# 0.1 Preliminaries  (Page 2/5)

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The Winograd Structure can be described in this manner also. Suppose $M\left(s\right)$ can be factored as $M\left(s\right)={M}_{1}\left(s\right){M}_{2}\left(s\right)$ where ${M}_{1}$ and ${M}_{2}$ have no common roots, then ${C}_{M}\sim \left({C}_{{M}_{1}},\oplus ,{C}_{{M}_{2}}\right)$ where $\oplus$ denotes the matrix direct sum. Using this similarity and recalling [link] , the original convolution is decomposed intodisjoint convolutions. This is, in fact, a statement of the Chinese Remainder Theoremfor polynomials expressed in matrix notation. In the case of circular convolution, ${s}^{n}-1={\prod }_{d|n}{\Phi }_{d}\left(s\right)$ , so that ${S}_{n}$ can be transformed to a block diagonal matrix,

${S}_{n}\sim \left[\begin{array}{cccc}{C}_{{\Phi }_{1}}& & & \\ & {C}_{{\Phi }_{d}}& & \\ & & \ddots & \\ & & & {C}_{{\Phi }_{n}}\end{array}\right]=\left(\underset{d|n}{\oplus },{C}_{{\Phi }_{d}}\right)$

where ${\Phi }_{d}\left(s\right)$ is the ${d}^{th}$ cyclotomic polynomial. In this case, each block represents a convolutionwith respect to a cyclotomic polynomial, or a `cyclotomic convolution'.Winograd's approach carries out these cyclotomic convolutions using the Toom-Cook algorithm.Note that for each divisor, $d$ , of $n$ there is a corresponding block on the diagonal of size $\phi \left(d\right)$ , for the degree of ${\Phi }_{d}\left(s\right)$ is $\phi \left(d\right)$ where $\phi \left(·\right)$ is the Euler totient function. This method is good for short lengths, butas $n$ increases the cyclotomic convolutions become cumbersome,for as the number of distinct prime divisors of $d$ increases, the operation described by ${\sum }_{k}{h}_{k}{\left({C}_{{\Phi }_{d}}\right)}^{k}$ becomes more difficult to implement.

The Agarwal-Cooley Algorithm utilizes the fact that

${S}_{n}={P}^{t}\left({S}_{{n}_{1}},\otimes ,{S}_{{n}_{2}}\right)P$

where $n={n}_{1}{n}_{2}$ , $\left({n}_{1},{n}_{2}\right)=1$ and $P$ is an appropriate permutation [link] . This converts the one dimensional circular convolutionof length $n$ to a two dimensional one of length ${n}_{1}$ along one dimension and length ${n}_{2}$ along the second.Then an ${n}_{1}$ -point and an ${n}_{2}$ -point circular convolution algorithm can be combined to obtain an $n$ -point algorithm. In polynomial notation, the mapping accomplished bythis permutation $P$ can be informally indicated by

$Y\left(s\right)=\phantom{\rule{3.33333pt}{0ex}}⟨\phantom{\rule{0.277778em}{0ex}}X\left(s\right)H\left(s\right){⟩}_{{s}^{n}-1}\stackrel{P}{⇔}Y\left(s,t\right)=\phantom{\rule{3.33333pt}{0ex}}⟨\phantom{\rule{0.277778em}{0ex}}X\left(s,t\right)H\left(s,t\right){⟩}_{{s}^{{n}_{1}}-1,{t}^{{n}_{2}}-1}.$

It should be noted that [link] implies that a circulant matrix of size ${n}_{1}{n}_{2}$ can be written as a block circulant matrix with circulantblocks.

The Split-Nesting algorithm [link] combines the structures of the Winograd and Agarwal-Cooley methods, so that ${S}_{n}$ is transformed to a block diagonalmatrix as in [link] ,

${S}_{n}\phantom{\rule{0.166667em}{0ex}}\sim \phantom{\rule{0.166667em}{0ex}}\underset{d|n}{\oplus }\Psi \left(d\right).$

Here $\Psi \left(d\right)={\otimes }_{p|d,p\in \mathcal{P}}{C}_{{\Phi }_{{H}_{d}\left(p\right)}}$ where ${H}_{d}\left(p\right)$ is the highest power of $p$ dividing $d$ , and $\mathcal{P}$ is the set of primes.

${S}_{45}\sim \left[\begin{array}{cccccc}1& & & & & \\ & {C}_{{\Phi }_{3}}& & & & \\ & & {C}_{{\Phi }_{9}}& & & \\ & & & {C}_{{\Phi }_{5}}& & \\ & & & & {C}_{{\Phi }_{3}}\otimes {C}_{{\Phi }_{5}}& \\ & & & & & {C}_{{\Phi }_{9}}\otimes {C}_{{\Phi }_{5}}\end{array}\right]$

In this structure a multidimensional cyclotomic convolution, represented by $\Psi \left(d\right)$ , replaces each cyclotomic convolution in Winograd's algorithm (represented by ${C}_{{\Phi }_{d}}$ in [link] . Indeed, if the product of ${b}_{1},\cdots ,{b}_{k}$ is $d$ and they are pairwise relatively prime, then ${C}_{{\Phi }_{d}}\sim {C}_{{\Phi }_{{b}_{1}}}\otimes \cdots \otimes {C}_{{\Phi }_{{b}_{k}}}$ . This gives a method for combining cyclotomic convolutionsto compute a longer circular convolution. It is like the Agarwal-Cooley method but requires feweradditions [link] .

## Prime factor permutations

One can obtain ${S}_{{n}_{1}}\otimes {S}_{{n}_{2}}$ from ${S}_{{n}_{1}{n}_{2}}$ when $\left({n}_{1},{n}_{2}\right)=1$ , for in this case, ${S}_{n}$ is similar to ${S}_{{n}_{1}}\otimes {S}_{{n}_{2}}$ , $n={n}_{1}{n}_{2}$ . Moreover, they are related by a permutation.This permutation is that of the prime factor FFT algorithms and is employed in nesting algorithmsfor circular convolution [link] , [link] . The permutation is described by Zalcstein [link] , among others, and it is his description we draw on in the following.

Let $n={n}_{1}{n}_{2}$ where $\left({n}_{1},{n}_{2}\right)=1$ . Define ${e}_{k}$ , ( $0\le k\le n-1$ ), to be the standard basis vector, ${\left(0,\cdots ,0,1,0,\cdots ,0\right)}^{t}$ , where the 1 is in the ${k}^{th}$ position. Then, the circular shift matrix, ${S}_{n}$ , can be described by

do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
Do somebody tell me a best nano engineering book for beginners?
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
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Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
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Porter
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Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
how did you get the value of 2000N.What calculations are needed to arrive at it
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