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function x = Kcrot(p,e,K,x) % Kronecker product of Cyclotomic Reduction Operations.% x = (G(p(1)^e(1)) kron ... kron G(p(K)^(K)))^t*x % (transpose)% p : p = [p(1),...,p(K)];% e : e = [e(1),...,e(K)];a = (p-1).*((p).^(e-1)); r = a; % r(i) = number of rows of G(i)c = 2*a-1; % c(i) = number of columns of G(i) m = 1;n = prod(r); for i = 1:Kn = n / r(i); x = IcrotI(p(i),e(i),m,n,x);m = m * c(i); end function y = IcrotI(p,e,m,n,x) % y = (eye(m) kron G(p^e)^t kron eye(n))*x% (transpose) a = (p-1)*(p^(e-1));c = a; r = 2*a-1;y = zeros(r*m*n,1); v = 0:n:(r-1)*n;u = 0:n:(c-1)*n; for i = 0:m-1for j = 0:n-1 y(v+i*r*n+j+1) = crot(p,e,x(u+i*c*n+j+1));end end function y = crot(p,e,x) % y = crot(p,x)% cyclotomic reduction matrix (transpose) % length(x) == 2*n-1% length(y) == n % where n = (p-1)*(p^(e-1))n = (p-1)*(p^(e-1)); y = zeros(2*n-1,1);if p == 2 n = p^(e-1);y(1:n) = x; y(n+1:2*n-1) = -x(1:n-1);else y(1:n) = x;L = p^(e-1); y(n+1:n+L) = -x(1:L);a = L; for k = 2:p-1y(n+1:n+L) = y(n+1:n+L) - x(a+1:a+L); a = a + L;end b = 2*n-1 - p*(p^(e-1));y(p*L+1:p*L+b) = x(1:b); end

The following programs tell the programs for code generation relevant information about the bilinear forms for cyclotomic convolution.Specifically, they indicates the linear convolution out of which these cyclotomic convolution are composed, and thedimensions of the corresponding matrices. See the appendix Bilinear Forms for Linear Convolution .

function [d,r,c,Q,Qt] = A_data(n)% A : A matrix in bilinear form for cyclotomic convolution % d : linear convolution modules used% r : rows % c : columns% Q : Q(i) = cost associated with D(d(i)) % Qt : Qt(i) = cost associated with D(d(i))'if n == 2, d = [1];elseif n == 4, d = [2];elseif n == 8, d = [2 2];elseif n == 16, d = [2 2 2];elseif n == 3, d = [2];elseif n == 9, d = [2 3];elseif n == 27, d = [2 3 3];elseif n == 5, d = [2 2];elseif n == 7, d = [2 3];end r = []; c = []; Q = []; Qt = [];for k = 1:length(d) [rk, ck, Qk, Qtk]= D_data(d(k)); r = [r rk]; c = [c ck]; Q = [Q Qk]; Qt = [Qt Qtk];end function [r,c,Q,Qt] = D_data(d);% D : D matrix in bilinear form for linear convolution % r : rows% c : columns % Q : cost associated with D(d)% Qt : cost associated with D(d)' if d == 1, r = 1; c = 1; Q = 0; Qt = 0;elseif d == 2, r = 3; c = 2; Q = 1; Qt = 2; elseif d == 3, r = 5; c = 3; Q = 7; Qt = 9;end function [f,r,c] = C_data(p,e)% f : length of linear convolution% r : rows % c : columnsf = prod((p-1).*(p.^(e-1))); % (Euler Totient Function)r = 2*f-1; c = F_data(f); function c = F_data(n) % c : columns of F matrixif n == 1, c = 1; elseif n == 2, c = 3;elseif n == 4, c = 9; elseif n == 8, c = 27;elseif n == 3, c = 5; elseif n == 6, c = 15;elseif n == 18, c = 75; end

