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the definition of polynomial
consisting of or characterized by two or more names or terms.
(in one variable) an
consisting of the sum of two or more terms each of
is the product of a constant and a variable raised to an integral power:
ax ^2
is a polynomial, where
a, b,
are constants and
is a variable.
a similar expression in more than one variable, as 4 x ^2 y ^3 − 3 xy + 5 x + 7.
Now Rare.. Also called multinomial.
any expression consisting of the sum of two or more terms, as 4
x ^3
+ cos
a polynomial
or term.
Biology. a species name containing more than two terms.
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Transforming between Mathematical Representations of Systems, Overview
This document is a compilation of all of the pages relating to the transformation between different mathematical representations of systems. It is useful for printing.
Transformation Description
CDE → 1DE Coupled Differential Equation to Single Differential Equation
1DE ↔ TF Single Differential Equation to/from Transfer Function
1DE ↔ SS Single Differential Equation to/from State Space
1DE ↔ PZ Single Differential Equation to/from Pole Zero
1DE ↔ SFG Single Differential Equation to/from Signal Flow Graph/Block Diagram
TF ↔ SS Transfer Function to/from State Space
TF ↔ PZ Transfer Function to/from State Space
TF ↔ SFG Transfer Function to/from State Space
SS ↔ SS State Space to/from State Space
SS ↔ PZ State Space to/from Pole Zero
SS ↔ SFG State Space to/from Signal Flow Graph/Block Diagram
PZ ↔ SFG Pole Zero to/from Signal Flow Graph/Block Diagram
Transformation: Coupled Diff Eq → Single Diff Eq
Methods for transforming from coupled differential equations to a single n^th order differential equation were discussed on the page "System Representation by Differential Equations," example 3 and
example 4. Another example is included below. It shows how to start with a set of coupled differential equations and transform them into a single n^th order differential equation.
Transformation: Single Diff Eq ↔ Transfer Function
Single Differential Equation to Transfer Function
If a system is represented by a single n^th order differential equation, it is easy to represent it in transfer function form. Starting with a third order differential equation with x(t) as input and
y(t) as output.
To find the transfer function, first take the Laplace Transform of the differential equation (with zero initial conditions). Recall that differentiation in the time domain is equivalent to
multiplication by "s" in the Laplace domain.
The transfer function is then the ratio of output to input and is often called H(s).
Note: This notation takes increasing subscripts for the a[n] and b[n] coefficients as the power of s (or order of derivative decreases) while some references use decreasing subscripts with decreasing
power. This notation was chosen here in part because it is consistent with MatLab's use.
For the general case of an n^th order differential equation with m derivatives of the input (superscripted numbers in parentheses indicate the order of the derivative):
This can be written in even more compact notation:
Transfer Function to Single Differential Equation
Going from a transfer function to a single nth order differential equation is equally straightforward; the procedure is simply reversed. Starting with a third order transfer function with x(t) as
input and y(t) as output.
To find the transfer function, first write an equation for X(s) and Y(s), and then take the inverse Laplace Transform. Recall that multiplication by "s" in the Laplace domain is equivalent to
differentiation in the time domain.
For the general case of an n^th order transfer function:
This can be written in even more compact notation:
Example: Transforming Between Single Differential Equation and Transfer Function
Transformation: Differential Equation ↔ State Space
Given a system differential equation it is possible to derive a state space model directly, but it is more convenient to go first derive the transfer function, and then go from the transfer function
to the state space model.
Transformation: Differential Equation ↔ Pole-Zero
Given a system differential equation it is possible to derive a pole-zero model directly, but it is more convenient to go first derive the transfer function, and then go from the transfer function to
the pole-zero model.
Transformation: Differential Equation ↔ Signal Flow Graph
Given a system differential equation it is possible to derive a signal flow graph directly, but it is more convenient to go first derive the transfer function, and then go from the transfer function
to the state space model, and then from the state space model to the signal flow graph.
Transformation: Transfer Function ↔ State Space
Two of the most powerful (and common) ways to represent systems are the transfer function form and the state space form. This page describes how to transform a transfer function to a state space
representation, and vice versa. Converting from state space form to a transfer function is straightforward because the transfer function form is unique. Converting from transfer function to state
space is more involved, largely because there are many state space forms to describe a system.
Consider the state space system:
Now, take the Laplace Transform (with zero initial conditions since we are finding a transfer function):
We want to solve for the ratio of Y(s) to U(s), so we need so remove Q(s) from the output equation. We start by solving the state equation for Q(s)
The matrix Φ(s) is called the state transition matrix. Now we put this into the output equation
Now we can solve for the transfer function:
Note that although there are many state space representations of a given system, all of those representations will result in the same transfer function (i.e., the transfer function of a system is
unique; the state space representation is not).
Recall that state space models of systems are not unique; a system has many state space representations. Therefore we will develop a few methods for creating state space models of systems.
Before we look at procedures for converting from a transfer function to a state space model of a system, let's first examine going from a differential equation to state space. We'll do this first
with a simple system, then move to a more complex system that will demonstrate the usefulness of a standard technique.
First we start with an example demonstrating a simple way of converting from a single differential equation to state space, followed by a conversion from transfer function to state space.
If we try this method on a slightly more complicated system, we find that it initially fails (though we can succeed with a little cleverness).
The process described in the previous example can be generalized to systems with higher order input derivatives but unfortunately gets increasingly difficult as the order of the derivative increases.
When the order of derivatives is equal on both sides, the process becomes much more difficult (and the variable "D" is no longer equal to zero). Clearly more straightforward techniques are necessary.
Two are outlined below, one generates a state space method known as the "controllable canonical form" and the other generates the "observable canonical form (the meaning of these terms derives from
Control Theory but are not important to us).
Probably the most straightforward method for converting from the transfer function of a system to a state space model is to generate a model in "controllable canonical form." This term comes from
Control Theory but its exact meaning is not important to us. To see how this method of generating a state space model works, consider the third order differential transfer function:
We start by multiplying by Z(s)/Z(s) and then solving for Y(s) and U(s) in terms of Z(s). We also convert back to a differential equation.
We can now choose z and its first two derivatives as our state variables
Now we just need to form the output
From these results we can easily form the state space model:
In this case, the order of the numerator of the transfer function was less than that of the denominator. If they are equal, the process is somewhat more complex. A result that works in all cases is
given below; the details are here. For a general n^th order transfer function:
the controllable canonical state space model form is
Observable Canonical Form (OCF)
Another commonly used state variable form is the "observable canonical form." This term comes from Control Theory but its exact meaning is not important to us. To understand how this method works
consider a third order system with transfer function:
We can convert this to a differential equation and solve for the highest order derivative of y:
Now we integrate twice (the reason for this will be apparent soon), and collect terms according to order of the integral:
Choose the output as our first state variable
Looking at the right hand side of the differential equation we note that y=q[1] and we call the two integral terms q[2]:
This is our first state variable equation.
Now let's examine q[2] and its derivative:
Again we note that y=q[1] and we call the integral terms q[3]:
This is our second state variable equation.
Now let's examine q[3] and its derivative:
This is our third, and last, state variable equation.
Our state space model now becomes:
In this case, the order of the numerator of the transfer function was less than that of the denominator. If they are equal, the process is somewhat more complex. A result that works in all cases is
given below; the details are here. For a general n^th order transfer function:
the observable canonical state space model form is
There are many other forms that are possible. For example MATLAB uses a variant of the controllability canonical form in its "ssdata" function.
Transfer Function → State Space (order of numerator=order of denominator)
If the order of the numerator is equal to the order of the denominator, it becomes more difficult to convert from a system transfer function to a state space model. This document shows how to do this
for a 3rd order system. The technique easily generalizes to higher order.
Consider the third order differential transfer function:
We start by multiplying by Z(s)/Z(s) and then solving for Y(s) and U(s) in terms of Z(s). We also convert back to a differential equation.
We can now choose z and its first two derivatives as our state variables
Now we just need to form the output
Unfortunately, the third derivative of z is not a state variable or an input, so this is not a valid output equation. However, we can represent the term as a sum of state variables and outputs:
From these results we can easily form the state space model:
In this case, the order of the numerator of the transfer function was less than that of the denominator. If they are equal, the process is somewhat more complex. A result that works in all cases is
given below; the details are here.
Observable Canonical Form (OCF)
Consider the third order differential transfer function:
We can convert this to a differential equation and solve for the highest order derivative of y:
Now we integrate twice (the reason for this will be apparent soon), and collect terms according to order of the integral (this includes bringing the first derivative of u to the left hand side):
Without an justification we choose y-b[0]u as our first state variable
Looking at the right hand side of the differential equation we note that y=q[1] and we call the two integral terms q[2]:
This isn't a valid state equation because it has "y" on the right side (recall that only state variables and inputs are allowed). We can get rid of it by noting:
This is our first state variable equation.
Now let's examine q[2] and its derivative:
Again we note that y=q[1]+b[0]u and we call the integral terms q[3]:
This is our second state variable equation.
Now let's examine q[3] and its derivative:
This is our third, and last, state variable equation.
Our state space model now becomes:
Here is a good reference that does the same derivations from a different perspective: http://www.ece.rutgers.edu/~gajic/psfiles/canonicalforms.pdf
Transformation: Transfer Function ↔ Pole Zero
The pole-zero and transfer function representations of a system are tightly linked. For example consider the transfer function:
If we rewrite this in a standard form such that the highest order term of the numerator and denominator are unity (the reason for this is explained below).
This is just a constant term (b[0]/a[0]) multiplied by a ratio of polynomials which can be factored.
In this equation the constant k=b[0]/a[0]. The z[i] terms are the zeros of the transfer function; as s→z[i] the numerator polynomial goes to zero, so the transfer function also goes to zero. The p[i]
terms are the poles of the transfer function; as s→p[i] the denominator polynomial is zero, so the transfer function goes to infinity.
In the general case of a transfer function with an mth order numerator and an nth order denominator, the transfer function can be represented as:
The pole-zero representation consists of the poles (p[i]), the zeros (z[i]) and the gain term (k).
Note: now the step of pulling out the constant term becomes obvious. With the constant term out of the polynomials they can be written as a product of simple terms of the form (s-zi). This would not
be possible if the highest order term of the polynomials was not equal to one.
Often the gain term is not given as part of the representation. The nature of the behavior of the system is given by the poles and zeros (e.g., does it oscillate? decay quickly? ...), the gain term
only determines the magnitude of the response. In many cases a plot is made of the s-plane that shows the locations of the poles and zeros, and the gain term (k) is not shown. See the example below.
Transformation: Transfer Function ↔ Signal Flow Graph
Given a system transfer function it is possible to derive a signal flow graph directly, but it is more convenient to first find a space model, and then move from the state space model to the signal
flow graph.
Transformation: State Space ↔ State Space
A state variable representation of a system is not unique. In fact there are infinitely many representations. Methods for transforming from one set of state variables to another are discussed below,
followed by an example.
Consider the state space representation
We can define a new set of independent variables (i.e., T is invertible)
Though it may not be obvious we can use this new set of variables as state variables. Start by solving for q
(note: some textbooks use the matrix P=T^-1 to define the transformation)
We can now rewrite the state space model by replacing q in the original equations
Multiply the top equation by T to solve for qˆ
We recognize this as a state space representation
Transformation: State Space ↔ Pole-Zero
Given a state space representation it is possible to derive a pole-zero model directly, but it is more convenient to go first derive the transfer function, and then go from the transfer function to
the pole-zero model model.
Transformation: State Space ↔ Signal Flow Graphs
This document is unfinished. It will detail how to transform back and forth from state space and signal flow graph/block diagram representations.
Transformation: Pole-Zero ↔ Signal Flow Graph
Given a pole-zero representation of a system it is possible to derive a signal flow graph directly, but it is more convenient to go first derive the transfer function, and then go from the transfer
function to the state space model, and then from the state space model to the signal flow graph.
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[FOM] The denial of '~p'
Richard Heck rgheck at brown.edu
Thu Aug 26 16:41:43 EDT 2010
On 8/26/10 9:53 AM, Alex Blum wrote:
> One anomaly in thinking of 'p' as the denial of '~p' is that while it is
> immediately clear that it must be the case that one of the two is true. It
> is not immediately clear that this disallows them from being jointly true.
> However, it is immediately clear that this disallows, the denial of each,
> i.e.,'~p' and '~~p', from being jointly true.
I'm confused. How can it be any less clear that `p' and `~p' cannot both
be true than it is that `~p' and `~~p' cannot both be true? Surely what
makes it clear that the latter cannot both be true (in so far, with a
nod to dialetheists, it is) is the fact that one is the negation
(denial, if you wish) of the other. But that is true in the other case,
too: whether `p' is the denial of `~p' or not, surely `~p' is the denial
of `p', and so they cannot both be true.
I would have thought, moreover, that if anything isn't clear here, it is
that one of `p' and `~p' must be true. There are plenty of relatively
sane views on which that need not hold, independently even of whether `p
v ~p' must be true.
Richard Heck
More information about the FOM mailing list
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Level curves
August 21st 2010, 02:25 PM #1
May 2010
Level curves
I have to find some level curves for: $f(x,y)=1-|x|-|y|$
So, if we call $S$ at the surface given by the equation $z=f(x,y)$, then
$z=1\Rightarrow{-|x|-|y|=0}\Rightarrow{x=y=0} \therefore P(0,0,1)\in{S}$
Now, that particular case its simple, cause it gives just a point, but if I go downwards I get:
I'm not sure how to represent this. How does this look on the xy plane?
I know that:
$|y|=\begin{Bmatrix} 1-|x| & \mbox{ si }& y\geq{0}\\-1+|x| & \mbox{si}& y<0\end{matrix}$
$|x|=\begin{Bmatrix} x & \mbox{ si }& x\geq{0}\\-x & \mbox{si}& x<0\end{matrix}$
But it don't helps me to visualize the "curve". I know actually that it looks like a parallelogram, but thats because I've used mathematica to compute the surface :P I don't know how to deduce it
Write $y=\pm(1-|x|)$. Can you graph $y=1-|x|$?
A level curve for f(x)= 1- |x|- |y| is given by 1- |x|- |y|= C, where C is a constant, or |x|+ |y|= 1- C.
Now, it should be obvious that, since the left side of that is never negative, C cannot be larger than 1.
If C= 1, the equation becomes |x|+ |y|= 0 which, since neither |x| nor |y| can be negative, is only true when x= y= 0. The level curve is the single point (0, 0).
For C< 1, as always with absolute values, the simplest thing to do is to break the problem into cases.
1. If x and y are both positive, |x|+ |y|= x+ y= 1- C. That is a straight line but remember to only draw it in the first quadrant.
2. If x< 0 and y> 0 then |x|+ |y|= -x+ y= 1- C. Again, a portion of a straight line but now in the second quadrant.
3. If x< 0 and y< 0 then |x|+ |y|= -x- y= 1- C. A portion of a straight line in the third quadrant.
4. If x> 0 and y< 0 then |x|+ |y|= x- y= 1- C. A portion of a straight line in the fourth quadrant.
If you draw those four segments for one value of C, say C= 0, it should be easy to see what the other level curves are.
August 21st 2010, 06:06 PM #2
Senior Member
May 2010
Los Angeles, California
August 22nd 2010, 03:58 AM #3
MHF Contributor
Apr 2005
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Uniqueness of distance realizing geodesic in hyperbolic surface.
up vote 0 down vote favorite
Possible Duplicate:
Hyperbolic surfaces
Given a hyperbolic surface S with geodesic boundary. Let a and b be two distinct simple closed geodesic boundaries. Does there exist a unique distance realizing geodesic in S? (1) for S is a pair of
pants. (2) S is any hyperbolic surface with boundary.
If you want to refine your question, you should edit the first one. There is an "edit" link below the tags. – S. Carnahan♦ May 2 '12 at 7:16
add comment
marked as duplicate by S. Carnahan♦ May 2 '12 at 7:15
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
1 Answer
active oldest votes
For the pants, yes. In general, no. To prove this for the pants, classify all geodesic arcs and just observe the result. There are many ways to find a "no" example in the general
case; the first one that came to my mind was taking a double cover.
up vote 1 down EDIT - I see that this is a near-duplicate of a closed question. You could improve your question by giving some motivation. Reading the FAQ will be very useful in writing questions
vote accepted that get good answers. In particular please see http://mathoverflow.net/faq#whatnot
well, I have got an example of hyperbolic surface with boundary where more than one (at least two) distance realizing geodesics between two distinct geodesic boundaries will exist.
I have a further question: Suppose p and q are two distance realizing geodesics between the boundary geodesics. Is it true that p and q are always disjoint? – Bidyut Sanki May 2
'12 at 8:26
They are always disjoint. This is proved by the usual "exchange and round-off" technique. See, if $p$ and $q$ intersect, then at the intersection point you can do a little cut and
paste to get two new arcs $p'$ and $q'$, still connecting the same boundary components, and slightly shorter, which is a contradiction. – Sam Nead Jan 3 at 11:00
add comment
Not the answer you're looking for? Browse other questions tagged hyperbolic-geometry or ask your own question.
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D - Linux and std.intrinsic
Nye <Nye_member pathlink.com>
Since I've seen that the intrinsic functions could "result in some surprising
speedups", I've been trying to replace some math functions with intrinsic ones
(like cos and sqrt).
However, at compile time dmd says that these functions do not exists and indeed,
intrinsic.d doesn't mention these at all.
First of all, did I forgot something obvious ? :)
Else, would these functions (cos and sqrt) be really faster than math's ones ?
And if yes, are you really sure I can't use them ? :)
Apr 26 2004
"Nye" <Nye_member pathlink.com> wrote in message
news:c6jukr$1gr3$1 digitaldaemon.com...
Since I've seen that the intrinsic functions could "result in some
speedups", I've been trying to replace some math functions with intrinsic
(like cos and sqrt).
Just use the std.math ones, and the compiler will make 'intrinsic' the ones that should be.
May 15 2004
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Strings, Vibrational modes
Overtones and just tempering.
Consider a taut string (or wire) fixed at both ends; a piano string for example, or a violin string: we'll ignore the effects of the bridges that happen to exist in both instruments. The string set
in motion can vibrate in a number of distinct ways called modes of vibration (or eigen modes or eigen functions). Generally, when the string is set in motion, it actually vibrates in some weighted
linear combination of these modes. But first it's easier to look at modes individually.
Modes and Nodes
The modes of vibration are related to the nodes or fixed points of the vibration; that is, excluding the end points which are given. In the fundamental mode, there are no nodes. The entire string
arches up and down with the maximum displacement from its position of rest being at the center. The maximum upward outline of the string and the maximum downward outline of the string form what is
called the envelope of vibration. As the string vibrates in this mode is causes compresional waves in air that we perceive as sound. The frequency of that sound is directly related to the frequency
of vibration of the string, so we can speak in terms of the string vibrating rather that in terms of the sound and still be talking about the sound. Say the fundamental mode of vibration is set up to
be the pitch called middle C on the piano.
The envelope of the second mode of vibration (sometimes called the first overtone), has arches, above and below which meet at the end points and at the center of the string. There is then a node of
the vibration at the center of the string, and the node has divided the string into two equal parts. The vibration produced is almost like cutting the string in half and starting over. The difference
being that when the right half of the string in up, the left half is down. It turns out that the frequency of vibration will be doubled; this corresponds also to a C pitch but an octave above middle
C; we'll call this C' and the apostrophe as a postfix operator that raises the designated pitch by an octave.
Violinists and flutists often use a technique called "playing in harmonics", which suppresses the fundamental mode of vibration and therefore sounds and octave higher. Describing how a flutist
accomplishes this is hard, but describing what a violinist does is easy: the pinky is extended so that it reaches into the middle of a stopped (or open string) and rests ever so lightly there to
produce the node and thus suppress the fundamental mode.
The envelope of the third mode of vibration has three arches, above and below, and 2 nodes, which divide the string into 3 parts equal in length. The actual vibration can be described as
up down up
| . . |
down up down
The pitch associated with this mode is G' which is 3 times the fundamental frequency of C. To translate this G' downward an octave to G, we must divide the frequency by 2. Thus symbolically of
G = (3/2) C
The musical interval defined by C-G is a perfect fifth in just tempering. The overtone series, in fact, defines what is meant by just tempering. This ratio for frequencies determining a fifth was
known even before Pythagoras.
The envelope of the fourth mode of vibration has four arches, and three nodes dividing the string into four equal parts.
up down up down
| . . . |
down up down up
Adjacent string segments vibrate oppositely, and the pitch here is four times that of the fundamental frequency, that is double the frequency of C', or C'', an octave above C' and two octaves above
Clearly one can continue dividing the string by an increasing number of nodes, thus generating the overtone series for any given pitch. From a musical point of view there is a point of diminishing
returns and one needs to consider only a finite number of overtones; 16 is not a bad place to stop, from a purely practical standpoint; I will list the first 20 pitches associated with a fundamental
of C.
NB (September 1, 2000): In the following, regarding the names of the pitches, the actual method of constructing the C major scale, and the tuning of a physical keyboard instrument in just
temperament, there are some very misleading and incomplete statements. Be wary. I hope to have these problems rersolved soon, along with more careful definitions of: overtones, harmonics, and (upper)
partials. Please bear with me.
C C' G' C'' E'' G'' Bb'' C''' D''' E'''
F''' G''' A''' Bb''' B''' C'''' C#'''' D'''' D#''''E
* * *
* Bad w.r.t. well tempering 7 is flat relative to keyboard Bb
In fact, all above 11 except the C at the 16th overtone
are not all that good relative to an equal tempered
There are two reasons to stop the series: 1) the frequencies exceed the range of human hearing 2) overtones get weaker as they get higher, which is to say that the fundamental mode of vibration is
dominant in determining the perceived pitch, while the overtones provide color or timbre to the sound. Trumpets have prominant odd harmonic number; The covering of an organ pipe (lieblich gedeckt)
damps the odd harmonics and creates a soft somewhat hollow sound. [Benade 1960], [Seashore 1938], Music & Mathematics [Link].
The first observation that can be made about the overtone series is that its lower elements together form a MAJOR seventh chord, lending weight to the words of taxonomic distinction "major" and
"minor". A major chord is somehow more natural in the context of simple vibrating things.
If we tune the C major scale according to the overtones of C, using also the overtones of the F (a perfect fifth below) and G (a perfect fifth above).
C = (1) C
D = (9/8) C
E = (5/4) C
F = (4/3) C
G = (3/2) C
A = (5/3) C
B = (15/8) C
C' = (2) C
This defines the relations between the frequencies of a justly tempered major scale; C major when the frequency of C is inserted into the formulas. This pattern of ratios is extensible in both
directions to tune all the white keys of the piano.
Keyboard and tuning to C, we are also ok for the harmonically closest keys of G major (with only F#) and F major (with only Bb). The next Generate from F and from G F F' C' F'' A'' C'' Eb'' From this
we get the ratios for A and F given above 1 2 3 4 5 6 7 G G' D' G'' B'' D'' F''
The just tempering scheme ulimately based on the perfect fifth runs into problems in constructing a circle of fifths because the circle does not close. The gap or discrepancy in frequency that
appears at 12 consecutive fifths and 7 octaves down is known as the Pythagorean Comma.
The various schemes to reconcile the Pythagorean Comma, that is to close the circle of fifths are called cyclic temperaments. Well or equal tempering is one of them. There have been other schemes to
divide the octave not only into 12 parts but also into 5, 14, 16, 19, 31 and 53 parts. These are all cyclic temperaments. The ancient Greek tuning was not cyclic and is one of the linear
termperaments. Various linear temperaments have been used, throughout musical history under circumstances where the music was not essentially harmonic but linear; hence there was no need to define
intervals of relative consonance and dissonance. [Helmholtz 1877] , appendix XX.
There is a mathematical story to tell associated with this construction of overtone series that centers around a very important theorem of Fourier. Cast into the current context the theorem says that
any vibration that the string is capable of can be expressed by a suitable addition of the fundamental modes each with some weighting coefficient. The actual mathematical theorem takes into account
that the vibrational modes can be given in terms of sinusoidal functions (sine and cosine) of trigonometry. Even among mathematicians, Fourier's theory is also called "Harmonic Analysis".
What is so bad, harmonically speaking if we approximate musical notes with an equal temperament?
