general problem of detecting an embedded component can be formulated as follows:
\begin{quote}
Consider an ideal $I\subset R$ and a prime ideal $P_0\supset I$. Let $P_1,\ldots,P_r\supset I$ be associated ideals of $I$ such that $\sqrt{P_i}\subsetneq P_0$.

Given generators of $I$ together with general points $y_0\in {\mathbb V}(P)$ and $y_i\in {\mathbb V}(P_i)$ ($i=1,\cdots,r$) determine whether $P$ is an associated prime of $R/I$.
\end{quote}

The problem that we solve here is more special: we consider the case when the variety is locally a curve, namely, $\dim _{y_0} I = 1$. That means $\dim P_i=1$ for $i\neq 0$ and ${\mathbb V}(P_0) = \{y_0\}$ is a point that may or may not be an embedded component.

\medskip
Let an ideal $I$ be given by its generators $F$ and suppose, without a loss of generality, that the point in question is $y_0=0$. Let the 1-dimensional primary components in the problem be $P_1,\ldots,P_r$ with $V_i = \Var(P_i)$ containing the origin.  Saturating $I$ by the ideal $\ideal{x_1}$ eliminates all the components of $I$ that contain $\ideal{x_1}$.  After a generic linear change of coordinates, we may assume that no $V_i$ is contained in the hyperplane $x_1 = 0$ except for $V_0=\Var(Q_0)$, so $I:\ideal{x_1}^\infty \neq I$ if and only if the origin is an embedded component.

This leads to the following algorithm that employs the eliminating dual spaces.
\begin{algorithm}\label{alg:saturation-test-for-a-curve} $B = \operatorname{IsOriginEmbeddedInCurve}(I)$
\begin{algorithmic}
\REQUIRE $I$, a 1-dimensional ideal of $R$ in regular position relative to $x_1$.
\ENSURE $B = \text{``origin is an embedded component of $I$''}$, a Boolean value.
\smallskip \hrule \smallskip

\STATE $r \gets \rho_0(I)$;
\STATE $m \gets \mu_0(I)$;
\STATE $k \gets \max(r,m-1)$;
\STATE $E \gets E_0^{k}[I,\{x_1\}]$;
\RETURN $x_1\cdot E \subsetneq E_0^{k-1}[I,\{x_1\}]$

\smallskip \hrule \smallskip
\end{algorithmic}
\end{algorithm}

Here $\mu_0(I)$ denotes the {\em multiplicity} (or {\em degree}) of $I$ at the origin.  For $I$ a curve, note the (local) Hilbert polynomial of $I$ is the constant polynomial $\HP_I(k) = \mu_0(I)$.  To compute $\rho_0(I)$ and $\mu_0(I)$ we refer to the algorithm given in \cite{Krone:dual-bases-for-pos-dim} which produces the Hilbert function of $I$ from a set of generators, and in the process the Hilbert regularity index and the Hilbert polynomial of $I$.

The following two lemmas are used in the proof of correctness of Algorithm~\ref{alg:saturation-test-for-a-curve}.

\begin{lemma}\label{prop:truncated-test}
 Suppose ideals $I,J \subset R_0$ satisfy $I \subseteq J$ and $\dim_{\mathbb C} J/I$ is finite.  Then $I = J$ if and only if
 \[ D^{r-1}_0[I] = D^{r-1}_0[J] \]
 where $r = \max\{\rho_0(I),\rho_0(J)\}$.
\end{lemma}
\begin{proof}
 Since $I \subseteq J$, to show $I = J$ it is enough to show that $H_I(k) = H_J(k)$ for all $k \geq 0$.  Because $\dim_{\mathbb C} J/I$ is finite, $\HP_I = \HP_J$ so the Hilbert functions agree for $k \geq r$.  If additionally $D^{r-1}_0[I] = D^{r-1}_0[J]$, then the Hilbert functions also agree for $0 \leq k < r$.
\end{proof}