Programs for inverse transpose reduction operations

function x = itKRED(P,E,K,x) % x = itKRED(P,E,K,x);% (inverse transpose) % P : P = [P(1),...,P(K)]; % E : E = [E(K),...,E(K)]; for i = 1:Ka = prod(P(1:i-1).^E(1:i-1)); c = prod(P(i+1:K).^E(i+1:K));p = P(i); e = E(i);for j = e-1:-1:0 x(1:a*c*(p^(j+1))) = itRED(p,a,c*(p^j),x(1:a*c*(p^(j+1))));end end function y = itRED(p,a,c,x) % y = itRED(p,a,c,x);% (inverse transpose) y = zeros(a*c*p,1);for i = 0:c:(a-1)*c for j = 0:c-1A = x(i*p+j+1); for k = 0:c:c*(p-2)A = A + x(i*p+j+k+c+1); endy(i+j+1) = A; for k = 0:c:c*(p-2)y(i*(p-1)+j+k+a*c+1) = p*x(i*p+j+k+1) - A; endend endy = y/p;

Programs for permutations

The permutation of Equation 18 from Preliminaries is implemented by pfp . It calls the function pfp2I . The transpose is implemented by pfpt and it calls pfpt2I .

function x = pfp(n,K,x) % x = P(n(1),...,n(K)) * x% n = [n(1),...,n(K)];% length(x) = prod(n(1),...,n(K)) a = prod(n);s = 1; for i = K:-1:2a = a / n(i); x = pfp2I(a,n(i),s,x);s = s * n(i); end function y = pfp2I(a,b,s,x) % y = kron(P(a,b),I(s)) * x;% length(x) = a*b*s n = a * b;y = zeros(n*s,1); k1 = 0;k2 = 0; for k = 0:n-1i1 = s * (k1 + b * k2); i2 = s * k;for i = 1:s y(i1 + i) = x(i2 + i);end k1 = k1 + 1;k2 = k2 + 1; if k1>= b k1 = k1 - b;end if k2>= a k2 = k2 - a;end end function x = pfpt(n,K,x) % x = P(n(1),...,n(K))' * x% (tanspose) % n = [n(1),...,n(K)]; % length(x) = prod(n(1),...,n(K))% a = prod(n); a = n(1);s = prod(n(2:K)); for i = 2:Ks = s / n(i); x = pfpt2I(a,n(i),s,x);a = a * n(i); end function y = pfpt2I(a,b,s,x) % y = P(a,b)' kron I(s) * x;% (transpose) % length(x) = a*b*sn = a * b; y = zeros(n*s,1);k1 = 0; k2 = 0;for k = 0:n-1 i1 = s * (k1 + b * k2);i2 = s * k; for i = 1:sy(i2 + i) = x(i1 + i); endk1 = k1 + 1; k2 = k2 + 1;if k1>= b k1 = k1 - b;end if k2>= a k2 = k2 - a;end end

The following Matlab programs implement Rader's permutation and its transpose.They require the primitive root to be passed to them as an argument.

function y = rp(p,r,x) % Rader's Permutation% p : prime % r : a primitive root of p% x : length(x) == p a = 1;y = zeros(p,1); y(1) = x(1);for k = 2:p y(k) = x(a+1);a = rem(a*r,p); end function y = rpt(p,r,x) % Rader's Permutation% (transpose) % p : prime% r : a primitive root of p % x : length(x) == pa = 1; y = zeros(p,1);y(1) = x(1); for k = 2:py(a+1) = x(k); a = rem(a*r,p);end function [R, R_inv] = primitive_root(N)% function [R, R_inv] = primitive_root(N)% Ivan Selesnick % N is assumed to be prime. This function returns R,% the smallest primitive root of N, and R_inv, the % inverse of R modulo N.R = 'Not Found'; m = 0:(N-2);for x = 1:(N-1) if ( 1:(N-1) == sort(rem2(x,m,N)) )R = x; breakend endR_inv = 'Not Found'; for x = 1:Nif rem(x*R,N) == 1 R_inv = x;break endend

Questions & Answers

what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
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da
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Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
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Application of nanotechnology in medicine
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Jyoti Reply
I only see partial conversation and what's the question here!
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RAW Reply
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Damian
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Professor
I think
Professor
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Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
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What is meant by 'nano scale'?
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What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
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Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
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Bob Reply
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Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
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?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
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biomolecules are e building blocks of every organics and inorganic materials.
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research.net
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sciencedirect big data base
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Source:  OpenStax, Automatic generation of prime length fft programs. OpenStax CNX. Sep 09, 2009 Download for free at http://cnx.org/content/col10596/1.4
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