mit mathematics for computer science lecture notes
Mathematics for Computer Science pot
Ngày tải lên :
05/03/2014, 23:20
... important proof technique in computer science. Generally,
induction is used to prove that some statement holds for all natural values of a variable.
For example, here is a classic formula:
46 Number ... 1) for all n ∈ N.
By the principle of induction, P (n) is true for all n ∈ N, which proves the claim.
This proof would look quite mysterious to anyone not privy to the scratchwork we did
beforehand. ... divisibility hold.
1. If a | b, then a | bc for all c.
2. If a | b and b | c, then a | c.
3. If a | b and a | c, then a | sb + tc for all s and t.
4. For all c = 0, a | b if and only if ca | cb.
Proof....
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Mathematics for Computer Science pptx
Ngày tải lên :
23/03/2014, 23:20
... — page i — #1
Mathematics for Computer Science
revised Thursday 10
th
January, 2013, 00:28
Eric Lehman
Google Inc.
F Thomson Leighton
Department of Mathematics
and the Computer Science and AI ... proposition for each
possible set of truth values for the variables. For example, the truth table for the
proposition “P AND Q” has four lines, since there are four settings of truth values
for the ... b for some s 2 Sg:
For example, if we let Œr; s denote set of numbers in the interval from r to s on the
real line, then f
1
.Œ1; 2/ D Œ1=4; 1.
For another example, let’s take the “search for...
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concrete mathematics a foundation for computer science phần 1 pdf
Ngày tải lên :
14/08/2014, 04:21
... defined for
SO
= ao;
all n 3 0
.)
S,
=
S-1
+ a,,
for n > 0.
(2.6)
Therefore we can evaluate sums in closed form by using the methods we
learned in Chapter 1 to solve recurrences in closed form.
For ... much happier. That is, we’d like a nice, neat,
“closed form” for
T,,
that lets us compute it quickly, even for large n. With
a closed form, we can understand what
T,,
really is.
So how do ... a,
6,
and y and trying to find a closed form for the more general recurrence
f(1) =
cc;
f(2n) = 2f(n) + fi,
for
n
3
1;
(1.11)
f(2n+1)=2f(n)+y,
for
n
3
1.
(Our original recurrence had...
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concrete mathematics a foundation for computer science phần 2 pptx
Ngày tải lên :
14/08/2014, 04:21
... exponents for rising factorial powers, analogous to
(2.52)? Use this to define
XC”.
10
The text derives the following formula for the difference of a product:
A(uv)
= uAv + EvAu.
How can this formula ... circle.
34 Let f(n) =
Et=,
[lgkl.
Find a closed form for f(n) , when n 3 1.
L
Provethatf(n)=n-l+f([n/2~)+f(~n/Z])foralln~l.
35
Simplify the formula \(n + 1
)‘n!
e]
mod n.
Simplify it,
but
36 ... to understand the
general formula for the sum of a geometric progression:
t
a<k<b
for c # 1.
Every time we encounter a function f that might be useful as a closed
form, we can compute its...
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concrete mathematics a foundation for computer science phần 3 ppsx
Ngày tải lên :
14/08/2014, 04:21
... (mod 4).
QED for the case m = 12.
QED: Quite
Easily
So far we’ve proved our congruence for prime m, for m = 4, and for m =
Done.
12. Now let’s try to prove it for prime powers. For concreteness ... reduced).
It is defined recursively by starting with
For N > 1, we form ?$,+I by inserting two symbols just before the kNth
symbol of ?N, for all k > 0. The two inserted symbols are
k-l
-
... -1;
p(pk)
= 0 for k > 1.
Therefore by (4.52), we have the general formula
ifm=pjpz p,;
if m is divisible by some p2.
(4.57)
That’s
F.
If we regard (4.54) as a recurrence for the function...
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Tài liệu Discrete Math for Computer Science Students doc
Ngày tải lên :
21/02/2014, 09:20
... original formula for q. Recall that our proof of the formula we had in
Exercise 1.4-5 did not explain why the product of three factorials appeared in the denominator,
it simply proved the formula ... distinct elements. There are n choices for the first number in the list.
For each way of choosing the first element, there are n −1choices for the second. For each choice
of the first two elements, ... members (assuming that the president is not on the advisory
board)?
10. By using the formula for
n
2
is is straightforward to show that
n
n − 1
2
=
n
2
(n − 2).
However this proof just uses...
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Vic broquard c++ for computer science and engineering
Ngày tải lên :
19/03/2014, 14:14
... Introduction to Programming
3
Computers have a fixed set of instructions that they can perform for us. The specific
instruction set depends upon the make and model of a computer. However, these instructions ... that the computer
always attempts to do precisely what you tell it to do. Say, for example, you tell the computer to
divide ten by zero, it tries to do so and fails at once. If you tell the computer ... instructions that tell the computer
every step to take in the proper sequence in order to solve a problem for a user. A programmer
is one who writes the computer program. When the computer produces a...
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Logic For Computer Science Foundations of Automatic Theorem Proving potx
Ngày tải lên :
29/03/2014, 09:20
... proof theory and procedures for constructing formal proofs of for-
mulae algorithmically.
This book is designed primarily for computer scientists, and more gen-
erally, for mathematically inclined ... Theorem for Formulae in NNF, 355
7.6.1 Skolem-Herbrand-Găodels Theorem in Unsatisability
Form, 355
7.6.2 Skolem Normal Form, 357
7.6.3 Compound Instances, 359
7.6.4 Half of a Herbrand-like Theorem for ... proposition is a Horn
formula iff it is a conjunction of basic Horn formulae.
(a) Show that every Horn formula A is equivalent to a conjunction of
distinct formulae of the form,
P
i
, or
ơP
1
...
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norbury. general relativity and cosmology for undergraduates (wisconsin lecture notes, 1997)(116s)
Ngày tải lên :
24/04/2014, 17:07
... (1.13)
It is customary to combine the equations of state into the form
p =
γ
3
ρ (1.14)
where γ ≡ 1 for radiation and γ ≡ 0 for matter. These equations of state
are needed in order to discuss ... the form A
à
, rank three A
à
etc. A mixed tensor, e.g. A
à
,
is partly covariant and partly contravariant.
In order for an object to be called a tensor it must satisfy the tensor
transformation ... itself which expands [19]. Even though this
equation is derived for matter, it is also true for radiation. (In fact it is also
true for vacuum, with Λ ≡ 8πGρ
vac
, where Λ is the cosmological constant
and...
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Mathematics for Life Science and Medicine docx
Ngày tải lên :
27/06/2014, 10:20
... Li
Department of Mathematics and
Physics,
Air Force Engineering University,
China
jianq_li@263.net
Wanbiao Ma
Department of Mathematics and
Mechanics,
School of Applied Science,
University of Science ... Sports, Science and Technology,
The Japanese Society for Mathematical Biology, The Society of Population
Ecology, Mathematical Society of Japan, Japan Society for Industrial and
Applied Mathematics, ... Competition: A Survey
H.I. Freedman 207
Index 225
Y. Takeuchi Y. Iwasa K. S ato (Eds.)
Mathematics
for Life Science
and Medicine
With 31 Figures
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