\begin{lemma}\label{prop:J-reg}
 If $J$ is a one-dimensional monomial ideal that is saturated at the origin ($J = J:{\mathfrak m}^\infty$), then
  \[ \rho_0(J) \leq \mu_0(J)-1. \]
\end{lemma}
\begin{proof}
 We consider a {\em monomial cone decomposition} of the standard monomials of $J$.  For monomial $m \in R_0$ and a set of variables $v = \{x_{i_1},\ldots,x_{i_k}\}$ the monomial cone $C_{m,v}$ is
  \[ C_{m,v} := \{x_{i_1}^{a_1}\ldots x_{i_k}^{a_k} m \mid (a_1,\ldots,a_k) \in {\mathbb N}^k\}. \]
 A monomial cone decomposition of $R_0/J$ is a finite list of pairs $$(m_1,v_1),\ldots,(m_s,v_s)$$ such that the standard monomials of $J$ are a disjoint union of the cones $$C_{m_1,v_1},\ldots,C_{m_s,v_s}.$$  The dimension of a cone $C_{m,v}$ is defined to be the size of $v$.  A cone decomposition is closely related to the Hilbert function of $J$: the maximum dimension of a cone in the decomposition is the dimension of the ideal; the number of maximal dimensional cones is the multiplicity $\mu_0(J)$; and the maximum degree of the monomials $m_1,\ldots,m_s$ bounds the regularity $\rho_0(J)$.  For $J$ a one-dimensional monomial ideal saturated at the origin, there is a cone decomposition $(m_1,\{x_{i_1}\}),\ldots,(m_s,\{x_{i_s}\})$ of $R_0/J$ consisting only of dimension 1 cones.

 Modify this decomposition slightly by letting $m'_j := m_j|_{x_{i_j} = 1}$, the monomial obtained from $m_j$ by removing $x_{i_j}$.  The cones $C_{m'_1,v_1},\ldots,C_{m'_s,v_s}$ also have the standard monomials of $J$ as their union, but are generally not disjoint.  To prove the proposition, it is sufficient show that for all $d \geq \mu_0(J) - 1$ each cone contains exactly one monomial of degree $d$ and these monomials are distinct, and therefore $H_J(d) = \mu_0(J)$.

 Let $M_k := \{m'_j \mid i_j = k \}$.  Note that $\sum_k |M_k| = \mu_0(J)$.  For each $k$, $M_k$ is closed under differentiation.  This follows from the fact that $M_k$ is the set of standard monomials of $\pi(J)$ where $\pi:{\mathbb C}[x_1,\ldots,x_N] \to {\mathbb C}[x_1,\ldots,\hat{x_k},\ldots,x_N]$ is the projection sending $x_k$ to 1.  If $M_k$ has a monomial of degree $d$, it also has at least one monomial of each degree $< d$.  Therefore
  \[ \max_{m \in M_k}\deg m \leq |M_k| - 1 \leq \mu_0(J) - 1. \]
 So for $d \geq \mu_0(J) - 1$, each cone contains a monomial of degree $d$.

 Suppose two cones in the decomposition intersect, so $n = m'_jx_{i_j}^a = m'_lx_{i_l}^b$ for some $j \neq l$ and $x_{i_j} \neq x_{i_l}$.  Then $x_{i_j}^a$ divides $m'_l$ so $a \leq \deg m'_l$.
  \[ \deg n = \deg m'_j + a \leq \deg m'_j + \deg m'_l \leq |M_{i_l}| + |M_{i_j}| - 2 \leq \mu_0 - 2. \]
 No two cones have a monomial in common of degree $d \geq \mu_0(J) - 1$.
\end{proof}

\begin{proof}[Proof of correctness of Algorithm~\ref{alg:saturation-test-for-a-curve}]
By Proposition~\ref{prop:E_0 of colon ideal} $x_1\cdot E = E_0^{k-1}[I:\ideal{x_1},\{x_1\}]$.  If this dual space is not equal to $E_0^{r-1}[I,\{x_1\}]$ then $I:\ideal{x_1} \neq I$.  This implies there is an embedded component at the origin.

Suppose instead $x_1\cdot E = E_0^{k-1}[I,\{x_1\}]$.  We will use Lemma~\ref{prop:truncated-test} to prove that $I:\ideal{x_1} = I$.  The truncated dual space of degree $r-1$ is contained in the eliminating dual space of degree $r$, so $D_0^{r-1}[I:\ideal{x_1}] = D_0^{r-1}[I]$.  We know that $I \subseteq I:\ideal{x_1}$.  Because they differ by at most a zero-dimensional component, $\dim_{{\mathbb C}}(I:\ideal{x_1})/I$ is finite.

