Method and apparatus for contour mapping using synthetic aperture radar

By using two SAR antennas spaced a known distance, B, and oriented at substantially the same look angle to illuminate the same target area, pixel data from the two antennas may be compared in phase to determine a difference delta phi from which a slant angle theta is determined for each pixel point from an equation Delta phi = (2 pi B/lambda)sin(theta - alpha), where lambda is the radar wavelength and alpha is the roll angle of the aircraft. The height, h, of each pixel point from the aircraft is determined from the equation h = R cos theta, and from the known altitude, a, of the aircraft above sea level, the altitude (elevation), a', of each point is determined from the difference a - h. This elevation data may be displayed with the SAR image by, for example, quantizing the elevation at increments of 100 feet starting at sea level, and color coding pixels of the same quantized elevation. The distance, d, of each pixel from the ground track of the aircraft used for the display may be determined more accurately from the equation d = R sin theta

Niccolo' Isouard

L-awtur jagħti tagħrif bijografiku dwar il-mużiċist u l-kompożistur Nicolò Isouard.N/

The observational limit of wave packets with noisy measurements

The authors consider the problem of recovering an observable from certain measurements containing random errors. The observable is given by a pseudodifferential operator while the random errors are generated by a Gaussian white noise. The authors show how wave packets can be used to partially recover the observable from the measurements almost surely. Furthermore, they point out the limitation of wave packets to recover the remaining part of the observable, and show how the errors hide the signal coming from the observable. The recovery results are based on an ergodicity property of the errors produced by wave packets

Gesture as a Window to Justification and Proof

The role of the body, particularly gesture, in supporting mathematical reasoning is an emerging area of research in mathematics education. In the present study, we examine undergraduate students providing a justification for a task about a system of alternating gears, which involves concepts of number relating to even/odd patterns. Some participants were directed to perform gestures relevant to alternation and parity before attempting their justification, while others were not. Although these directed actions did not seem to influence the gestures participants used to solve the problem, we found an important relationship between gesture and mathematical reasoning. In particular, certain types of gestures during problem solving were associated with valid justifications. This research provides insight into the link between action and mathematical reasoning, and has implications for supporting students’ proof activities

Being Mathematical Relations: Dynamic Gestures Support Mathematical Reasoning

In mathematics classrooms, body-based actions, including gestures, offer an important way for students to become mathematical ideas as they engage in mathematical practices. In particular, a type of gesture that we call a dynamic depictive gesture allows learners to model and represent fluid transformations of mathematical objects with their bodies. In this paper, we report on two empirical studies – one in which dynamic gestures were observed, and one where these gestures were directed. We conclude that dynamic gestures are a key element in successful justification and proof activities in mathematics

Strategically Chosen Examples Leading to Proof Insight: A Case Study of a Mathematician’s Proving Process.

Examples play a critical role in the exploration and proving of conjectures. Although proof has been studied extensively, the precise ways in which examples might facilitate successful proofs are not well documented or understood. Working within a larger set of studies that argue for the value of examples in proof-related activity, in this paper we present a case study of one mathematician’s work on a conjecture in which his strategic, intentional use of examples led to a proof of that conjecture. By examining his work in detail, we highlight specific mechanisms by which the mathematician’s examples led to successful proof production. These mechanisms shed light on precise ways in which examples can directly lead to proof and inform our understanding of the conceptual landscape of the interplay between examples and proof