It is clear that each string of a piano will have its set of overtones whose fequencies are determined by simple ratios of whole numbers. To get the G above C as above, multiply the frequency of C by
(3/2)-1.500. The equal tempered G that is now there would use a factor of 1.498. So now the the equal tempered G and its overtones will beat with those of the C, where this would be minimized if the
G were tuned using the 1.5 factor. Beats happen when two tones are played together that are just slightly different in frequency. The double angle formulae from trigonometry are:
sin( A + B ) = sin( A ) cos( B ) + cos( A ) sin( B )
and then also
sin( A - B ) = sin( A ) cos( B ) - cos( A ) sin( B )
Adding these two formulas gives
sin( A + B ) + sin( A - B ) = 2 sin( A ) cos( B )
Change variables by letting
u = A + B, v = A - B
so that inverting and solving for A and B in terms of u and v
A = (1/2)(u + v)
B = (1/2)(u - v)
Then substituting in the last trigonometric equation gives
sin(u) + sin(v) = 2 sin[(1/2)(u + v)] cos[(1/2)(u - v)]
The expression sin[(1/2)(u + v)] is a sine wave with a frequency
that is the average of the two frequencies u and v. If u and v
are close then (u-v) will be very small and the factor
cos[(1/2)(u - v)] modulates the average sine wave with a
frequency that is low with respect to the average frequency.
The amplitude of the sine wave, hence it's loudness swells
and dimishes. This swelling and diminishing of loudness is
called beating.
The more beating going on, the less consonant is the perception
to the ear. Therefore, equal temperament makes fifths and fourths
(the inversions of fifths) less consonant. Since the very basis
of harmony is the interval of the fifth, some of the consonance
of all harmony has been compromised by equal tempering.
Question: If the close frequencies u and v are both above the
range of human hearing, while their difference is within the
range of human hearing, will the modulating beat frequency be
heard as a sound?
Answer: yes.
With the advent of computers and computer software powerful enough to handle the digital and analog manipulations of sound, the music of the future, providing there is one, can be free of the equal
temperament that has been imposed on western music by the piano keyboard and still allow free modulation to maintain natural harmonic relations when wanted, and at the same time allow for music that
can also be expressed using the rich nonharmonic linear language that has been created in many other cultures. Should this revolution come about, music may itself still not be universal, but at least
it will have a much richer and universal alphabet.
Return to Home Page
Return to Music Page
Cirle of Fifths
The Octave
Medieval Modes
Greek Modes
Traditional Harmony
Patterns, Transformations and Groups in Musical Composition
Evolving Dodecaphony
Mathematical Groups
Pitch Sets in Composition
Email me, Bill Hammel at
bhammel@graham.main.nc.us READ WARNING BEFORE SENDING E-MAIL
The URL for this document is:
Created: September 1997
Last Updated: September 1, 2000
Last Updated: May 27, 2011
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When is a $\ast$-algebra a $C^{\ast}$-algebra?
up vote 3 down vote favorite
The purpose of this question is to collect sufficient conditions on a unital $\ast$-subalgebra $\mathcal{A}$ of the algebra of bounded linear operators $B(\mathcal{H})$ on a separable Hilbert space $
\mathcal{H}$ that guarantee that $\mathcal{A}$ is actually a $C^{*}$ algebra (is closed in the operator norm). Please provide links and references. At least, I'd like a reference or proof for the
"Thm:" If $\mathcal{A}$ is a unital $\ast$-subalgebra of $B(\mathcal{H})$ and whenever $A\in\mathcal{A}$ is self-adjoint it follows that $A_{+}$ and $A_{-}$ both lie in $\mathcal{A}$, then $\mathcal
{A}$ is norm-closed.
(Here, $A_{+}$ and $A_{-}$ live naturally in the $C^{*}$-algebra generated by $A$ and $I$, isomorphic to $C(\sigma(A)))$, where $A$ corresponds to the function $f(x)=x$, $A_{+}$ corresponds to $max
[f,0\]$ and $A_{-}$ to $min[f,0]$.)
(Edit: Nik has pointed out that the "Thm" is false. The broader question stands: Is there any other interesting abstract characterization of a C*-algebra that doesn't obviously say the algebra is
oa.operator-algebras reference-request
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1 Answer
active oldest votes
Jon, I think your "Theorem" is false. For example, $A$ could be the algebra of complex-valued Lipschitz functions on $[0,1]$, acting as multiplication operators on $L^2[0,1]$. That's a
up vote 4 unital *-subalgebra of $B(H)$ which is stable under lattice operations, but not closed in operator norm.
down vote
Aha! Thank you! – Jon Bannon Aug 16 '12 at 18:08
2 Here's another example: let $A$ be the set of all eventually constant sequences of complex numbers. This is a unital $*$-subalgebra of $l^\infty$ that is not norm closed. Not only is
it stable under lattice operations, it's stable under the continuous functional calculus. – Nik Weaver Aug 17 '12 at 1:34
2 ... the Borel functional calculus, even. – Nik Weaver Aug 17 '12 at 4:25
In light of your comment, it seems like an answer to my broader question seems unlikely...my hope was that closure under the functional calculi plus some other condition should force
the algebra to be C*...Thanks again for the examples! – Jon Bannon Aug 17 '12 at 10:34
You are welcome. – Nik Weaver Aug 17 '12 at 14:18
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analysis questions
Dmitri Zaykin zaykin at statgen.ncsu.edu
Thu Feb 6 23:32:10 EST 1997
smallm at PBS.DFO.CA wrote:
> Has anyone had conflicting results when comparing GENEPOP to CHIHW.
> Has anyone had populations out of equilibrium test out within
> equilibrium with CHIHW?
I did a number of simulations (with one and more loci) where
conditional probability test was consistently exhibiting slightly
greater power than conditional X2. However, my simulations were
limited to the case of drift with migration, and equal starting allele
frequencies. It is easy to generate a particular case, where either
Cond_X2 or Cond_Pr would give much smaller p-value than the other.
In Genetica, 96:169-178 we have a table illustrating the opposite
case, where Cond_X2 p-value is about 0.01, but Cond_Pr is 1.
Together with CHIHW
I have another program, that performs a test for disequilibrium for
many loci (so that HW is its special case). The test is described in
Zaykin D., L. Zhivotovsky, B.S. Weir. 1995. Exact tests for
association between alleles at arbitrary number of loci. Genetica,
96:169-178, or in Bruce Weir's book (GDA 2, 1996). It does both
conditional X2 (like CHIHW), and the test based on the conditional
probability 1/Cond_Pr (like GENEPOP). The program can analyze each
locus separately, all pairs at once, triples, etc. When each locus
contributes a little to HWD, the power is greatest if all loci are
treated simultaneously.
Self-extracting executables are on:
ftp://brooks.statgen.ncsu.edu/pub/zaykin/xaloci16.exe [Win3.1]
ftp://brooks.statgen.ncsu.edu/pub/zaykin/xaloci32.exe [Win95]
Source code is available upon request. It is portable enough to be
compiled under UNIX without modifications.
(I'm finishing putting a similar code into the GDA package maintained
by Paul Lewis: http://biology.unm.edu/~lewisp/gda.html)
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Tensorial decomposition of $B(H)$
up vote 9 down vote favorite
Let $H$ be an infinite-dimensional Hilbert space and let $\mathcal{B}(H)$ be the (C*/W*-)algebra of bounded operators on it. Actually, you may forget about the involution in $\mathcal{B}(H)$ because
I am interested in $\mathcal{B}(H)$ only as a Banach algebra (operator algebra).
Do there exist two infinite-dimensional Banach algebras $A, B$ such that $\mathcal{B}(H)$ is isomorphic as a Banach algebra to the projective tensor product $A\otimes_\gamma B$?
You may also replace the projective tensor product by any other Banach algebra tensor product which arises from a reasonable crossnorm (so the vNA tensor product is not good).
No. Even the Banach algebra structure is irrelevant here. ${\cal B}(H)$ (and its kin) is not Banach isomorphic to a reasonable tensor product of two infinite-dimensional Banach spaces. The proof
involves a few deep known facts about Banach space structure of ${\cal B}(H)$ and this margin is too small to contain it. – Narutaka OZAWA Feb 6 at 8:42
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Confused, has to be something simple.
October 18th 2009, 12:44 PM #1
Junior Member
Sep 2009
Confused, has to be something simple.
Must prove using different identities that the left side equals the right side, meaning you don't change the right side at ALL
I tried using double angle identities after factoring it (Difference of squares right?) But it didnt seem to work?
Any ideas?
$Cos^2 2u - Sin^2 2u = 2Cos^2 2u -1$
on the left side, change $\sin^2(2u)$ to $[1 - cos^2(2u)]$
This is just the composite angle formula for the cosine of twice an angle. Don't let the $2u$ confuse you. They are using it as the argument on all the functions, so it's equivalent to just x. So
this is actually very simple. The function:
Can be substituted into the left side of the euqation, to obtain the right side.
Haha thanks guys
October 18th 2009, 12:48 PM #2
October 18th 2009, 12:49 PM #3
Super Member
Jun 2009
United States
October 18th 2009, 12:52 PM #4
Junior Member
Sep 2009
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Re: List all different equations for a given one
in reply to List all different equations for a given one
If you are trying to solve a permutation problem (which is what it sounds like to me), I think you are applying yourself incorrectly. Rather than figuring out how to map your answer to their answer,
your parser should generate a canonical read that will necessarily map to yours. For your simple case, it might look like:
my $eq = {op => '+',
terms => [1,
where a canonical sorting algorithm is used to order terms. Note that, if you want to accept complex expressions, the sort is non-trivial since you'll need to rank complex references, so
won't be enough. This type of format also supports nested operations:
my $eq = {op => '+',
terms => [{ op => '*',
terms => [3,
You can then do a recursive descent in order to check equivalence. Of course, this isn't going to help you with distributivity.
#11929 First ask yourself `How would I do this without a computer?' Then have the computer do it the same way.
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Euler's mathematics in terms of modern theories?
up vote 13 down vote favorite
Some aspects of Euler's work were formalized in terms of modern infinitesimal theories by Laugwitz, McKinzie, Tuckey, and others. Referring to the latter, G. Ferraro claims that "one can see in
operation in their writings a conception of mathematics which is quite extraneous to that of Euler." Ferraro concludes that "the attempt to specify Euler's notions by applying modern concepts is only
possible if elements are used which are essentially alien to them, and thus Eulerian mathematics is transformed into something wholly different"; see http://dx.doi.org/10.1016/S0315-0860(03)00030-2.
Meanwhile, P. Reeder writes: "I aim to reformulate a pair of proofs from [Euler's] "Introductio" using concepts and techniques from Abraham Robinson's celebrated non-standard analysis (NSA). I will
specifically examine Euler's proof of the Euler formula and his proof of the divergence of the harmonic series. Both of these results have been proved in subsequent centuries using epsilontic
(standard epsilon-delta) arguments. The epsilontic arguments differ significantly from Euler's original proofs." Reeder concludes that "NSA possesses the tools to provide appropriate proxies of the
inferential moves found in the Introductio"; see http://philosophy.nd.edu/assets/81379/mwpmw_13.summaries.pdf (page 6).
Historians and philosophers thus appear to disagree sharply as to the relevance of modern theories to Euler's mathematics. Can one meaningfully reformulate Euler's infinitesimal mathematics in terms
of modern theories?
Note 1. There is a related thread at Would Euler's proofs get published in a modern math Journal, especially considering his treatment of the Infinite?
3 Euler's mathematical theories are easily understood in their own right without any so called modern improvements. Euler's main fault is unsatisfactory acknowledgement and explanation of previous
authorities. Solving the Basel problem needs help from the Newtonian formulae which are to be found in D.T. Whiteside's Mathematical Papers of Isaac Newton vol 5 pages 358-359. Euler was not able
to give such a precise reference. – user37007 Jul 12 '13 at 15:40
2 Given that "Historians and philosophers thus appear to disagree sharply," this would appear to be a question for which there are opinions rather than answers, thus poorly-suited to MO. And when OP
writes in a comment about "scholars like Ferraro who are ill-equipped to deal with the mathematics beyond $\epsilon,\delta$, that tips the balance over for me. – Gerry Myerson Feb 6 at 5:22
1 @GerryMyerson, deleting a question based on a comment (since deleted) does not seem too friendly a procedure. – katz Feb 6 at 15:46
1 For example, would "synthetic differential geometry" count as an answer? – Daniel Moskovich Feb 6 at 16:27
Dear @Daniel, I originally posed this question in the early stages of a current joint text on Euler's infinitesimal mathematics and its interpretation where we address some of the issues that came
3 up here. The paper is currently being considered at a leading philosophy journal; I have recently submitted a revised draft. I can send you a current version if you are interested. – katz Feb 6 at
show 11 more comments
1 Answer
active oldest votes
[Converted from comment to answer per Yemon Choi's suggestion.]
From a casual run-through of the Ferraro paper, it seems like Euler's ideas about infinitesimals were, unsurprisingly, not formalized to modern standards and therefore don't map exactly onto
up vote modern concepts. He apparently didn't think of a line segment as a point set, which would be more similar to smooth infinitesimal analysis than to NSA. But other aspects of Ferraro's
3 down description do seem more like NSA than SIA. Infinite numbers are imagined as infinitely increasing sequences, whereas not all models of SIA have invertible infinitesimals. I assume Euler
vote used Aristotelian logic.
"He apparently didn't think of a line segment as a point set." Sounds good, please tell us more. – Paul Taylor Feb 6 at 9:12
1 @PaulTaylor, Ben has apparently not been around recently, so I would comment that most historians agree that Euler was not working with a punctiform continuum that we are used to in the
post-Cantor era. Rather, points were locations marked on an unanalyzed continuum. Some have argued that this makes the Eulerian continuum closer to an intuitionistic continuum. I am
somewhat sceptical about this claim. – katz Feb 6 at 16:53
1 I am of the view that the "punctiform continuum that we are used to in the post-Cantor era" was vandalism on his part and hope to see the end of his "era". So I would like to hear more of
how Euler saw the continuum prior to this damage. I would also like to see the answers to your question, but the Thought Police have moved in again. – Paul Taylor Feb 6 at 17:44
4 @PaulTaylor Name-calling is neither appropriate nor constructive. – S. Carnahan♦ Feb 7 at 0:44
@PaulTaylor I am joining Scott Carnahan to ask you to please stop referring to those who vote to close questions as 'Thought Police'. Closing and reopening questions is a normal part of
3 the operation of MO, and is not done to shut down 'thought' but rather to help bring questions into the form for which MO was created. This has been explained before. MO is not a board
for posting people's opinions and getting in arguments; it is a site for people to ask focused questions and get focused and definitive answers. Anyway, further rude references to
'Thought Police' will henceforth be pruned out. – Todd Trimble♦ Feb 7 at 15:19
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[Numpy-discussion] MemoryError for computing eigen-vector on 10, 000*10, 000 matrix
[Numpy-discussion] MemoryError for computing eigen-vector on 10, 000*10, 000 matrix
Zhenxin Zhan andyjian430074@gmail....
Wed Apr 29 00:19:38 CDT 2009
Thanks for your reply.
My os is Windows XP SP3. I tried to use array(ojb, dtype=float), but it didn't work. And I tried 'float32' as you told me. And here is the error message:
File "C:\Python26\Lib\site-packages\numpy\linalg\linalg.py", line 791, in eig
a, t, result_t = _convertarray(a) # convert to double or cdouble type
File "C:\Python26\Lib\site-packages\numpy\linalg\linalg.py", line 727, in _con
a = _fastCT(a.astype(t))
Zhenxin Zhan
发件人： David Cournapeau
发送时间： 2009-04-29 00:04:28
收件人： Discussion of Numerical Python
主题： Re: [Numpy-discussion] MemoryError for computing eigen-vector on 10,000*10, 000 matrix
Zhenxin Zhan wrote:
> Hello,
> I am a new learner of Numpy. From 'numpybook', I use
> numpy.linalg.eig(A) to calculate a matrix 2,000*2,000 and it works well.
> But, when I calculate eigen vector for 10,000*10,000 matrix, there is
> 'MemoryError' error message in statement numpy.array(...). My laptop
> has 4GB memory.
Which OS are you using ? If your OS is 32 bits, you won't be able to use
4 Gb for you python process, and a 10000x10000 matrix is big (the matrix
alone takes ~750 Mb of memory). If that's an option, you could also try
single precision instead of the default double precision:
import numpy as np
A = np.array(...., dtype=np.float32)
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Studying The Innate and Cultural Cognitive Origins of Math
Mathematics or Math evolved from counting, measurement, study of shapes and motions of physical objects. One definition of math is that it is the study of quantity, structure, space, and change.
The practical application of math has been ingrained into human activity since the discovery of writing and communication.
The primary investigation into the origin of math and its discoveries and methods can first be found ancient documents such as the following:
• Plimpton 322 - Babylon c. 1900 BC
• Rhind Mathematical Papyrus - Egypt c. 2000-1800 BC
• Moscow Mathematical Papyrus - Egypt c. 1890 BC
All of these texts cover the Pythagorean theorem, which seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry.
Some would argue that the discovery of the Pythagorean theorem in 6th century BC is where the study of mathematics begins. The theorem, the square of both sides of a right triangle equals the square
of the hypotenuse, is attributed to Greek mathematician, Pythagorias. Even the word "Mathematics" started with the Greeks. It means "subject of instruction. The Greeks expanded and refined math
methods through deductive reasoning and mathematical rigor in proofs.
Aside from the Greeks, Chinese mathematics made early contributions as well, including a place value system. The Hindu-Arabic numeral system and its rules likely evolved over the course of the first
millennium AD in India and was transmitted to the west via Islamic mathematics. Islamic mathematics, in turn, developed and expanded the mathematics known to these civilizations. Many Greek and
Arabic texts on mathematics were then translated into Latin, which led to further development of mathematics in medieval Europe.
Looking past written records, scientists and researchers have wondered how the fundamental concept of math began with the human race.
Study finds twist to the story of the number line
Tape measures. Rulers. Graphs. The gas gauge in your car, and the icon on your favorite digital device showing battery power. The number line and its cousins – notations that map numbers onto space
and often represent magnitude – are everywhere. Most adults in industrialized societies are so fluent at using the concept, we hardly think about it. We don't stop to wonder: Is it "natural"? Is it
Now, challenging a mainstream scholarly position that the number-line concept is innate, a study suggests it is learned.
The study, published in PLoS ONE, is based on experiments with an indigenous group in Papua New Guinea. It was led by Rafael Nunez, director of the Embodied Cognition Lab and associate professor of
cognitive science in the UC San Diego Division of Social Sciences.
"Influential scholars have advanced the thesis that many of the building blocks of mathematics are 'hard-wired' in the human mind through millions of years of evolution. And a number of different
sources of evidence do suggest that humans naturally associate numbers with space," said Nunez, coauthor of "Where Mathematics Comes From" and co-director of the newly established Fields Cognitive
Science Network at the Fields Institute for Research in Mathematical Sciences.
Video: History of Math
"Our study shows, for the first time, that the number-line concept is not a 'universal intuition' but a particular cultural tool that requires training and education to master," Nunez said. "Also, we
document that precise number concepts can exist independently of linear or other metric-driven spatial representations."
Nunez and the research team, which includes UC San Diego cognitive science doctoral alumnus Kensy Cooperrider, now at Case Western Reserve University, and Jurg Wassmann, an anthropologist at the
University of Heidelberg who has studied the indigenous group for 25 years, traveled to a remote area of the Finisterre Range of Papua New Guinea to conduct the study.
The upper Yupno valley, like much of Papua New Guinea, has no roads. The research team flew in on a four-seat plane and hiked in the rest of the way, armed with solar-powered equipment, since the
valley has no electricity.
The indigenous Yupno in this area number some 5,000, spread over many small villages. They are subsistence farmers. Most have little formal schooling, if any at all. While there is no native writing
system, there is a native counting system, with precise number concepts and specific words for numbers greater than 20. But there doesn't seem to be any evidence of measurement of any sort, Nunez
said, "not with numbers, or feet or elbows."
Neither Hard-Wired nor "Out There"
Courtesy of Embodied Cognition Lab, UC San Diego.
Nunez and colleagues asked Yupno adults of the village of Gua to complete a task that has been used widely by researchers interested in basic mathematical intuitions and where they come from. In the
original task, people are shown a line and are asked to place numbers onto the line according to their size, with "1" going on the left endpoint and "10" (or sometimes "100" or "1000") going on the
right endpoint. Since many in the study group were illiterate, Nunez and colleagues followed previous studies and adapted the task using groups of one to 10 dots, tones and the spoken words instead
of written numbers.
After confirming the Yupno participants' understanding of numbers with piles of oranges, the researchers gave the number-line task to 14 adults with no schooling and six adults with some degree of
formal schooling. There was also a control group of participants in California.
The researchers found that unschooled Yupno adults placed numbers on the line (or mapped numbers onto space), but they did it in a categorical manner, using systematically only the endpoints: putting
small numbers on the left endpoint and the mid-size and large numbers on the right, ignoring the extension of the line — an essential component of the number-line concept. Schooled Yupno adults used
the line's extension but not quite as evenly as adults in California.
"Mathematics all over the world – from Europe to Asia to the Americas – is largely taught dogmatically, as objective fact, black and white, right/wrong," Nunez said. "But our work shows that there
are meaningful human ideas in math, ingenious solutions and designs that have been mediated by writing and notational devices, like the number line. Perhaps we should think about bringing the human
saga to the subject – instead of continuing to treat it romantically, as the 'universal language' it's not. Mathematics is neither hardwired, nor 'out there.'"
Out-of-Body Time
The researchers ran several experiments while in Gua, Papua New Guinea, including those that examine another fundamental concept: time.
When talking about past, present and future, people all over the world show a tendency to conceive of these notions spatially, Nunez said. The most common spatial pattern is the one found in the
English-speaking world, in which people talk about the future as being in front of them and the past behind, encapsulated, for example, in expressions such as the "week ahead" and "way back when."
(In earlier research, Nunez found that the Aymara of the Andes seem to do the reverse, placing the past in front and the future behind.)
In their time study with the Yupno, now in press at the journal Cognition, Nunez and colleagues find that the Yupno don't use their bodies as reference points for time – but rather their valley's
slope and terrain. Analysis of their gestures suggests they co-locate the present with themselves, as do all previously studied groups. (Picture for a moment how you probably point down at the ground
when you talk about "now.") But, regardless of which way they are facing at the moment, the Yupno point uphill when talking about the future and downhill when talking about the past.
Interestingly and also very unusually, Nunez said, the Yupno seem to think of past and future not as being arranged on a line, such as the familiar "time line" we have in many Western cultures, but
as having a three-dimensional bent shape that reflects the valley's terrain.
"These findings suggest that how we think about abstract concepts is even more flexible than previously thought and is profoundly affected by language, culture and environment," said Nunez.
"Our familiar notions on 'fundamental' concepts such as time and number are so deeply ingrained that they feel natural to us, as though they couldn't be any other way," added former graduate student
Cooperrider. "When confronted with radically different ways of construing experience, we can no longer take for granted our own. Ultimately, no way is more or less 'natural' than the Yupno way."
University of California - San Diego PLoS ONE Fields Cognitive Science Network Fields Institute for Research in Mathematical Sciences Case Western Reserve University University of Heidelberg What is
String Theory? Weekend Video: Bikini Calculus: A Fun Way To Learn Calculus Studying The Physics Behind An Investment Bubble Physics Technology and Education Key Factor To Advances In Medical Sciences
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By what number must you multiply each side of the equation 2/3 x = 10 to produce the equivalent equation x = 15?
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Just multiply by the recipricol of 2/3 which is 3/2 example, recip of 1/2 is 2/1 or just 2
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thanks :)
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The Fluoride Ion Selective Electrode Experiment
Direct Potentiometry and Standard Addition Methods
Dr. David L. Zellmer
Revised February 15, 1999
The fluoride ion selective electrode experiment consists of the following parts:
• Choose a real world unknown and prepare it for analysis. Suggestions for sample preparation may be found in Skoog, West, and Holler, Fundamentals of Analytical Chemistry, 7th Ed., Saunders, 1996,
Section 36I-5 "The Direct Potentiometric Determination of Fluoride Ion.", pp. 850-852. Note how Total Ionic Strength Adjustment Buffer (TISAB) is used to maintain constant ionic strength and to
remove certain interferences.
• Obtain background information on probable fluoride levels in your unknown. Toothpaste and mouthwash must contain fluoride concentrations on their labels. For levels in drinking water you could
try 1 mg F/L as a starting point, or check web sources such as Water Fluoride, or use a web search engine for additional information. Call your local water district if you plan to test your own
drinking water.
• Prepare a series of standard fluoride solutions with a constant amount of TISAB in each. Use these solutions to determine if your electrode is operating properly by plotting E vs. log C and
checking for Nernstian behavior.
