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 Riemannian Geometry (PDF)
R3 and the Riemann sphere Ĉ are diffeomorphic. Exercise 2.8. Find a proof of Proposition 2.24. Exercise 2.9. Let the spheres S1, S3 and the Lie groups SO(n), O(n), SU(n), U(n) be equipped with their standard differentiable structures introduced above. Use Proposition 2.24 to prove the f...
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R3 and the Riemann sphere Ĉ are diffeomorphic. Exercise 2.8. Find a proof of Proposition 2.24. Exercise 2.9. Let the spheres S1, S3 and the Lie groups SO(n), O(n), SU(n), U(n) be equipped with their standard differentiable structures introduced above. Use Proposition 2.24 to prove the fol- lowing diffeomorphisms S1 ∼= SO(2), S3 ∼= SU(2), SO(n)×O(1) ∼= O(n), SU(n)×U(1) ∼= U(n). Exercise 2.10. Find a proof of Corollary 2.28. Exercise 2.11. Let (G, ∗) and (H, ·) be two Lie groups. Prove that the product
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http://www.matematik.lu.se/matematiklu/personal/sigma/Riemann.pdf#page=22
www.matematik.lu.se/matematiklu/personal/sigma/Riemann.pdf#page=22
R3 and the Riemann sphere Ĉ are diffeomorphic. Exercise 2.8. Find a proof of Proposition 2.24. Exercise 2.9. Let the spheres S1, <span class="highlight">S3</span> and the Lie groups SO(n), O(n), SU(n), U(n) be equipped with their standard differentiable structures introduced above. Use Proposition 2.24 to prove the fol- lowing diffeomorphisms S1 ∼= SO(2), <span class="highlight">S3</span> ∼= SU(2), SO(n)×O(1) ∼= O(n), SU(n)×U(1) ∼= U(n). Exercise 2.10. Find a proof of Corollary 2.28. Exercise 2.11. Let (G, ∗) and (H, ·) be two Lie groups. Prove that the product
NYTimes People: Watanabe, Junya
a Japanese designer who broke out from under the shadow of the towering presence of Rei Kawakubo, the creative force behind Comme des Garçons. A former protégé of Ms. Kawakubo and a knitwear designer, Mr. Watanabe launched his own line in 1993. Within years, his clothes...
Regents Exam Prep: Physics
C.Dynamics C.Magnetism C.Models of the Atom D.Momentum D.Electromagnetic Induction D.Standard Model of Particle Physics Videos Videos 2.Energy4.Waves Additional Review Resources: A.Work A.Wave Characteristics 1
science_pub2003
Vermont Department of Education (Science Grade Expectations) S19 Science — Scientific Inquiry: Vermont Standards and Evidences—Representing Data and Analysis 7.1 c, cc, d, dd, ddd, e, f, g; 1.17a, aa, aaa, b, bb, bbb, c, cc, ccc, d, dd; 1.18, 1.20; 2.2c, cc, d, dd Grades 3-4 Grades 5-...
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Vermont Department of Education (Science Grade Expectations) S19 Science — Scientific Inquiry: Vermont Standards and Evidences—Representing Data and Analysis 7.1 c, cc, d, dd, ddd, e, f, g; 1.17a, aa, aaa, b, bb, bbb, c, cc, ccc, d, dd; 1.18, 1.20; 2.2c, cc, d, dd Grades 3-4 Grades 5-6 Representing Data and Analysis S3-4:6 Students demonstrate their ability to ANALYZE DATA by… • Interpreting patterns or trends in data. AND • Relating data to the original question and prediction
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http://education.vermont.gov/new/pdfdoc/pubs/grade_expectations/science.pdf#page=12
education.vermont.gov/new/pdfdoc/pubs/grade_expectations/science.pdf#page...
Vermont Department of Education (Science Grade Expectations) S11 Science — Scientific Inquiry: Vermont Standards and Evidences—Scientific Questioning 7.1 a, aa, aaa; 2.1 a, b, c, d; Predicting and Hypothesizing 7.1 b, bb, bbb; Designing Experiments 7.1 c, cc, 7.2 a, aa, b, bb; 2.2 a, aa, aaa, b, bb, 3.10 Grades 3-4 Grades 5-6 Scientific Questioning <span class="highlight">S3</span>-4:1 Students demonstrate their understanding of SCIENTIFIC QUESTIONING by… • Identifying at least one variable that affects a system
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http://education.vermont.gov/new/pdfdoc/pubs/grade_expectations/science.pdf#page=16
education.vermont.gov/new/pdfdoc/pubs/grade_expectations/science.pdf#page...
