text stringlengths 30 4k | source stringlengths 60 201 |
|---|---|
. ε < |y| < xn, where K cancels on whole spheres.
2. xn < |y| < R for some large R, which is a bounded region on which K is well-behaved.
3. |y| > R, on which the hemisphere cancellation of K is useful.
The details of the argument are left to the homework.
3
MIT OpenCourseWare
http://ocw.mit.edu
18.156 Differential An... | https://ocw.mit.edu/courses/18-156-differential-analysis-ii-partial-differential-equations-and-fourier-analysis-spring-2016/091bf989a474217c56f80bbb84cead6e_MIT18_156S16_lec7.pdf |
Linear Spaces
we have seen (12.1-12.3 of Apostol) that n-tuple space
has the following properties :
V,
Addition:
1. (Commutativity) A + B = B + A.
2. (Associativity) A + (B+c) = (A+B) + C.
3. (Existence of zero) There is an element -0
such 'that A + -0 = A for all A.
4. (Existence of negatives) Given A, there is... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/0924c923e90a09cf95778ba4543644d6_MIT18_024s11_ChAnotes.pdf |
will be concerned only with n-tuple space
and with certain of its subsets called "linear subspaces" :
Vn
-Definition. Let W be a non-empty subset of Vn ; suppose W
is closed under vector addition and scalar multiplication. Then W is
called a linear subspace of V n (or sometimes simply a subspace of Vn .)
To say W ... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/0924c923e90a09cf95778ba4543644d6_MIT18_024s11_ChAnotes.pdf |
subset of Vn
.
consisting of all vectors X of the form X = cA is a subspace of
'n
It is called the subspace spanned by A. In the case n = 2 or 3 , it can
be pictured as consisting of all vectors lying on a line through the origin.
Example 3. Let A and B be given non-zero vectors that are not
1
parallel. The su... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/0924c923e90a09cf95778ba4543644d6_MIT18_024s11_ChAnotes.pdf |
cl+dl)A1 f * * * + (ck+dk)Akr
ax = (ac ) A + * * * + (ack)Akf
1
1
so both X + Y and ax belong to W by definition. Thus W
is a subspace of Vn. -
Giving a spanning set for W is one standard way of specifying W.
Different spanning sets can of course give the same subspace. Fcr example,
it is intuitively clear that, f... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/0924c923e90a09cf95778ba4543644d6_MIT18_024s11_ChAnotes.pdf |
so are X + Y and
cX. It is easy to see that W is the linear span of (1,0,0)
I
and (O,l,O).
Example 6. The subset of V3 consisting of all vectors of the
form X = (3a+2b,a-b,a+7b) is a subspace of V3. It consists
of all vectors of the form
X = a(3,1,1) + b(2,-1,7),
so it is the linear span of 3 1 , and (2,-1,7). ... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/0924c923e90a09cf95778ba4543644d6_MIT18_024s11_ChAnotes.pdf |
?
n n
n
2. In each of the following, let W denote the set of
I
I
all vectors (x,y,z) in Vj satisfying the condition given.
(Here we use (x,y,z) instead of (xl,x2,x3) for the general
element of V3.) Determine whether W is a subspace of V j .
If it is, draw a picture of it or describe it geometrically,
and find ... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/0924c923e90a09cf95778ba4543644d6_MIT18_024s11_ChAnotes.pdf |
c A + ... + ck+
1 1
for some scalars ci. If S spans the vector X, we say that S spans X
uniquely if the equations
X =
ciAi
i=l
and
imply that ci = di for all i.
It is easy to check the following:
Theorem 1, Let S = < A ~ ,.. .
be a set of vectors of Vn; let
X be a vector in L(S). Then S spans X uniquely if a... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/0924c923e90a09cf95778ba4543644d6_MIT18_024s11_ChAnotes.pdf |
A] ,...,%J of vectors of V is said to
n
be linearly independent (or simply, independent) if it spans the zero vector
I
uniquely. The vectors themselves are also said to be independent in this
situation.
If a set is not independent, it is said to be dependent.
Banple 8. If a subset T of a set S is dependent, the... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/0924c923e90a09cf95778ba4543644d6_MIT18_024s11_ChAnotes.pdf |
set of vectors is called an orthonormal set. The coordinate
by the vectors
Ai
vectors E
l form such a set.
n
B.mple
A set ccnsisting of a single vector A is independent
if A # Q.
A set consisting of two non-zero vectors ARB is independent if and
I
only if the vectors are not parallel. More generally, one ha... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/0924c923e90a09cf95778ba4543644d6_MIT18_024s11_ChAnotes.pdf |
of
the vectors Ai .
as follows:
We do so, using a "double-indexing" notation for the coefficents,
j
Multiplying the equation by x and summing over j, and collecting terms, we
j
have the equation
In order for <x .B to equal 2 , it will suffice if we can choose the x
j
so that coefficient of each vector Ai in t... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/0924c923e90a09cf95778ba4543644d6_MIT18_024s11_ChAnotes.pdf |
less than n, then the solution space con-
tains some vector other t2ian 0.
I'ronf.. We are concer~tcd here only v i t h proving the existence of some
solutioli otlicr tJta11 0, not with nctt~nlly fitidirtg such a solution it1 practice,
nor wit11 finditig all possildd solutiot~s. (We 11-ill study the practical prob-... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/0924c923e90a09cf95778ba4543644d6_MIT18_024s11_ChAnotes.pdf |
the coefiicielits of st is
nonzero, attd 1t.e rncly s~tj~poscfor cortvetlier~ce that the equations have
beerr arranged so that this happetls ill the first ec~uation, with the result
that 0 1 1 + 0. We rnultiply the first crlt~ation 1)y the scalar azl/afl and then
suhtract i t from the second, eli~nitiat~itlg tghe X... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/0924c923e90a09cf95778ba4543644d6_MIT18_024s11_ChAnotes.pdf |
io~lo t l ~ c rthan the zcro vcctor). $Ire prove this as follows:
Sr~ppose(d2, . . . , d.,) is a solutiolr of t l ~ o stna1.ller system, different from
, . . . ,. We su1)stitutc itito tllc first equation and solve for XI,thereby
obtailiirlg the follo~ving vector,
w1ricI1 yo11 may verify is a s o l ~ ~ t ~ i o nof t... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/0924c923e90a09cf95778ba4543644d6_MIT18_024s11_ChAnotes.pdf |
it to you to show that this statement holds.(Be sure you
ccnsider the case where one or more or all of the coefficents are zero.) a
-E2ample 13. We have already noted that the vectors El,...,E span all
It. follows, for example, that any three vectors in V2 are dependent,
n
of Vn.
that is, one of them equals a lin... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/0924c923e90a09cf95778ba4543644d6_MIT18_024s11_ChAnotes.pdf |
depend ecce of A , ..A
for A
i
# 0, we can solve this equation
contradicting the fact that Ai+l does not belong to L(Alt....A,).
r while if c ~ + ~
1
Cc~ntinuing the process just described, we can find larger and larger
independent sets of vectors in W. The process stops only when the set we obtain
spans W. Does... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/0924c923e90a09cf95778ba4543644d6_MIT18_024s11_ChAnotes.pdf |
0 alone, then the dimension of
W:
is zero.