• Using direct potentiometry, measure the potential of your unknown and read its provisional concentration from your E vs. log C graph. You will need both the slope of the electrode and the
provisional concentration to do the standard addition portion of the experiment.
• Use standard addition to make a more careful measurement of your unknown. The mathematical details are given below.
Direct Potentiometry
In this method a series of fluoride standards are prepared in a background matrix of TISAB (Total Ionic Strength Adjustment Buffer). The unknown is prepared using TISAB in the hope that the matrix
will be similar to the standards.
Since E = K + S log C
A plot of E vs. log C should yield a working curve that can be used to measure an unknown.
The first thing we learn from this plot is that our electrode is working properly, and has the expected linear response of E vs. Log C with slope measured at -59.4 mV, very near to the theoretical
-59.2 mV at 25^o C. A Linear Least Squares analysis of this plot allows us to measure the concentration of our unknown directly, and to compute the error of this determination. In the figure above
the unknown was found to be 2.28 +or- 0.05 mg/L (one-sigma error).
Standard Addition
The problem with direct potentiometry is the possible matrix effect on the unknown. The presence of iron(III), for example, might complex part of the fluoride in spite of the TISAB which is supposed
to combat this. To check for this possibility we will need to use Standard Addition. [Note that the standard addition method described will not correct for the presence of other ions to which the
electrode also responds; where E = K + S log(Cunk + kCintf)].
The graphics above show two types of standard addition: constant volume and variable volume. (For constant volume standard addition see the tutorial at www.csufresno.edu/chem/~davidz/Chem106/StdAddn/
The discussion below will be for variable volume standard addition. In this method we have a large volume of unknown to which small amounts of a standard solution are added. Although the total volume
will not change a great deal, we will have to compensate for the volume changes produced. Note that a special pipet will have to be used that can dispense small volumes with high precision. (As an
alternative, you could measure the density of the standard solution, then dispense the standard by mass using a weight buret.)
For an ion selective electrode with slope S that responds only to the analyte ion with concentration C,
Now, if we wish to remove any effect that ties up part of the analyte ion, we need to divide the response with standard addition by the response without standard addition.
The final equation is in the form of y = mx + b, where y is the electrode response to the amount x of standard added. Note that x is not expressed as a concentration, but rather as the total amount
CstdVstd of standard added. The slope of this line will be m=1/Cunk, from which we could compute Cunk directly. If we wish to use the more traditional standard addition approach using an x-axis
intercept, then when y=0, the x-axis intercept will be at x=-b/m or CstdVstd=-CunkVo. Divide the amount CstdVstd at the x-axis intercept by Vo and you have Cunk.
Confused? The following example may help.
An unknown solution is prepared by working up a sample and buffering it with TISAB. 100.0 mL of this solution is placed in a beaker and its potential measured. This potential is called E1. Then 0.300
mL of a 300 mg/L standard is added, the solution mixed, and the potential measured. This potential is called E2. Additional standard is added, and additional values of E2 are measured. A plot of
(Vo+Vstd)*10^((E2-E1)/S) is plotted vs. CstdVstd. The following graph was produced.
We can see that at y=0 a value of around 220 micrograms of fluoride is found. Since this was in 100 mL of solution the unknown must be around 220 micrograms/100 mL = 2.2 micrograms/mL or 2.2 mg/L. We
are then told that the unknown solution was actually 2.15 mg/L, so we are pretty close.
Another important feature of this graph is the amount of increase shown in the response function for each addition of standard. Each amount of standard added should increase the fluoride level about
30% above the concentration of fluoride present due to the unknown. If the amount of standard added is too small, we will be left with a very large extrapolation to the x-axis. If the amount added is
too large, the size of the x-axis intercept will be very small compared to the rest of the graph. Both of these extremes will produce unacceptable levels of relative error.
The spreadsheet that produced the graph above includes a careful linear least squares analysis of the results.
Some portions of this spreadsheet were blocked out because they represent "secret" knowledge about the unknown or about the response function of the electrode. The remaining numbers are similar to
those found in a real experiment, however, and give a final result of 2.16 +or- 0.05 mg/L for the unknown solution. Compare this to the actual value of 2.15 mg/L, which is given as "secret"
information in Cell B10. In a real analysis you would not have this value, of course.
Note that the value of "S" given in Cell B6 is the -59.4 mV found from the direct potentiometric method given above. Don't confuse this with the slope of the response function as computed by LLS
analysis of the data from the standard addition runs. If a poor value of "S" is chosen, the Response function will become nonlinear, making it useless for a standard addition determination.
Note that the formula in Cell D14 is the Response function (Vo+Vstd)*10^((E2-E1)/S) which should be linear with respect to the number of micrograms of fluoride added to the system. When Vstd=0, the
exponential term contains (E1-E1)=0 which makes the Response function equal to Vo or 100.0 mL. When filled down, the value of E1 stays fixed on Cell C14, while the value of E2 changes to the new
potentials measured as each standard addition is made.
When doing an x-axis intercept at y=0, the value of M is assumed to be infinite because the value of y=0 is assumed to be perfectly known. The traditional formula for "s sub c" is modified by
removing the 1/M term. The formula given in row 38 for Cell C30 contains an error which has confused a few people. The reference to cell C19 points to a blank cell. This happens to work out OK, since
C19 refers to the y-value of the "unknown" which in this case of standard addition would be zero (the X-axis intercept is at y=0). The calculation comes out OK, since Excel assigned a value of zero
to this blank cell. To correct the equation shown in row 38, replace "C19" with 0.
(See www.csufresno.edu/chem/~davidz/Stat/LLSTutorial/LLSmodel/LLSmodelSp99.html for details about the "traditional" LLS formulas.)
In Cell C33 the final one-sigma reported error for the unknown concentration is computed by multiplying the computed concentration of the unknown by the relative error of the x-axis intercept. The
ABS() function is used to remove any minus signs that may result.
In case you may not have noted the pertinent information needed in your report to describe the electrodes used, the following images may prove useful: Corning1, Corning2, Orion1, Orion2, OrionSJR1,
and OrionSJR2.
For current information, try websites for Corning and for Orion.
This page was last updated on May 18, 1999.
David L. Zellmer, Ph.D.
Department of Chemistry
California State University, Fresno
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trying to find an infliction point. the rest i'll make neat.
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in an mRNA molecule, a group of three nitrogenous bases makes up a codon, and each codon codes for an amino acid. there are 4 diff nitrogenous bases that can be arranged into different codons, but
there are only 20 diff type of amino acids. based on this info what conclusionn can you make about the genetic code?
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A.during translation nitrogenous n]bases are rearanged into a sequence tht corresponds to an amino acid
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each coden can be translated into more than one possible amino acid
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C, many possible combos
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D.several diff codens result in proction of the same amino acid
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The total number of combinations of 3 bases is 4 raised to the power of 3 or 64. So there could be 64 different codons.There are only 20amino acids, so I'll let you figure out the answer. As a
hint I'll suggest you look at the genetic code: http://en.wikipedia.org/wiki/Genetic_code The RNA codon table should be the most relevant bit for this question. Best wishes
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Department of Mathematics, U.Va.
Commutative algebra was born out of three classical fields: number theory, algebraic geometry, and invariant theory, but now is used to study many other topics in mathematics. My own focus is the
study of solutions of polynomial or power series equations in many variables. I am especially interested in a method called "reduction to characteristic p." Here are a few of my papers:
The structure of linkage (with B. Ulrich), Annals of Math. 126 (1987), 277-334.
Tight closure, invariant theory, and the Briançon-Skoda theorem (with M. Hochster), J. Amer. Math. Soc. 3 (1990), 31-116.
Infinite integral extensions and big Cohen-Macaulay algebras (with M. Hochster), Annals of Math 135 (1992), 53-89.
Direct methods for primary decomposition (with D. Eisenbud and W. Vasconcelos), Inventiones Math. 110 (1992), 207-236.
Uniform bounds in noetherian rings, Inventiones Math. 107 (1992), 203-223.
Comparison of symbolic and ordinary powers of ideals (with M. Hochster), Invent. Math. 147 (2002), 349-369.
The regularity of Tor and graded Betti numbers (with D. Eisenbud and B. Ulrich), Amer. J. Math. 128 (2006), 573-605.
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What is the relationship between "translation" and time complexity?
up vote 8 down vote favorite
Consider the problem of deciding a language $L$; for concreteness, say that this is the graph isomorphism problem. That is, $L$ consists of pairs of graphs $(G, H)$ such that $G\simeq H$. Now the
time complexity of deciding this problem as stated depends on how the graphs are encoded. For example, if one were to have a "canonical" encoding of graphs (such that encoding strings are in
bijective correspondence with isomorphism classes of graphs) the problem would be $O(n)$, as we could decide whether $G\simeq H$ simply by comparing the string representing $G$ to the string
representing $H$.
On the other hand, if we represent a graph via its adjacency matrix, the best known algorithm (according to Wikipedia) gives only a subfactorial bound. Now consider the time complexity of converting
from one language to another. If we let $T_1, T_2$ be the time complexity of deciding languages $L_1$ and $L_2$ respectively, and $T_{ij}$ be the time it takes a Turing machine to take a string $S$
and output another string $S'$ which is in language $j$ if and only if $S$ is in language $i$. We have
$$T_1\leq T_2+T_{12}$$ $$T_2\leq T_1+ T_{21}$$
as given a string that we want to test for its belonging to $L_i$, we may run it through the translation $L_i \to L_j$ and then decide language $j$. Indeed, this is a special case of a trivial
"triangle inequality" for translation; the time it takes to translate from $L_1$ to $L_2$ plus the time it takes to translate from $L_2$ to $L_3$ is greater than or equal to the time it takes to
translate from $L_1$ to $L_3$. (I say it is a special case because a decision problem is the same as converting a language $L$ to the language $\{ 1 \}$.)
What I want to know is:
Can we better quantify the relationship between the time complexity of a decision problem and the nature of the encoding?
So that this question is not prohibitively vague, let us say that I am looking for (1) related references, and (2) a measure of the complexity of an encoding which more tightly relates to time
complexity of the "underlying" decision problem.
Added (7/19/2010): The answers below, particularly Ryan Williams' excellent survey of the dependence of the time complexity of various problems on their encoding, get at the motivation to my question
but not at my question itself. In particular, it's clear that every problem may be re-encoded to allow (say) $O(\log n)$ time complexity, by padding. My question is whether there's a reasonable way
to measure this dependence.
For example, say the decision problem for $L_1$ is reducible to the decision problem for $L_2$, and vice versa, so that $L_1$ and $L_2$ in some sense represent the same problem. Is there a way to
formalize this last statement (about "representing the same problem")? I am imagining, for example, a measure $C_i$ of the complexity of a language so that if $T_i$ is the time complexity of the
language, and $L_1$ and $L_2$ are, say, easily reducible to one another, then $T_1/C_1\sim T_2/C_2$. (Of course $C_i=T_i$ works, but ideally $C_i$ would be somehow a property of the language, rather
than the decision problem.) This is unfortunately becoming quite speculative, so again, related references would be a great answer.
computational-complexity computer-science
Usually, I think people say L1 and L2 represent the same problem precisely when there are sufficiently efficient reductions back and forth between them. (Whatever "sufficiently efficient" means
depends on what you want to do with the problems.) It sounds like your idea (of taking the ratio of time over some other complexity measure) could be a definition for some other complexity measure.
I haven't seen such a concept. I'm not sure why you distinguish between "decision problem" and "language"; for all intents and purposes these are the same. (I certainly have been using them
interchangably.) – Ryan Williams Jul 19 '10 at 21:59
@Ryan Williams: I'm asking if such a complexity measure exists. And as far as I can tell I am using the terms "decision problem" and "language" interchangeably; I think I'm (unfortunately) using
the word "problem" to refer to a class of languages which "represent the same thing" (again, I'm being vague because the question is asking if such a concept exists). – Daniel Litt Jul 20 '10 at
@Daniel: I was referring to your phrase: "ideally C_i would be somehow a property of the language, rather than the decision problem." I've updated my answer to include a reference that you may find
interesting. – Ryan Williams Jul 20 '10 at 3:48
@Ryan: Oh, good call. I am being imprecise; what I mean is that ideally the word "Turing machine" should not appear in the definition of $C_i$. – Daniel Litt Jul 20 '10 at 13:06
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When it comes to the time complexity of problems, the encoding of the problem can be totally crucial. In general, the encoding of the problem cannot be separated from the complexity of
the problem itself.
The first canonical example of this (as mentioned before in answering another question) can be seen with the following two problems:
(1) Given a deterministic Turing machine $M$, string $x$, and integer $k$ written in binary, does $M$ accept $x$ within $k$ steps?
Problem (1) is $EXPTIME$-complete. However the following problem is $P$-complete:
(2) Given a deterministic Turing machine $M$, string $x$, and integer $k$ written as a string of $k$ ones, does $M$ accept $x$ within $k$ steps?
So already, the way in which $k$ is represented in an instance completely determines the complexity of the problem. (Note if I wrote $k$ in ternary, $4$-ary, etc., problem (1) remains
Another interesting example comes from circuit complexity. Consider the following two problems:
(3) Given a truth table of $2^n$ bits for a function $f:$ {$0,1$}$^n \rightarrow ${$0,1$}, return a circuit with AND/OR/NOT gates that computes $f$ and contains a minimum number of gates.
(4) Given a function $f:$ {$0,1$}$^n \rightarrow ${$0,1$} represented as a circuit with AND/OR/NOT gates, return a circuit that also computes $f$ and contains a minimum number of gates.
Problem (3) can be easily seen to be in $NP$, since the minimum circuit for $f$ needs at most $O(2^n/n)$ gates, and checking that a given circuit works for $f$ takes $2^{O(n)}$ steps.
However (3) is not known to be in $P$, nor is it clear that it's $NP$-complete. The curious status of (3) is discussed in
up vote 9
down vote Valentine Kabanets, Jin-yi Cai: Circuit minimization problem. STOC 2000: 73-79
What about problem (4)? It is not known to be in $NP$! It is known to be in $\Sigma_2 P$ of the polynomial time hierarchy, but not known to be complete for that class. However the version
where you use the representation of formulas instead of circuits is known to be $\Sigma_2 P$-complete under Turing reductions:
David Buchfuhrer, Christopher Umans: The Complexity of Boolean Formula Minimization. ICALP (1) 2008: 24-35
Examples of this sort are everywhere in complexity theory, simply because the encoding can really matter if the relative sizes of encodings (or the complexities of encodings) are
different enough. Luckily, most "natural" encodings (for which there are polynomial time mappings from one encoding to another) do not seem to affect the overall complexity of a problem
(e.g. whether or not a problem is in $NP$). This is another reason why the notion of polynomial time is one of the main focuses in complexity. It is a "robust" notion that isn't affected
by whether you use e.g. adjacency lists versus adjacency matrices to represent a graph in your graph problem. Related to this, there is a recent and thought-provoking reference that
outlines a complexity theory for succinctly represented graphs (graphs whose adjacency matrices are the truth tables of small size circuits):
Sanjeev Arora, David Steurer, Avi Wigderson: Towards a Study of Low-Complexity Graphs. ICALP (1) 2009: 119-131
Finally, concerning your proposed "isomorphism-respecting" encoding of graphs: while it would be very neat to have, it would not be considered natural, since we don't know how to
efficiently obtain such an encoding from any of the other encodings that have already been deemed natural.
UPDATE TO ADDRESS YOUR REVISED QUESTION: I think it is a neat idea to try to study "problems" as classes of languages that "represent the same thing" in some strong sense. I'm not aware
of significant prior work on this (other than the cheap reply that "all NP-complete problems represent the same thing", which I don't think is what you are driving at). The closest
reference I can think of is a related attempt to define "algorithm" in a similar way. See Blass, Dershowitz, and Gurevich's cool paper: http://research.microsoft.com/en-us/um/people/
Probably the question becomes less interesting this way, but I’d go as far as to say that the encoding is itself part of the statement of the problem (particularly when we describe
decision problems as languages). – Antonio E. Porreca Jul 19 '10 at 16:06
@Antonio: That is one way of defining the question away, but it seems to run counter to the philosophy of "reducing" one problem to another. – Daniel Litt Jul 19 '10 at 16:10
Indeed, while the encoding of a problem matters, we'd rather have that it doesn't matter, as much as that is possible. Sometimes this is unavoidable (e.g. in algorithms for planar
1 graphs, using an adjacency matrix kills your linear time algorithm). It would be very annoying if the "true" complexity of Boolean satisfiability (say) depended on the precise manner
in which you encoded formulas. The reason that it doesn't (up to "natural" encodings) is due to the robustness of the definition of NP. One encoding may put it in NTIME[n] or another
in NTIME[n^2], but it's still NP. – Ryan Williams Jul 19 '10 at 17:13
Interesting paper! I think a similar philosophical argument can be made against the existence of a useful equivalence relation on languages "representing the same problem." This
doesn't of course rule out a more quantitative complexity measure. – Daniel Litt Jul 20 '10 at 13:18
add comment
The general abstract setting for the issue driving your question is the notion of reduction of equivalence relations. The idea of this is that one equivalence relation $E$ reduces to another
$F$ with respect to some complexity concept if there is a function $f$ in this class such that
• $x\, E\, y$ if and only if $f(x)\, F\, f(y)$
You can imagine that $E$ is the equivalence relation arising from one way of representing mathematical objects (graphs, algebraic structures, whatever) and $F$ is the relation corresponding
to an alternative method. The reduction is saying that equivalence with respect to the $E$ way of representing the objects is no more difficult than equivalence with respect to the $F$ way
of representing them.
I claim that understanding this reducibility relation amounts to understanding exactly what your question is aimed at, the question of how one manner of representing the same objects can be
simpler than another. More generally, this reducibility relation provides a very precise way to understand what it means to say that one classification problem is strictly harder than
another, even when the objects in the two cases seem totally unrelated at first.
up vote
2 down In the case you seem most interested, you could regard $E$ and $F$ as NP equivalence relations and insist that $f$ is polynomial time computable. This is a case that has been recently
vote investigated by Sy Friedman, and this MO question arose out of a talk he gave on this topic here in New York, and discusses as motivation some of the relevant general theory. This appears to
be a completely new research area, ripe for progress. I would encourage anyone to enter into it.
Much of that theory is inspired by the enormous successes of the much more developed instance of this concept, occurring when $E$ and $F$ are Borel relations on the reals and $f$ is a Borel
function. This case is the emerging-but-possibly-now-mature field of Borel equivalence relation theory (see Greg Hjorth's survey article and Simon Thomas' notes). The theory of Borel
equivalence relation theory has to deal explicitly with the Borel analogues of the precise issues you mention in your question, and has made huge illuminating progress in understanding the
structure of Borel equivalence relations under Borel reducibility. I mention some of the basic results in this MO answer.
In general, for each notion of complexity, the goal is to study the whole hierarchy of equivalence relations, to discover its features and general structural results. The Borel case is quite
well developed by now, exhibiting many fascinating features, but the NP case is much less well developed. I do know personally, however, that other researchers are working on several other
natural contexts of this idea.
Excellent! Do you know any preprints/papers on the NP case? – Daniel Litt Jul 22 '10 at 3:55
You can look at Sy Friedman's web page at logic.univie.ac.at/~sdf and also in particular at this preprint: logic.univie.ac.at/~sdf/papers/joint.sam.joerg.moritz.yijia.pdf – Joel David
Hamkins Jul 22 '10 at 4:10
add comment
A legit encoding would give two different codings for two different objects. The problem of, for instance, graph isomorphism only makes sense if one considers two isomorphic graphs to be
different. There are tons of problems for which changing the encoding leads to a classification in a smaller complexity class. Take for example (Garey & Johnson, p. 159), LINEAR
DIVISIBILITY, which is, given integers $a$ and $c$ the problem of answering $(\exists x)[ax +1|c]$. This problem is $\gamma$-complete, but trivially in P if the inputs are given in unary. G&
J add "[The] supposed intractability [of LINEAR DIVISIBILITY] depends heavily on the convention that numbers be represented by strings having length logarithmic in their magnitudes."
up vote
0 down You should in particular check the notion of "pseudo-polynomial time" and section 4.2 of Garey & Johnson.
Hope this helps.
Thanks! This is sort of a restatement of the motivation for the question; but I will take a look at the text you mention. – Daniel Litt Jul 17 '10 at 23:21
add comment
Your question reminded me of matroid problems. With these it is of great importance to specify how the input is given, as translating between input forms can increase the size of the
input exponentially. There is a survey of this issue here:
up vote 0 down
vote http://arxiv.org/abs/math/0702567
add comment
Not the answer you're looking for? Browse other questions tagged computational-complexity computer-science or ask your own question.
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Least Squares Fitting (Regression)
In our time it is easy to show a positive correlation between any pair of things...
How to Lie with Statistics by Darrell Huff (1954)
Click here for least squares data entry
Click here for least squares data entry with plot option
If you go into a forest you find yourself surrounded by many types and sizes of trees. If you consider just one species of tree a natural assumption is that the largest trees are the oldest. While we
expect that there is a correlation between the size of a tree and its age, the relationship between between these two variables is probably not exact: you would expect that genes and environment
would also play a role. A tree that happened to have access to more light or better soil or proper moisture or whose parents were unusually large will probably grow more per year than a less "lucky"
tree. Furthermore, not all years are the same: we expect trees to grow little during years of drought. Even though we expect that each individual tree has its own growth history, we expect on average
there will be a relationship between tree size and tree age.
The age of a tree can be determined by counting each annual growth ring in the trunk of the tree. (You need not convert the tree into a stump to count these growth rings: a thin core of wood --
reaching from bark to dead center -- can be extracted from a living tree using a borer.) A common measure of the size of a tree is the "diameter at breast height" DBH. "Breast height" is defined as 4
½ feet above the uphill side of the tree. Since a tree trunk is not a perfect circle, "diameter" is defined as the circumference divided by
Note that the DBH of a tree is easy to determine: it just takes a measuring tape whereas the age of a tree requires specialized instruments and additional work and time. This is one reason why it can
be helpful to know the average relationship between size and age: we can then use an easy measurement (DBH) and some calculations to determine a hard-to-measure quantity (age). Of course, this easy
measure of age is also only an approximate measure of age.
Consider the following data on 12 northern red oaks from an unthinned stand in southwestern Wisconsin:
│ Age │ DBH │
│(years) │(inch) │
│ 97│ 12.5│
│ 93│ 12.5│
│ 88│ 8.0│
│ 81│ 9.5│
│ 75│ 16.5│
│ 57│ 11.0│
│ 52│ 10.5│
│ 45│ 9.0│
│ 28│ 6.0│
│ 15│ 1.5│
│ 12│ 1.0│
│ 11│ 1.0│
We can display this data in an x-y scatter plot. One of the first decisions that needs to be made is which variable (age or DBH) to put on the x (horizontal) axis and which on the y (vertical) axis.
One rule is to put the least precisely measured variable on the y axis. The counting of annual growth rings should be precise; of course the trees are probably a bit older than the count as it took a
few years for the tree to grow to the height at which the core was taken. The DBH could have been measured to greater accuracy: the students were told to measure to the nearest ½ inch. The errors in
age (a systematic understatement of age by a few years) and in DBH (roundoff error of ± ¼ inch) are difficult to compare since they have different units and different natures. If we compare them in
percent terms using typical values (say 2 out of 50 years = 4% or ¼ inch out of 8 inches = 3%) the errors are similar.
Another rule is to put the "controlling" variable on the x axis and the dependent variable on the y axis. I think of age causing growth rather than growth causing age, so I would put age on the x
On the other hand, if the aim of this process is to come up with a formula predicting age based on DBH then we must put age on the y axis.
The choice of which variable goes where is not just a matter of display: different "trendlines" will be generated by different choices. On the other hand the "correlation coefficient" r and its
associated P value (see below) will not depend on this choice.
You can see below which choice I made (this time):
It should be clear that there is a general trend for the old trees to be big trees. On the other hand, there is a lot of variation: For example, the the biggest tree is not the oldest. I hope it is
clear that the relationship between age and DBH is not that given by "connecting the dots". (It is almost always wrong to produce such "connect the dots" plots!).
