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How to Juggle the Proof for Every Theorem an Exploration of the Mathematics Behind Juggling
How to juggle the proof for every theorem An exploration of the mathematics behind juggling Mees Jager 5965802 (0101) 3 0 4 1 2 (1100) (1010) 1 4 4 3 (1001) (0011) 2 0 0 (0110) Supervised by Gil Cavalcanti Contents 1 Abstract 2 2 Preface 4 3 Preliminaries 5 3.1 Conventions and notation . .5 3.2 A mathematical description of juggling . .5 4 Practical problems with mathematical answers 9 4.1 When is a sequence jugglable? . .9 4.2 How many balls? . 13 5 Answers only generate more questions 21 5.1 Changing juggling sequences . 21 5.2 Constructing all sequences with the Permutation Test . 23 5.3 The converse to the average theorem . 25 6 Mathematical problems with mathematical answers 35 6.1 Scramblable and magic sequences . 35 6.2 Orbits . 39 6.3 How many patterns? . 43 6.3.1 Preliminaries and a strategy . 43 6.3.2 Computing N(b; p).................... 47 6.3.3 Filtering out redundancies . 52 7 State diagrams 54 7.1 What are they? . 54 7.2 Grounded or Excited? . 58 7.3 Transitions . 59 7.3.1 The superior approach . 59 7.3.2 There is a preference . 62 7.3.3 Finding transitions using the flattening algorithm . 64 7.3.4 Transitions of minimal length . 69 7.4 Counting states, arrows and patterns . 75 7.5 Prime patterns . 81 1 8 Sometimes we do not find the answers 86 8.1 The converse average theorem . 86 8.2 Magic sequence construction . 87 8.3 finding transitions with flattening algorithm . -
Yosemite Valley Hiking Map U.S
Yosemite National Park National Park Service Yosemite Valley Hiking Map U.S. Department of the Interior To To ) S k Tioga n Tioga m e To o e k w r Road 10 Shuttle Route / Stop Road 7 Tioga . C Ranger Station C 4 n 3.I mi (year round) 6.9 mi ( Road r e i o 5.0 km y I e II.I km . 3.6 mi m n 6 k To a 9 m 5.9 km 18 Shuttle Route / Stop . C Self-guiding Nature Trail Tioga North 0 2 i Y n ( . o (summer only) 6 a Road 2 i s . d 6 m e 5.0 mi n m k i I Trailhead Parking ( 8.0 km m Bicycle / Foot Path I. it I.3 0 e ) k C m (paved) m re i ( e 2 ) ) k . Snow I Walk-in Campground m k k m Creek Hiking Trail .2 k ) Falls 3 Upper e ( e Campground i r Waterfall C Yosemite m ) 0 Fall Yosemite h I Kilometer . c r m 2 Point A k Store l 8 6936 ft . a ) y 0 2II4 m ( m I Mile o k i R 9 I. m ( 3. i 2 5 m . To Tamarack Flat North m i Yosemite Village 0 Lower (5 .2 Campground . I I Dome 2.5 mi Yosemite k Visitor Center m 7525 ft 0 Fall 3.9 km ) 2294 m . 3 k m e Cre i 2.0 mi Lower Yosemite Fall Trail a (3 To Tamarack Flat ( Medical Royal Mirror .2 0 y The Ahwahnee a m) k . -
Gunnar's Arrival in the Golden State a Painting That Has Always Intrigued
April 2014 It was a very busy winter for me, which delayed this edition of the newsletter. Last year was an extremely productive year for work on the Gunnar Widforss project. I added many new works to the catalgoue raisonné and located a wealth of materials pertaining to Gunnar’s life and work, including letters written by the artist to his family. Th e highlight of my year was meeting the current members of Gunnar’s family in Sweden last summer. I spent six weeks in Sweden, Norway, Denmark, and Germany conducting research and photographing paintings for the catalogue. My work was again supported by a generous grant from the Barbro Osher Pro Suecia Foundation and I would like to extend a very grateful note of appreciation for their continued support. My sabbatical leave for teaching at Coconino Community College also continued through June and this allowed lots of time to devote to the project. 1921 - Gunnar’s Arrival in the Golden State Gunnar Widforss (far right center) at the Mt. Lowe Railway, January 1921. Gunnar may have made the painting. Climbing one of the trails to the site of an old inn and tavern I was able to fi nd the exact spot where Widforss made the painting. Th e view in the painting is west towards the Verdugo Mountains, Burbank, and the Santa Monica Mountains. In letters to his mother from Los Angeles Gunnar mentions painting at Mt. Lowe for a couple of weeks. California Scene, or View from Mt. Lowe, was one the very fi rst works he painted following his arrival in the United States in January 1921. -