Example 14. The space Vn has a "naturaln basis consisting of the
vectors E1,...,E
. It follows that Vn has dimension n. (Surprise!) There
n
are many other bases for Vn., For instance, the vectors
form a basis for V,,
as you can check.
I
I
1
i
~xercises
1. Consi... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/0924c923e90a09cf95778ba4543644d6_MIT18_024s11_ChAnotes.pdf |
ose All.. .,A,, is a basis for V; let B1p.afBk be arbitrary
vectors of W. (a) Slim? there exists a linear transformation T : V + W
such that T(Ai) = Bi fc>r all i. (b) Show this linear transformation is unique.
7. L e t W be a subspace of Vn: let Al,...,% be a basis for W.
Let X, Y be vectors of W. Then X = 2 xjAi ... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/0924c923e90a09cf95778ba4543644d6_MIT18_024s11_ChAnotes.pdf |
1 + "
CmBm
Bnt+l
Show that Bmcl is different from 2 and that L(B1,. ..,B ,B
L(B~,...,B ,A
orthogonal to each of B ..,B, .
m m+l
+
) - Then show that the ci mmy be so chosen that Bm+l is
m m+l) =
Steep 2. Show that if W is a subspace of Vn of positive dimension.
then W has a basis consisting of vectors that are m... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/0924c923e90a09cf95778ba4543644d6_MIT18_024s11_ChAnotes.pdf |
of another row,
say rcm m.
(3) Multiply row i of A by a non-zero scalar.
These operations are called the elcmentary & operations. Their usefulness cclmes
from the following fact:
Theorem 6. Suppose B is the matrix obtained by applying a sequence
of elementary row operations to A,successively. Then the row spaces ... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/0924c923e90a09cf95778ba4543644d6_MIT18_024s11_ChAnotes.pdf |
first column of your matrix.
I
(I) If this column consists entirely of zeros, nothing needs to ba
I
done. Restrict your attention now to the matrix obtained by deleting the
first column, and begin again.
%
(11) If this column has a non-zero entry, exchange rows if necessary
to bring it to the top row. Then add ... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/0924c923e90a09cf95778ba4543644d6_MIT18_024s11_ChAnotes.pdf |
appear at the "inside cornersw of the stairsteps
are often called the pivots in the echelon form.
Yclu can check readily that the non-zero rows of the matrix B are
independent. (We shall prove this fact later.) It follows that the non-zero rows
of the matrix B form a basis for the row space of B, and hence a basis ... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/0924c923e90a09cf95778ba4543644d6_MIT18_024s11_ChAnotes.pdf |
The final step is to multiply each non-zero
row by an appropriate non-zero scalar, chosen so as to make the pivot entry
into 1. This we can do, because the pivots are non-zero. At the end of
this process, the matrix is in what is called reduced
echelon form.
The reduced
echelon form of the matrix C above is the m... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/0924c923e90a09cf95778ba4543644d6_MIT18_024s11_ChAnotes.pdf |
Proof. Let D and D 1 be two reduced
echelon matrices, w5ose
rows span the same subspace W of Vn. We show that D = D'.
Let R
be the non-zero rows of D ; and suppose that the
pivots (first non-zero entries) in these rows occur in columns jl,...,j
k t
respectively.
( a ) =ow that the pivots of D 1 wrur in the colw... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/0924c923e90a09cf95778ba4543644d6_MIT18_024s11_ChAnotes.pdf |
,, '
The equation ( * ) is often called a parametric equation
for the line, and t is called the parameter in this equation.
As t ranges over all real numbers, the corresponding point X
ranges over all points of the line L. When t = 0, then X = P; when
t = 1, then X = P + A; when t = $, then X = P + %A; and so on. ... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/0924c923e90a09cf95778ba4543644d6_MIT18_024s11_ChAnotes.pdf |
0 . We can
solve these equations for P in terms of Q and B:
P = Q + t 2 B - tlA = Q + (t2-tl&JB.
Now, given any point X = P + tA of the line L(P;A), WE can write
X = P + tA = Q + (t2-tlc)B+ tcB.
Thus X belongs to the line L (Q;B).
Thus every point of L (P;A) belongs to L (Q:B) . The
symmetry of the argument show... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/0924c923e90a09cf95778ba4543644d6_MIT18_024s11_ChAnotes.pdf |
i s p a r a l l e l t o L. By Theorem 8 , any
o t h e r l i n e c o n t a i n i n g Q
and p a r a l l e l t o L
i s e q u a l t o t h i s
one. 0
Theorem - 11. Given - two d i s t i n c t p o i n t s P - and Q ,
t h e r e -i s e x a c t l y --one l i n e c o n t a i n i n g -them.
P r o o f . L e t A = Q - P ;
t... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/0924c923e90a09cf95778ba4543644d6_MIT18_024s11_ChAnotes.pdf |
i s a p o i n t of Vn
B a r e independent v e c t o r s of V n r w e def i n e t h e p l a n e through
P determined 2 A -and B
t o be t h e s e t of a l l p o i n t s X of
and i f . A
and
t h e form
where s and t run through a l l r e a l numbers. We denote t h i s
p l a n e by M ( P ; A , B ) .
The e q u a t ... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/0924c923e90a09cf95778ba4543644d6_MIT18_024s11_ChAnotes.pdf |
subspace
of Vn span&: by A and B.
J u s t as f o r l i n e s , a p l a n e has many d i f f e r e n t p a r a m e t r i c
r e p r e s e n t a t i o n s . More p r e c i s e l y , one h a s t h e following theorem:
Theorem 12. - The p l a n e s M ( P ; A , B ) - and M(Q:C,D) - are
have a p o i n t - i n common and ... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/0924c923e90a09cf95778ba4543644d6_MIT18_024s11_ChAnotes.pdf |
of C and D. Symmetry shows that
C and D lie in the linear span of A and B as well. Thus these linear
spans are the same.