On the other hand, the relationship between age and the average DBH, might be a smooth curve that misses individual data points (some high and some low), but instead hits some sort of average between
the points like this:
If we do a least squares analysis of the data the following results are reported:
y = a + bx where:
a= 1.29 ([a] = 1.0 )
b= 0.128 ([b] = 2.11E-02 )
degrees of freedom = 10
r = 0.830 (p = 0.001)
We are given a line (displayed above) that represents an average relationship between age and DBH (the parameters that describe that line, the y intercept a and the slope b are given along with
estimates of the expected range of variation correlation coefficient, r, with an associated probability p. The small value of p indicates that is highly unlikely that the apparent relationship
between age and DBH came about by chance. Do not be highly impressed by small p values: they are not uncommon particularly in larger datasets. Instead focus on r, which will always be between -1 and
1. The fact that r is positive for this data indicates that larger age generally goes along with larger DBH -- a "direct" relationship. If r is negative more x goes along with less y -- a negatively
sloping "inverse" relationship. Values of r near zero indicate no particular relationship between the variables. It is often said that r^2 is the fraction of the variation in y that is explained by
its relationship with x. What this means is the standard deviation of the data's deviation from the trendline (the blue lines shown below) divided by the the standard deviation of the y data is 1-r^
1-r^2 = (deviations from trendline)/(standard deviation of y data)
Clearly, if r is near 1 or -1, the deviations from the trendline must be "small".
Thus if r is near 1 or -1, there must be relatively small deviations from the line.
Please guard against the not uncommon situation of "statistically significant" correlations (i.e., small p values) that explain miniscule variations in the data (i.e., small r^2 values). For example,
with 100 data points a correlation that explains just 4% of the variation in y (i.e., r=.2) would be considered statistically significant (i.e., p<.05). Here is what such data looks like in a scatter
The correlation may be statistically significant, but it is probably not important in understanding the variation in y.
The name "least square" comes from the process of defining a trendline. The line is adjusted until the sum of the squares of the y deviations from the line (shown above in blue) are as small as
possible. Note that there are other ways to produced such trendlines (a topic addressed in greater detail here).
The solid line, which does a very good job matching most of the data but leaves 4 points well off the line, is based on minimizing the length of the horizontal deviations from the line (shown above
in red). It has a significantly steeper slope than the least squares line (about 4½× [b] more than the least squares b). The dotted curve, which badly misses only 3 points, is a parabola chosen to
minimize the square of the y deviations. There really is not a way of selecting the best trendline from among all the possible trendlines. You may be guided by the suggestions of known theory, by the
requirements of a particular instructor, by standard practice (usually a least squares line), by knowledge of which points are most likely to be anomalous, or (unfortunately) by a desire to produce a
particular answer. The option to push an answer onto the data -- to Lie with Statistics -- comes from the relatively large deviations seen in this data. If the relationship were "tighter" all
possibilities would be quite close.
Trendlines are often used just to "guide the eye": to display an average trend. They may also be used to make quantitative predictions. You can answer questions like "how big will my oak tree be in
20 years?" or "How old is this 10 inch diameter tree likely to be?". It is safest to use the predictive abilities of trendlines only within the range of the data that defined the trendline (this is
basically interpolation). When used outside the tested range (extrapolation) trendlines may well give wrong or even crazy answers. For example the above solid line (with minimum horizontal
deviations) suggests a 1 year old tree has a negative diameter. The parabolic trendline suggests trees actually start to shrink for ages beyond about 80 years and would have negative diameters when
older than about 150 years.
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Elements of Intuitionism (Oxford
- Journal of Automated Reasoning , 1989
"... Isabelle [28, 30] is an interactive theorem prover that supports a variety of logics. It represents rules as propositions (not as functions) and builds proofs by combining rules. These
operations constitute a meta-logic (or `logical framework') in which the object-logics are formalized. Isabelle is ..."
Cited by 422 (47 self)
Add to MetaCart
Isabelle [28, 30] is an interactive theorem prover that supports a variety of logics. It represents rules as propositions (not as functions) and builds proofs by combining rules. These operations
constitute a meta-logic (or `logical framework') in which the object-logics are formalized. Isabelle is now based on higher-order logic --- a precise and well-understood foundation. Examples
illustrate use of this meta-logic to formalize logics and proofs. Axioms for first-order logic are shown sound and complete. Backwards proof is formalized by meta-reasoning about object-level
entailment. Higher-order logic has several practical advantages over other meta-logics. Many proof techniques are known, such as Huet's higher-order unification procedure. Key words: higher-order
logic, higher-order unification, Isabelle, LCF, logical frameworks, meta-reasoning, natural deduction Contents 1 History and overview 2 2 The meta-logic M 4 2.1 Syntax of the meta-logic
......................... 4 2.2 ...
- A CHAPTER IN THE FORTHCOMING VOLUME VI OF HANDBOOK OF LOGIC IN COMPUTER SCIENCE , 1995
"... ..."
, 1990
"... This paper introduces a new higher-order typed constructive predicate logic for fixpoint computations, which exploits the categorical semantics of computations introduced by Moggi [8] and
contains a strong version of Martin-Lof's `iteration type' [11]. The type system enforces a separation of comput ..."
Cited by 12 (4 self)
Add to MetaCart
This paper introduces a new higher-order typed constructive predicate logic for fixpoint computations, which exploits the categorical semantics of computations introduced by Moggi [8] and contains a
strong version of Martin-Lof's `iteration type' [11]. The type system enforces a separation of computations from values. The logic contains a novel form of fixpoint induction and can express partial
and total correctness statements about evaluation of computations to values. The constructive nature of the logic is witnessed by strong metalogical properties which are proved using a
category-theoretic version of the `logical relations' method. 1 Computation types It is well known that primitive recursion at higher types can be given a categorical characterisation in terms of
Lawvere's concept of natural number object [6]. We show that a similar characterisation can be given for general recursion via fixpoint operators of higher types, in terms of a new concept---that of
a fixpoint object in ...
"... The theorem prover Isabelle and several of its object-logics are described. Where ..."
"... This paper introduces a new higher-order typed constructive predicate logic for fixpoint computations, which exploits the categorical semantics of computations introduced by Moggi [8] and
contains a strong version of Martin-Lof’s ‘iteration type ’ [ll]. The type system enforces a separation of compu ..."
Add to MetaCart
This paper introduces a new higher-order typed constructive predicate logic for fixpoint computations, which exploits the categorical semantics of computations introduced by Moggi [8] and contains a
strong version of Martin-Lof’s ‘iteration type ’ [ll]. The type system enforces a separation of computations from values. The logic contains a novel form of fixpoint induction and can express partial
and total correctness statements about evaluation of computations to values. The constructive nature of the logic is witnessed by strong metalogical properties which are proved using a
category-theoretic version of the ‘logical relations ’ method. 1 Computation types It is well known that primitive recursion at higher types can be given a categorical characterisation in terms of
Lawvere’s concept of natural number object [6]. We show that a similar characterisation can be given for general recursion via fixpoint operators of higher types, in terms of a new concept-that of a
fixpoint object in a suitably
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Relation Question
June 9th 2013, 06:29 AM
Relation Question
Relation R defined on the set of seven-bit strings by s[1]Rs[2], provided that the first four bits of s[1] and s[2] coincide
(i) Show that R is anequivalence relation.
(iii) List one (1) member of each equivalence class.
June 9th 2013, 07:25 AM
Re: Relation Question
Here coincide is synonymous with same as. As such almost all such relations are equivalence relations.
You post your efforts on showing the three required properties.
For the second part, there will be sixteen strings.
June 9th 2013, 08:20 AM
Re: Relation Question
hey plato thanks for the reply, I just have a doubt here....if the first four bits coincide how about the rest three? how i want to proof in matrix form if i can't determine the rest three bits?
June 9th 2013, 08:45 AM
Re: Relation Question
The last three bits have nothing whatsoever to do with this question.
$1101000~R~1101111$but not $1100000~{R}~1101000$ WHY?
In all of these strings , does each string have the same first four bits as itself? What property would that prove?
June 9th 2013, 08:53 AM
Re: Relation Question
June 9th 2013, 08:56 AM
Re: Relation Question
June 9th 2013, 09:00 AM
Re: Relation Question
but is this relation valid
1101010 R 1101000?
June 9th 2013, 09:11 AM
Re: Relation Question
June 9th 2013, 09:13 AM
Re: Relation Question
thanks dude, for the reply.....i'm just confused with the last four bits....now i'm fine with it ...thanks again dude....sorry for taking your time....thank you
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Next Article
Contents of this Issue
Other Issues
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ELibM Home
EMIS Home
Projectivity and flatness over the endomorphism ring of a finitely generated comodule
T. Guédénon
1120 avenue Fournier, Québec, QC, G1V 2H8, Canada, e-mail:guedenth@yahoo.ca
Abstract: Let $k$ be a commutative ring, $A$ a $k$-algebra, $\cal C$ an $A$-coring that is projective as a left $A$-module, $^*{\cal C}$ the dual ring of ${\cal C}$ and $\Lambda$ a right $\cal
C$-comodule that is finitely generated as a left $^*{\cal C}$-module. We give necessary and sufficient conditions for projectivity and flatness of a module over the endomorphism ring $End^{\cal C}(\
Lambda)$. If $\cal C$ contains a grouplike element, we can replace $\Lambda$ with $A$.
Full text of the article:
Electronic version published on: 18 Sep 2008. This page was last modified: 28 Jan 2013.
© 2008 Heldermann Verlag
© 2008–2013 FIZ Karlsruhe / Zentralblatt MATH for the EMIS Electronic Edition
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Big Questions Online
I enjoyed the read.
I'd first like to consider a very egocentric hybrid that can eliminate the need for MWI. Perhaps it is only "my" observation that can collapse the wavefunction and the rest of nature just follows
Schoedingers Equation. This would make "me" a sort of god of some closed system, but into necessarily God. I guess that is similar to MWI but with no forks in the road.
We could also treat humanity as a collective. If our minds are together one perfectly coupled wave collapsing detector than there need only be one world, and one mind... collective free will? That
would fit nicely with original sin and Christian dogma.
Stephen M. Barr
July 10, 2012
jrd261,Your suggestions have merit
Dear jrd261,
What you are suggesting, namely solipsism, the idea that there is only one mind, is a perfectly consistent way of interpreting quantum mechanics. I quoted Eugene Wigner in my essay as saying, that
materialism is “logically consistent with present quantum mechanics." The full quote was actually, "Solipsism may be logically consistent with present quantum mechanics, but materialism is not." In
the solipsistic interpretation, there would still be "forks in the road": Each time the one observer (you) made an observation, the wavefunction would collapse, i.e. the universe would take one road
at the fork.
Your other suggestion --- all minds being a collective --- also can be made consistent. In particular one could take the view that whenever there is a branching of the wavefunction (which happens
when different parts of the wavefunction "decohere" from each other, in the technical jargon) all consciousness in the universe proceeds down just one branch. The wavefunction would continually
branch, exactly as MWI says, but there would never be a situation where the same observer existed in several conscious versions in distinct branches. In this picture, the wavefunction itself is
constantly branching, like train tracks; and what happens at the "collapse of the wavefunction" is not really any change in the wavefunction --- all the tracks are still there --- but rather all
consciousness proceeds down a single track, so to speak. (What I have just described is my own speculative view of quantum mechanics, for what it's worth.)
There are different ways that mind could be imagined to come into the story. But unless mind comes in somehow at a fundamental level, then everything is just matter governed by Schrodinger's
equation, and one is left with all parts of the wavefunction being equally real (even after decoherence). That is, one is left with MWI, and each observer existing in a huge number of equally real
Joshua Weiner
July 10, 2012
God as observer
Excellent and succinct essay on the problem, Stephen! I believe I asked this naive question before but perhaps it is worthwhile to raise here again. Given the nature of quantum mechanics, many events
will remain probabilistic; that is, they will never have an observer, and thus the wavefunction will never collapse and there will be no "jump". While this obviously is outside the main question, in
that I am simply assuming an omniscient God, my question is: could God be the ultimate observer who can collapse any wave function? Would God's assumed omniscience preclude this (that is, would he
see all wavefunctions at once and thus there would be no need for him to "turn his attention" to any particular set of quantum events), or, philosophically might it remain a possibility? I suppose
one problem with this is that, while God could interact with the physical world "at a distance" in this way (which seems attractive in that one objection atheists have is how some kind of
supernatural being can possibly interact with the physical world), the result would not be determined by God but rather the wavefunction could collapse either way (if there is a way for the observer
to "choose" which direction the collapse goes, I've never heard of such). So God would be necessary, in this sense, for anything to happen (consistent with traditional theism), but he would still not
necessarily know the outcome (clearly inconsistent with traditional theism). I realize this is nebulous but I still find some aspect of the argument compelling, in that observation of quantum events
could be a way to explain God's continual interaction with our world in a way that is intellectually defensible. I would love to hear your opinion, and I apologize if you gave it previously but I
forgot what you said!
As to the main point of the essay I certainly agree that QM does NOT directly prove anything about God, but does prevent, pretty clearly, it seems to me, an purely materialistic (or deterministic)
worldview. It leaves open the door. The MWI is as ludicrous as any desperate attempt to square the circle is: sometimes one's instinctual resistance to the data leads one to propound ever-more
improbable explanations. It seems there was a similar resistance to the Big Bang, which still exists but has become fringe, as I understand.
An interesting question is the relationship of quantum uncertainty to events in the brain (or rather, as the very fact that they are there is trivial, the outward effects on how the mind works). I
don't think I've seen anything that isn't fairly silly on this, pace Penrose. It does seem to me that QM undermines our traditional understanding of free will as much as any purely materialistic
world view would. So working those issues out will be a key area--I'd love to hear any sources you know of on this topic that are worth reading.
Stephen M. Barr
July 10, 2012
Reply to Joshua Weiner
Hi Josh,
Good questions. I don't think God (as understood in what is sometimes called "classical theism") can play the role of an "observer" in quantum mechanics, for several reasons.
In the traditional (or "Copenhagen") interpretation of quantum mechanics, as expounded by Wigner,
Peierls, and others, and explained in my essay, the collapse of the wavefunction corresponds to a change in the knowledge of the observer. But in classical theism, God is outside of time, and his
knowledge does not change.
Another way of looking at all this is that the wavefunction gives the probability correlations between the results of one set of measurements and the results of another (usually later) set of
measurements. For example, if I measure that an electron is located in a cubical box whose sides are of length L and that it has a certain energy, then that gives me an initial wavefunction that I
can use to calculate the relative chances of finding various values for the electron's position or momentum at a later time. In other words, in practice the wavefunction is a way of correlating the
of various measurements. But, in classical theism, God does not make measurements. His knowledge is not "a posteriori" but "a priori". That is, he does not learn about things as a result of their
happening, but rather they happen because he knows them, so speak.
The wavefunction encodes what some observer is in a position to assert about the physical world given some knowledge that he already possesses by virtue of other measurements or observations made by
him.. So the observer is, practicaly by definition, some being who acquires information by means of physicallly interacting with the rest of the universe. God, however, (according
to traditional theism) does not physically interact with things in the universe, as he is not a physical entity. Physical entities cause things to happen in the universe by means of physically
interacting with other entities. But God causes things in an utterly different way: by willing the reality or being of the whole universe. An analogy: a character in a novel has effects within the
novel's plot by means of acting within the plot in accordance with the internal rules of the novel. The author of the novel, on the other hand, causes things to happen as they do in the novel not by
being'an actor in the plot, but by simply by conceiving of the plot (and of the rules that givern it).
As far as free will goes, and possible connection to "quantum indeterminacy", I would suggest the relevant sections of my book "Modern Physics and Ancient Faith".
Joshua Weiner
July 10, 2012
Yes. . .but
There is something I still do not understand. My understanding is that until someone observes a quantum system, the wavefunction remains probabilistic. If God has already observed everything, because
he is out of time, then there would be no uncollapsed wavefunctions. If not, however, then there must still be quantum events that even God does not know the result of, because they are logically
unknowable. God can do anything that is logical, but not that which is by its nature illogical, correct? So I guess in response to your statement: "That is, he does not learn about things as a result
of their happening, but rather they happen because he knows them"; I would ask, doesn't this preclude the possibility of there being any QM events that remain probabilistic? If all events (including,
say, particle decays) happen because he knows they are going to happen (and where, and when, and how), then his very knowing would preclude any uncollapsed wavefunctions. . . and we would not have
made the observations that led to the formulation of QM. Probably I missed something here. . .
I guess where this my argument may go wrong is in your point that God does not "physically" interact with the physical world, but rather through his will. But in traditional theism, God can, if he
wills, cause a physical event such as a flood. Though the distal cause is his "will" (and it is difficult for us to understand quite what this means), the proximal cause is rainfall, which presumably
is not some special kind of supernatural precipitation but the same kind of physical rainfall we know (and this summer, wish for). So I don't quite understand why the action-at-a-distance of
observation by God (or we can call it willing) could not be possible; Even in the case of the flood, there must be some interface between God's will and a physical event. If it is not collapse of
wavefunctions, it is presumably something else. Am I way off base?
Wallace Forman
July 10, 2012
Determined, Non MWI
Hi Dr. Barr,
Thank you for the interesting article, however I am afraid I am not convinced. Here is my objection:----------------------------------------If god observed all the quantum variables outside of time
and then disappeared, the universe would be determined, but still appear probabilistic to us as we observed the variables. If we can imagine such a probabilistic, determined universe as a possibly
existing thing, we can also imagine it existing without the need for determination by observation. If process Y will create a quantifiable object X with characteristics A and B, we can also just
imagine quantifiable object X with characteristics A and B, not necessarily created by process Y (determination by observation), a process which is itself fundamentally incapable of our observation.
(Wouldn't we prefer this under Occam's Razor?) This reasoning tells me I don't need to choose between MWI and determination by observation.----------------------------------------Since I composed the
above I see you have included some discussion about whether God can observe the variables in the comments section. What you say there seems more to be a discussion of theology than how we should
think about probability so let me offer just one more thought about my objection: It doesn't really matter for my point whether god exists, whether he can observe the variables, or determine them by
doing so. The point is that we can imagine a universe with variables that look probabilistic to its inhabitants, but are not really undetermined. If we can imagine such a universe, then we need not
choose between MWI and determination by observation. Moreover, if there is no way for us to test between the imagineably "determined" and untestable "determined, by observation," Occam's Razor tells
us to prefer "determined."
July 11, 2012
Measurement in MWI
Prof Barr,
How does a many-world wavefunction know when to branch?.
That is, the measurement is supposed to be external to the wavefunction but if everything is internal, what would measurement mean anyway?.
Also, quantum mechanics was initially formulated as a theory of interaction of microscopic systems interacting with measuring devices. Now the physicists define and use wavefunctions of the entire
universe. How is this extrapolation justified?. Or are they doing it as a matter of faith in atomism that all macroscopic objects are fully explainable as interactions of fundamental microscopic
In Thomism, the parts are only defined with respect to the whole and can not be fully understood
without reference to the whole. This whole-part loop is called the Formal Cause of the thing.
Perhaps this is what quantum mechanics paradoxes require?
July 11, 2012
Limits of language?
A problem and a possibility. The problem: Jane’s relationship to the french exam is accidental to Jane – it is the relationship which is probable, not Jane. It is easy to verbalise this relationship.
In quantum physics probability is of the essence of the subject, a fact which mathematics reveals consistently and persistently. Verbalization of probability in this sense of essence, to use
Aristotelian language, seems impossible to adopt within the necessary constraints of grammar and syntax. Is it possible to verbalize quantum mechanics without gross distortion of its fundamentals?
The possibility: Panentheism ‘God in the world’ finds ‘in’ difficult. If probability is of the essence of all created things then probability is at the heart of ‘in’. A creator God might therefore be
said to be unchangeably committed to a probability inspired universe.
Stephen M. Barr
July 11, 2012
Reply to Wallace Forman
Dear Wallace,
First, the logical structure of the argument against materialism that I outlined in my essay is based on what happens when a measurement is made and how we are to understand the famous "collapse of
the wavefunction" that occurs when a measurement gives one definite result. The argument is to the effect that the mind of the measurer (or "observer") cannot be completely described by wavefunctions
governed by the laws of quantum physics. So, it is the mind of measurers that is in question, not the mind of God or gods. Let us therefore leave them out of the discussion.
Second, you say that we "might imagine a world with variables that look probabilistic to its inhabitants, but are not really undetermined." That is true, and it is exactly what happens in classical
physics. In classical physics, all measureable quantities have definite values at all times, but because we don't know most of those values, we must use probabilities. Many physicists, including
Einstein, have suggested that the same may be true in quantum mechanics, i.e. there are "hidden variables," whose values we don't know and that this is why probabilities seem to be needed in quantum
mechanics. The most successful attempt to construct a hidden variables theory was made by David Bohm. While such theories can reproduce many of the successes of standard quantum mechanics, they are
unable to deal with full-blown "quantum field theory." Most physicists, therefore, think the hidden variables approach is extremely unlikely to succeed.
Third, the structure of the argument I explained in the essay was not so much about the values that variables truly have, but rather about what a wavefunction means. Basically, a wavefunction is one
of two things: it is either (a) a straightforward description of the world as it is, or (b) it encodes what some observer knows or is in a position to assert about the world. If it is (a), one is led
perforce to the MWI. The reason is simple: The wavefunction contains many branches describing alternative histories of the world. So if the wavefunction is just "the world as it is", then such
histories are really all going on at once, and that is the essence of MWI. On the other hand, if (b) is correct, then "what some observer knows" --- and thus mind --- enters into how our fundamental
theories of physics. That is why Wigner wrote about 50 years ago
“While it may be premature to imagine that the present philosophy of quantum mechanics will remain a permanent feature of future physical theories, it will remain remarkable, in whatever way our
future concepts may develop, that the very study of the external world led to the conclusion that the content of the consciousness is an ultimate reality."
Stephen M. Barr
July 11, 2012
Reply to Vishmehr24
Dear Vishmehr24
The branching of a wavefunction takes place, roughly speaking, when some microscopic system (e.g. a radioactive nucleus) interacts with a macroscopic system (e.g. a Geiger counter). The wavefunction
has various parts describing the nucleus decaying at different times. In these different parts of the wavefunction the Geiger counter is also doing different things (because it is affected by what
the nucleus does). Since a Geiger counter is made up of a huge number of
particles, a Geiger counter "doing different things" entails a huge number of particles behaving differently in the different parts of the wavefunction. That means that the different parts of the
wavefunction differ very greatly from each other, and therefore they "decohere" from each other, to use the technical jargon. That means that for all practical purposes, they no longer can affect
each other and are like different worlds.
In other words, the splitting apart of the branches happens whenever macroscopic objects are involved. The big question that distinguishes MWI from the traditional interpretation of quantum mechanics
is whether all the branches are equally real. In MWI, they are equally real. In the traditional interpretation, they are not. E.g. if the observer sees the nucleus decay at 1 PM (because the Geiger
counter clicked at 1 PM) then (according to the traditional view) that is what really happened and all the other branches of the wavefunction (for example the one where the nucleus
decayed at 5 PM) are just unrealized possibilities.
You are right that one is doing a great extrapolation in supposing that oen could (in principle) write down a wavefunction of the entire physical universe. Speaking just for myself, I see no problem
in making that extrapolation. Of course one is always making similar extrapolations when doing physics. When one assumes that nature obeys universal laws, so that every electron in a piece of metal
obeys the same Dirac equation, that is an extrapolation. (I think a reasonable and justified one. )
Stephen M. Barr
July 11, 2012
Reply to Josh's "yes ... but"
I think I know what is bothering you, Josh. As I said in reply to Wallace Forman (in the third paragraph), it all comes down to what the wavefunction of a system is. One would like to be able to
say that it is just a straightforward description of what is happening in the world, of the world as it really is, apart from what you or I know about it. That leads straight to the Many Worlds
picture, because the wavefunction typically contains descriptions of many alternative branches. In the traditional or Copenhagen interpretation, one has a more modest view of what the wavefunction
is: It is not simply "the world as it is", but rather it encodes what some observers know or are in a position to assert about the world. that is why heisenberg himself said that the mathematics of
quantum mechanics "represents no longer the behavior of elementary particles, but rather our knowledge of this behavior". And it is why Rudolf peierls said, "the quantum mechanical description is in
terms of knowledge."
That raises a very important question --- which, I think, is your question: What DOES describe "the world as it really is"? Even if the wavefunction does not describe it, there must be some
comprehensiove and complete and accurate description of physical reality --- call it the "God's eye view of things" (even thouigh I don't want to drag God back into the discussion).
In other words, what IS really going on when no one is looking? What if beings such as ourselves had never evolved? What about regions of the universe that no human or other sentient organism is
ever going to observe or make measurements of? What about what will be happening in the universe after all life has died out? Good questions! The wimpy answer is that science cannot speak about
things that cannot be observed, and what is going on in places that will never be observed is, by definition, something that cannot be observed! But that seems a pretty unsatisfactory answer. The
traditional Copehagen interpretation doesn't give an answer. I have an answer that satisfies me, and I give a very brief sketch of it in my reply to jrd261. Here I will only say that I think that
even in the context of the traditional interpretation of quantum mechanics there does exist an answer to the question "what is really going on in the world even when no observers are looking". In
other words, the traditional interpretation does NOT commit one to some form of subjectivism or Berkeleyan idealism, but can be consistent with a robust philosophical "realism". But this is a tricky
business, and probably beyond what can be discussed in such a forum.
