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Micro-droplets are widely used in the fields of chemical and biological research, such as drug delivery, material synthesis, point-of-care diagnostics, and digital PCR. Droplet-based microfluidics has many advantages, such as small reagent consumption, fast reaction time, and independent control of each droplet. Therefore, various micro-droplet generation methods have been proposed, including T-junction breakup, capillary flow-focusing, planar flow-focusing, step emulsification, and high aspect (height-to-width) ratio confinement. In this study, we propose a microfluidic device for generating monodisperse micro-droplets, the microfluidic channel of which has an asymmetric cross-sectional shape and high hypotenuse-to-width ratio (HTWR). It was fabricated using basic MEMS processes, such as photolithography, anisotropic wet etching of Si, and polydimethylsiloxane (PDMS) molding. Due to the geometric similarity of a Si channel and a PDMS mold, both of which were created through the anisotropic etching process of a single crystal Si, the microfluidic channel with the asymmetric cross-sectional shape and high HTWR was easily realized. The effects of HTWR of channels on the size and uniformity of generated micro-droplets were investigated. The monodisperse micro-droplets were generated as the HTWR of the asymmetric channel was over 3.5. In addition, it was found that the flow direction of the oil solution (continuous phase) affected the size of micro-droplets due to the asymmetric channel structures. Two kinds of monodisperse droplets with different sizes were successfully generated for a wider range of flow rates using the asymmetric channel structure in the developed microfluidic device.

We study the dissection of a square into congruent convex polygons. Yuan et al. [Dissecting the square into five congruent parts, Discrete Math. 339 (2016) 288-298] asked whether, if the number of tiles is a prime number$\\geq 3$ , it is true that the tile must be a rectangle. We conjecture that the same conclusion still holds even if the number of tiles is an odd number$\\geq 3$ . Our conjecture has been confirmed for triangles in earlier works. We prove that the conjecture holds if either the tile is a convex$q$ -gon with$q\\geq 6$or it is a right-angle trapezoid. Comment: 19 pages, 11 figure

Spherical trigonometry was at the heart of astronomy and ocean-going navigation for two millennia. The discipline was a mainstay of mathematics education for centuries, and it was a standard subject in high schools until the 1950s. Today, however, it is rarely taught.Heavenly Mathematicstraces the rich history of this forgotten art, revealing how the cultures of classical Greece, medieval Islam, and the modern West used spherical trigonometry to chart the heavens and the Earth. Glen Van Brummelen explores this exquisite branch of mathematics and its role in ancient astronomy, geography, and cartography; Islamic religious rituals; celestial navigation; polyhedra; stereographic projection; and more. He conveys the sheer beauty of spherical trigonometry, providing readers with a new appreciation for its elegant proofs and often surprising conclusions. Heavenly Mathematicsis illustrated throughout with stunning historical images and informative drawings and diagrams that have been used to teach the subject in the past. This unique compendium also features easy-to-use appendixes as well as exercises at the end of each chapter that originally appeared in textbooks from the eighteenth to the early twentieth centuries.

Based on extensive research in Sanskrit sources, Mathematics in India chronicles the development of mathematical techniques and texts in South Asia from antiquity to the early modern period. Kim Plofker reexamines the few facts about Indian mathematics that have become common knowledge--such as the Indian origin of Arabic numerals--and she sets them in a larger textual and cultural framework. The book details aspects of the subject that have been largely passed over in the past, including the relationships between Indian mathematics and astronomy, and their cross-fertilizations with Islamic scientific traditions. Plofker shows that Indian mathematics appears not as a disconnected set of discoveries, but as a lively, diverse, yet strongly unified discipline, intimately linked to other Indian forms of learning.

