Transcript Exciting Applications of Polynomials

Exciting Applications of
Polynomials
Why Learning Polynomials?
What is a Polynomial?
• Example:
5x 2  2 x  7
• Building Blocks:
2
5x
2x
Monomials
• Parts of a Monomial:
Exponent
Variable
Coefficient
• A Monomial is
considered a
Polynomial
7
Why Study Polynomials? (1)
• Many formulas in math, science, and everyday
life are polynomials
• Examples:
2
E

mc
energy of mass
3
V L
volume of a cube
M
body mass index
BMI  2
• Mastery of polynomials
H
is needed for a deeper
understanding of formulas
Why Study Polynomials? (2)
• Modern electronic devices rely on ICs
(Integrated Circuits), which can have over
a million components (e.g. transistors).
The design and production of ICs require
many complex tests to ensure the ICs do
what they are designed to do. Automated
testing relies on LFSRs (Linear Feedback
Shift Registers). Why & how LFSRs work
is explained in the theory of polynomials
Why Study Polynomials? (3)
• Encryption is used in military, business,
and personal operations. Information are
often encrypted for efficient transmission
and to protect secrecy. (Cell phone calls
are encrypted for these reasons.) To
understand and to exploit why and how
encryption work requires an elaborate
study of polynomials.
Modern Electronic Devices (1)
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Flash Drive (1)
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USB connector
USB mass storage
controller device
Test points
Flash memory chip
Crystal oscillator
LED
Write-protect switch
Space for second
flash memory chip
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7.
8.
Modern Electronic Devices (1)
• Flash Drive (2)
• The mass storage controller
is an ASIC (Application
Specific Integrated Circuit) for
directing the movement of
data
• This and many other ASICs
have embedded a LFSR for
BIST (Built-in Self test), which
warns the user if the ASIC is
not fully functional
Modern Electronic Devices (2)
• Cell Phone (1)
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A microscopic microphone
A speaker
An LCD or plasma display
A keyboard
An antenna
A battery
A circuit board containing many ASICs:
DSP (Digital Signal Processor)
handles all the signal compression and
decompression
Microprocessor (and memory) handle
operations of the keyboard and display
Modern Electronic Devices (2)
• Cell Phone (2)
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A cell phones use ASICs for
its operations. (Ericsson phone
shown uses the ASIC version
of Z-80). The design and
production of ASICs reply on
the use of LFSRs.
The calls to or from a cell
phone are encrypted for
efficient transmission
and to protect secrecy.
LFSRs are used in encryption.
Modern Electronic Devices (3)
• MP3 (Music Player)(1)
• MP3 is the name of a file format,
just as PPT is for Powerpoint format
• MP3 players are basically just
USB flash drives, and use
USB connector + flash memory
• Music can be downloaded onto an
MP3 if it is connected to a computer
with supporting software like iTunes
• Key to this operation is compression.
A CD track occupies about 12 times the storage
as the same track in MP3, allowing MP3 more
music or video in less space.
Modern Electronic Devices (3)
• MP3 (Music Player)(2)
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Shown is the inside of an iPod
50% space is used by the HD
(SanDisk MP3 players use the
more compact flash memory)
Recent MP3 players play audio &
video; older MP3 plays only audio
MP3 uses a CPU (e.g. ARMTDM1)
and/or a DSP, both are ASICs,
for compressing/decompressing
digital signals
LFSRs and Polynomials are
vital to MP3 developers
Modern Electronic Devices (4)
• Digital Camera (1)
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Takes still photographs and video
digitally by recording images via an
electronic image sensor
Incorporated into many devices, e.g.
PDAs, mobile phone, & vehicles.
(The Hubble Space Telescope is
basically a digital camera.)
Has an image sensor that converts light
into electrical charges  often a CCD
(Charge Coupled Device) or a CMOS
(Complementary Metal Oxide Semiconductor)
Modern Electronic Devices (4)
• Digital Camera (2)
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The amount of detail that the camera
can capture is called the resolution, and
it is measured in pixels.
The more pixels a camera has, the more
detail it can capture. Larger pictures can
remain sharp.
Each photosite on a sensor keeps track
only of the total intensity of the light that
strikes its surface. In order to get a full
color image, most sensors use filtering
to look at the light in its three primary colors.
Once the camera records all three colors, it
combines them to create the full spectrum.
Modern Electronic Devices (4)
• Digital Camera (3)
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Picture images can be stored in
many devices: SmartMedia cards,
CompactFlash cards, Memory Sticks,
Hard disks, Writable CDs and DVDs
Two features of digital images make
compression possible:
repetition – patterns that repeat can
be reconstructed w/o loss of content
irrelevancy – dicard details that the
human eyes can detect
Use ASICs to handle color processing,
data compression, storage and
housekeeping.
LFSR & Polynomials are relevant!