July 11, 2012
I'm curious as to your take on how the concept of retrocausality impacts your discussion. Very good article as well, thank you!
Stephen M. Barr
July 11, 2012
two replies
Dear Ageingcrofter,
I would agree with you that attempts to discuss mathematical subjects "verbally" (if by that is meant "natural languages" rather than mathematical symbols) can lead to distortion. This is especially
the case with subatomic physics, where the concepts needed are far removed from everyday experience. For example, the idea that the wavefunction is a "vector in a Hilbert space" involves a high level
of mathematical abstraction. On the other hand, it is impossible to learn either mathematics or physics without using natural language in addition to mathematical symbolism. One starts with ordinary
experience, and everything one learns must be somehow connected to ordinary experience if one is to grasp it. One cannot understand complex numbers, for instance, or develop intuition about them,
unless one first has some experience with ordinary counting numbers.
Dear jarruda,
If by retrocausality you mean effects coming before their causes, then this is not allowed in present theories of physics. Theories that "violate causality" are regarded as "pathological" and
But do you mean something else?
July 11, 2012
Another option
So I follow the thread of the argument you've made, Stephen, but it seems that you're making a probably false assumption - namely, that the universe admits nonzero probabilities of infinitely small
value. One way of explaining why wave functions "jump" in some interactions but not in others would be to say that the universe can only sustain probabilities that are of a sufficiently high
magnitude (because it only measures things to a finite accuracy), so that once a single wave function is tested enough times it necessarily collapses one way or the other. To make a very rough
analogy, this would be along the lines of a calculator rounding 2/3 up to 0.666667: because the calculator can't display (or even really calculate) infinitely many digits, at some point it has to
terminate its operations and simply decide to put a 6 or a 7 at the end. If the engine that's running the physics of our universe is only finitely powerful, it would make perfect sense for minds to
be sufficient (in most cases) but unnecessary for the collapse of wave functions: minds are very complex but not only minds are very complex, and eventually time alone would guarantee a wave
function's collapse.
Granted, this explanation isn't obvious and it leaves some things to be answered (though if you can give me a physical theory that leaves *no* questions to be answered then I'll be very impressed).
But it would at least explain the data without requiring the positing either of magical properties for minds or the reality of the many worlds hypothesis, and (so far as I'm aware) it is at least
plausible. We obviously know that the universe measures things very precisely indeed, but I've not read anything that would necessitate the universe measuring things infinitely precisely.
Stephen M. Barr
July 11, 2012
reply to larryniven
Dear larryniven,
You make an interesting suggestion, but I am not sure it can be made to work. The "collapses" of wavefunctions (i.e. jumps in probabilities) don't only happen when the relevant probabilities are
close to 0 or 1. Consider, for example, the case of a (hypothetical) radioactive nucleus that has a half-life of one hour. If the nucleus is there at noon, it has 25% chance of decaying during the
interval between 1 PM and 2 PM. During that interval, the survival probability of the nucleus varies continuously between 0.5 and 0.25 (and the decay probability varies in the corresponding way).
At no point in this time interval are the decay or survival probabilities that are given by the wavefunction close to 0 or 1. Typically the probabilities jump by an amount that is finite and not at
all small.
Your idea that probabilities are discrete rather than continuous is nevertheless interesting to contemplate. One would have to modify the rules of quantum mechanics in a radical way for it to work,
however, as those rules use mathematical operations that would not be consistent with discretized probabilities.
You wouldn't happen to be the science fiction writer Larry Niven?
Ethan Blaylock
July 11, 2012
Several questions
Thanks for the interesting essay. I have several questions.
1) Why do you say that the Copenhagen interpretation requires a human observer; are there other ways of understanding the Copenagen interpretation that do not require this?
2) Are Copenhagen and Many Worlds the only tenable options; what about objective collapse theory (which does not require a human observer)?
3) How do you differentiate your position from a "God of the Gaps" argument? Can't we just say that our physics models and our interpretations of them currently only incompletely describe the
universe as we experience it (as has always been the case)? For example, the math that describes quantum mechanics does not cleanly mesh with general relativity, but I think the most reasonable
conclusion to draw is that the math is not complete. Even if the math we currently use to describe quantum mechanics has hit something of a dead end, we already know that quantum mechanics is only
an incomplete explanation of the universe. It seems strange to make so much of its quirks.
Stephen M. Barr
July 11, 2012
Reply to Mr. Blaylock
Dear Ethan,
Excellent questions. I will take them in the order presented.
Question 1: The traditional interpretation of quantum mechanics (also called the Copenhagen or standard interpretation) makes a distinction between the "system" being measured and the
"observer" doing the measurement. The wavefunction describes the system. The wavefunction changes in two ways: (a) When the system is left alone, its wavefunction evolves in a continuous and unique
way in accordance with the Schrodinger equation. (b) When the observer makes a measurement of some observable property of the system and gets a definite result, the wavefunction
undergoes a "collapse" that reflects the result of the measurement. Unlike the Schrodinger evolution (a), the collapse (b) is sudden and unpredictable.
The problem I described in my essay is that if the observer is included completely in the system, then the wavefunction changes only in way (a) and there is no "collapse of the wavefunction". There
is thus no unique and definite outcome, but all possibilities remain in play --- thus MWI.
So the question arises, what or who can play the role of "observer"? The first point --- emphasised by Peierls, whom I quoted in the essay --- is that the observer cannot be a purely physical entity,
for if it were one could consider the larger system which comprised both the original system and the (purely
physical) observer. But then one would have the problem that the wavefunction of this larger system could collapse ---- unless there were some other observer outside of IT. But then, if THAT observer
is purely physical, one can expand the system yet further to include IT. An infinute regress.
So it seems that the observer has to be something that is not purely physical. Moreover, since a measuremet is only completed when the observer KNOWS the result, it seems that
the observer has to be a knower. That is why many people --- such as Wigner and Peierls ---- have argued that the "consciousness" or the "mind" or the "knowledge" of the observer plays an essential
Does the observer have to be "human"? Well, certainly humans are observers: we make measurements and get definite results. But that doesn't mean that only human beings are observers.
Could inteligent space alliens be "observers"? Obviously, if there are any. Could chimpanzees or dogs? That depends on whether they can make measurements and whether they are completely describable
by physics.
Question 2: What about objective collapse theory? Well, if one keeps to the mathematical structure of quantum mechanics as we have had it since the days of Heisenberg and Schrodinger, then there is
no collapse without an observer, as explained above. (Some people --- including some physicists --- have the misconception that what is called "decoherence" is the same thing as wavefunction
collapse. And since decoherence only requires the involvement of macroscopic systems (as I explained in answer to another person's question) not conscious observers, they
think they have explained wavefunction collapse without invoking observers. But they are wrong.)
So, to get a collapse without an observer, one would have to change the mathematics of quantum mechanics. There have been attempts to do this. (Wigner tried, for example.) Perhaps that is
what is needed --- many people have suggested that in a more complete or improved theory, all the puzzles of quantum mechanics will be resolved and wavefunction collapse will be explained. It should
be noted, however, that quantum mechanics has passed hundreds of thousands of tests with stunning accuracy over the last 85 years or so. It is hard to modify it without destroying its
self-consistency or its consistency with experiment. This leads us directly into question 3.
Question 3: Can't we say that quantum mechanics is incomplete? Isn't the difficulty of meshing it with General Relativity evidence of this? Well, many people have argued this. However,
it is believed that superstring theory is a consistent theory that incorporates both the principles of quantum mechanics and General Relativity. So that takes away a major argument' for the
inadequacy of quantum mechanics. One never knows what the future will bring in science, but right now quantum mechanics looks like it is in perfect health. It is simply not true to say "that we
already know that quantum mechanics is only an incomplete explanation of the universe." We know no such thing!
Stephen M. Barr
July 11, 2012
tyoes in reply to Mr. Blaylock
There is a bad typo in line 6 of paragraph 4 of my Reply to Mr. Blaylock. It should read that "the wavefunction of this larger system could NOT collapse"
And of course intelligent has two l's and alien has one l.
July 11, 2012
Nope, just a fan
"You make an interesting suggestion, but I am not sure it can be made to work. The "collapses" of wavefunctions (i.e. jumps in probabilities) don't only happen when the relevant probabilities are
close to 0 or 1."
Sure - but I didn't say that they would only collapse when near 0 or 1, just that they would always do so. Again, think of it like it was a simulation that you were running on your own computer: you
could easily write a "random" decay function so that it would have a 50% chance of decaying at time N, a 25% chance of decaying at time N+1 and so on. What you could not do is write a decay function
that stays smooth indefinitely, because eventually your machine would run out of digits with which to calculate, and the probability would effectively become 0.
"One would have to modify the rules of quantum mechanics in a radical way for it to work, however, as those rules use mathematical operations that would not be consistent with discretized
Really? I'd like to see some examples of what you mean by this, because I'm not at all sure that it's true. I mean, yeah, if the discrete increments were (relatively) large, we'd see something very
different in the universe and the theory would never have come up. But if they were (relatively) small? I haven't done the proof yet, but I suspect we'd have to look very closely in order to notice
the difference. And it's not like probability theory requires continuous distributions; in fact, most math that deals with continuous distributions is just a clever abstraction of math that deals
with discrete distributions (to take an easy example, calculus is like this). So which operations would not be consistent with what I'm suggesting, exactly?
July 12, 2012
Dear Prof Barr,
Thanks for your reply but I am still not convinced that the quantum mechanical extrapolation of the wavefunction to the entire universe is of the same character as the extrapolation of homogenity of
the universe such that same laws apply on Mars as on the Earth.
The quantum mechanics demands observers; the formulation is expressed in terms of measurement and measuring devices and observables. These presupposed things are external to the system described by
the wavefunction. What possible meaning could be attributed to the wavefunction of the universe?
Am I correct in supposing that the measurement process has been resolved to the decoherence pheneomenon?. But decoherence is governed by Schrondinger's equation and thus by the Copenhagen picture or
principles can not be the full story. Measurement is required to break the natural evolution of the wavefunction. The decoherence, I believe, is asymptotic; the superpositions never die 100%
And you say that MWI keeps alive the possibiliies that are not realized in the Copenhagen picture. But the decoherence should give unique answer since it is just the evolution of Schrodinger's
equation. So how does MWI keep the possibilites alive?
Chris Thompson
July 12, 2012
Believing in God
When I want to believe in something, I am quite able to use any reason whatsoever to do so. My question back to the OP is where are you going with this question? Are you paving a road and making a
case to "believe in God" or what?
Stephen M. Barr
July 12, 2012
Reply 2 to larry niven
Dear larryniven,
I can give two examples.
First example. Suppose that a particle X has three possible "decay modes", A,B, and C, and that a "symmetry principle" makes these three decay modes equally probable. (Symmetry principles
play an important role in fundamental theories of physics.) Then the "amplitude" for X to decay in each of these modes must be EXACTLY equal to the square root of 1/3. (The "amplitudes" are what
appear in the Schrodinger equation. To get the probabilities one takes the square of these amplitudes.) The exactness of the answer logically follows from two unavoidable requirements: the
probabilitiies of all outcomes must add up to 1 (100%), and the three decay amplitudes have to be equal by the symmetry principle. So, there is no way for these numbers to be "rounded off" in some
high decimal place.
Second example. In the decay of an isolated radioactive nucleus, the decay amplitude and the decay probability are given EXACTLY by' an exponential function of time. This can be shown to follow
from "time tranlation invariance" of the laws of physics (a symmetry principle).
The point is that quantum probabilities follow from a precise mathematical formalism, which involves a number of fundamental principles. These probabilities cannot depart from the values
given by the formalism without violating the basic principles and assumptions of the theory.
I am not saying that your idea couldn't be true, only that for it to be true, the rules of quantum mechanics would have to be changed in a fundamental way.
Stephen M. Barr
July 12, 2012
reply 2 to vishmehr
Dear Vishmehr,
If I understand your argument correctly, it is as follows: The collapse of the wavefunction of a system requires the observer to be outside the system. No observer (measurer) could be outside of the
entire universe. Therefore, the "wavefunction of the entire universe" could never collapse. And if the wavefunction of the entire universe never collapses, one ends up with the Many Worlds
All of that is true. It is one of the aguments that proponents of the MWI use: they say that since there ought to be a wavefunction of the entire universe, one is forced to accept MWI. Turning this
around, it would seem that to reject MWI, one must also reject the possibility of a "wavefunction of the entire universe".
This logic is all sound. I would distinguish, however, between the entire universe and the entire PHYSICAL universe. Is everything in the universe physical? According to physics, as we have
understood physics for a long time, a physical system is completely characterized by some set of variables. There presumably is some set of variables that completely characterize the whole physical
universe. I see no compelling reason why one could not (in principle) construct a wavefunction for the physical universe. If one considers a human observer,however, that observer would be ( in part)
"outside" the description provided by that wavefunction to the extent that there is an aspect of that human being that is non-physical. In other words, there are two issues, (a) does it make sense to
speak of a wavefunction of the entire physical universe, and (b) is the universe the same thing as the PHYSICAL universe? It is not clear, therefore, that accepting a wavefunction of the universe as
a meaningful concept necessarily commits one to MWI.
July 12, 2012
Sorry, still not buying it
"The exactness of the answer logically follows from two unavoidable requirements: the probabilitiies of all outcomes must add up to 1 (100%), and the three decay amplitudes have to be equal by the
symmetry principle."
Equality is trivial in the sort of thing I'm talking about: because the system is going to have the same limitations on "memory" in every case, it'll truncate at the same place in every instance and
they will be equal. (No matter how many times you divide 2 into 3 on your calculator, you'll still get .6666667.) Adding up to one is a little trickier, because the additive margin of error is not
necessarily within the original margin of error - though it could be (it depends, among other things, on the additive algorithm being used). On the other hand, though, I find it awfully strange to
think that the universe first generates the odds of decay and then "adds them up." That seems more like a convenient way of talking about this to make it more intuitive than a description of any
real event. It has to be more plausible to say that the individual probabilities are parasitic on the fact that one of them must happen, not the other way around - or even that our talk of "outcomes
having probabilities" is really shorthand for something else. But then that's entirely compatible with having a finitely powerful physics engine.
Also, like the decay thing you mention next, the power of this argument depends on evidence, does it not? I readily admit that the predictive law is infinitely exact - math is nice that way - but it
begs the question to say that reality must be infinitely precise just because a predictive law that we've come up with is infinitely precise. So neither of these examples is tremendously persuasive,
at least so far as I can see.
At any rate, I did about an hour's research last night and discovered that there's actually already a theory not unlike the one I'm proposing. The Penrose version of objective collapse theory
evidently hypothesizes (and apologies if I'm misusing technical terms here, I'm just going for the gist) that there's a threshold amount of energy past which quantum phenomena collapse into more
standard phenomena, which is not quite what I've been saying but has enough similarities to be at least analogous (e.g. energy becomes the analog of hard memory). I see that you've discussed
objective collapse theories in this comment thread already, but you seem not to have addressed the Penrose variation (in that the thing about confusing decoherence with collapse seems not to apply to
it). Given that this is still a live theory, even to the point of being (apparently) (semi-)endorsed by the people at Stanford's philosophy encyclopedia, and given that it's fairly close to what
I've been saying, I'm beginning to feel like you gave an incomplete picture of the current state of affairs.
Stephen M. Barr
July 12, 2012
Reply to Chris Thompson
You ask where I am going with the question.
I am simply explaining a very important line of argument that goes back to the physicists John von Neumann, Fritz London, and Edmund Bauer, and that has been accepted by many other physicists
including Eugene P. Wigner and Rudolf Peierls. I think the argument presents one with three choices: (a) present quantum mechanics is wrong or incomplete in some fundamental way, (b) the Many Worlds
Interpretation is right, or (c) materialism is false, at least as regards the minds of observers. Pick your poison.
Why is God mentioned in the title of the essay? The answer is simple: I was asked by the people who run Big Questions Online whether I would be willing to write an article answering the question
"Does Quantum Physics Make it Easier to Believe in God?" The question was given to me, and I answered it.
I suspect that a certain line of thought may underlie your question. (I base this on your talking about how "wanting to believe something" can affect one's reasoning.) Tell me if I am wrong. The
line of thought is the follwoing: Religious people are driven not be reason but by wishful thinking. People who engage in wishful thinking bend reason to their desires and look for arguments to
sustain beliefs that are actually based on completely non-rational --- if not irrational --- grounds. I would make several comments about this. First, the validity of an argument does not at all
depend on the motives of the one making it. That is why it does not matter in science whether the person proposing a theory is a liberal, a conservative, a Marxist, a Buddhist, a Hindu, an atheist
or a Christian. What matters is the strength of his arguments. The same is true in philosophy. I have co-authored research papers with colleagues whose religious, political, and philosophical ideas
are diametricallly opposed to mine. For example, I have written a well known paper on the "multiverse" idea with three co-athors, one of whom is strongly atheist, and one of whom is (like me)
religious. Someone who worries about the ulterior motives of the person proposing a scientific theory or philosophical theory has already betrayed reason. Second, wishful thinking is a danger to
which all people are subject. Some people want to believe in God, some people want not to believe in God. Both thesm and atheism can be comforting beliefs, though in diferent ways. Third, all
people have fundamental convictions and intuitions about reality that make them more disposed to accept certain conclusions and reject others. For example, one of my deepest convictions is that
there is such a thing as objective moral truth. Another is that the world makes sense. To have basic presuppositions is not irrational. Indeed, one cannot be a rational person without them. Every
argument requires premises.
July 12, 2012
Comment on Discretized Probabilities
Hi Dr. Barr and larryniven,
I might add that if the values that the wavefunction could take on were universally discrete (at least in the way larryniven suggests), then any continuous function of the probability would inherit
the discrete nature of the wavefunction. Thus the discretization ought to be manifested in macroscopic ways that can be measured (example below). To my knowledge, this has not been seen in general
cases, which suggests that wavefunctions do not indeed take on only discrete values.
First, consider a Stern-Gerlach apparatus. That is, let an ensemble of electrons pass through a strong non-uniform magnetic field pointed along a given axis (call it the z-axis). Then the trajectory
of the ensemble will be split in two as it passes through the field, with equal numbers of electrons in each. We label each trajectory by 'up' or 'down'. Thus electrons in one branch have 100%
probability of having the property 'up', and those in the other have 100% probability of having the property 'down', and we may reasonably say that a randomly selected electron from the initial
ensemble has a 50% chance of having the property 'up', and a 50% chance of having the property 'down'.
Next, pass only the 'up' ensemble through another Stern-Gerlach apparatus, but this time oriented along a different axis. Let the angle this axis makes with the first z-axis be called T. We find the
initial 'up' ensemble is again split in two. We label one beam 'UP', and the other 'DOWN'. We also find that the relative number of electrons in each path depends on the angle T. In particular, it is
found that the relative number of electrons in one beam is cos^2(T/2), while for the other beam it is sin^2(T/2). We may again reasonbly say that an electron randomly sampled from the beam of 'up'
electrons in the middle of the apparatus has a chance of having the property 'UP' of cos^2(T/2), and a chance of 'DOWN' of sin^2(T/2).
Therefore, if the wavefunction could only take on discrete values (labeled by Pn), then the angle between the first and second apparatus can also only take on the discrete values Tn=2*arccos(sqrt
(Pn)) and Tn=2*arcsin(sqrt(Pn)). While this is not impossible, this seems unlikely, and there has been no experimental evidence of it to my knowledge.
Furthermore, a Stern-Gerlach apparatus applied to atomic Sodium will split the initial ensemble into three, whereas when it is applied to atomic Beryllium that number is four. Indeed, given any
positive integer N, it is in principle possible to find (or construct) an atom for which an ensemble will split into N beams. When a double-magnet apparatus is used on such ensembles, one sees the
same qualitative features as with the simple electron case described above: the relative numbers of atoms in each trajectory are a continuous function of the angle T between the magnets, and these
relative numbers are precisely the probabilities.
If the discretization of probability is to be universal (eg, if only probabilities of 1/10, 2/10, 3/10, ... were allowed for any system), then it cannot depend on the N of the atomic species being
passed through the apparatus. Thus a given apparatus must be able to be oriented at all angles Tn for all possible atomic species at the same time, even if it never has anything but electron
ensembles pass through it.
To be self-consistent, it is therefore required that the discretized probablities are such that the discretized angles for one ensemble are in general the same as the discretized angles for another.
But the function relating angles to probability is in general very complicated, with complexity increasing with N. For example, given a discretized set of probabilities (0, 1/10, 2/10, ..., 1), the
set of allowed angles for electrons differs from the set of allowed angles for atomic Sodium, which itself differs from the set of allowed angles for atomic Beryllium, such that if the apparatus were
adjusted to say the 8th allowed angle for Beryllium while electrons were passing through, it would produce a relative abundance of 'UP' electrons that is not equal to any of (0, 1/10, 2/10, etc),
which is inconsistent with the original idea that the only allowed probabilities are (0, 1/10, 2/10, etc).
While I have no proof, I suspect that the only set of probabilities that is consistent with all values of N is not discrete at all, but is continuous (or something very close to it), which leads us
back to a traditional understanding of the wavefunction.
This line of thought is extremelty suggestive, especially when one considers the relatively simple nature of the Stern-Gerlach/spin problem, where there exists continuous and bounded functions
relating macroscopic variables to probabilities. The analysis is likely more forbidding to discretized probability when one considers position and momentum, where the wavefunction values and the
underlying operators are not in general bounded.
Stephen M. Barr
July 12, 2012
Reply 3 toi larryniven
Dear Mr. Niven,
I think you may be missing the qualifiers embedded in my previous statements. I don't deny that a scheme such as you propose might conceivably be the correct description of the world. (Though I
think it unlikely in the extreme.) All I am saying is that such a scheme is NOT standard quantum mechanics. It entails a MODIFICATION of the mathematical formalism of quantum mechanics. By the same
token, Penrose is not claiming that his theory of collapse is something that is contained in our present quantum mechanical formalism; rather, he is proposing a modification of that formalism. (By
the way, this is clear from the following consideration: In the standard quantum mechanical formalism, the evolutiojn of wavefunction given by the Schrodinger equation is "unitary". The collapse of
the wavefunction, on the other hand, is a non-unitary process. Thus, one must modify the equations somehow to get a collapse from them.)
You also are proposing a modification of the equations, though you apparently do not appreciate this fact. You seem to be envisioning the equations of the universe being run on a computer of some
sort. (You speak of a "system" that has "memory" and that "truncates".) In any event, the process of "truncating" numbers or mathematical expressions is a mathematical operation, just like
multiplication and division. Such truncation operations are simply not a part of the mathematical formalism of quantum mechanics as we have had it since the 1920's.
It is perfectly OK to modify a theory, as long as one remains consistent with known experimental facts. You are perfectly entitled to propose such modifications. But they ARE modifications.
I am not giving an "incomplete picture of the current state of affairs", if by that you meant a misleading picture. (Is that what you meant?). I not only clearly stated that people have made
attempts to construct objective collapse theories, but I cited Wigner's own attempts to do so as an example. I could have mentioned Penrose's attempt as another example. I suppose that if I do not
mention every speculative attempt to modify quantum mechanics that has ever been proposed --- and there have been many --- you will say that I am not giving a complete account. In some trivial and
sense that is true. A complete account would require a review article of several hundred pages length.
Penrose's idea is interesting (most of his ideas are interesting); but has not attracted much of a following in the physics community. Penrose thinks that unifying quantum mechanics with General
Relativity (GR) is likely to require a modification of quantum mechanics. That is one reason he thinks that wavefunction collapse will end up being explained by gravitational effects. The backstory
of this is that for decades physicists were unable to find a consistent way to "quantize gravity", and this led many people to suspect that GR would force a modification of quantum mechanics. Some
still think that, but far fewer than previously, because superstring theory has demonstrated that GR can be quantized without changing the postulates of quantum mechanics.