Conversely, suppose t h a t t h e p l a n e s i n t e r s e c t i n a p o i n t
R and t h a t L(A,B) = L(C,D). Then
P + s l A + t B =
1
R = Q + s 2 C + t 2 D
for some scalars si ard ti. We can... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/0924c923e90a09cf95778ba4543644d6_MIT18_024s11_ChAnotes.pdf |
s a y t h e p l a n e s M(P;A,B) and M(Q;C,D) a r e .
p a r a l l e l i f L(A,B) = L(C,D).
Corollary 13. TKO distinct parallel planes cannot intersect.
corollary - 14.
Given - a plane M - -
and a point Q, there -is
exactly one - plane containinq Q --
that is parallel - to M .
Proof. Suppose M = M ( P ; A , B )... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/0924c923e90a09cf95778ba4543644d6_MIT18_024s11_ChAnotes.pdf |
P and B = R - P; then
A and B are independent. The plane M(P; A,B) cGntains P and P + A = Q
and P + B = R *
Now suppose M(S;C,D) is 'another plane containing P,
Q, and R. Then
'I
. Subtracting, we see that the vectors
for some scalars s
Q - P = A and R - P = B belong to the linear span ~f f2and D. By
and ti
i... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/0924c923e90a09cf95778ba4543644d6_MIT18_024s11_ChAnotes.pdf |
L be the line in Vj through the points P = (1.0.2) and
Q = (1,13). Let L1 be the line through 2 parallel to the vector
A = ( 3 - 1 ) Find parametric equations for the line that intersects both L
I
and Lf and is orthogonal to both of them.
Parametric equations for k-planes
Vn.
432
Following t h e p a t t e r n f... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/0924c923e90a09cf95778ba4543644d6_MIT18_024s11_ChAnotes.pdf |
t h e k - p l a n e M (P;Al,. . .,Ak'
i f and o n l y i f X - P
i s i n t h e l i n e a r s p a n o f A1, ...,Ak. -
Note that if P = 2, then t h i s k-plane is just the k- dimensional
1
linear subspace of Vn s p ~ e dby A1, ...,%.
J u s t a s w i t h t h e c a s e o f l i n e s
( 1 - p l a n e s ) and p l a n e s... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/0924c923e90a09cf95778ba4543644d6_MIT18_024s11_ChAnotes.pdf |
t h e r e -i s e x a c t l y -one k - p l a n e -i n V,
Q ,
p a r a l l e l -
t o
M..
Vn
5 p o i n t
c o n t a i n i n g Q
.
..-
4
-- Lemma 19. Given points POt..,Pk
in Vn,
they are contained i n
a plane of dimension l e s s than k
i f and only i f the vectors
I
P1 - Po,..., Pk- Po are dependent.
Theor... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/0924c923e90a09cf95778ba4543644d6_MIT18_024s11_ChAnotes.pdf |
both the point P and the point Q = (-1,0,2)?
4. Given the 2-plane MI. in V4 containing the points P = (1,-I, 2,-1)
and Q = (O.l,l,O)wd R = (lrl,0,3).Find parametric equations for a 3-plane
in Vq that containsthe point S = (l,l,l.) and is parallel to MI.
Is it unique? Can you find such a 3-plane thatcontains both S ... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/0924c923e90a09cf95778ba4543644d6_MIT18_024s11_ChAnotes.pdf |
Verilog
L5: Simple Sequential Circuits and Verilog
L5: Simple Sequential Circuits and
Acknowledgements:
Materials in this lecture are courtesy of the following sources and are used with
permission.
Nathan Ickes
Rex Min
L5: 6.111 Spring 2006
Introductory Digital Systems Laboratory
1
Key Points from L4 (Sequential Bl... | https://ocw.mit.edu/courses/6-111-introductory-digital-systems-laboratory-spring-2006/09793dcc124f0226e93eb01591762630_l5_seql_verilog.pdf |
logic,cd: contamination or
minimum delay
through logic network
L5: 6.111 Spring 2006
Introductory Digital Systems Laboratory
3
System Timing (I): Minimum Period
System Timing (I): Minimum Period
CLout
D
Q
Clk
In
D
Q
Combinational
Logic
CLK
IN
FF1
CLout
Clk
Th
Tsu
Tcq
Tlogic
Tcq,cd
Tl,cd
Th
Tsu
Tcq
Tcq,cd
Tsu2
T > Tc... | https://ocw.mit.edu/courses/6-111-introductory-digital-systems-laboratory-spring-2006/09793dcc124f0226e93eb01591762630_l5_seql_verilog.pdf |
Spring 2006
Introductory Digital Systems Laboratory
6
Importance of the Sensitivity List
Importance of the Sensitivity List
(cid:132) The use of posedge and negedge makes an always block sequential
(edge-triggered)
(cid:132) Unlike a combinational always block, the sensitivity list does
determine behavior for synthes... | https://ocw.mit.edu/courses/6-111-introductory-digital-systems-laboratory-spring-2006/09793dcc124f0226e93eb01591762630_l5_seql_verilog.pdf |
ilog supports two types of assignments within always blocks, with
subtly different behaviors.
