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Register a new userThrowing a Ball as Far as Possible, Revisited
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What initial trajectory angle maximizes the arc length of an ideal projectile? We show the optimal angle, which depends neither on the initial speed nor on the acceleration of gravity, is the solution x to a surprising transcendental equation: csc(x) = coth(csc(x)), i.e., x = arccsc(y) where y is the unique positive fixed point of coth. Numerically, $x approx 0.9855 approx 56.47^circ$. The derivation involves a nice application of differentiation under the integral sign.
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Suppose there is a message generated at a node $v$ in a network and $v$ decides to pass the message to one of the neighbors $u$, and $u$ next decides to pass the message to one of its own neighbors, and so on. How to relay the message as far as possible with local decisions? To the best of our knowledge no general solution other than randomly picking available adjacent node exists. Here we report some progress. Our first contribution is a new framework called tp-separate chain decomposition for studying network structures. Each tp-separate chain induces a ranking of nodes. We then prove that the ranks can be locally and distributively computed via searching some stable states of certain dynamical systems on the network and can be used to search long paths of a guaranteed length containing any given node. Numerical analyses on a number of typical real-world networks demonstrate the effectiveness of the approach.
This note tries to show that a re-examination of a first course in analysis, using the more sophisticated tools and approaches obtained in later stages, can be a real fun for experts, advanced students, etc. We start by going to the extreme, namely we present two proofs of the Extreme Value Theorem: the programmer proof that suggests a method (which is practical in down-to-earth settings) to approximate, to any required precision, the extreme values of the given function in a metric space setting, and an abstract space proof (the level-set proof) for semicontinuous functions defined on compact topological spaces. Next, in the intermediate part, we consider the Intermediate Value Theorem, generalize it to a wide class of discontinuous functions, and re-examine the meaning of the intermediate value property. The trek reaches the final frontier when we discuss the Uniform Continuity Theorem, generalize it, re-examine the meaning of uniform continuity, and find the optimal delta of the given epsilon. Have fun!
It is well-known that the Continuum Hypothesis (CH) is independent of the other axioms of Zermelo-Fraenkel set theory with choice (ZFC). This raises the question of whether an intuitive justification exists for CH as an additional axiom, or conversely whether it is more intuitive to deny CH. Freilings Axiom of Symmetry (AS) is one candidate for an intuitively justifiable axiom that, when appended to ZFC, is equivalent to the denial of CH. The intuition relies on a probabalistic argument, usually cast in terms of throwing random darts at the real line, and has been defended by researchers as well as popular writers. In this note, the intuitive argument is reviewed. Following William Abram, it is suggested that while accepting CH leads directly to a counterexample to AS, this is not necessarily fatal to our intuition. Instead, we suggest, it serves to alert us to the error in a naive intuition that leaps too readily from the near-certainty of individual events to near-certainty of a joint event.
Mathematicians have traditionally been a select group of academics that produce high-impact ideas allowing substantial results in several fields of science. Throughout the past 35 years, undergraduates enrolling in mathematics or statistics have represented a nearly constant rate of approximately 1% of bachelor degrees awarded in the United States. Even within STEM majors, mathematics or statistics only constitute about 6% of undergraduate degrees awarded nationally. However, the need for STEM professionals continues to grow and the list of needed occupational skills rests heavily in foundational concepts of mathematical modeling curricula, where the interplay of measurements, computer simulation and underlying theoretical frameworks takes center stage. It is not viable to expect a majority of these STEM undergraduates would pursue a double-major that includes mathematics. Here we present our solution, some early results of implementation, and a plan for nationwide adoption.
The online homework system WeBWorK has been successfully used at several hundred colleges and universities. Despite its popularity, the WeBWorK system does not provide detailed metrics of student performance to instructors. In this article, we illustrate how an analysis of the log files of the WeBWorK system can provide information such as the amount of time students spend on WeBWorK assignments and how long they persist on problems. We estimate the time spent on an assignment by combining log file events into sessions of student activity. The validity of this method is confirmed by cross referencing with another time estimate obtained from a learning management system. As an application of these performance metrics, we contrast the behaviour of students with WeBWorK scores less than 50% with the remainder of the class in a first year Calculus course. This reveals that on average, the students who fail their homework start their homework later, have shorter activity sessions, and are less persistent when solving problems. We conclude by discussing the implications of WeBWorK analytics for instructional practices and for the future of learning analytics in undergraduate mathematics education.
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