July 12, 2012
Dear Dr. Barr, Thanks for a thought provoking article and interesting clarifying follow-up discussion. As you have already emphasized in the answer to one of the questions, decoherence by environment
DOES NOT solve the measurement problem (or wavefunction collapse). Instead, it only removes interference of different possibilities, while someone still has to select the single outcome of a
measurement. For example, the Schrodinger cat is certainly either dead or alive, and not in the superposition of alive + dead, but observer has to open the box and see what is the really inside (in
the technical language, decoherence produced improper rather than proper mixed quantum states, and the latter ones are required to understand classical outcomes of the measurement). In this respect,
there are two questions about the relation of observing consciousness and a physical world which is being collapsed by it into a single outcome: 1. Modern neuroscience states that all of our thoughts
and actions have neural correlates. The wetware of the brain is described by classical physics and many attempts to find quantum-mechanical superpositions inside have failed due to extremely short
decoherence times in such a "dirty" (biochemical) environment. However, the brain itself must obey laws of quantum mechanics, so the final single neural correlate must be a result of some measurement
(i.e., wavefunction collapse). The questions is who is doing the collapse? H. Stapp pushes the idea of quantum Zeno effect, which is impervious to any decoherence, where repeated observation freezes
the quantum system into a particular state (neural correlate here). The physically efficacious consciousness (or "abstract ego" in the words of von Neumann), which gives free will beyond classical
determinism, then seems to be outside of the physical world which it is observing. Stapp again goes here (in the Chapter in "Information and the Nature of Reality: From Physics to Metaphysics",
edited by P. Davies and N. H. Gregersen) beyond the conclusion of your article to argue that there is a direct connection to God here involved. What is your opinion on this? 2. In the recent
discussions in cosmology, the so-called top-down view (advocated by Hawking) claims that conscious observers are actually changing the multiverse evolution (not from the many world interpretation of
quantum mechanics, but simply the one encoded by some kind of master path integral of the Feynman formulation of quantum mechanics) by selecting only one of many possible choices (i.e., by
continuously collapsing the wavefunction of the Universe). The problem here again is in the fact that observer is the part of the Universe (with laws of physics which support their own existence). Is
it possible that an observer within the system can perform the collapse of its wavefunction, or we are forced again to think about consciousness doing the collapse while residing outside of the whole
observable Universe?
July 12, 2012
But "an ultimate Mind" isn't a modification?
"In any event, the process of 'truncating' numbers or mathematical expressions is a mathematical operation, just like multiplication and division. Such truncation operations are simply not a part of
the mathematical formalism of quantum mechanics as we have had it since the 1920's."
As I've already conceded, it has never been a part of any physics at all, since well before the '20s. I think, however, that you're missing my point somewhat: exactness in reality is not required
just because the best explanatory models that we have are exact, nor does reality have to work in the same way that our stories about it work. The "adding up" thing is a perfect example of this.
Again, you seem to think that I must be wrong because (e.g.) .333... + .333... + .333... wouldn't add up to 1 under my system. But doesn't this presume something odd about reality (i.e., that the
"outcomes" and their "probabilities" are, as it were, ontologically primary, and the fact that the probabilities add up to 1 is only a sort of accident)? There are ways of structuring random events
so that "probability" is just a way of descriptively talking about the results or the architecture of the system, not something that's actually in the system. You should be able to appreciate this,
because you're making a similar point about my usage of computer language. I fully understand that my analogy is just an analogy - that, in other words, the "systems" and "memory" and so on are more
suggestive than literal. But do you understand that your various linguistic approximations (about "adding up" and so on?) are likewise just linguistic approximations? I'm beginning to doubt it.
As for my theory being a modification, I was under the impression that any additional theory would require a modification of some sort, because an explanatorily incomplete theory is one that by
definition requires modification in order to become (more) complete. Positing the causal influence of "transcendent" minds, for instance, means writing a whole new set of laws that aren't causally
closed, which - unless I'm very much mistaken - is also well outside the traditions of physics; along similar lines, it requires defining minds in a coherent way, which will be very difficult to do
when (1) they're meant to be "transcendent" (and so, presumably, outside our capacity to study) and (2) the evidence we do have about minds is converging rapidly on the conclusion that they are just
matter. Maybe, then, you're really just concerned about changing one thing as opposed to another? But in the absence of evidence to decide which thing ought to be changed, that's no more than an
aesthetic consideration (i.e., you find one thing more displeasing than I do). I mean, *are* you arguing that the evidence absolutely rules out MWI or the Penrose interpretation (or any of the other
alternatives that I guess are out there)? My reading of you (which, in fairness, coheres with what I read from other physicists) was that you just didn't like those other ones, in the aesthetic sense
- but that, I'm afraid, is neither something that "make[s] it easier to believe in God" (except, of course, for you personally) nor "an argument against the philosophy called materialism."
Stephen M. Barr
July 12, 2012
Reply to bknikolic
I am very happy to hear from my esteemed colleague! (I should mention for the rest of the audience, that bknikolic is a theoretical physicist.) My short answer to question 1 is that I don't think
God has anything to do with the problems of how QM is to be interpreted. (I have read some of Stapp's work, but am not familar with this aspect of his ideas.) As for 2, I don't see how something
that is entirely physical and thus described by the wavefunction of the universe could collapse the wavefunction, unless the laws of QM were modified in some way.
I appreciate the point about the brain being a hot dense system in which decoherence times are very short. Some argue from this that the brain acts in an essentially classical way. That may be so. I
have my own (VERY TENTATIVE) speculative hypothesis on how wavefunction collapse is related to the mind. I say a few words about it in my answer to jrd261 above.
My basic idea, in a nutshell, is that whenever there is decoherence produced by any process anywhere in the universe, all consciousness in the universe travels down one path. { I hasten to say that
I am assuming many distinct conscious beings, not some collective consciousness. I am just supposing that the consciousness of every conscious being in the universe go together down one branch of the
wavefunction of the universe, at each branching.) The wavefunction of the universe keeps branching just as in MWI. But among all those branches just one path contains all consciousness. This may seem
strange, in that in some other branch there are sets of degrees of freedom that look like you and me, but they have no consciousness associated with them. (In the whimsical jargon used by
"philosophers of mind" nowadays, they would be "zombies".) But this assumption violates nothing that we know empirically. There is no way to to derive (logically and mathematically) from the
physical description of a system anything about whether it has consciousness associated with it. (This is connected with the well-known "problem of other minds") Nor can we examine those other
versions of us, since they are in other branches that have decohered from our branch. So the idea that all consciousness is in one branch violates nothing we know. For this idea to be consistent,
one would need the decoherence time between any two branches that would look different to our senses to be shorter than the time it takes fro the brain to perform a perceptual act. (Otherwise, during
the branching, when there was still significant coherence, it would be ill-defined to say that minds wer in a particular branch.) But here my idea is actually HELPED by the fact that decoherence
times involving the brain are so short. The brain cannot perform any perceptual act facter than, say, a microsecond (to take a safely short time). But the decoherence times of any relevant process
would be much shorter than that.
In the picture I am outlining, one has (it seems to me) the best features of both MWI and Copenhagen. It preserves the nice feature of MWI that the wavefunction changes in just one way --- according
to the unitary Schrodinger evolution. One never has to "saw off branches". One can talk unambiguously about what the physical world is doing at all times: it is described by the wavefunction of the
universe. But I avoid the bad feature of MWI: there is only one copy of each conscious being. I can also make sense of the "probability rule" of QM, which is somewhat problematic in MWI: The
probability given by the absolute square of an amplitude is the probability that consciousness will take the corresponding branch rather than the others. Like the Copenhagen interpretation (as
understood by Wigner, Peierls and others) my interpretation posits consciousness as a reality connected to, but distinct from physical entities.
I would like to discuss this further with you (off-line, it would be too hard to do it in this forum) to see if you can punch any holes in this idea. It seems to me consistent with everything we
know, gives an account of wavefunction collapse, and entails no modification of the postulates of QM.
Stephen M. Barr
July 12, 2012
Replay to larryniven on modifying theories
While there is much disagreement about how quantum mechanics {QM} should be interpreted, everyone agrees about what the mathematical formalism of QM is. That formalism is precise and well-defined.
Everyone agree how to calculate physically measureable quantities in QM and everyone gets the same answers. Neither the MWI interpretation nor the traditional (Copenhagen) interpretation propose
any modification of that formalism.
The fact that you call Penrose's idea an "interpretation" bespeaks some confusion. "Interpretations" of QM assume the standard mathematical formalism is true, and then try to make philosophical sense
of it. That is to be distinguished from various suggestions for modifying the formalism, which includes Bohmian mechanics, and Penrose's idea --- and your idea.
I wonder if you are reading my answers. You ask, for example, whether I am arguing that MWI or Penrose's idea is wrong. Nowhere have I argued that they are wrong. How did you get that idea? In my
article, and in every subject response I have given, I have only tried to explain the alternatives. which, basically, are these: (a) The MWI interpretation, (b) the traditional interpretaion, and (c)
modifying the formalism in some way. As I said in a response to another person: pick your poison.
I have argued that if the traditional interpretation is right, then materialsim is wrong, and if materialism is right then one must choose either (a) or (c). That is all I have claimed.
July 13, 2012
Universal Wavefunction
Prof Barr,
Your solution is ingenious but leads to doubts about the physics that can be obtained from such a procedure.
If agents can not be described quantum mechanically and agents are part physical, then it leads to less confidence in the postulate of universal wavefunction.
Do you have the same degree of confidence in quantum cosmological applications as you would have on solid state?
Stephen M. Barr
July 13, 2012
Reply to Vishmehr24
In my (tentatively proposed) scheme, the physical bodies of human beings would be characterized by measureable variables (e.g. the positions of particles) and these variables would be included in the
wavefunction of the universe. That is, the wavefunction would provide an accurate physical description of a human being --- but only a physical description. I agree with Peierls that not everything
about a human being, including his knowledge and his consciousness, is describable in this way.
I should emphasize that even if my own tentaive ideas on this subject are wrong, the general discussion given in my essay is unaffected.
July 13, 2012
God's Causal Actions
I thank Professor Barr for such a clear and illuminating article. He certainly keeps up the excellent standard set by his book, Modern Physics and Ancient Faith.
I begin with a question for Professor Barr. In the standard model, is the idea that taking the measurement or making the observation causes the wave function to collapse? So that, for example, when I
check whether a certain radioactive nucleus has decayed, my observation somehow causes the nucleus to have either decayed or not decayed at the time of the observation? If this is the way it is, then
human knowledge of the physical world is just as much a cause as an effect of physical reality, and the traditional understanding of a posteriori empirical knowledge is quite wrong.
Second, an observation. In a traditional theistic theology, God is, of course, the ultimate cause of all things. So, how is he the cause of the collapse of a given wave function? If some hidden
variable theory were true, then we would say that God acted indirectly through the hidden variables to collapse the wave function, but Professor Barr has advised me on previous occasions that there
are powerful reasons to think that hidden variable theories are untenable. If the standard model gives us the true picture of reality, and if further the idea is, as I say above, that the observer’s
observation causes the wave function to collapse, then God would be acting indirectly through the observer to cause that collapse, and, presumably, with respect to wave functions for which no
observations are ever made, they simply never collapse and so there is nothing there of which God needs be the cause. If the many-worlds interpretation is correct, then God is the cause of these many
worlds (“O Lord my God, when I in awesome wonder, consider all the worlds thy hands have made….”), and again there is no problem.
But there is another possible interpretation of quantum mechanics, or, at least, it seems to me that there is, and I will be very interested in Professor Barr’s thoughts on it. It seems to me that a
theist could say that God made physical reality such that it is accurately described by the equations of quantum mechanics (or the ultimate Grand Unified Theory, etc.). In particular, wave functions
do not collapse because of “physical causes.” But, as we know from experience, wave functions do in fact collapse all the time—e.g., radioactive nuclei sometimes actually decay, etc. On the
interpretation I am suggesting, God acts directly (whatever that means) on physical systems to collapse wave functions, though this direct action of his is, barring the occasional miracle (no pun
intended, Steve), in accordance with the probabilities established by the equations of quantum mechanics so that the frequency distributions of events will tend strongly to confirm the equations. For
example, if a certain radioactive isotope has a half-life of one hour, then every hour God causes only about half of the nuclei in a sample to decay. If this is correct, then when a human being makes
an observation and sees that the wave function has collapsed, his observation in no way causes the collapse of the wave function: God has already done that. Human empirical knowledge is thus pretty
much as we always thought—an effect and not a cause of physical reality. Regardless of whether observations are made, wave functions collapse when God acts to collapse them, and so there will be
some—indeed, a great many—wave functions that collapse “unobserved.” Physical systems are basically never actually in superpositions because immediately, or almost immediately, collapses them. Call
this the theological interpretation of quantum mechanics. What do you think, Professor Barr? Is this consistent with what physics know about the physical world?
Robert T. Miller
Stephen M. Barr
July 13, 2012
reply to Robert T. Miller
Hi, Robert,
The traditional interpretation (or Copenhagen interpretation, or standard interpretation) seems to be (have been) held in somewhat different form by different people. But let's tke its defining
characteristic to be that it posits a "collapse of the wavefunction" whenever a measurement gives a definite result.
I think that in the traditional interpretation one is practicaly forced to say that the wavefunction is NOT just a description of "the world as it is," but rather that it encodes what some observer
(or class of observers) knows (or is in a position to assert) about the world. If that is the case, then the observer's observation does not NECESSARILY cause a change in extramental reality; it
may only cause a change in the observer's state of knowledge.
There is a severe difficulty in saying that the observer's observation changes extramental reality: what if there are several observers of the same system? This is the basis of the famous "Wigner's
friend paradox". Suppose that both Wigner and his friend (graduate student?) are waiting for a nucleus to decay. Wigner get tired and goes home for the evening, leaving his friend on watch. At 8 PM,
Wigner's friend hears the Geiger counter click. At 9 AM the next morning, Wigner arrives at the lab and asks his friend whether the nucleus has decayed, and gets the answer "yes". One can consider
Wigner's friend as the observer, in which case the collapse happened at 8 PM. Or one can consider Wigner as the observer and his friend as part of the measuring apparatus, like the Geiger counter.
Then the wavefunction collapse happens at 9 AM. Which is right? If "the wavefunction" is just the world as it objectively is, then one is caught in a dilemma. If, on the other hand, one says that
the wavefunction encodes the observer's state of knowledge, then there are two wavefunctions involved here --- that encoding Wigner's knowledge and that encoding his friend's. The former collapses at
9 AM, the latter at 8 PM.
So, it is not clear that the traditional interpretation forces one to say that "the world as it is" is changed by an observation. It surely does force one to say that the observer's knowledge is
changed by the observation --- but that is obvious anyway.
Now, that being said, I think there are some adherents of the traditional interpretation who would indeed say that the observer IS changing extramental reality just by virtue of his coming to know
the result of his measurement. That, as you say, is a radical idea. But, it is far from clear that the traditional interpretation compels one to accept it. This is a very murky area!
Another thing is worth saying. Consider Schrodinger's cat. Suppose the observer opens the box at Noon and sees that the cat is dead. The wavefunction collapses at Noon. Does that mean that the cat
died at Noon according to the traditional interpretation? Not at all! The observer may decide after opening the box to do some forensic pathology on the cat: measure body temperature, insect
actovity, rigor mortis, state of decomposition, etc. He may be able to infer that the cat actually died at 9 PM the night before. What to make of this? Before he opened the box, all he was in the
position to assert on the basis of his past observations is that the cat had such-and-such probability of having died in such-and-such a time interval, this-and-so probability of having died in
this-and-so other time interval, etc. and some probability of still being alive Then he opens the box and sees the cat dead --- at that point the wavefinction collapses to "cat dead" state. Then he
does his forensic tests, and the wavefunction further collapses to the "cat died at 9 PM" state.
Now, if we retreat to the modest view that the wavefunction is merely someone's state of knowledge, and NOT "things as they are", then how should one give a complete and objective mathematical
description of reality as it is in itself? (The God's-eye view.) That is far from clear.
Your idea, as I understand it, is to consider the wavefunction as a description of the world as it is and then say that God acts directly (in some sense) to prune away all but one branch when
"wavefunction collapse" occurs. I think this is a consistent view, as long as one modifies it slightly.
If you say that the pruning happens whenever branches of the wavefunction "decohere" from each other, then I think it probably is OK. Decoherence generally happens when macroscopic objects are
involved (e.g. a Geiger counter) --- you don't need the macroscopic object to be conscious or have a mind. (Of course, your idea does not need to invoke God. You can just say that some law of
physics comes into play when decoherence happens and causes the unwanted branches to wither in accordance with some new equation, or modification of the Schrodinger equation. This would then be what
people call an "objective collapse theory". Whether such a theory can be found that is mathematically elegant and consistent is another question. In any event, it would be a change of the
mathematical formalism of QM.) Your version would not require the mathematical formalism to be modified, but would require God to doa "manually over-ride" of the equations.
Your idea is a bit like my idea, which I explained in my answer to bknokolic above. Except that I invoke neither God nor a new law of physics. I posit what one might call a psycho-physical law,
which is that all consciousness in the universe follows just one branch at any point where branches split apart and decohere. This law is probabilistic and says that which branch is followed is
governed by the quantum mechanical probabilities. In this view, the wavefunction itself does not collapse, it keeps ramifying as in MWI. But consciousness selects one branch (or rather one branch is
selected for it to follow, in accordance with the QM probabilities). One could look at this in two ways. (A) The wavefunction represents potentialities, not actualities. Only the branch that
consciousness follows is actual. Or (B) The wavefunction, with all its branches, have merely "physical reality". But the branch down which consciousness goes has a stronger kind of reality --- call
it Berkeleyan reality: a reality actually experienced by a sentient being.
Long answer. Hope it is not totally confusing.
July 14, 2012
A sceptical take
"If, on the other hand, we accept the more traditional understanding of quantum mechanics that goes back to von Neumann, one is led by its logic (as Wigner and Peierls were) to the conclusion that
not everything is just matter in motion, and that in particular there is something about the human mind that transcends matter and its laws. It then becomes possible to take seriously certain
questions that materialism had ruled out of court: If the human mind transcends matter to some extent, could there not exist minds that transcend the physical universe altogether? And might there not
even exist an ultimate Mind?"
Certainly it leaves the way open for that. But you could also say 'all knowledge of phenomena is incomplete. It is simply approximative, and always will be'. In this case it is not something about
'the human mind' that transcends laws, but something about the nature of reality that transcends the human mind.
July 14, 2012
Mind Body Problem
From the perspective of the philosophical 'Mind-Body Problem' mainstrean quantum physics makes a 'dualistic' solution more or less certain. It proves the existance of the soul. Mainstream
philosophers of mind regard this view as absurd, clinging as they do to any form of materialism they can find. Also, this dualism does not involve the non-material mind somehow emerging from the
material brain. Rather mind and matter are both equally ontologically 'first' in the scheme of things. Mind and brain exist in tandem. And 'mind' does not simply observe matter but interacts with
it, altering its nature. Mind is there from the begining. Our minds are outcrops of this Mind and we return to it. Such a 'God' is the ulltimate power in the universe but is not All Powerful in
the usually understood sense. Thus solving the 'Problem of Evil'...............?
July 15, 2012
Freedom of the Will
Traditional Philosophy of Mind argues strongly for determinism. Essentially basing this view on a Newtonian Physical analysis. A backdrop of quantum physics might not make contra-causality certain
but would make determinism almost absurd. Only 'God the Father Allmighty' can have 'free will'. In a Newtonian universe all is pre-determined while in (most, as in the majority viewpoint) quantum
models the possibility of being free to make choices is more likely to be the case than the opposite view.
July 15, 2012
Free Will?
A simple example. The Young's Slits experiment 'proved' that light was composed of waves. Some time later Einstein's Photoelectric Effect experiment 'proved' that light was composed of discrete
particles. Both are true yet diametrically opposite. Wave-particle duality in light is actualised as either a wave or a particle answer depending on the choice of experimental device.
There is an 'observer', a 'macroscopic measurnment device' devised by the observer and the 'observed microscopic entity'.
Could it be the case (certainly within the context of the Copenhagen Interpretation) that the above observation would point to the likelihood of 'contra-causality' at the macroscopic level?
Is the human will accordingly capable of innitiating choices? 'Free will', I would suggest, would be possible only for the God of traditional theism. Not for the Mind I postulated earlier which
would be capable of choice but not infinate choice uninhibited by circumstances.
Again most mainstream Philosophy of Mind would regard the possibility of contra-causality as being absurd.
Stephen M. Barr
July 16, 2012
Reply to Brendan
Dear Brendan,
I won't comment on your purely theological ideas, as my concern here is only with the possible implications of quantum mechancis.
As far as free will goes, that is a distinct issue from the one that I discussed in my essay (though obviously related to it). It is one thing to argue that the traditional interpretation of QM has
anti-materialist implications, and that consciousness or mind is as fundamental as matter. To argue, however, that quantum mechanics creates an opening for human free will (as many also have) is to
go further.
You refer to "mainstream quantum mcehanics". On this I will make several comments:
(1) There is no consensus among physicists on the best "interpretation" of quantum mechanics. Most physicists are dissatisfied with both the traditional Copenhagen interpretation and the Many Worlds
Interpretation. Indeed, msot physicists simply ignore the issues, and among those who have thought about them, probably most would say they don't have any idea what the right way to resolve them is.
To the extent that there is a trend, it is away from Copenhagen and toward Many Worlds. Probably most quantum cosmologists favor Many Worlds. So, it is hard to speak really of what the mainstream
is here.
(2) The anti-materialist conclusions drawn from the traditional interpretation of QM by Wigner, Peierls and others (which I find compelling) are not embraced by many physicists. Most are unaware of
the antimaterialist argument and. those who are aware of it are not pursuaded by it. On the other hand, most also find Many Worlds hard to take. That is why the prevailing attiitude is a baffled
(3) The term of "wavefunction collapse" is now shied away from by many people. Nevertheless, most quantum mechanics textbooks in effect or implicitly teach the traditional interpretation, which
involves wavefunction collapse, even if they may not call it that. That is, they speak as though the result obtained by a measurement is the one true state of affairs (which is the essence of
wavefunction collapse and the traditional interpretation). At least, they do not teach that there are many equally real branches in which all measurement results are realized. So ,implicitly, the
traditional interpretation is the way most physicists still actually have learned and teach QM and think about it, even if not explicitly aware of that fact.
(4) While the traditional interpretation logically leads to anti-materialist implications, as has been argued by Wigner, Peierls and others, that does not mean that most physicists are willing to be
led by that logic.
July 16, 2012
The Measurement Problem
Weak Measurements support the idea of the Wavefunciton being "real". From what I've gleaned from reading papers by Yakir Aharonov (I'm not a physist) the probalistic nature is due to the fact that
in most cases one is only controlling 1/2 of the Wave Function, i.e. the retarded half. If measurements are made where both the advanced and retarded waves are defined then the result is certain.
Or in other words, the other half of the wave function moving backwards from the tuture completely defines the outcome.
So if one could control the boundary conditons at the beginning and end of time, then the Universe could be completely deterministic. But then how would one explain our minds from this perspective?
Our free will would have to operate from that future boundary, how could that be?
July 16, 2012
I find that the most natural interpretation of quantum mechanics is idealism.
Given the success of quantum mechanics we know beyond reasonable doubt that the physical phenomena we observe are such that observations will not contradict the probabilistic rules of quantum
mechanics (in the same way that on the long run the throwing of dice will not contradict the respective probabilistic rules either). But according to idealism what produces that observable order is
not, say, some concrete material reality interacting with the observer’s mind to collapse the wavefunction, but rather a universal mind which directly guides and produces for us that concrete
observed order. Finally, contrary to what many believe, it is not the case that according to idealism physical things do not exist. Whether apples or electrons or wavefunctions - they all exist,
albeit the nature of their existence consists in being patterns present, really present, in our experience of physical phenomena.
Idealism I think very elegantly and economically solves all paradoxes of quantum mechanics, including the paradoxes related to non-locality or the paradoxes related to the Copenhagen interpretation
such as “Wigner’s friend”. Indeed the idea that physical reality is basically a big wavefunction with little bubbles of collapsed regions where conscious observers happen to take a look is a weird
picture to say the least.
Stephen M. Barr
July 16, 2012
Reply to Tidy Tim
To answer your last question first, I did not say anything about the free will to quantum mechanics in my essay. But since you ask: a scheme in which our decisions are "determined" by the future
state of the universe would not contradict free will even in the strong "incompatibilist" understanding of free will. Consider the statement, "if X pulls the level for candidate Y at time T, then X
chose at some t < T to do this." That is practically an empty statement and certainly doesn't imply that the decision at t was not free. Take an example: The current state of the universe is one
where history books say that John Wilkes Booth assassinated President Lincoln. Does that tell us that the assassination that happened in 1865 was not free?