(cid:132) Blocking assignment: evaluation and assignment are immediate
always @ (a or b or c)
begin
x = a | b;
y = a ^ b ^ c;
z = b & ~c;
1. Evaluate a | b, assign result to x
2. Evaluate a^b^c, assign result to y
3. Evaluate... | https://ocw.mit.edu/courses/6-111-introductory-digital-systems-laboratory-spring-2006/09793dcc124f0226e93eb01591762630_l5_seql_verilog.pdf |
6
Introductory Digital Systems Laboratory
10
Use Use Nonblocking
for Sequential Logic
Nonblocking for Sequential Logic
always @ (posedge clk)
begin
always @ (posedge clk)
begin
q1 <= in;
q2 <= q1;
out <= q2;
end
q1 = in;
q2 = q1;
out = q2;
end
“At each rising clock edge, q1, q2, and out
simultaneously receive the o... | https://ocw.mit.edu/courses/6-111-introductory-digital-systems-laboratory-spring-2006/09793dcc124f0226e93eb01591762630_l5_seql_verilog.pdf |
c;
end
endmodule
Nonblocking Behavior
a b c x y Deferred
module nonblocking(a,b,c,x,y);
(Given) Initial Condition
a changes;
always block triggered
x <= a & b;
y <= x | c;
Assignment completion
1 1 0 1 1
0 1 0 1 1
0 1 0 1 1 x<=0
0 1 0 1 1 x<=0, y<=1
0 1 0 0 1
input a,b,c;
output x,y;
reg x,y;
always @ (a or b or c)
... | https://ocw.mit.edu/courses/6-111-introductory-digital-systems-laboratory-spring-2006/09793dcc124f0226e93eb01591762630_l5_seql_verilog.pdf |
Introductory Digital Systems Laboratory
14
Verilog
The Ripple Counter in Verilog
The Ripple Counter in
Single D Register with Asynchronous Clear:
module dreg_async_reset (clk, clear, d, q, qbar);
input d, clk, clear;
output q, qbar;
reg q;
always @ (posedge clk or negedge clear)
begin
if (!clear)
q <= 1'b0;
else q <... | https://ocw.mit.edu/courses/6-111-introductory-digital-systems-laboratory-spring-2006/09793dcc124f0226e93eb01591762630_l5_seql_verilog.pdf |
will retained by a D Register
(cid:132) Next value of counter (N) computed by combinational logic
C3 C2 C1 N3 N2 N1
0
1
0
0
0
1
0
0
1
1
1
0
1
1
1
0
0
0
1
1
0
0
1
1
0
1
0
1
0
1
0
1
0
1
1
0
0
1
1
0
0
0
0
1
1
1
1
0
N1 := C1
N2 := C1 C2 + C1 C2
:= C1 xor C2
N3 := C1 C2 C3 + C1 C3 + C2 C3
:= C1 C2 C3 + (C1 + C2 ) C3
:= (C1 ... | https://ocw.mit.edu/courses/6-111-introductory-digital-systems-laboratory-spring-2006/09793dcc124f0226e93eb01591762630_l5_seql_verilog.pdf |
1111 (conditioned by T);
used for cascading counters
Synchronous CLR and LOAD
If CLRb = 0 then Q <= 0
Else if LOADb=0 then Q <= D
Else if P * T = 1 then Q <= Q + 1
Else Q <= Q
7
10
P
T
163
2
6
5
4
3
CLK
RCO
D
C
B
A
QD
QC
QB
QA
15
11
12
13
14
9
1
LOAD
CLR
74163 Synchronous
4-Bit Upcounter
L5: 6.111 Spring 2006
Introduc... | https://ocw.mit.edu/courses/6-111-introductory-digital-systems-laboratory-spring-2006/09793dcc124f0226e93eb01591762630_l5_seql_verilog.pdf |
Introductory Digital Systems Laboratory
19
Simulation
Simulation
Notice the glitch on RCO!
L5: 6.111 Spring 2006
Introductory Digital Systems Laboratory
20
Output Transitions
Output Transitions
(cid:132) Any time multiple bits
change, the counter output
needs time to settle.
(cid:132) Even though all flip-flops
sh... | https://ocw.mit.edu/courses/6-111-introductory-digital-systems-laboratory-spring-2006/09793dcc124f0226e93eb01591762630_l5_seql_verilog.pdf |
fine up to 8’b11101111:
VDD
1 1 1 1
QA QB QC QD
T
‘163
RCO
DA DB DC DD
CL LD
P
VDD
1
P
0 1 1 1
QA QB QC QD
T
‘163
RCO
DA DB DC DD
CL LD
0 0 0 0
QA QB QC QD
T
‘163
RCO
DA DB DC DD
0
P
CL LD
CLK
Problem at 8’b11110000: one of the RCOs is now stuck high for 16 cycles!
VDD
0 0 0 0
QA QB QC QD
T
‘163
RCO
DA DB DC DD
CL LD
P... | https://ocw.mit.edu/courses/6-111-introductory-digital-systems-laboratory-spring-2006/09793dcc124f0226e93eb01591762630_l5_seql_verilog.pdf |
24
Summary
Summary
(cid:132) Use blocking assignments for combinational
always blocks
(cid:132) Use non-blocking assignments for sequential
always blocks
(cid:132) Synchronous design methodology usually used in
digital circuits
(cid:134)Single global clocks to all sequential elements
(cid:134)Sequential elements al... | https://ocw.mit.edu/courses/6-111-introductory-digital-systems-laboratory-spring-2006/09793dcc124f0226e93eb01591762630_l5_seql_verilog.pdf |
OpenCourseWare
1. Overview and some basics
9 August 2006
Spoken language conveys not only words, but a wide range of other information about timing,
intonation, prominence, phrasing, voice quality, rhythm etc. that is often collectively called
spoken prosody. These aspects of an utterance are sometimes called supr... | https://ocw.mit.edu/courses/6-911-transcribing-prosodic-structure-of-spoken-utterances-with-tobi-january-iap-2006/0986334a8a593bba602eaaf8ab13de2f_chap1.pdf |
. 1992, Lehiste 1987). On the one hand, this string can be produced with
a prominence on broke and a phrase boundary just after that word, corresponding to the
orthographic representation It BROKE, out in WASHington. <broke1> On the other hand, it can
be produced with a prominence on out and a phrase boundary just a... | https://ocw.mit.edu/courses/6-911-transcribing-prosodic-structure-of-spoken-utterances-with-tobi-january-iap-2006/0986334a8a593bba602eaaf8ab13de2f_chap1.pdf |
the intonation of an utterance, i.e.
changes in the pitch that are caused by changes in the frequency of vibration of the vocal folds,
often called f0. An f0 that is high in a speaker’s range on a salient syllable can mark a pitch
accent, but so can a lower (but still high-for-this-speaker... | https://ocw.mit.edu/courses/6-911-transcribing-prosodic-structure-of-spoken-utterances-with-tobi-january-iap-2006/0986334a8a593bba602eaaf8ab13de2f_chap1.pdf |
. At this writing, the nature of
variation for prosodic categories (and thus for ToBI labels) is not fully understood, and this
results in some reasonable disagreement in prosodic parses in some renditions of some utterances,
particularly in spontaneous speech. ToBI labelling is a new endeavor compared to, say, phon... | https://ocw.mit.edu/courses/6-911-transcribing-prosodic-structure-of-spoken-utterances-with-tobi-january-iap-2006/0986334a8a593bba602eaaf8ab13de2f_chap1.pdf |
for the needs of particular sites.) The four
labelling tiers each appear in their own window:
(1) the Tone tier, for transcribing tonal events
(2) the Orthographic tier, for transcribing words
(3) the Break-Index tier, for transcribing boundaries between words
(4) the Miscellaneous tier, for recording addition... | https://ocw.mit.edu/courses/6-911-transcribing-prosodic-structure-of-spoken-utterances-with-tobi-january-iap-2006/0986334a8a593bba602eaaf8ab13de2f_chap1.pdf |
a constriction in the vocal tract), and zero (or nearly zero) when there is no speech
signal.