IJA Enewsletter Editor Don Lewis (Email: [email protected]) Renew At
THE INTERNATIONAL JUGGLERS’ ASSOCIATION June 2015 IJA eNewsletter editor Don Lewis (email: [email protected]) Renew at http://www.juggle.org/renew IJA eNewsletter IJA Festival July 20 - 26, 2015 Quebec City, QC, Canada Register online soon! Discounts on Contents: Event Packages end June 30! IJA Pre-Reg Deadline Only Days Away Full info is on our website: Chairman’s Message www.juggle.org/festival IJA Election - New, Vote Online! Candidates’ Statements After June 30th, Stage Championships Finalists Register in Person at the Festival IJA Festival Information Online IJA Fest’s Special Guests See fest details starting on page 4, Festival Checklist where the Championships Finalists are listed! WJD shirts, YJA badges in IJA Store What’s New at eJuggle Coming Soon to eJuggle... Juggling Festivals Juggling Festivals: Lincolnshire, UK Eugene, OR Quebec City, Quebec, Canada (IJA) Collinée en Bretagne, France Bruneck, South Tyrol, Italy (EJC) Montpeyroux, France Garsington, Oxfordshire, UK Cleveland, OH Portland, OR Kansas City, MO Philadelphia, PA Fukushima, Japan Ottumwa, IA WWW.JUGGLE.ORG Page 1 THE INTERNATIONAL JUGGLERS’ ASSOCIATION June 2015 Chairman’s Message, by Nathan Wakefield - Obstacle course: $500 - Waterballoon slip and slide: $200 - Drinks and flair bartender: $200 - Onsite massage therapist: $1,000 - Cardboard box castle building contest: $60 - Pinata filled with juggling props: $250 - Tye Dye $60 "To render assistance to fellow jugglers." - Food. $1,630 and the remainder of any additional funds. Special thanks to donor Unna Med and all those who Less than one month until the 2015 IJA Festival in contributed towards this fund of awesomeness! Quebec City! It's been a long road of hard work for our festival team If logistics is an issue for you, we have rideboards and officers, but everything is in place for this year's available on both our festival forum and on Facebook. -
Juggling – 15013000/15013100 HOPE PE 1506320G
Course: Juggling – 15013000/15013100 HOPE PE 1506320G Credit for Graduation: 1.0 Credit – HOPE elective credit / Physical Education Credit Pre-requisite: Desire to explore and develop the skill set needed for juggling. Description: Expectations: The purpose of this course is to provide students Students will be expected to embrace the many with the knowledge, skills, and values they need to challenges involved in developing a unique skill become healthy and physically active for a lifetime. set. In addition to developing skills with assorted This course addresses both the health and skill- props, students will be expected to learn about related components of physical fitness which are the greater juggling community, learning theory, critical for students' success. The SAIL juggling program exists to promote self-expression and to prop building, routine development etc. encourage creativity. Students will have the Participation and movement are key components opportunity to form new friendships and develop for being successful in this course. skills that will last a lifetime. In addition to conventional ball, club and ring juggling, students will be exposed to a plethora of props likely including but not limited to unicycle, cigar box manipulation, rolla bolla, contact juggling, card throwing, throw top, yo-yo, diabolo, hacky sack, passing, kendama and ball spinning. Resources/Materials: Assorted jugging equipment Website: https://www.leonschools.net/Domain/2453 Course: Juggling 15013000 / 15013000 HOPE PE: #15063200 Credit for Graduation: 1.0 Credit - HOPE elective credit Pre-requisite: Description: Expectations: HOPE PE: The purpose of this course is to develop and enhance healthy behaviors that influence lifestyle choices and student health and fitness. -
