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Transformations of Functions III01:20

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Transformations modify the graphical representation of a function without changing its fundamental form. One common transformation is reflection, which flips the graph across a designated axis. When the vertical coordinates of all points are multiplied by the negative one, the entire graph is mirrored over the horizontal axis. This transformation reverses the vertical orientation of peaks and troughs, akin to signal inversion in electrical systems, where a waveform is flipped, but the timing of...
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Transformations of Functions II01:29

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Transformations in mathematics alter the position or orientation of a function’s graph while preserving its fundamental shape. One important type of transformation is the horizontal shift, which involves modifying the input variable within a function’s equation. This operation affects where outputs occur along the horizontal axis but does not alter the function’s overall structure.A horizontal shift is achieved by replacing the input variable x with either x + c or x - c,...
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Transformations of Functions I01:29

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A function's graph can be modified by changing its position or size without altering its overall shape. These transformations allow the graph to be moved across the coordinate plane while preserving its pattern and structure. One of the most common transformations is shifting, which repositions the graph without distorting it.When the output of a function is adjusted by adding or subtracting a constant, the graph shifts vertically. A positive value moves the graph upward, while a negative value...
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Coordinates and map projections are essential tools in accurately representing the Earth's surface for various applications, ranging from navigation to spatial analysis. The latitude and longitude coordinate system is a universally recognized framework for defining locations. Latitude specifies the distance of a point north or south of the equator, measured in degrees from 0° at the equator to 90° at the poles. Longitude indicates a location's position east or west of the prime meridian,...
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Transformation of Plane Strain01:12

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When analyzing elongated structures like bars subjected to uniformly distributed loads, it is essential to understand the transformation of plane strain when coordinate axes are rotated. This transformation helps to assess how material deformation characteristics vary with orientation, which is crucial in materials science and structural engineering.
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Geographic Information Systems (GIS) rely on two core types of data: spatial data and attribute data.Spatial DataSpatial data defines the physical location of features within a coordinate system, typically expressed in terms of latitude and longitude. It provides precise positioning for elements like roads, rivers, or buildings.Attribute DataAttribute data complements spatial data by adding descriptive information about these features. For example, a road's spatial data includes its start and...
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Two cases of spatial transformations.

Mário G Silveirinha1, Cristian Della Giovampaola2, Nader Engheta3

  • 1Department of Electrical and Systems Engineering, University of Pennsylvania, Philadelphia, PA 19104, USA Department of Electrical Engineering, University of Coimbra, Instituto de Telecomunicações, 3030 Coimbra, Portugal.

Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences
|July 29, 2015
PubMed
Summary
This summary is machine-generated.

This study extends transformation optics to electronics and digital metamaterials. Researchers demonstrate tailoring electron effective mass in semiconductor superlattices and engineering electromagnetic permittivity using subwavelength building blocks.

Keywords:
electron wavesmetamaterialssemiconductor superlatticestransformations

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Area of Science:

  • Wave theory
  • Condensed matter physics
  • Materials science

Background:

  • Transformation optics enables wave manipulation by controlling material properties.
  • Extending these concepts to other physical phenomena is an active research area.
  • Metamaterials offer unique electromagnetic properties through engineered structures.

Purpose of the Study:

  • To explore the extension of spatial transformation concepts to electronics and digital metamaterials.
  • To demonstrate the tailoring of effective mass for charged carriers in semiconductor superlattices.
  • To show the engineering of electromagnetic permittivity using subwavelength building blocks.

Main Methods:

  • Designing semiconductor superlattices with varying electron effective mass layers.
  • Combining deeply subwavelength building blocks with opposite permittivity signs.
  • Analyzing plasmonic resonance effects in composite structures.

Main Results:

  • Achieved extreme anisotropy in electron effective mass within semiconductor superlattices.
  • Demonstrated the ability to engineer electromagnetic permittivity outside the range of constituent materials.
  • Observed plasmonic resonance enhancing effective permittivity values.

Conclusions:

  • Spatial transformation principles are applicable beyond optics, including electronics.
  • Digital metamaterials can be engineered for desired electromagnetic responses.
  • These advancements open new avenues for controlling wave phenomena.