The rate of vibration of the vocal folds is what we hear as the pitch, and this is represented in the
second window as a semi-continuous blue line superimposed on a different representation of the
speech signal, the spectr... | https://ocw.mit.edu/courses/6-911-transcribing-prosodic-structure-of-spoken-utterances-with-tobi-january-iap-2006/0986334a8a593bba602eaaf8ab13de2f_chap1.pdf |
perienced ToBI labellers often use the spectrogram to help find the location of
successive sounds, syllables and words. This in turn helps them to keep track of
where changes in the pitch track occur across the words and syllables of the utterance.
Learning how to use this information may take some time, so at the b... | https://ocw.mit.edu/courses/6-911-transcribing-prosodic-structure-of-spoken-utterances-with-tobi-january-iap-2006/0986334a8a593bba602eaaf8ab13de2f_chap1.pdf |
end of each word for the subjective strength of its association with the next word, on a scale from
0 (for weakest perceived boundary/strongest perceived conjoining, as in doncha for don’t you) to
4 (for the most disjoint boundary, i.e. at the end of the highest-level intonationally marked
phrase). These categories ... | https://ocw.mit.edu/courses/6-911-transcribing-prosodic-structure-of-spoken-utterances-with-tobi-january-iap-2006/0986334a8a593bba602eaaf8ab13de2f_chap1.pdf |
correspond to which words in
the utterance.
The Miscellaneous tier is the bottom white box in this display. It is essentially a 'comment' tier
that can be used to mark events such as breaths, coughs, laughter, long silences and other non-
speech events. These are traditionally marked with angle brackets (e.g. <cough... | https://ocw.mit.edu/courses/6-911-transcribing-prosodic-structure-of-spoken-utterances-with-tobi-january-iap-2006/0986334a8a593bba602eaaf8ab13de2f_chap1.pdf |
of prosody which are
predictable from other parts of the transcription or from auxiliary tools, such as dictionaries, that
can be used to determine the location of lexical stress within words.
The categorical aspects of prosody which we try to capture completely (according to the first
principle) are of two types. ... | https://ocw.mit.edu/courses/6-911-transcribing-prosodic-structure-of-spoken-utterances-with-tobi-january-iap-2006/0986334a8a593bba602eaaf8ab13de2f_chap1.pdf |
aspect of prosody which we leave out (in accordance with the third principle)
because it should be fairly predictable is the marking of the lexically stressed and unstressed
syllables within each word. By this level of stress we mean the word-internal alternation
between more stressed and less stressed syllables, wh... | https://ocw.mit.edu/courses/6-911-transcribing-prosodic-structure-of-spoken-utterances-with-tobi-january-iap-2006/0986334a8a593bba602eaaf8ab13de2f_chap1.pdf |
MIT OpenCourseWare
http://ocw.mit.edu
6.013/ESD.013J Electromagnetics and Applications, Fall 2005
Please use the following citation format:
Markus Zahn, 6.013/ESD.013J Electromagnetics and Applications, Fall
2005. (Massachusetts Institute of Technology: MIT OpenCourseWare).
http://ocw.mit.edu (accessed MM DD, YYYY... | https://ocw.mit.edu/courses/6-013-electromagnetics-and-applications-fall-2005/099a51f799422c0747c7e8d691e42401_lec9.pdf |
��� 0
k ¯ × k ¯ × E ¯ˆ = k ¯ k ¯ · E ¯ˆ − E¯ˆ(k ¯ · k¯) = ωµ k ¯ × H ¯ˆ = −ω2�µE ¯ˆ
�
�
2
|k¯ | = kx
2 = ω2�µ
A ¯ × (B ¯ × C¯) = B¯(A ¯ C¯) − C¯(A ¯ B¯)
·
2 + ky
·
2 + kz
S ¯ˆ =
ˆ¯
S =
1 ¯ˆ H¯ˆ ∗, H ¯ˆ =
E ×
2
�
1
ωµ
(¯ E¯ˆ)
k ×
ˆ
E ¯ ×
1
2
�
1
k × Eˆ∗ =
¯
¯
ωµ
1
2ωµ
�
¯ E ¯ · E¯∗) − ¯ Eˆ · k¯)
k( ... | https://ocw.mit.edu/courses/6-013-electromagnetics-and-applications-fall-2005/099a51f799422c0747c7e8d691e42401_lec9.pdf |
j(ωt−kxr x+kzrz)¯iy
�
H¯ r = Re
�
�
Eˆr (cos(θr)¯ix + sin(θr)¯iz)ej(ωt−kxr x+kzrz)
η
Boundary conditions require that
kxr = k sin(θr), kzr = k cos(θr)
2
From Electromagnetic Field Theory: A Problem Solving Approach, by Markus Zahn, 1987. Used with permission.
Eˆy(x, z = 0) = 0 = Eˆyi(x, z = 0) + Eˆyr(x, z = 0... | https://ocw.mit.edu/courses/6-013-electromagnetics-and-applications-fall-2005/099a51f799422c0747c7e8d691e42401_lec9.pdf |
ix − sin(θ) e−jkz z + e +jkz z ¯iz ej(ωt−kxx)
� �
�
�
�
�
�
= 2Ei [cos(θ) sin(kzz) sin(ωt − kxx)¯ix − sin(θ) cos(kzz) cos(ωt − kxx)¯iz]
H ¯ = Re
�
ˆ �
Ei e−jkz z + e +jkz z ej(ωt−kxx)¯iy
η
�
�
=
2Ei cos(kzz) cos(ωt − kxx)¯iy
η
Kx(x, z = 0) = Hy(x, z = 0) =
2Ei cos(ωt − kxx)
η
σs(x, z = 0) = −�Ez(x, z = 0) =... | https://ocw.mit.edu/courses/6-013-electromagnetics-and-applications-fall-2005/099a51f799422c0747c7e8d691e42401_lec9.pdf |
= −
Eˆt cos(θt)e−jkxtx
1
η2
kxi = kxr = kxt ⇒ k1 sin(θi) = k1 sin(θr) = k2 sin(θt)
θi = θr
k1
k2
sin(θt) =
sin(θi) =
ωc2
�
ωc1
�
sin(θi) =
c2
c1
sin(θi)
(Snell’s Law)
Index of refraction:
Reflection Coefficent:
R =
Transmission Coefficent: T =
Eˆt
Eˆi
B. Brewster’s Angle of No Reflection
�rµr
√
√
η2
√
... | https://ocw.mit.edu/courses/6-013-electromagnetics-and-applications-fall-2005/099a51f799422c0747c7e8d691e42401_lec9.pdf |
)
c1
�
2
2c2
η1
sin2(θi)
2
c1
�
µ1 �1µ1
��
�1 �2µ2
��
sin2(θi)
2 cos2(θt) = η1
�
− η2 = η1
2
�
2 − η2
2
−
µ2
�2
=
µ1
�1
−
µ2
�2
sin2(θi) = sin2(θB) =
�
µ
1 − �
2
1
µ
1
2
� �2
µ
1 − µ
1
2
θB is called the Brewster angle. There is no Brewster angle for TE polarization if
µ1 = µ2.