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Cairns, John William (1976) The strength of lapped joints in reinforced concrete columns. PhD thesis. http://theses.gla.ac.uk/1472/ Copyright and moral rights for this thesis are retained by the author A copy can be downloaded for personal non-commercial research or study, without prior permission or charge This thesis cannot be reproduced or quoted extensively from without first obtaining permission in writing from the Author The content must not be changed in any way or sold commercially in any format or medium without the formal permission of the Author When referring to this work, full bibliographic details including the author, title, awarding institution and date of the thesis must be given Glasgow Theses Service http://theses.gla.ac.uk/ [email protected] THE STRENGTHOF LAPPEDJOINTS IN REINFORCEDCONCRETE COLUMNS by J. Cairns, B. Sc. A THESIS PRESENTEDFOR THE DEGREEOF DOCTOROF PHILOSOPHY Y4 THE UNIVERSITY OF GLASGOW by JOHN CAIRNS, B. Sc. October, 1976 3 !ý I CONTENTS Page ACKNOWLEDGEIvIENTS 1 SUMMARY 2 NOTATION 4 Chapter 1 INTRODUCTION 7 Chapter 2 REVIEW OF LITERATURE 2.1 Bond s General 10 2.2 Anchorage Tests 12" 2.3 Tension Lapped Joints 17 2.4 Compression Lapped Joints 21 2.5 End Bearing 26 2.6 Current Codes of Practice 29 Chapter 3 THEORETICALSTUDY OF THE STRENGTHOF LAPPED JOINTS 3.1 Review of Theoretical Work on Bond Strength 32 3.2 Theoretical Studies of Ultimate Bond Strength 33 3.3 Theoretical Approach 42 3.4 End Bearing 48 3.5 Bond of Single Bars Surrounded by a Spiral 51 3.6 Lapped Joints 53 3.7 Variation of Steel Stress Through a Lapped Joint 59 3.8 Design of Experiments 65 Chapter 4 DESCRIPTION OF EXPERIMENTALWORK 4.1 General Description of Test Specimens 68 4.2 Materials 72 4.3 Fabrication of Test Specimens 78 4.4 Test Procedure 79 4.5 Instrumentation 80 4.6 Push-in Test Specimens 82 CONTENTScontd. -
Yosemitenational Park
THE MADDEN HOUSE Welcome to Yosemite national park Yosemite AHWAHNEE the local Native Americans. Some tribes violently INTO THE WILDERNESS Before it was Yosemite, the area protested the move and the The park is 747, 956 was originally known as battalion responded in force acres, almost 1,169 Ahwahnee, named after the square miles, with 95% to remove them from the land. Ahwahnee Indian tribe who of the park being The battalion who entered the designated as settled in that area. valley made record of their wilderness. Ahwanee translates to “big breathtaking surroundings. As mouth” in english, referring to word of the park’s natural BEAUTIFUL WATERFALLS the Yosemite Valley, with its beauty spread, tourists began Yosemite is home to 21 expansive views of the tall arrive. The park is named in waterfalls, with some mountains on both sides of honor of one of the area only existing during tribes, the Yosemite. peak seasons of the valley. In the 1800s, the gold rush made the area a snowmelt. The Ahwahnee tribe is an popular destination. The important piece of Yosemite LUSH VEGETATION sudden increase in people history. As you wandered the flooding into the area for the The national park has park, did you see any five vegetation zones; gold rush bred tensions with references to this tribe? Did chaparral, lower the Ahwahnee, who did not montane forrest, upper you learn anything new about want to share their land. With montane forest, them? subalpine, and alpine. increased government interest in the area, the Mariposa ___________________________ Battalion was sent to relocate ___________________________ Yosemite | www.themaddenhouse.com | Page !1 THE YOSEMITE GRANT As a means of early protection of the area, a The protections given to the area by the Yosemite Grant were given to California to bill was passed in 1864 by oversee on the state level and managed by a President Abraham Lincoln government commission. -
Yosemite Forest Dynamics Plot