Vermont Department of Education (Science Grade Expectations) S15 Science — Grades 3-4 Grades 5-6 Scientific Inquiry: Vermont Standards and Evidences—Conducting Experiments 7.1 c, cc, i; 7.2 c, cc, d; 2.2 e, ee, f, g; Rep- resenting Data and Analysis 7.1 c, cc, d, dd, ddd, e, f, g; 1.17a, aa, aaa, b, bb, bbb, c, cc, ccc, d, dd; 1.18, 1.20; 2.2c, cc, d, dd, 3.10, 3.11, 3.12, 3.13 Conducting Experiments <span class="highlight">S3</span>-4:4 Students demonstrate their ability to CONDUCT EXPERI- MENTS by… • Referring to and
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http://education.vermont.gov/new/pdfdoc/pubs/grade_expectations/science.pdf#page=20
education.vermont.gov/new/pdfdoc/pubs/grade_expectations/science.pdf#page...
Vermont Department of Education (Science Grade Expectations) S19 Science — Scientific Inquiry: Vermont Standards and Evidences—Representing Data and Analysis 7.1 c, cc, d, dd, ddd, e, f, g; 1.17a, aa, aaa, b, bb, bbb, c, cc, ccc, d, dd; 1.18, 1.20; 2.2c, cc, d, dd Grades 3-4 Grades 5-6 Representing Data and Analysis <span class="highlight">S3</span>-4:6 Students demonstrate their ability to ANALYZE DATA by… • Interpreting patterns or trends in data. AND • Relating data to the original question and prediction
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http://education.vermont.gov/new/pdfdoc/pubs/grade_expectations/science.pdf#page=24
education.vermont.gov/new/pdfdoc/pubs/grade_expectations/science.pdf#page...
Vermont Department of Education (Science Grade Expectations) S23 Science — Physical Science: Vermont Standards and Evidences—Properties of Matter 7.12 a, aa, aaa, b, bb, bbb, c, cc, ccc, 3.10, 3.11, 3.13 [See also connection with S: 14 Physical Science—Physical Change] Grades 3-4 Grades 5-6 Properties of Matter <span class="highlight">S3</span>-4:9 Students demonstrate their understanding of the Properties of Matter by… • Investigating and measuring how the total weight of the parts of a substance, no matter how they are
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http://education.vermont.gov/new/pdfdoc/pubs/grade_expectations/science.pdf#page=28
education.vermont.gov/new/pdfdoc/pubs/grade_expectations/science.pdf#page...
Vermont Department of Education (Science Grade Expectations) S27 Science — Physical Science: Vermont Standards and Evidences—Properties of Matter 7.12 a, aa, aaa, b, bb, bbb, c, cc, ccc, 3.10, 3.11, 3.13 [See also connection with S: 14 Physical Science—Physical Change] Grades 3-4 Grades 5-6 Properties of Matter <span class="highlight">S3</span>-4:12 Students demonstrate their understanding of the States of Matter by… • Identifying , describing and comparing the properties of selected solids, liquids and gases. Science
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http://education.vermont.gov/new/pdfdoc/pubs/grade_expectations/science.pdf#page=32
education.vermont.gov/new/pdfdoc/pubs/grade_expectations/science.pdf#page...
Vermont Department of Education (Science Grade Expectations) S31 Science — Physical Science: Vermont Standards and Evidences—Energy and Energy Transformation 7.12 b, bb, bbb; Chemical Change 7.12 b, bb, bbb, e, ee, eee, 2.2, 3.10, 3.11 [See also connection with S: 48 Universe, Earth, Environment—Weather] Grades 3-4 Grades 5-6 Properties of Matter <span class="highlight">S3</span>-4:14 Students demonstrate their understanding of Physical Change by… • Investigating and explaining what happens to liquids in open containers
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http://education.vermont.gov/new/pdfdoc/pubs/grade_expectations/science.pdf#page=44
education.vermont.gov/new/pdfdoc/pubs/grade_expectations/science.pdf#page...
Vermont Department of Education (Science Grade Expectations) S43 Science — Grades 3-4 Grades 5-6 Physical Science: Vermont Standards and Evidences—Force 7.12 d, dd, ddd, 2.2a, aa, aaa, 3.11 Force <span class="highlight">S3</span>-4:21 Students demonstrate their understanding of Force by… • Investigating and describing how different amounts of force can change the direction and speed of an object in motion. Science Concepts: a. Changes in speed or direction of motion are caused by forces. b. The greater the force
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find the least squares regression line by estimating visually and by calculating the equation of the regression line. Interpret the slope of the equation for a regression line. S2.2.2 Use the equation of the least squares regression line to make appropriate predictions. StanDarD S3: SaMPLeS,...