If c2 > c1,
... | https://ocw.mit.edu/courses/6-013-electromagnetics-and-applications-fall-2005/099a51f799422c0747c7e8d691e42401_lec9.pdf |
1
1 + �1
�2
θB + θt =
π
2
+ 1
⇒
θC > θB
1
sin2(θB )
=
1
sin2(θC )
8 | https://ocw.mit.edu/courses/6-013-electromagnetics-and-applications-fall-2005/099a51f799422c0747c7e8d691e42401_lec9.pdf |
6 . 2 7 0 : A U T O N O M O U S R O B O T
D E S I G N C O M P E T I T I O N
• Assignment 2: General
Comments
• More on sensors
Servos
•
• RF receiver
• Robot control and state
machines
Threads
Assignment 3 handed out
•
•
LECTURE3: Advanced Techniques
Delinquent Teams
• Assignment 2 teams not finished:
–... | https://ocw.mit.edu/courses/6-270-autonomous-robot-design-competition-january-iap-2005/09c931ae703e564cd4a5f3f559d49987_lecture3_slides.pdf |
• p = 1800 W
– Thirty 60-watt light bulbs
• Lesson: ensure battery
leads are well-insulated!
i
v
R
Phototransistors
• It’s an art
• Need to figure out an effective way of reading the color
off the board or object
– Factors: glossiness, ambient lighting
– It’s not really color; it’s grayscale
– Contest nigh... | https://ocw.mit.edu/courses/6-270-autonomous-robot-design-competition-january-iap-2005/09c931ae703e564cd4a5f3f559d49987_lecture3_slides.pdf |
kΩ Pull-Up
• Cut traces
5 4
2
(make
sure you
know
where!)
Distance Sensor
• Range: 15-150 cm
– 6-60 in
Distance Sensor
• You probably
don’t need more
than 3
• But if you’re really
thatneedy, cut
port 0 or 1
IN_16
IN_17
IN_18
IN_19
IN_20
IN_21
IN_22
IN_23
v = 5 V
R = 47 kΩ
to ADC
VDD
IN_0
GND ... | https://ocw.mit.edu/courses/6-270-autonomous-robot-design-competition-january-iap-2005/09c931ae703e564cd4a5f3f559d49987_lecture3_slides.pdf |
match begins, and stop it when the
match ends
Voting Information
• rf_vote_red
• rf_vote_green
• rf_vote_winner
– You need some way of determining a winner
when there is a tie
– All automagically updated
Position Information
• rf_x0, rf_y0
• rf_x1, rf_y1
– Tell you the x and y coordinates of the two robots
–... | https://ocw.mit.edu/courses/6-270-autonomous-robot-design-competition-january-iap-2005/09c931ae703e564cd4a5f3f559d49987_lecture3_slides.pdf |
at a given point in
time
– Each state has predefined outputs
– Transitions to other states depend on inputs
• Why?
– Effective way of thinking about your strategy
– Define what to do for any combination of inputs
Implementing a State Machine
• Each action is a state
– Moving forward
– Turning
• Actuators are o... | https://ocw.mit.edu/courses/6-270-autonomous-robot-design-competition-january-iap-2005/09c931ae703e564cd4a5f3f559d49987_lecture3_slides.pdf |
on
off
drive
wheel
steering
wheel
steering
and drive
wheel
phototransistor
LED
wall follow
w all follo w
Drive Mechanisms
• Differential Drive
• Synchro Drive (servos)
• Rack-and-Pinion Drive (car)
• Independent Drive (gearboxes; Assignment 2)
differential
steering
synchro
drive
wheel
steering
wh... | https://ocw.mit.edu/courses/6-270-autonomous-robot-design-competition-january-iap-2005/09c931ae703e564cd4a5f3f559d49987_lecture3_slides.pdf |
charge
phototransistor
LED
Shaft Encoding
• Works better on some ports:
– Ports 7 and 8 have hardware counters (faster, more
accurate)
– Others use software counters
– If you need more than 2, try using ports 2-6
• Both wheels may not turn at same speed
• Use revolutions for feedback
• Determine difference in... | https://ocw.mit.edu/courses/6-270-autonomous-robot-design-competition-january-iap-2005/09c931ae703e564cd4a5f3f559d49987_lecture3_slides.pdf |
to see these things in advance
• Test and debug incrementally
Hints
• Test sensors before mounting
• Test small pieces of code before
combining into larger procedures
• Use the LCD screen
• Remember mechanical reliability
Error Detection
• Your robot will mess up
• How can it find out what’s wrong?
• Timeouts ... | https://ocw.mit.edu/courses/6-270-autonomous-robot-design-competition-january-iap-2005/09c931ae703e564cd4a5f3f559d49987_lecture3_slides.pdf |
true) {
go forward
wait until sensor
pressed
go backward
wait until sensor
pressed
}
}
main() {
while (true) {
while (vote is tied)
play tone 1
while (red is winning)
play tone 2
while (green is winning)
play tone 3
}
}
Example
move() {
while (true) {
go forward
wait until sensor
pressed
go backward
wait unti... | https://ocw.mit.edu/courses/6-270-autonomous-robot-design-competition-january-iap-2005/09c931ae703e564cd4a5f3f559d49987_lecture3_slides.pdf |
{
/* check for anything (like sensor inputs) */
if (you_really_need_to_leave_the_while_loop)
break;
}
Your Winning Strategy
• Sufficient sensors and AI to determine location of robot
• Be able to react to potential problems that the robot
might face
• Be aware of your limitations
– Amount of LEGO
– Power and ... | https://ocw.mit.edu/courses/6-270-autonomous-robot-design-competition-january-iap-2005/09c931ae703e564cd4a5f3f559d49987_lecture3_slides.pdf |
Heinrich Hencky (1885-1952)
• Natural logarithmic strain measure;
•
Biography:
(
!(t) = ln L(t) L0
)
High School (Humanistic Gymnasium),
Speyer am Rhein, Germany
Technical University, Munich; Dipl. Eng. 1908
Technical University, Darmstadt; D. Eng. 1913
•
Professor of Mechanical Engineering, M.I.T. 1930-
1933.