REFERENCE COPY - USE for xeroxing historic resource siuay VOLUME 3 OF 3 discussion of historical resources, appendixes, historical base maps, bibliography YOSEMITE NATIONAL PARK / CALIFORNIA Historic Resource Study YOSEMITE: THE PARK AND ITS RESOURCES A History of the Discovery, Management, and Physical Development of Yosemite National Park, California Volume 3 of 3 Discussion of Historical Resources, Appendixes, Historical Base Maps, Bibliography by Linda Wedel Greene September 1987 U.S. Department of the Interior / National Park Service b) Frederick Olmsted's Treatise on Parks ... 55 c) Significance of the Yosemite Grant .... 59 B. State Management of the Yosemite Grant .... 65 1. Land Surveys ......... 65 2. Immediate Problems Facing the State .... 66 3. Settlers' Claims ........ 69 4. Trails ........%.. 77 a) Early Survey Work ....... 77 b) Routes To and Around Yosemite Valley ... 78 c) Tourist Trails in the Valley ..... 79 (1) Four-Mile Trail to Glacier Point ... 80 (2) Indian Canyon Trail ..... 82 (3) Yosemite Fall and Eagle Peak Trail ... 83 (4) Rim Trail, Pohono Trail ..... 83 (5) Clouds Rest and Half (South) Dome Trails . 84 (6) Vernal Fall and Mist Trails .... 85 (7) Snow Trail ....... 87 (8) Anderson Trail ....... (9) Panorama Trail ....... (10) Ledge Trail 89 5. Improvement of Trails ....... 89 a) Hardships Attending Travel to Yosemite Valley . 89 b) Yosemite Commissioners Encourage Road Construction 91 c) Work Begins on the Big Oak Flat and Coulterville Roads ......... 92 d) Improved Roads and Railroad Service Increase Visitation ......... 94 e) The Coulterville Road Reaches the Valley Floor . 95 1) A New Transportation Era Begins ... 95 2) Later History 99 f) The Big Oak Flat Road Reaches the Valley Floor . -
Siteswap-Notes-Extended-Ltr 2014.Pages
Understanding Two-handed Siteswap http://kingstonjugglers.club/r/siteswap.pdf Greg Phillips, [email protected] Overview Alternating throws Siteswap is a set of notations for describing a key feature Many juggling patterns are based on alternating right- of juggling patterns: the order in which objects are hand and lef-hand throws. We describe these using thrown and re-thrown. For an object to be re-thrown asynchronous siteswap notation. later rather than earlier, it needs to be out of the hand longer. In regular toss juggling, more time out of the A1 The right and lef hands throw on alternate beats. hand means a higher throw. Any pattern with different throw heights is at least partly described by siteswap. Rules C2 and A1 together require that odd-numbered Siteswap can be used for any number of “hands”. In this throws end up in the opposite hand, while even guide we’ll consider only two-handed siteswap; numbers stay in the same hand. Here are a few however, everything here extends to siteswap with asynchronous siteswap examples: three, four or more hands with just minor tweaks. 3 a three-object cascade Core rules (for all patterns) 522 also a three-object cascade C1 Imagine a metronome ticking at some constant rate. 42 two juggled in one hand, a held object in the other Each tick is called a “beat.” 40 two juggled in one hand, the other hand empty C2 Indicate each thrown object by a number that tells 330 a three-object cascade with a hole (two objects) us how many beats later that object must be back in 71 a four-object asynchronous shower a hand and ready to re-throw. -
Understanding Siteswap Juggling Patterns a Guide for the Perplexed by Greg Phillips, [email protected]
Understanding Siteswap Juggling Patterns A guide for the perplexed by Greg Phillips, [email protected] Core rules (for all patterns) Synchronous siteswap (simultaneous throws) C1 We imagine a metronome ticking at some constant rate. Some juggling patterns involve both hands throwing at the Each tick is called a “beat.” same time. We describe these using synchronous siteswap. C2 We indicate each thrown object by a number that tells us how many beats later that object must be back in a hand S1 The right and left hands throw at the same time (which and ready to re-throw. counts as two throws) on every second beat. We group C3 We indicate a sequence of throws by a string of numbers the simultaneous throws in parentheses and separate the (plus punctuation and the letter x). We imagine these hands with commas. strings repeat indefinitely, making 531 equivalent to S2 We indicate throws that cross from hand to hand with an …531531531… So are 315 and 153 — can you see x (for ‘xing’). why? C4 The sum of all throw numbers in a siteswap string divided Rules C2 and S1 taken together demand that there be only by the number of throws in the string gives the number of even numbers in valid synchronous siteswaps. Can you see objects in the pattern. This must always be a whole why? The x notation of rule S2 is required to distinguish number. 