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find the least squares regression line by estimating visually and by calculating the equation of the regression line. Interpret the slope of the equation for a regression line. S2.2.2 Use the equation of the least squares regression line to make appropriate predictions. StanDarD S3: SaMPLeS, SurveyS, anD exPeriMentS Students understand and apply sampling and various sampling methods, examine surveys and experiments, identify bias in methods of conducting surveys, and learn strategies to minimize bias
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http://www.michigan.gov/documents/Math11-14-open1_142202_7.pdf#page=5
www.michigan.gov/documents/Math11-14-open1_142202_7.pdf#page=5
HIGH SCHOOL MATHEMATICS CONTENT EXPECTATIONS page 3 of 19 11/07 MICHIGAN DEPARTMENT OF EDUCATION STANDARDS (and number of core expectations in each standard) A1: Expressions, Equations, and Inequalities (16) A2: Functions (16) A3: Families of Functions (27) G1: Figures and Their Properties (29) G2: Relationships Between Figures (10) G3: Transformations of Figures in the Plane (5) S1: Univariate Data- Examining Distributions (9) S2: Bivariate Data— Examining Relationships (6) <span class="highlight">S3</span>: Samples
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http://www.michigan.gov/documents/Math11-14-open1_142202_7.pdf#page=20
www.michigan.gov/documents/Math11-14-open1_142202_7.pdf#page=20
find the least squares regression line by estimating visually and by calculating the equation of the regression line. Interpret the slope of the equation for a regression line. S2.2.2 Use the equation of the least squares regression line to make appropriate predictions. StanDarD <span class="highlight">S3</span>: SaMPLeS, SurveyS, anD exPeriMentS Students understand and apply sampling and various sampling methods, examine surveys and experiments, identify bias in methods of conducting surveys, and learn strategies to minimize bias
Standard 1: Students engage in the research process
North Dakota English Language Arts April 2005 Content and Achievement Standards 101 Standard 5: Students understand media. PROFICIENCY DESCRIPTOR Benchmark Expectations ADVANCED PROFICIENT PROFICIENT PARTIALLY PROFICIENT NOVICE USING MEDIA FOR A PURPOSE 9.5.2. Access media (e.g., television, film,...
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North Dakota English Language Arts April 2005 Content and Achievement Standards 101 Standard 5: Students understand media. PROFICIENCY DESCRIPTOR Benchmark Expectations ADVANCED PROFICIENT PROFICIENT PARTIALLY PROFICIENT NOVICE USING MEDIA FOR A PURPOSE 9.5.2. Access media (e.g., television, film, music, electronic databases, videos, DVDs, comics, visual and performing arts, newspapers, and periodicals
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http://www.dpi.state.nd.us/standard/content/ELA/ELA.pdf#page=100
www.dpi.state.nd.us/standard/content/ELA/ELA.pdf#page=100
North Dakota English Language Arts April 2005 Content and Achievement Standards 100 Standard 5: Students understand media. PROFICIENCY DESCRIPTOR Benchmark Expectations ADVANCED PROFICIENT PROFICIENT PARTIALLY PROFICIENT NOVICE USING MEDIA FOR A PURPOSE 8.5.2. Access media (e.g., television, film, music, electronic databases, <span class="highlight">videos</span>, DVDs, comics, visual and performing arts, newspapers, and periodicals
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http://www.dpi.state.nd.us/standard/content/ELA/ELA.pdf#page=101
www.dpi.state.nd.us/standard/content/ELA/ELA.pdf#page=101
North Dakota English Language Arts April 2005 Content and Achievement Standards 101 Standard 5: Students understand media. PROFICIENCY DESCRIPTOR Benchmark Expectations ADVANCED PROFICIENT PROFICIENT PARTIALLY PROFICIENT NOVICE USING MEDIA FOR A PURPOSE 9.5.2. Access media (e.g., television, film, music, electronic databases, <span class="highlight">videos</span>, DVDs, comics, visual and performing arts, newspapers, and periodicals
Microsoft Word - sci_standards_Aug07b.docsci_standards_Aug07b.pdf
? 2. reads short stories and views films or videos; listens to guest speakers who are involved in science. Teacher Notes: This standard should be integrated with physical science, life science, and earth and space science standards. Technology – application of knowledge through inv...