... | https://ocw.mit.edu/courses/2-341j-macromolecular-hydrodynamics-spring-2016/09d1e70558b8678c9712932a5fc55beb_MIT2_341JS16_Lec02-slides.pdf |
Elasticity (normal stress differences)
“Second order fluids” (SOF)
Boger fluids
$11 %$22
!" 2
!1( !") #
• Fluid Memory (stress relaxation)
Relaxation time λ
!0
s
s
e
r
t
s
r
a
e
h
S
time
G(t) =
!12 (t)
"0
~ G0e#t $
3
Natural Time Scale of Complex Fluids
• Natural time scale
• The Deborah number is a dimensio... | https://ocw.mit.edu/courses/2-341j-macromolecular-hydrodynamics-spring-2016/09d1e70558b8678c9712932a5fc55beb_MIT2_341JS16_Lec02-slides.pdf |
8.701
Introduction to Nuclear
and Particle Physics
Markus Klute - MIT
0. Introduction
0.5 Early History and
People in Nuclear and
Particle Physics
1
Early Developments in Nuclear & Particle Physics
~1820s: geologists and biologists have come to believe that the Earth is much older than 10s of
thousands of year, p... | https://ocw.mit.edu/courses/8-701-introduction-to-nuclear-and-particle-physics-fall-2020/09d46adb32b9af809d01687fc861ee9b_MIT8_701f20_lec0.5.pdf |
1856-1950
Marie Curie
1867-1934
Pierre Curie
1859-1906
3
Early Developments in Nuclear & Particle Physics
1899: Paul Villard discovers a third component of radiation from uranium
and calls them ɣ rays.
1901: The Curie’s measure the energy emitted by radioactive elements and discover that one gram of
radium gives off... | https://ocw.mit.edu/courses/8-701-introduction-to-nuclear-and-particle-physics-fall-2020/09d46adb32b9af809d01687fc861ee9b_MIT8_701f20_lec0.5.pdf |
© Source unknown. All rights reserved. This content is
excluded from our Creative Commons license. For more information,
see https://ocw.mit.edu/fairuse.
Eugene Marsden
1882-1936
Paul Dirac
1902-1984
This photo is in the public domain.
Early Developments in Nuclear & Particle Physics
1931: Pauli and Fermi propose t... | https://ocw.mit.edu/courses/8-701-introduction-to-nuclear-and-particle-physics-fall-2020/09d46adb32b9af809d01687fc861ee9b_MIT8_701f20_lec0.5.pdf |
he
1906-2005
7
MIT OpenCourseWare
https://ocw.mit.edu
8.701 Introduction to Nuclear and Particle Physics
Fall 2020
For information about citing these materials or our Terms of Use, visit: https://ocw.mit.edu/terms. | https://ocw.mit.edu/courses/8-701-introduction-to-nuclear-and-particle-physics-fall-2020/09d46adb32b9af809d01687fc861ee9b_MIT8_701f20_lec0.5.pdf |
:28)(cid:16)%(cid:31)
(cid:2)(cid:3)(cid:4)(cid:4)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10)(cid:11)(cid:12)(cid:8)(cid:6)(cid:13)
:
2(cid:23)(cid:12)(cid:4)(cid:11)(cid:2)(cid:21)(cid:11)(cid:12)(cid:3)(cid:11)(cid:13)(cid:14)(cid:4)(cid:10)(cid:15)(cid:10)(cid:16)(cid:12)(cid:4)(cid:10)(cid:6)(cid:3)(cid:11)... | https://ocw.mit.edu/courses/6-0002-introduction-to-computational-thinking-and-data-science-fall-2016/0a353b26f1c6bd161b28b3f249aa05d1_MIT6_0002F16_lec1.pdf |
cid:29)(cid:10)(cid:9)(cid:10)(cid:20)(cid:3)(cid:9)/
def greedy(items, maxCost, keyFunction):
itemsCopy = sorted(items, key = keyFunction,
result = []
totalValue, totalCost = 0.0, 0.0
reverse = True)
for i in range(len(itemsCopy)):
if (totalCost+itemsCopy[i].getCost()) <= maxCost:
result.append(itemsCopy[i])
totalCost... | https://ocw.mit.edu/courses/6-0002-introduction-to-computational-thinking-and-data-science-fall-2016/0a353b26f1c6bd161b28b3f249aa05d1_MIT6_0002F16_lec1.pdf |
:2)(cid:3)(cid:4)(cid:4)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10)(cid:11)(cid:12)(cid:8)(cid:6)(cid:13)
&(cid:4)
MIT OpenCourseWare
https://ocw.mit.edu
6.0002 Introduction to Computational Thinking and Data Science
Fall 2016
For information about citing these materials or our Terms of Use, visit: https://ocw.... | https://ocw.mit.edu/courses/6-0002-introduction-to-computational-thinking-and-data-science-fall-2016/0a353b26f1c6bd161b28b3f249aa05d1_MIT6_0002F16_lec1.pdf |
AN EXPOSITION OF BRETAGNOLLE AND MASSART’S
PROOF OF THE KMT THEOREM FOR THE UNIFORM
EMPIRICAL PROCESS
R. M. Dudley
February 23, 2005
Preface
These lecture notes, part of a course given in Aarhus, August 1999, treat the classical
empirical process defined in terms of empirical distribution functions. A proof, exp... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/0a472bc75921bd9a7a12c37bb261d572_bretagn_massart.pdf |
. . . . . . . .
1.4 Proof of Lemma 1.4
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5 Proof of Lemma 1.2
. . . . . . . . . . . . . . . . . . . . . . .
1.6
1.7 Proof of Theorem 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.8 Another way of defining the KMT construct... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/0a472bc75921bd9a7a12c37bb261d572_bretagn_massart.pdf |
0 ≤ t ≤ 1, with EB(t) = 0 and EB(t)B(u) = t(1 − u) for 0 ≤ t ≤ u ≤
1. Donsker (1952) proved (neglecting measurability problems) that αn(t) converges in law to
a Brownian bridge B(t) with respect to the sup norm. Koml´
ady (1975)
stated a sharp rate of convergence, namely that on some probability space there exist Xi... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/0a472bc75921bd9a7a12c37bb261d572_bretagn_massart.pdf |
with specific constants, Theorem
1.1 below. Bretagnolle and Massart’s proof was rather compressed and some readers have
had difficulty following it. Cs¨ o and Horv´
ath (1993), pp. 116-139, expanded the proof while
making it more elementary and gave a proof of Lemma 1.4 for n ≥ n0 where n0 is at least 100.