As far as Aharonov's formulation of quantum mechanics, it does not sem to me, from what I understand of it, to give a third option between MWI and the collapse of the wavefunction. Suppose that
given the initial state of a system a measurement can give more than outcome. After the measurement is seen to give one result, a question arises: Is that result the one and only true state of
affairs post-measurement? Or did the other results also happen in other brancjes of reality? In the first case, one has by definition wavefunction collapse, which is a process not described by the
Schrodinger equation. Thus it either has to involve some non-physical process, or the mathematical formalism has to be modified. If the other results actually do happen in other branches, then one
has MWI. To the extent that Aharonov's idea is a reformulation of standard QM it gives nothing new. If, however, like Bohmian mechanics, it is a modification of the rules of QM, then it might.
Stephen M. Barr
July 16, 2012
Reply to Tidy Tim
To answer your last question first, I did not say anything about the free will to quantum mechanics in my essay. But since you ask: a scheme in which our decisions are "determined" by the future
state of the universe would not contradict free will even in the strong "incompatibilist" understanding of free will. Consider the statement, "if X pulls the level for candidate Y at time T, then X
chose at some t < T to do this." That is practically an empty statement and certainly doesn't imply that the decision at t was not free. Take an example: The current state of the universe is one
where history books say that John Wilkes Booth assassinated President Lincoln. Does that tell us that the assassination that happened in 1865 was not free?
As far as Aharonov's formulation of quantum mechanics, it does not sem to me, from what I understand of it, to give a third option between MWI and the collapse of the wavefunction. Suppose that
given the initial state of a system a measurement can give more than outcome. After the measurement is seen to give one result, a question arises: Is that result the one and only true state of
affairs post-measurement? Or did the other results also happen in other brancjes of reality? In the first case, one has by definition wavefunction collapse, which is a process not described by the
Schrodinger equation. Thus it either has to involve some non-physical process, or the mathematical formalism has to be modified. If the other results actually do happen in other branches, then one
has MWI. To the extent that Aharonov's idea is a reformulation of standard QM it gives nothing new. If, however, like Bohmian mechanics, it is a modification of the rules of QM, then it might.
July 16, 2012
I find that the most natural interpretation of quantum mechanics is idealism.
Given the success of quantum mechanics we know beyond reasonable doubt that the physical phenomena we observe are such that observations will not contradict the probabilistic rules of quantum
mechanics (in the same way that on the long run the throwing of dice will not contradict the respective probabilistic rules either). But according to idealism what produces that observable order is
not, say, some concrete material reality interacting with the observer’s mind to collapse the wavefunction, but rather a universal mind which directly guides and produces for us that concrete
observed order. Finally, contrary to what many believe, it is not the case that according to idealism physical things do not exist. Whether apples or electrons or wavefunctions - they all exist,
albeit the nature of their existence consists in being patterns present, really present, in our experience of physical phenomena.
Idealism I think very elegantly and economically solves all paradoxes of quantum mechanics, including the paradoxes related to non-locality or the paradoxes related to the Copenhagen interpretation
such as “Wigner’s friend”. Indeed the idea that physical reality is basically a big wavefunction with little bubbles of collapsed regions where conscious observers happen to take a look is a weird
picture to say the least.
July 16, 2012
The Measurement Problem
Weak Measurements support the idea of the Wavefunciton being "real". From what I've gleaned from reading papers by Yakir Aharonov (I'm not a physist) the probalistic nature is due to the fact that
in most cases one is only controlling 1/2 of the Wave Function, i.e. the retarded half. If measurements are made where both the advanced and retarded waves are defined then the result is certain.
Or in other words, the other half of the wave function moving backwards from the tuture completely defines the outcome.
So if one could control the boundary conditons at the beginning and end of time, then the Universe could be completely deterministic. But then how would one explain our minds from this perspective?
Our free will would have to operate from that future boundary, how could that be?
Stephen M. Barr
July 16, 2012
Reply to Brendan
Dear Brendan,
I won't comment on your purely theological ideas, as my concern here is only with the possible implications of quantum mechancis.
As far as free will goes, that is a distinct issue from the one that I discussed in my essay (though obviously related to it). It is one thing to argue that the traditional interpretation of QM has
anti-materialist implications, and that consciousness or mind is as fundamental as matter. To argue, however, that quantum mechanics creates an opening for human free will (as many also have) is to
go further.
You refer to "mainstream quantum mcehanics". On this I will make several comments:
(1) There is no consensus among physicists on the best "interpretation" of quantum mechanics. Most physicists are dissatisfied with both the traditional Copenhagen interpretation and the Many Worlds
Interpretation. Indeed, msot physicists simply ignore the issues, and among those who have thought about them, probably most would say they don't have any idea what the right way to resolve them is.
To the extent that there is a trend, it is away from Copenhagen and toward Many Worlds. Probably most quantum cosmologists favor Many Worlds. So, it is hard to speak really of what the mainstream
is here.
(2) The anti-materialist conclusions drawn from the traditional interpretation of QM by Wigner, Peierls and others (which I find compelling) are not embraced by many physicists. Most are unaware of
the antimaterialist argument and. those who are aware of it are not pursuaded by it. On the other hand, most also find Many Worlds hard to take. That is why the prevailing attiitude is a baffled
(3) The term of "wavefunction collapse" is now shied away from by many people. Nevertheless, most quantum mechanics textbooks in effect or implicitly teach the traditional interpretation, which
involves wavefunction collapse, even if they may not call it that. That is, they speak as though the result obtained by a measurement is the one true state of affairs (which is the essence of
wavefunction collapse and the traditional interpretation). At least, they do not teach that there are many equally real branches in which all measurement results are realized. So ,implicitly, the
traditional interpretation is the way most physicists still actually have learned and teach QM and think about it, even if not explicitly aware of that fact.
(4) While the traditional interpretation logically leads to anti-materialist implications, as has been argued by Wigner, Peierls and others, that does not mean that most physicists are willing to be
led by that logic.
July 15, 2012
Free Will?
A simple example. The Young's Slits experiment 'proved' that light was composed of waves. Some time later Einstein's Photoelectric Effect experiment 'proved' that light was composed of discrete
particles. Both are true yet diametrically opposite. Wave-particle duality in light is actualised as either a wave or a particle answer depending on the choice of experimental device.
There is an 'observer', a 'macroscopic measurnment device' devised by the observer and the 'observed microscopic entity'.
Could it be the case (certainly within the context of the Copenhagen Interpretation) that the above observation would point to the likelihood of 'contra-causality' at the macroscopic level?
Is the human will accordingly capable of innitiating choices? 'Free will', I would suggest, would be possible only for the God of traditional theism. Not for the Mind I postulated earlier which
would be capable of choice but not infinate choice uninhibited by circumstances.
Again most mainstream Philosophy of Mind would regard the possibility of contra-causality as being absurd.
July 15, 2012
Freedom of the Will
Traditional Philosophy of Mind argues strongly for determinism. Essentially basing this view on a Newtonian Physical analysis. A backdrop of quantum physics might not make contra-causality certain
but would make determinism almost absurd. Only 'God the Father Allmighty' can have 'free will'. In a Newtonian universe all is pre-determined while in (most, as in the majority viewpoint) quantum
models the possibility of being free to make choices is more likely to be the case than the opposite view.
July 14, 2012
Mind Body Problem
From the perspective of the philosophical 'Mind-Body Problem' mainstrean quantum physics makes a 'dualistic' solution more or less certain. It proves the existance of the soul. Mainstream
philosophers of mind regard this view as absurd, clinging as they do to any form of materialism they can find. Also, this dualism does not involve the non-material mind somehow emerging from the
material brain. Rather mind and matter are both equally ontologically 'first' in the scheme of things. Mind and brain exist in tandem. And 'mind' does not simply observe matter but interacts with
it, altering its nature. Mind is there from the begining. Our minds are outcrops of this Mind and we return to it. Such a 'God' is the ulltimate power in the universe but is not All Powerful in
the usually understood sense. Thus solving the 'Problem of Evil'...............?
July 14, 2012
A sceptical take
"If, on the other hand, we accept the more traditional understanding of quantum mechanics that goes back to von Neumann, one is led by its logic (as Wigner and Peierls were) to the conclusion that
not everything is just matter in motion, and that in particular there is something about the human mind that transcends matter and its laws. It then becomes possible to take seriously certain
questions that materialism had ruled out of court: If the human mind transcends matter to some extent, could there not exist minds that transcend the physical universe altogether? And might there not
even exist an ultimate Mind?"
Certainly it leaves the way open for that. But you could also say 'all knowledge of phenomena is incomplete. It is simply approximative, and always will be'. In this case it is not something about
'the human mind' that transcends laws, but something about the nature of reality that transcends the human mind.
Stephen M. Barr
July 13, 2012
reply to Robert T. Miller
Hi, Robert,
The traditional interpretation (or Copenhagen interpretation, or standard interpretation) seems to be (have been) held in somewhat different form by different people. But let's tke its defining
characteristic to be that it posits a "collapse of the wavefunction" whenever a measurement gives a definite result.
I think that in the traditional interpretation one is practicaly forced to say that the wavefunction is NOT just a description of "the world as it is," but rather that it encodes what some observer
(or class of observers) knows (or is in a position to assert) about the world. If that is the case, then the observer's observation does not NECESSARILY cause a change in extramental reality; it
may only cause a change in the observer's state of knowledge.
There is a severe difficulty in saying that the observer's observation changes extramental reality: what if there are several observers of the same system? This is the basis of the famous "Wigner's
friend paradox". Suppose that both Wigner and his friend (graduate student?) are waiting for a nucleus to decay. Wigner get tired and goes home for the evening, leaving his friend on watch. At 8 PM,
Wigner's friend hears the Geiger counter click. At 9 AM the next morning, Wigner arrives at the lab and asks his friend whether the nucleus has decayed, and gets the answer "yes". One can consider
Wigner's friend as the observer, in which case the collapse happened at 8 PM. Or one can consider Wigner as the observer and his friend as part of the measuring apparatus, like the Geiger counter.
Then the wavefunction collapse happens at 9 AM. Which is right? If "the wavefunction" is just the world as it objectively is, then one is caught in a dilemma. If, on the other hand, one says that
the wavefunction encodes the observer's state of knowledge, then there are two wavefunctions involved here --- that encoding Wigner's knowledge and that encoding his friend's. The former collapses at
9 AM, the latter at 8 PM.
So, it is not clear that the traditional interpretation forces one to say that "the world as it is" is changed by an observation. It surely does force one to say that the observer's knowledge is
changed by the observation --- but that is obvious anyway.
Now, that being said, I think there are some adherents of the traditional interpretation who would indeed say that the observer IS changing extramental reality just by virtue of his coming to know
the result of his measurement. That, as you say, is a radical idea. But, it is far from clear that the traditional interpretation compels one to accept it. This is a very murky area!
Another thing is worth saying. Consider Schrodinger's cat. Suppose the observer opens the box at Noon and sees that the cat is dead. The wavefunction collapses at Noon. Does that mean that the cat
died at Noon according to the traditional interpretation? Not at all! The observer may decide after opening the box to do some forensic pathology on the cat: measure body temperature, insect
actovity, rigor mortis, state of decomposition, etc. He may be able to infer that the cat actually died at 9 PM the night before. What to make of this? Before he opened the box, all he was in the
position to assert on the basis of his past observations is that the cat had such-and-such probability of having died in such-and-such a time interval, this-and-so probability of having died in
this-and-so other time interval, etc. and some probability of still being alive Then he opens the box and sees the cat dead --- at that point the wavefinction collapses to "cat dead" state. Then he
does his forensic tests, and the wavefunction further collapses to the "cat died at 9 PM" state.
Now, if we retreat to the modest view that the wavefunction is merely someone's state of knowledge, and NOT "things as they are", then how should one give a complete and objective mathematical
description of reality as it is in itself? (The God's-eye view.) That is far from clear.
Your idea, as I understand it, is to consider the wavefunction as a description of the world as it is and then say that God acts directly (in some sense) to prune away all but one branch when
"wavefunction collapse" occurs. I think this is a consistent view, as long as one modifies it slightly.
If you say that the pruning happens whenever branches of the wavefunction "decohere" from each other, then I think it probably is OK. Decoherence generally happens when macroscopic objects are
involved (e.g. a Geiger counter) --- you don't need the macroscopic object to be conscious or have a mind. (Of course, your idea does not need to invoke God. You can just say that some law of
physics comes into play when decoherence happens and causes the unwanted branches to wither in accordance with some new equation, or modification of the Schrodinger equation. This would then be what
people call an "objective collapse theory". Whether such a theory can be found that is mathematically elegant and consistent is another question. In any event, it would be a change of the
mathematical formalism of QM.) Your version would not require the mathematical formalism to be modified, but would require God to doa "manually over-ride" of the equations.
Your idea is a bit like my idea, which I explained in my answer to bknokolic above. Except that I invoke neither God nor a new law of physics. I posit what one might call a psycho-physical law,
which is that all consciousness in the universe follows just one branch at any point where branches split apart and decohere. This law is probabilistic and says that which branch is followed is
governed by the quantum mechanical probabilities. In this view, the wavefunction itself does not collapse, it keeps ramifying as in MWI. But consciousness selects one branch (or rather one branch is
selected for it to follow, in accordance with the QM probabilities). One could look at this in two ways. (A) The wavefunction represents potentialities, not actualities. Only the branch that
consciousness follows is actual. Or (B) The wavefunction, with all its branches, have merely "physical reality". But the branch down which consciousness goes has a stronger kind of reality --- call
it Berkeleyan reality: a reality actually experienced by a sentient being.
Long answer. Hope it is not totally confusing.
July 13, 2012
God's Causal Actions
I thank Professor Barr for such a clear and illuminating article. He certainly keeps up the excellent standard set by his book, Modern Physics and Ancient Faith.
I begin with a question for Professor Barr. In the standard model, is the idea that taking the measurement or making the observation causes the wave function to collapse? So that, for example, when I
check whether a certain radioactive nucleus has decayed, my observation somehow causes the nucleus to have either decayed or not decayed at the time of the observation? If this is the way it is, then
human knowledge of the physical world is just as much a cause as an effect of physical reality, and the traditional understanding of a posteriori empirical knowledge is quite wrong.
Second, an observation. In a traditional theistic theology, God is, of course, the ultimate cause of all things. So, how is he the cause of the collapse of a given wave function? If some hidden
variable theory were true, then we would say that God acted indirectly through the hidden variables to collapse the wave function, but Professor Barr has advised me on previous occasions that there
are powerful reasons to think that hidden variable theories are untenable. If the standard model gives us the true picture of reality, and if further the idea is, as I say above, that the observer’s
observation causes the wave function to collapse, then God would be acting indirectly through the observer to cause that collapse, and, presumably, with respect to wave functions for which no
observations are ever made, they simply never collapse and so there is nothing there of which God needs be the cause. If the many-worlds interpretation is correct, then God is the cause of these many
worlds (“O Lord my God, when I in awesome wonder, consider all the worlds thy hands have made….”), and again there is no problem.
But there is another possible interpretation of quantum mechanics, or, at least, it seems to me that there is, and I will be very interested in Professor Barr’s thoughts on it. It seems to me that a
theist could say that God made physical reality such that it is accurately described by the equations of quantum mechanics (or the ultimate Grand Unified Theory, etc.). In particular, wave functions
do not collapse because of “physical causes.” But, as we know from experience, wave functions do in fact collapse all the time—e.g., radioactive nuclei sometimes actually decay, etc. On the
interpretation I am suggesting, God acts directly (whatever that means) on physical systems to collapse wave functions, though this direct action of his is, barring the occasional miracle (no pun
intended, Steve), in accordance with the probabilities established by the equations of quantum mechanics so that the frequency distributions of events will tend strongly to confirm the equations. For
example, if a certain radioactive isotope has a half-life of one hour, then every hour God causes only about half of the nuclei in a sample to decay. If this is correct, then when a human being makes
an observation and sees that the wave function has collapsed, his observation in no way causes the collapse of the wave function: God has already done that. Human empirical knowledge is thus pretty
much as we always thought—an effect and not a cause of physical reality. Regardless of whether observations are made, wave functions collapse when God acts to collapse them, and so there will be
some—indeed, a great many—wave functions that collapse “unobserved.” Physical systems are basically never actually in superpositions because immediately, or almost immediately, collapses them. Call
this the theological interpretation of quantum mechanics. What do you think, Professor Barr? Is this consistent with what physics know about the physical world?
Robert T. Miller
Stephen M. Barr
July 13, 2012
Reply to Vishmehr24
In my (tentatively proposed) scheme, the physical bodies of human beings would be characterized by measureable variables (e.g. the positions of particles) and these variables would be included in the
wavefunction of the universe. That is, the wavefunction would provide an accurate physical description of a human being --- but only a physical description. I agree with Peierls that not everything
about a human being, including his knowledge and his consciousness, is describable in this way.
I should emphasize that even if my own tentaive ideas on this subject are wrong, the general discussion given in my essay is unaffected.
July 13, 2012
Universal Wavefunction
Prof Barr,
Your solution is ingenious but leads to doubts about the physics that can be obtained from such a procedure.
If agents can not be described quantum mechanically and agents are part physical, then it leads to less confidence in the postulate of universal wavefunction.
Do you have the same degree of confidence in quantum cosmological applications as you would have on solid state?
Stephen M. Barr
July 12, 2012
Replay to larryniven on modifying theories
While there is much disagreement about how quantum mechanics {QM} should be interpreted, everyone agrees about what the mathematical formalism of QM is. That formalism is precise and well-defined.
Everyone agree how to calculate physically measureable quantities in QM and everyone gets the same answers. Neither the MWI interpretation nor the traditional (Copenhagen) interpretation propose
any modification of that formalism.
The fact that you call Penrose's idea an "interpretation" bespeaks some confusion. "Interpretations" of QM assume the standard mathematical formalism is true, and then try to make philosophical sense
of it. That is to be distinguished from various suggestions for modifying the formalism, which includes Bohmian mechanics, and Penrose's idea --- and your idea.
I wonder if you are reading my answers. You ask, for example, whether I am arguing that MWI or Penrose's idea is wrong. Nowhere have I argued that they are wrong. How did you get that idea? In my
article, and in every subject response I have given, I have only tried to explain the alternatives. which, basically, are these: (a) The MWI interpretation, (b) the traditional interpretaion, and (c)
modifying the formalism in some way. As I said in a response to another person: pick your poison.
I have argued that if the traditional interpretation is right, then materialsim is wrong, and if materialism is right then one must choose either (a) or (c). That is all I have claimed.
Stephen M. Barr
July 12, 2012
Reply to bknikolic
I am very happy to hear from my esteemed colleague! (I should mention for the rest of the audience, that bknikolic is a theoretical physicist.) My short answer to question 1 is that I don't think
God has anything to do with the problems of how QM is to be interpreted. (I have read some of Stapp's work, but am not familar with this aspect of his ideas.) As for 2, I don't see how something
that is entirely physical and thus described by the wavefunction of the universe could collapse the wavefunction, unless the laws of QM were modified in some way.
I appreciate the point about the brain being a hot dense system in which decoherence times are very short. Some argue from this that the brain acts in an essentially classical way. That may be so. I
have my own (VERY TENTATIVE) speculative hypothesis on how wavefunction collapse is related to the mind. I say a few words about it in my answer to jrd261 above.
My basic idea, in a nutshell, is that whenever there is decoherence produced by any process anywhere in the universe, all consciousness in the universe travels down one path. { I hasten to say that
I am assuming many distinct conscious beings, not some collective consciousness. I am just supposing that the consciousness of every conscious being in the universe go together down one branch of the
wavefunction of the universe, at each branching.) The wavefunction of the universe keeps branching just as in MWI. But among all those branches just one path contains all consciousness. This may seem
strange, in that in some other branch there are sets of degrees of freedom that look like you and me, but they have no consciousness associated with them. (In the whimsical jargon used by
"philosophers of mind" nowadays, they would be "zombies".) But this assumption violates nothing that we know empirically. There is no way to to derive (logically and mathematically) from the
physical description of a system anything about whether it has consciousness associated with it. (This is connected with the well-known "problem of other minds") Nor can we examine those other
versions of us, since they are in other branches that have decohered from our branch. So the idea that all consciousness is in one branch violates nothing we know. For this idea to be consistent,
one would need the decoherence time between any two branches that would look different to our senses to be shorter than the time it takes fro the brain to perform a perceptual act. (Otherwise, during
the branching, when there was still significant coherence, it would be ill-defined to say that minds wer in a particular branch.) But here my idea is actually HELPED by the fact that decoherence
times involving the brain are so short. The brain cannot perform any perceptual act facter than, say, a microsecond (to take a safely short time). But the decoherence times of any relevant process
would be much shorter than that.
In the picture I am outlining, one has (it seems to me) the best features of both MWI and Copenhagen. It preserves the nice feature of MWI that the wavefunction changes in just one way --- according
to the unitary Schrodinger evolution. One never has to "saw off branches". One can talk unambiguously about what the physical world is doing at all times: it is described by the wavefunction of the
universe. But I avoid the bad feature of MWI: there is only one copy of each conscious being. I can also make sense of the "probability rule" of QM, which is somewhat problematic in MWI: The
probability given by the absolute square of an amplitude is the probability that consciousness will take the corresponding branch rather than the others. Like the Copenhagen interpretation (as
understood by Wigner, Peierls and others) my interpretation posits consciousness as a reality connected to, but distinct from physical entities.
I would like to discuss this further with you (off-line, it would be too hard to do it in this forum) to see if you can punch any holes in this idea. It seems to me consistent with everything we
know, gives an account of wavefunction collapse, and entails no modification of the postulates of QM.
July 12, 2012
But "an ultimate Mind" isn't a modification?
"In any event, the process of 'truncating' numbers or mathematical expressions is a mathematical operation, just like multiplication and division. Such truncation operations are simply not a part of
the mathematical formalism of quantum mechanics as we have had it since the 1920's."
As I've already conceded, it has never been a part of any physics at all, since well before the '20s. I think, however, that you're missing my point somewhat: exactness in reality is not required
just because the best explanatory models that we have are exact, nor does reality have to work in the same way that our stories about it work. The "adding up" thing is a perfect example of this.
Again, you seem to think that I must be wrong because (e.g.) .333... + .333... + .333... wouldn't add up to 1 under my system. But doesn't this presume something odd about reality (i.e., that the
"outcomes" and their "probabilities" are, as it were, ontologically primary, and the fact that the probabilities add up to 1 is only a sort of accident)? There are ways of structuring random events
so that "probability" is just a way of descriptively talking about the results or the architecture of the system, not something that's actually in the system. You should be able to appreciate this,
because you're making a similar point about my usage of computer language. I fully understand that my analogy is just an analogy - that, in other words, the "systems" and "memory" and so on are more
suggestive than literal. But do you understand that your various linguistic approximations (about "adding up" and so on?) are likewise just linguistic approximations? I'm beginning to doubt it.
As for my theory being a modification, I was under the impression that any additional theory would require a modification of some sort, because an explanatorily incomplete theory is one that by
definition requires modification in order to become (more) complete. Positing the causal influence of "transcendent" minds, for instance, means writing a whole new set of laws that aren't causally
closed, which - unless I'm very much mistaken - is also well outside the traditions of physics; along similar lines, it requires defining minds in a coherent way, which will be very difficult to do
when (1) they're meant to be "transcendent" (and so, presumably, outside our capacity to study) and (2) the evidence we do have about minds is converging rapidly on the conclusion that they are just
matter. Maybe, then, you're really just concerned about changing one thing as opposed to another? But in the absence of evidence to decide which thing ought to be changed, that's no more than an
aesthetic consideration (i.e., you find one thing more displeasing than I do). I mean, *are* you arguing that the evidence absolutely rules out MWI or the Penrose interpretation (or any of the other
alternatives that I guess are out there)? My reading of you (which, in fairness, coheres with what I read from other physicists) was that you just didn't like those other ones, in the aesthetic sense
- but that, I'm afraid, is neither something that "make[s] it easier to believe in God" (except, of course, for you personally) nor "an argument against the philosophy called materialism."