531 is a three-ball pattern, since (5+3+1)/3 patterns like (4,4) (synchronous fountain) from (4x,4x) equals three; however, 532 cannot be juggled since (synchronous crossing). -
BEDROCK GEOLOGY of the YOSEMITE VALLEY AREA YOSEMITE NATIONAL PARK, CALIFORNIA Prepared by N
DEPARTMENT OF THE INTERIOR TO ACCOMPANY MAP I-1639 U.S. GEOLOGICAL SURVEY BEDROCK GEOLOGY OF THE YOSEMITE VALLEY AREA YOSEMITE NATIONAL PARK, CALIFORNIA Prepared by N. King Huber and Julie A. Roller From the writings of Frank C. Calkins and other sources PREFACE INTRODUCTION Frank Calkins' work in Yosemite was preceded by Yosemite Valley, one of the world's great natural works Henry W. Turner, also of the U.S. Geological Survey, of rock sculpture, is carved into the west slope of the who began mapping the Yosemite and Mount Lyell 30- Sierra Nevada. Immense cliffs, domes, and waterfalls minute quadrangles in 1897 and laid the foundation that tower over forest, meadows, and a meandering river, Calkins' work was built on. Although Turner never creating one of the most scenic natural landscapes in completed this sizable assignment, he recognized the North America (fig. 1). In Yosemite Valley and the differing types of plutonic rocks and, for example, named adjoining uplands, the forces of erosion have exposed, the El Capitan Granite. with exceptional clarity, a highly complex assemblage of Calkins mapped the valley and adjacent areas of granitic rocks. The accompanying geologic map shows the Yosemite National Park during the period 1913 through distribution of some of the different rocks that make up 1916, at the same time that Francois Matthes was this assemblage. This pamphlet briefly describes those studying the glacial geology of Yosemite. Calkins rocks and discusses how they differ, both in composition summarized the bedrock geology of part of Yosemite in and structure, and the role they played in the evolution the appendix of Matthes' classic volume "Geologic History of the valley. -
John Muir Writings Picturesque California and the Region West of the Rocky Mountains, from Alaska to Mexico (1888-1890) Edited by John Muir
John Muir Writings Picturesque California and the Region West of the Rocky Mountains, from Alaska to Mexico (1888-1890) Edited by John Muir Contents 1. Peaks and Glaciers of the High Sierra (by John Muir) 2. The Passes of the High Sierra (by John Muir) 3. A Glimpse of Monterey (by J. R. Fitch) 4. The Yosemite Valley (by John Muir) 5. Totokonulu (poem by J. Vance Cheney) 6. A Visit to the Lick Observatory (by Edward S. Holden) 7. About the Bay of San Francisco (by W. C. Bartlett) 8. Game Regions of the Upper Sacramento (by Joaquin Miller) 9. The Heart of Southern California (by Jeanne C. Carr) 10. Mount Shasta (by John Muir) 11. Southernmost California (by T. S. Van Dyke) 12. North of the Golden Gate (by Kate Field) 13. Early California Mining and the Argonauts (by Joaquin Miller) 14. The Santa Clara Valley and Santa Cruz Mountains (by George Hamlin Fitch) 15. The San Joaquin Valley (by Joaquin Miller) 16. The Foothill Region of the Northern Coast Range (by Charles Howard Shinn) 17. The New City by the Great Sea—San Francisco (by Joaquin Miller) 18. On the Height (poem by J. Vance Cheney) 19. The Sacramento Valley (by John P. Irish) 20. The Tule Region (by Charles Howard Shinn) 21. Monterey to Ventura (by WIlliam L. Ogle) 22. The Plain of Oaks (by Albert E. Gray) 23. The Land of the Redwoods (by Charles Howard Shinn) 24. Washington and the Puget Sound (by John Muir) 25. The Basin of the Columbia River (by John Muir) 26.