PPT-Equiangular Lines in R

d Moshe Rosenfeld University of Washington Shanghai Jiao Tong University July 1 2013 How wide and how even can you spread your chopsticks Equiangular Lines Definition A set of lines through the origin in R. Tropp Mathematics Department The University of Michigan 530 Church St Ann Arbor MI 481091043 USA ABSTRACT A complex equiangular tight frame ETF is a tight frame cons isting of unit vectors in whose absolute inner products are identical One may Tropp Institute for Computational Engineering and Sciences The University of Texas at Austin Inderjit S Dhillon and Robert W Heath Jr The University of Texas at Austin Thomas Strohmer The University of California at Davis brPage 2br Equiangular Tigh Mixon and Janet C Tremain Department of Mathematics and Statistics Air Force Institute of Technology WrightPatterson AFB OH Program in Applied and Computational Mathematics Princeton University Princeton NJ Department of Mathematics University of Mi Casazza Department of Mathematics The University of Missouri Columbia Missouri 652114100 Email petemathmissouriedu Dan Redmond Department of Mathematics The University of Missouri Columbia Missouri 652114100 Email dredmondmathmissouriedu Janet C Tre Introduction to Drafting and Design. What is the Alphabet of Lines?. Developed by the American Society of Mechanical Engineers (ASME).. The Alphabet of Lines is used to make a drawing neater and clearer to understand.. Classifying Triangles by Angles. ACUTE. OBTUSE. RIGHT. EQUIANGULAR. ACUTE TRIANGLE. All interior angles are acute (or have a measure less than 90°). Interior Angle. Example of Acute Triangle. Phineas’s. Robert Minchin. Spectral Lines. What is a Spectral Line?. Types of Spectral Line. Molecular Transitions. Linewidths. Reference Frames. When a Coconut falls…. Bang!. h. Energy released = . mgh. When an electron falls…. Warm-up: Quote. “. A line is a dot that went for a walk.” ~Paul Klee. Create a sketch that illustrates the quote.. Elements of Art: . Line. Joseph Stella. . The Voice of the City of New York Interpreted: The Bridge.. Q: What types of lines do you see?. Artist info:. Joan . Miró. Spanish, 1893–1983. The Farm. , 1921–1922. oil on canvas, 123.8 x 141.3 x 3.3 cm (48 3/4 x 55 5/8 x 1 5/16 in.). National Gallery of Art, Gift of Mary Hemingway. Many applications of statistical analysis involves a continuous variable as dependent variable (DV) but both continuous and categorical variables as independent variables (IV). . Relationship between DV and continuous IVs is linear and the slope remains the same in different groups: ANCOVA.. Pgs. 26, 28, 30. Warmup. (pg. 23) . The measures of 2 Vertical Angles are 90 and (5x 10). Find the value of x.. The measure of an angle is twice the measure of its compliment. . What’s a Transversal? . Warm-up: Quote. “. A line is a dot that went for a walk.” ~Paul Klee. Create a sketch that illustrates the quote.. Elements of Art: . Line. Joseph Stella. . The Voice of the City of New York Interpreted: The Bridge.. ALPHABET. OF . LINES. VISABLE (OBJECT) LINES. THICK, SOLID LINES. SHOWS THE OUTLINE OF THE OBJECT . VISIBLE TO THE EYE. HIDDEN (INVISIBLE) LINES. SHORT, EVENLY SPACED DASHES. REPRESENT SURFACES THAT ARE NOT DIRECTLY VISIBLE TO THE EYE.. i.e. . they do not meet).. What is the shortest distance between them?.  .  .  .  . Also find the co-ordinates of . and . ..  . shortest distance. must be perpendicular to both lines.. Let . Then .