The purpose... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/0a472bc75921bd9a7a12c37bb261d572_bretagn_massart.pdf |
Tusn´ady’s lemmas
The main result of the present chapter is:
Theorem 1.1. (Bretagnolle and Massart) The approximation (1.1) of the empirical process
by the Brownian bridge holds with c = 12, K = 2 and λ = 1/6 for n ≥ 2.
The rest of this chapter will give a proof of the theorem. In a preprint, Rio (1991, Theorem
5.... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/0a472bc75921bd9a7a12c37bb261d572_bretagn_massart.pdf |
and 0 < t < 1 let
F −1(t) := inf{x : F (x) ≥ t}. Here is one of Tusn´ady’s lemmas (Lemma 4 of Bretagnolle and
Massart (1989)).
Lemma 1.2. Let Φ be the standard normal distribution function and Y a standard normal
random variable. Let Φn be the distribution function of B(n, 1/2) and set Cn := Φ−1(Φ(Y )) −
n/2. Then ... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/0a472bc75921bd9a7a12c37bb261d572_bretagn_massart.pdf |
that 0 ≤ j ≤ n and n + j is even, we
have
√
P (βn ≥ (n + j)/2) ≥ P ( nY /2 ≥ n(1 − 1 − j/n)),
P (βn ≥ (n + j)/2) ≤ P ( nY /2 ≥ (j − 2)/2).
(1.5)
(1.6)
√
(cid:4)
Remarks. The restriction that n + j be even is not stated in the formulation of the lemma
by Bretagnolle and Massart (1989), but n + j is always even ... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/0a472bc75921bd9a7a12c37bb261d572_bretagn_massart.pdf |
lemma
for n ≥ 8. We have An = exp((12n)−1 − θn/(360n3 )) where 0 < θn < 1, see Whittaker and
Watson (1927), p. 252 or Nanjundiah (1959). Then by Taylor’s theorem with remainder,
(cid:5)
An =
1 +
1
12n
+
1
288n2 +
1
6(12n)3 φne
(cid:6)
1/12n exp(−θn/(360n
3 ))
where 0 < φn < 1. Next,
(cid:7)
(cid:5)
βn... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/0a472bc75921bd9a7a12c37bb261d572_bretagn_massart.pdf |
2
288n
(cid:6)
.
= 1 +
1
24n
−
1
30n2
1 +
1
12n
+
1
288n
2
Thus lim inf n→∞ βn ≥ 1 and βn → 1 as n → ∞. To prove βn ≥ βn+1 for n ≥ 8 it will suffice
to show that
1 +
1
24(n + 1)
+
e1/108
6 · 144n2
≤ 1 +
1
24n
−
1
30n2
(cid:7)
1
1 + +
96
(cid:8)
1
288 · 82
3
or
e1/108
6 · 144n2
+
1
30n2
(... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/0a472bc75921bd9a7a12c37bb261d572_bretagn_massart.pdf |
(n
k ) will be
approximated via Stirling’s formula with correction terms as in Lemma 1.5. To that end, let
:= 0 for n + i odd. The factorials in (n
:= i/n. Define pni
k )/2n and xi
CS(u, v, w, x, n) :=
1 + u/(12n)
.
(1 + v/[6n(1 − x)])(1 + w/[6n(1 + x)])
By Lemma 1.5, we can write for 0 ≤ i < n and n + i even
pn... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/0a472bc75921bd9a7a12c37bb261d572_bretagn_massart.pdf |
n) ≤ CS(βn, βn, βn, 0, n)
≤ CS(1, 1, 1, 0, n) = 1 +
(cid:5)
(cid:6) (cid:7)
1
12n
1
1 + +
3n
(cid:8)−1
.
1
2
36n
It will be shown next that log(1 + y) − 2 log(1 + 2y) ≤ −3y + 7y2/2 for y ≥ 0. Both sides
vanish for y = 0. Differentiating and clearing fractions, we get a clearly true inequality. Setting
y := 1/(... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/0a472bc75921bd9a7a12c37bb261d572_bretagn_massart.pdf |
. Let j2 ≥ 2n, in other words xj ≥ 2/n. Recall that for t > 0 we have P (Y > t) ≤
(t 2π)−1 exp(−t2/2), e.g. Dudley (1993), Lemma 12.1.6(a). Then (1.10) follows easily when
j = n and n ≥ 5. To prove it for j = n − 2 it is enough to show
n(2 − log 2) − 4 2n + log(n + 1) + 4 + log[2 2π( n − 2)] ≥ 0, n ≥ 5.
√
√ √
√
Th... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/0a472bc75921bd9a7a12c37bb261d572_bretagn_massart.pdf |
1.13)
(1.14)
Clearly fn > 0. To see that fn(x) is decreasing in x for 2/n ≤ x ≤ 1 − 4/n, note that
(cid:13)
2(1 − x)f (cid:4)
√
√
n/fn = 1 − 4n[ 1 − x − 1 + x],
√
so fn is decreasing where 1 − x − (1 − x) > 1/(4n). We have √y − y ≥ y for y ≤ 1/4, so
√
y − y > 1/(4n) for 1/(4n) < y ≤ 1/4. Let y := 1 − x. Also ... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/0a472bc75921bd9a7a12c37bb261d572_bretagn_massart.pdf |
√
J(x) := 4(1 − 1 − x)2 − g(x).
5
(1.15)
(1.16)
(1.17)
Then J is increasing for 0 < x < 1, since its first and second derivatives are both 0 at 0, while
its third derivative is easily checked to be positive on (0, 1). In light of (1.9), to prove (1.16) it
suffices to show that
(cid:5)
1 +
(cid:6)
1
12n
nJ (x... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/0a472bc75921bd9a7a12c37bb261d572_bretagn_massart.pdf |
) (0) = 7/2, so that
the right side of (1.19) is the Taylor series of J around 0 through fourth order. One then shows
straightforwardly that J (5) (x) > 0 for 0 ≤ x < 1.
It follows since nx2 ≥ 2 and n ≥ 8 that nJ (x)/2 ≥ x/2 + 7/24n. Let K(x)
:=
:= (K(x) − 1)/x2 . We will next see that κ(·) is decreasing
exp(x/2)/... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/0a472bc75921bd9a7a12c37bb261d572_bretagn_massart.pdf |
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