Finally it must be shown that $k \geq \max(\rho_0(I),\rho_0(I:\ideal{x_1}))$.  It is clear that $k \geq \rho_0(I)$.  To show $k \geq \rho_0(I:\ideal{x_1})$, let $J = \initial(I):{\mathfrak m}^{\infty}$, which has the same Hilbert polynomial as $I$ and $I:\ideal{x_1}$ and satisfies
 \[ \initial(I) \subseteq \initial(I:\ideal{x_1}) \subseteq J, \]
 \[ H_I \geq H_{I:\ideal{x_1}} \geq H_J. \]
By Lemma~\ref{prop:J-reg}, $\rho_0(J) \leq \mu_0(I) - 1$.  Since $H_{I:\ideal{x_1}}$ is sandwiched between $H_I$ and $H_J$, once they stabilize to $\mu_0(I)$, so must $H_{I:\ideal{x_1}}$.  This implies the regularity of $I:\ideal{x_1}$ is bounded by $k$.
\end{proof}

\begin{example}\label{example:cusp}
 Let $I = \ideal{x^2 - z^3,y-z^2} \subset {\mathbb C}[x,y,z]$ which defines a curve in ${\mathbb C}^3$ with a singular point at the origin.  The deflation algorithm from \cite{Leykin:NPD} will identify the origin as a possible embedded component.  Note that $\rho_0(I) = 1$, $\mu_0(I) = 2$ and no irreducible component of $\Var(I)$ is contained in the plane $x = 0$.  To test whether the origin is embedded, we compute the eliminating dual $E_0^1[I,\{x\}]$.  This is the set of all dual functionals with all terms having ${\partial}_x$-degree $\leq 1$.
  \[ E_0^1[I,\{x\}] = \operatorname{span}\{ 1, {\partial}_z^2 + {\partial}_y, {\partial}_z, {\partial}_x, {\partial}_x{\partial}_z^2 + {\partial}_x{\partial}_y, {\partial}_x{\partial}_z \}, \]
  \[ x\cdot E_0^1[I,\{x\}] = \operatorname{span}\{ 1, {\partial}_z^2 + {\partial}_y, {\partial}_z \}. \]
  Since $x\cdot E_0^1[I,\{x\}] = E_0^0[I,\{x\}]$ we conclude that the origin is not an embedded component of $I$.
\end{example}

\begin{example}
 For this example we compute with an implementation of Algorithm \ref{alg:saturation-test-for-a-curve} in {\em Macaulay2}.  Let $I$ be the ideal of the cyclic4 system, generated by
 \[ \{x_1 + x_2 + x_3 + x_4,\; x_1x_2 + x_2x_3 + x_3x_4 + x_4x_1, \]
 \[ x_2x_3x_4 + x_1x_3x_4 + x_1x_2x_4 + x_1x_2x_3,\; x_1x_2x_3x_4 - 1\}. \]
 $I$ is a curve with several singular points, which are discovered using the algorithm described in \cite{Leykin:NPD} up to some numerical precision.  One such point is $p =$
 \[ (-0.0000000000000000122 + 1.000000000000000222i, \]
 \[ -0.0000000000000001128 + 0.999999999999999889i, \]
 \[  0.0000000000000000459 - 0.999999999999999889i, \]
 \[ -0.0000000000000000935 - 1.000000000000000000i), \]
 approximately $(i,i,-i,-i)$.  Let $I'$ denote the ideal obtained from $I$ by a random affine change of coordinates that fixes $p$.  This ensures that $I'$ is in general position with respect of $x_1$.  Using the algorithm of \cite{Krone:dual-bases-for-pos-dim}, the regularity index is $\rho_p(I') = 2$ and the multiplicity is $\mu_p(I') = 1$, so $k = \max(2,0) = 2$.

 Computing $E_p^1[I',\{x_1\}]$ and $x_1\cdot E_p^2[I',\{x_1\}]$ the dimensions are 3 and 2 respectively, so they are not equal.  Therefore the point being approximated by $p$ is an embedded component of $I$.  The code for this example can be found at \cite{ECTwww}.
\end{example}

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