本研究旨在瞭解自行閱讀與閱讀不同編排順序的文本對國一學生有關勾股定理的概念、程序與解題表現之影響。本研究編擬有關勾股定理的「證明優先」與「應用優先」文本，這兩種文本的內容相同但編排順序不同；以及編製概念、程序與解題表現前後測試卷。在蒐集前後測資料後，主要以多變量與單變量變異數分析處理實驗資料。研究結果顯示，閱讀不同文本與學生程度之間存在顯著的交互作用；而且並非只要透過自行閱讀文本（測驗時也未收回文本），就一定有利於提升高、中、低三種程度的學生在概念、程序和解題等數學能力的測驗成績。 The purpose of this study aims at understanding the effects of reading texts by oneself with different layouts on 7th graders’ mathematical performance about “Pythagorean Theorem.” This study designed “proof-first” and “application-first” texts, and instruments for testing mathematical performance about “Pythagorean Theorem.” After collecting pre-test and post-test data, the statistical method of ANCOVA was mainly used to analyze experimental data. The results showed that the interaction between text layouts and students’ mathematical abilities was statistically significant, and that students, who read a mathematical text by themselves as testing in mathematics related to the text, did not necessarily perform better than students, who did not read the mathematical text by themselves as testing.

本研究旨在瞭解自行閱讀與閱讀不同編排順序的文本對國一學生有關勾股定理的概念、程序與解題表現之影響。本研究編擬有關勾股定理的「證明優先」與「應用優先」文本，這兩種文本的內容相同但編排順序不同；以及編製概念、程序與解題表現前後測試卷。在蒐集前後測資料後，主要以多變量與單變量變異數分析處理實驗資料。研究結果顯示，閱讀不同文本與學生程度之間存在顯著的交互作用；而且並非只要透過自行閱讀文本（測驗時也未收回文本），就一定有利於提升高、中、低三種程度的學生在概念、程序和解題等數學能力的測驗成績。

Despite what we may sometimes imagine, popular mathematics writing didn't begin with Martin Gardner. In fact, it has a rich tradition stretching back hundreds of years. This entertaining and enlightening anthology--the first of its kind--gathers nearly one hundred fascinating selections from the past 500 years of popular math writing, bringing to life a little-known side of math history. Ranging from the late fifteenth to the late twentieth century, and drawing from books, newspapers, magazines, and websites,A Wealth of Numbersincludes recreational, classroom, and work mathematics; mathematical histories and biographies; accounts of higher mathematics; explanations of mathematical instruments; discussions of how math should be taught and learned; reflections on the place of math in the world; and math in fiction and humor. Featuring many tricks, games, problems, and puzzles, as well as much history and trivia, the selections include a sixteenth-century guide to making a horizontal sundial; \"Newton for the Ladies\" (1739); Leonhard Euler on the idea of velocity (1760); \"Mathematical Toys\" (1785); a poetic version of the rule of three (1792); \"Lotteries and Mountebanks\" (1801); Lewis Carroll on the game of logic (1887); \"Maps and Mazes\" (1892); \"Einstein's Real Achievement\" (1921); \"Riddles in Mathematics\" (1945); \"New Math for Parents\" (1966); and \"PC Astronomy\" (1997). Organized by thematic chapters, each selection is placed in context by a brief introduction. A unique window into the hidden history of popular mathematics,A Wealth of Numberswill provide many hours of fun and learning to anyone who loves popular mathematics and science. Some images inside the book are unavailable due to digital copyright restrictions.

The Calculus of Friendship is the story of an extraordinary connection between a teacher and a student, as chronicled through more than thirty years of letters between them. What makes their relationship unique is that it is based almost entirely on a shared love of calculus. For them, calculus is more than a branch of mathematics; it is a game they love playing together, a constant when all else is in flux. The teacher goes from the prime of his career to retirement, competes in whitewater kayaking at the international level, and loses a son. The student matures from high school math whiz to Ivy League professor, suffers the sudden death of a parent, and blunders into a marriage destined to fail. Yet through it all they take refuge in the haven of calculus--until a day comes when calculus is no longer enough. Like calculus itself, The Calculus of Friendship is an exploration of change. It's about the transformation that takes place in a student's heart, as he and his teacher reverse roles, as they age, as they are buffeted by life itself. Written by a renowned teacher and communicator of mathematics, The Calculus of Friendship is warm, intimate, and deeply moving. The most inspiring ideas of calculus, differential equations, and chaos theory are explained through metaphors, images, and anecdotes in a way that all readers will find beautiful, and even poignant. Math enthusiasts, from high school students to professionals, will delight in the offbeat problems and lucid explanations in the letters.