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? 2. reads short stories and views films or videos; listens to guest speakers who are involved in science. Teacher Notes: This standard should be integrated with physical science, life science, and earth and space science standards. Technology – application of knowledge through inventions.
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http://www.ksde.org/LinkClick.aspx?fileticket=YgH0PnTTzS4%3d&tabid=144&mid=8019&forcedownload=true#page=25
www.ksde.org/LinkClick.aspx?fileticket=YgH0PnTTzS4%3d&tabid=144&mid=8019&...
? 2. reads short stories and views films or <span class="highlight">videos</span>; listens to guest speakers who are involved in science. Teacher Notes: This standard should be integrated with physical science, life science, and earth and space science standards. Technology – application of knowledge through inventions.
English Language Arts Content Standards - Curriculum Frameworks (CA Dept of Education)
conventional style for that type of document (e.g., résumé, memorandum) and use page formats, fonts, and spacing that contribute to the readability and impact of the document. 2.6 Deliver multimedia presentations: a. Combine text, images, and sound and draw information from many source...
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conventional style for that type of document (e.g., résumé, memorandum) and use page formats, fonts, and spacing that contribute to the readability and impact of the document. 2.6 Deliver multimedia presentations: a. Combine text, images, and sound and draw information from many sources (e.g., television broadcasts, videos, films, newspapers, magazines, CD-ROMs, the Internet, electronic media-generated images). b. Select an appropriate medium for each element of the presentation. c. Use the selected media
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http://www.cde.ca.gov/be/st/ss/documents/elacontentstnds.pdf#page=79
www.cde.ca.gov/be/st/ss/documents/elacontentstnds.pdf#page=79
conventional style for that type of document (e.g., résumé, memorandum) and use page formats, fonts, and spacing that contribute to the readability and impact of the document. 2.6 Deliver multimedia presentations: a. Combine text, images, and sound and draw information from many sources (e.g., television broadcasts, <span class="highlight">videos</span>, films, newspapers, magazines, CD-ROMs, the Internet, electronic media-generated images). b. Select an appropriate medium for each element of the presentation. c. Use the selected media
Microsoft Word - sci_std_Aug07_k2.docsci_std_Aug07_k2.pdf
? 2. reads short stories and views films or videos; listens to guest speakers who are involved in science. Teacher Notes: This standard should be integrated with physical science, life science, and earth and space science standards. Technology – application of knowledge through inv...
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? 2. reads short stories and views films or videos; listens to guest speakers who are involved in science. Teacher Notes: This standard should be integrated with physical science, life science, and earth and space science standards. Technology – application of knowledge through inventions.
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http://www.ksde.org/LinkClick.aspx?fileticket=ePaWqyOd2Ko%3d&tabid=144&mid=8019&forcedownload=true#page=10
www.ksde.org/LinkClick.aspx?fileticket=ePaWqyOd2Ko%3d&tabid=144&mid=8019&...
? 2. reads short stories and views films or <span class="highlight">videos</span>; listens to guest speakers who are involved in science. Teacher Notes: This standard should be integrated with physical science, life science, and earth and space science standards. Technology – application of knowledge through inventions.
Academic Standards-English/Language Arts Grade 12
personal property pursuant to the provisions of I.C. 29-1-15-8. The same sentence without legal language would say: As the person appointed to handle the estate of someone who has died, I am asking the court for permission to sell some property that person owned. 12.5.8 Deliver multimedia presentati...
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personal property pursuant to the provisions of I.C. 29-1-15-8. The same sentence without legal language would say: As the person appointed to handle the estate of someone who has died, I am asking the court for permission to sell some property that person owned. 12.5.8 Deliver multimedia presentations that: • combine text, images, and sound and draw information from many sources, including television broadcasts, videos, films, newspapers, magazines, CD-ROMs, the Internet, and electronic media- generated images
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http://dc.doe.in.gov/Standards/AcademicStandards/PrintLibrary/docs-english/2006-06-ela-grade12.pdf#page=7
dc.doe.in.gov/Standards/AcademicStandards/PrintLibrary/docs-english/2006-...
personal property pursuant to the provisions of I.C. 29-1-15-8. The same sentence without legal language would say: As the person appointed to handle the estate of someone who has died, I am asking the court for permission to sell some property that person owned. 12.5.8 Deliver multimedia presentations that: • combine text, images, and sound and draw information from many sources, including television broadcasts, <span class="highlight">videos</span>, films, newspapers, magazines, CD-ROMs, the Internet, and electronic media- generated images
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