July 12, 2012
Dear Dr. Barr, Thanks for a thought provoking article and interesting clarifying follow-up discussion. As you have already emphasized in the answer to one of the questions, decoherence by environment
DOES NOT solve the measurement problem (or wavefunction collapse). Instead, it only removes interference of different possibilities, while someone still has to select the single outcome of a
measurement. For example, the Schrodinger cat is certainly either dead or alive, and not in the superposition of alive + dead, but observer has to open the box and see what is the really inside (in
the technical language, decoherence produced improper rather than proper mixed quantum states, and the latter ones are required to understand classical outcomes of the measurement). In this respect,
there are two questions about the relation of observing consciousness and a physical world which is being collapsed by it into a single outcome: 1. Modern neuroscience states that all of our thoughts
and actions have neural correlates. The wetware of the brain is described by classical physics and many attempts to find quantum-mechanical superpositions inside have failed due to extremely short
decoherence times in such a "dirty" (biochemical) environment. However, the brain itself must obey laws of quantum mechanics, so the final single neural correlate must be a result of some measurement
(i.e., wavefunction collapse). The questions is who is doing the collapse? H. Stapp pushes the idea of quantum Zeno effect, which is impervious to any decoherence, where repeated observation freezes
the quantum system into a particular state (neural correlate here). The physically efficacious consciousness (or "abstract ego" in the words of von Neumann), which gives free will beyond classical
determinism, then seems to be outside of the physical world which it is observing. Stapp again goes here (in the Chapter in "Information and the Nature of Reality: From Physics to Metaphysics",
edited by P. Davies and N. H. Gregersen) beyond the conclusion of your article to argue that there is a direct connection to God here involved. What is your opinion on this? 2. In the recent
discussions in cosmology, the so-called top-down view (advocated by Hawking) claims that conscious observers are actually changing the multiverse evolution (not from the many world interpretation of
quantum mechanics, but simply the one encoded by some kind of master path integral of the Feynman formulation of quantum mechanics) by selecting only one of many possible choices (i.e., by
continuously collapsing the wavefunction of the Universe). The problem here again is in the fact that observer is the part of the Universe (with laws of physics which support their own existence). Is
it possible that an observer within the system can perform the collapse of its wavefunction, or we are forced again to think about consciousness doing the collapse while residing outside of the whole
observable Universe?
Stephen M. Barr
July 12, 2012
Reply 3 toi larryniven
Dear Mr. Niven,
I think you may be missing the qualifiers embedded in my previous statements. I don't deny that a scheme such as you propose might conceivably be the correct description of the world. (Though I
think it unlikely in the extreme.) All I am saying is that such a scheme is NOT standard quantum mechanics. It entails a MODIFICATION of the mathematical formalism of quantum mechanics. By the same
token, Penrose is not claiming that his theory of collapse is something that is contained in our present quantum mechanical formalism; rather, he is proposing a modification of that formalism. (By
the way, this is clear from the following consideration: In the standard quantum mechanical formalism, the evolutiojn of wavefunction given by the Schrodinger equation is "unitary". The collapse of
the wavefunction, on the other hand, is a non-unitary process. Thus, one must modify the equations somehow to get a collapse from them.)
You also are proposing a modification of the equations, though you apparently do not appreciate this fact. You seem to be envisioning the equations of the universe being run on a computer of some
sort. (You speak of a "system" that has "memory" and that "truncates".) In any event, the process of "truncating" numbers or mathematical expressions is a mathematical operation, just like
multiplication and division. Such truncation operations are simply not a part of the mathematical formalism of quantum mechanics as we have had it since the 1920's.
It is perfectly OK to modify a theory, as long as one remains consistent with known experimental facts. You are perfectly entitled to propose such modifications. But they ARE modifications.
I am not giving an "incomplete picture of the current state of affairs", if by that you meant a misleading picture. (Is that what you meant?). I not only clearly stated that people have made
attempts to construct objective collapse theories, but I cited Wigner's own attempts to do so as an example. I could have mentioned Penrose's attempt as another example. I suppose that if I do not
mention every speculative attempt to modify quantum mechanics that has ever been proposed --- and there have been many --- you will say that I am not giving a complete account. In some trivial and
sense that is true. A complete account would require a review article of several hundred pages length.
Penrose's idea is interesting (most of his ideas are interesting); but has not attracted much of a following in the physics community. Penrose thinks that unifying quantum mechanics with General
Relativity (GR) is likely to require a modification of quantum mechanics. That is one reason he thinks that wavefunction collapse will end up being explained by gravitational effects. The backstory
of this is that for decades physicists were unable to find a consistent way to "quantize gravity", and this led many people to suspect that GR would force a modification of quantum mechanics. Some
still think that, but far fewer than previously, because superstring theory has demonstrated that GR can be quantized without changing the postulates of quantum mechanics.
July 12, 2012
Comment on Discretized Probabilities
Hi Dr. Barr and larryniven,
I might add that if the values that the wavefunction could take on were universally discrete (at least in the way larryniven suggests), then any continuous function of the probability would inherit
the discrete nature of the wavefunction. Thus the discretization ought to be manifested in macroscopic ways that can be measured (example below). To my knowledge, this has not been seen in general
cases, which suggests that wavefunctions do not indeed take on only discrete values.
First, consider a Stern-Gerlach apparatus. That is, let an ensemble of electrons pass through a strong non-uniform magnetic field pointed along a given axis (call it the z-axis). Then the trajectory
of the ensemble will be split in two as it passes through the field, with equal numbers of electrons in each. We label each trajectory by 'up' or 'down'. Thus electrons in one branch have 100%
probability of having the property 'up', and those in the other have 100% probability of having the property 'down', and we may reasonably say that a randomly selected electron from the initial
ensemble has a 50% chance of having the property 'up', and a 50% chance of having the property 'down'.
Next, pass only the 'up' ensemble through another Stern-Gerlach apparatus, but this time oriented along a different axis. Let the angle this axis makes with the first z-axis be called T. We find the
initial 'up' ensemble is again split in two. We label one beam 'UP', and the other 'DOWN'. We also find that the relative number of electrons in each path depends on the angle T. In particular, it is
found that the relative number of electrons in one beam is cos^2(T/2), while for the other beam it is sin^2(T/2). We may again reasonbly say that an electron randomly sampled from the beam of 'up'
electrons in the middle of the apparatus has a chance of having the property 'UP' of cos^2(T/2), and a chance of 'DOWN' of sin^2(T/2).
Therefore, if the wavefunction could only take on discrete values (labeled by Pn), then the angle between the first and second apparatus can also only take on the discrete values Tn=2*arccos(sqrt
(Pn)) and Tn=2*arcsin(sqrt(Pn)). While this is not impossible, this seems unlikely, and there has been no experimental evidence of it to my knowledge.
Furthermore, a Stern-Gerlach apparatus applied to atomic Sodium will split the initial ensemble into three, whereas when it is applied to atomic Beryllium that number is four. Indeed, given any
positive integer N, it is in principle possible to find (or construct) an atom for which an ensemble will split into N beams. When a double-magnet apparatus is used on such ensembles, one sees the
same qualitative features as with the simple electron case described above: the relative numbers of atoms in each trajectory are a continuous function of the angle T between the magnets, and these
relative numbers are precisely the probabilities.
If the discretization of probability is to be universal (eg, if only probabilities of 1/10, 2/10, 3/10, ... were allowed for any system), then it cannot depend on the N of the atomic species being
passed through the apparatus. Thus a given apparatus must be able to be oriented at all angles Tn for all possible atomic species at the same time, even if it never has anything but electron
ensembles pass through it.
To be self-consistent, it is therefore required that the discretized probablities are such that the discretized angles for one ensemble are in general the same as the discretized angles for another.
But the function relating angles to probability is in general very complicated, with complexity increasing with N. For example, given a discretized set of probabilities (0, 1/10, 2/10, ..., 1), the
set of allowed angles for electrons differs from the set of allowed angles for atomic Sodium, which itself differs from the set of allowed angles for atomic Beryllium, such that if the apparatus were
adjusted to say the 8th allowed angle for Beryllium while electrons were passing through, it would produce a relative abundance of 'UP' electrons that is not equal to any of (0, 1/10, 2/10, etc),
which is inconsistent with the original idea that the only allowed probabilities are (0, 1/10, 2/10, etc).
While I have no proof, I suspect that the only set of probabilities that is consistent with all values of N is not discrete at all, but is continuous (or something very close to it), which leads us
back to a traditional understanding of the wavefunction.
This line of thought is extremelty suggestive, especially when one considers the relatively simple nature of the Stern-Gerlach/spin problem, where there exists continuous and bounded functions
relating macroscopic variables to probabilities. The analysis is likely more forbidding to discretized probability when one considers position and momentum, where the wavefunction values and the
underlying operators are not in general bounded.
Stephen M. Barr
July 12, 2012
Reply to Chris Thompson
You ask where I am going with the question.
I am simply explaining a very important line of argument that goes back to the physicists John von Neumann, Fritz London, and Edmund Bauer, and that has been accepted by many other physicists
including Eugene P. Wigner and Rudolf Peierls. I think the argument presents one with three choices: (a) present quantum mechanics is wrong or incomplete in some fundamental way, (b) the Many Worlds
Interpretation is right, or (c) materialism is false, at least as regards the minds of observers. Pick your poison.
Why is God mentioned in the title of the essay? The answer is simple: I was asked by the people who run Big Questions Online whether I would be willing to write an article answering the question
"Does Quantum Physics Make it Easier to Believe in God?" The question was given to me, and I answered it.
I suspect that a certain line of thought may underlie your question. (I base this on your talking about how "wanting to believe something" can affect one's reasoning.) Tell me if I am wrong. The
line of thought is the follwoing: Religious people are driven not be reason but by wishful thinking. People who engage in wishful thinking bend reason to their desires and look for arguments to
sustain beliefs that are actually based on completely non-rational --- if not irrational --- grounds. I would make several comments about this. First, the validity of an argument does not at all
depend on the motives of the one making it. That is why it does not matter in science whether the person proposing a theory is a liberal, a conservative, a Marxist, a Buddhist, a Hindu, an atheist
or a Christian. What matters is the strength of his arguments. The same is true in philosophy. I have co-authored research papers with colleagues whose religious, political, and philosophical ideas
are diametricallly opposed to mine. For example, I have written a well known paper on the "multiverse" idea with three co-athors, one of whom is strongly atheist, and one of whom is (like me)
religious. Someone who worries about the ulterior motives of the person proposing a scientific theory or philosophical theory has already betrayed reason. Second, wishful thinking is a danger to
which all people are subject. Some people want to believe in God, some people want not to believe in God. Both thesm and atheism can be comforting beliefs, though in diferent ways. Third, all
people have fundamental convictions and intuitions about reality that make them more disposed to accept certain conclusions and reject others. For example, one of my deepest convictions is that
there is such a thing as objective moral truth. Another is that the world makes sense. To have basic presuppositions is not irrational. Indeed, one cannot be a rational person without them. Every
argument requires premises.
July 12, 2012
Sorry, still not buying it
"The exactness of the answer logically follows from two unavoidable requirements: the probabilitiies of all outcomes must add up to 1 (100%), and the three decay amplitudes have to be equal by the
symmetry principle."
Equality is trivial in the sort of thing I'm talking about: because the system is going to have the same limitations on "memory" in every case, it'll truncate at the same place in every instance and
they will be equal. (No matter how many times you divide 2 into 3 on your calculator, you'll still get .6666667.) Adding up to one is a little trickier, because the additive margin of error is not
necessarily within the original margin of error - though it could be (it depends, among other things, on the additive algorithm being used). On the other hand, though, I find it awfully strange to
think that the universe first generates the odds of decay and then "adds them up." That seems more like a convenient way of talking about this to make it more intuitive than a description of any
real event. It has to be more plausible to say that the individual probabilities are parasitic on the fact that one of them must happen, not the other way around - or even that our talk of "outcomes
having probabilities" is really shorthand for something else. But then that's entirely compatible with having a finitely powerful physics engine.
Also, like the decay thing you mention next, the power of this argument depends on evidence, does it not? I readily admit that the predictive law is infinitely exact - math is nice that way - but it
begs the question to say that reality must be infinitely precise just because a predictive law that we've come up with is infinitely precise. So neither of these examples is tremendously persuasive,
at least so far as I can see.
At any rate, I did about an hour's research last night and discovered that there's actually already a theory not unlike the one I'm proposing. The Penrose version of objective collapse theory
evidently hypothesizes (and apologies if I'm misusing technical terms here, I'm just going for the gist) that there's a threshold amount of energy past which quantum phenomena collapse into more
standard phenomena, which is not quite what I've been saying but has enough similarities to be at least analogous (e.g. energy becomes the analog of hard memory). I see that you've discussed
objective collapse theories in this comment thread already, but you seem not to have addressed the Penrose variation (in that the thing about confusing decoherence with collapse seems not to apply to
it). Given that this is still a live theory, even to the point of being (apparently) (semi-)endorsed by the people at Stanford's philosophy encyclopedia, and given that it's fairly close to what
I've been saying, I'm beginning to feel like you gave an incomplete picture of the current state of affairs.
Stephen M. Barr
July 12, 2012
reply 2 to vishmehr
Dear Vishmehr,
If I understand your argument correctly, it is as follows: The collapse of the wavefunction of a system requires the observer to be outside the system. No observer (measurer) could be outside of the
entire universe. Therefore, the "wavefunction of the entire universe" could never collapse. And if the wavefunction of the entire universe never collapses, one ends up with the Many Worlds
All of that is true. It is one of the aguments that proponents of the MWI use: they say that since there ought to be a wavefunction of the entire universe, one is forced to accept MWI. Turning this
around, it would seem that to reject MWI, one must also reject the possibility of a "wavefunction of the entire universe".
This logic is all sound. I would distinguish, however, between the entire universe and the entire PHYSICAL universe. Is everything in the universe physical? According to physics, as we have
understood physics for a long time, a physical system is completely characterized by some set of variables. There presumably is some set of variables that completely characterize the whole physical
universe. I see no compelling reason why one could not (in principle) construct a wavefunction for the physical universe. If one considers a human observer,however, that observer would be ( in part)
"outside" the description provided by that wavefunction to the extent that there is an aspect of that human being that is non-physical. In other words, there are two issues, (a) does it make sense to
speak of a wavefunction of the entire physical universe, and (b) is the universe the same thing as the PHYSICAL universe? It is not clear, therefore, that accepting a wavefunction of the universe as
a meaningful concept necessarily commits one to MWI.
Stephen M. Barr
July 12, 2012
Reply 2 to larry niven
Dear larryniven,
I can give two examples.
First example. Suppose that a particle X has three possible "decay modes", A,B, and C, and that a "symmetry principle" makes these three decay modes equally probable. (Symmetry principles
play an important role in fundamental theories of physics.) Then the "amplitude" for X to decay in each of these modes must be EXACTLY equal to the square root of 1/3. (The "amplitudes" are what
appear in the Schrodinger equation. To get the probabilities one takes the square of these amplitudes.) The exactness of the answer logically follows from two unavoidable requirements: the
probabilitiies of all outcomes must add up to 1 (100%), and the three decay amplitudes have to be equal by the symmetry principle. So, there is no way for these numbers to be "rounded off" in some
high decimal place.
Second example. In the decay of an isolated radioactive nucleus, the decay amplitude and the decay probability are given EXACTLY by' an exponential function of time. This can be shown to follow
from "time tranlation invariance" of the laws of physics (a symmetry principle).
The point is that quantum probabilities follow from a precise mathematical formalism, which involves a number of fundamental principles. These probabilities cannot depart from the values
given by the formalism without violating the basic principles and assumptions of the theory.
I am not saying that your idea couldn't be true, only that for it to be true, the rules of quantum mechanics would have to be changed in a fundamental way.
Chris Thompson
July 12, 2012
Believing in God
When I want to believe in something, I am quite able to use any reason whatsoever to do so. My question back to the OP is where are you going with this question? Are you paving a road and making a
case to "believe in God" or what?
July 12, 2012
Dear Prof Barr,
Thanks for your reply but I am still not convinced that the quantum mechanical extrapolation of the wavefunction to the entire universe is of the same character as the extrapolation of homogenity of
the universe such that same laws apply on Mars as on the Earth.
The quantum mechanics demands observers; the formulation is expressed in terms of measurement and measuring devices and observables. These presupposed things are external to the system described by
the wavefunction. What possible meaning could be attributed to the wavefunction of the universe?
Am I correct in supposing that the measurement process has been resolved to the decoherence pheneomenon?. But decoherence is governed by Schrondinger's equation and thus by the Copenhagen picture or
principles can not be the full story. Measurement is required to break the natural evolution of the wavefunction. The decoherence, I believe, is asymptotic; the superpositions never die 100%
And you say that MWI keeps alive the possibiliies that are not realized in the Copenhagen picture. But the decoherence should give unique answer since it is just the evolution of Schrodinger's
equation. So how does MWI keep the possibilites alive?
July 11, 2012
Nope, just a fan
"You make an interesting suggestion, but I am not sure it can be made to work. The "collapses" of wavefunctions (i.e. jumps in probabilities) don't only happen when the relevant probabilities are
close to 0 or 1."
Sure - but I didn't say that they would only collapse when near 0 or 1, just that they would always do so. Again, think of it like it was a simulation that you were running on your own computer: you
could easily write a "random" decay function so that it would have a 50% chance of decaying at time N, a 25% chance of decaying at time N+1 and so on. What you could not do is write a decay function
that stays smooth indefinitely, because eventually your machine would run out of digits with which to calculate, and the probability would effectively become 0.
"One would have to modify the rules of quantum mechanics in a radical way for it to work, however, as those rules use mathematical operations that would not be consistent with discretized
Really? I'd like to see some examples of what you mean by this, because I'm not at all sure that it's true. I mean, yeah, if the discrete increments were (relatively) large, we'd see something very
different in the universe and the theory would never have come up. But if they were (relatively) small? I haven't done the proof yet, but I suspect we'd have to look very closely in order to notice
the difference. And it's not like probability theory requires continuous distributions; in fact, most math that deals with continuous distributions is just a clever abstraction of math that deals
with discrete distributions (to take an easy example, calculus is like this). So which operations would not be consistent with what I'm suggesting, exactly?
Stephen M. Barr
July 11, 2012
tyoes in reply to Mr. Blaylock
There is a bad typo in line 6 of paragraph 4 of my Reply to Mr. Blaylock. It should read that "the wavefunction of this larger system could NOT collapse"
And of course intelligent has two l's and alien has one l.
Stephen M. Barr
July 11, 2012
Reply to Mr. Blaylock
Dear Ethan,
Excellent questions. I will take them in the order presented.
Question 1: The traditional interpretation of quantum mechanics (also called the Copenhagen or standard interpretation) makes a distinction between the "system" being measured and the
"observer" doing the measurement. The wavefunction describes the system. The wavefunction changes in two ways: (a) When the system is left alone, its wavefunction evolves in a continuous and unique
way in accordance with the Schrodinger equation. (b) When the observer makes a measurement of some observable property of the system and gets a definite result, the wavefunction
undergoes a "collapse" that reflects the result of the measurement. Unlike the Schrodinger evolution (a), the collapse (b) is sudden and unpredictable.
The problem I described in my essay is that if the observer is included completely in the system, then the wavefunction changes only in way (a) and there is no "collapse of the wavefunction". There
is thus no unique and definite outcome, but all possibilities remain in play --- thus MWI.
So the question arises, what or who can play the role of "observer"? The first point --- emphasised by Peierls, whom I quoted in the essay --- is that the observer cannot be a purely physical entity,
for if it were one could consider the larger system which comprised both the original system and the (purely
physical) observer. But then one would have the problem that the wavefunction of this larger system could collapse ---- unless there were some other observer outside of IT. But then, if THAT observer
is purely physical, one can expand the system yet further to include IT. An infinute regress.
So it seems that the observer has to be something that is not purely physical. Moreover, since a measuremet is only completed when the observer KNOWS the result, it seems that
the observer has to be a knower. That is why many people --- such as Wigner and Peierls ---- have argued that the "consciousness" or the "mind" or the "knowledge" of the observer plays an essential
Does the observer have to be "human"? Well, certainly humans are observers: we make measurements and get definite results. But that doesn't mean that only human beings are observers.
Could inteligent space alliens be "observers"? Obviously, if there are any. Could chimpanzees or dogs? That depends on whether they can make measurements and whether they are completely describable
by physics.
Question 2: What about objective collapse theory? Well, if one keeps to the mathematical structure of quantum mechanics as we have had it since the days of Heisenberg and Schrodinger, then there is
no collapse without an observer, as explained above. (Some people --- including some physicists --- have the misconception that what is called "decoherence" is the same thing as wavefunction
collapse. And since decoherence only requires the involvement of macroscopic systems (as I explained in answer to another person's question) not conscious observers, they
think they have explained wavefunction collapse without invoking observers. But they are wrong.)
So, to get a collapse without an observer, one would have to change the mathematics of quantum mechanics. There have been attempts to do this. (Wigner tried, for example.) Perhaps that is
what is needed --- many people have suggested that in a more complete or improved theory, all the puzzles of quantum mechanics will be resolved and wavefunction collapse will be explained. It should
be noted, however, that quantum mechanics has passed hundreds of thousands of tests with stunning accuracy over the last 85 years or so. It is hard to modify it without destroying its
self-consistency or its consistency with experiment. This leads us directly into question 3.
Question 3: Can't we say that quantum mechanics is incomplete? Isn't the difficulty of meshing it with General Relativity evidence of this? Well, many people have argued this. However,
it is believed that superstring theory is a consistent theory that incorporates both the principles of quantum mechanics and General Relativity. So that takes away a major argument' for the
inadequacy of quantum mechanics. One never knows what the future will bring in science, but right now quantum mechanics looks like it is in perfect health. It is simply not true to say "that we
already know that quantum mechanics is only an incomplete explanation of the universe." We know no such thing!
Ethan Blaylock
July 11, 2012
Several questions
Thanks for the interesting essay. I have several questions.
1) Why do you say that the Copenhagen interpretation requires a human observer; are there other ways of understanding the Copenagen interpretation that do not require this?
2) Are Copenhagen and Many Worlds the only tenable options; what about objective collapse theory (which does not require a human observer)?
3) How do you differentiate your position from a "God of the Gaps" argument? Can't we just say that our physics models and our interpretations of them currently only incompletely describe the
universe as we experience it (as has always been the case)? For example, the math that describes quantum mechanics does not cleanly mesh with general relativity, but I think the most reasonable
conclusion to draw is that the math is not complete. Even if the math we currently use to describe quantum mechanics has hit something of a dead end, we already know that quantum mechanics is only
an incomplete explanation of the universe. It seems strange to make so much of its quirks.
Stephen M. Barr
July 11, 2012
reply to larryniven
Dear larryniven,
You make an interesting suggestion, but I am not sure it can be made to work. The "collapses" of wavefunctions (i.e. jumps in probabilities) don't only happen when the relevant probabilities are
close to 0 or 1. Consider, for example, the case of a (hypothetical) radioactive nucleus that has a half-life of one hour. If the nucleus is there at noon, it has 25% chance of decaying during the
interval between 1 PM and 2 PM. During that interval, the survival probability of the nucleus varies continuously between 0.5 and 0.25 (and the decay probability varies in the corresponding way).
At no point in this time interval are the decay or survival probabilities that are given by the wavefunction close to 0 or 1. Typically the probabilities jump by an amount that is finite and not at
all small.
Your idea that probabilities are discrete rather than continuous is nevertheless interesting to contemplate. One would have to modify the rules of quantum mechanics in a radical way for it to work,
however, as those rules use mathematical operations that would not be consistent with discretized probabilities.
You wouldn't happen to be the science fiction writer Larry Niven?
July 11, 2012
Another option
So I follow the thread of the argument you've made, Stephen, but it seems that you're making a probably false assumption - namely, that the universe admits nonzero probabilities of infinitely small
value. One way of explaining why wave functions "jump" in some interactions but not in others would be to say that the universe can only sustain probabilities that are of a sufficiently high
magnitude (because it only measures things to a finite accuracy), so that once a single wave function is tested enough times it necessarily collapses one way or the other. To make a very rough
analogy, this would be along the lines of a calculator rounding 2/3 up to 0.666667: because the calculator can't display (or even really calculate) infinitely many digits, at some point it has to
terminate its operations and simply decide to put a 6 or a 7 at the end. If the engine that's running the physics of our universe is only finitely powerful, it would make perfect sense for minds to
be sufficient (in most cases) but unnecessary for the collapse of wave functions: minds are very complex but not only minds are very complex, and eventually time alone would guarantee a wave
function's collapse.
Granted, this explanation isn't obvious and it leaves some things to be answered (though if you can give me a physical theory that leaves *no* questions to be answered then I'll be very impressed).
But it would at least explain the data without requiring the positing either of magical properties for minds or the reality of the many worlds hypothesis, and (so far as I'm aware) it is at least
plausible. We obviously know that the universe measures things very precisely indeed, but I've not read anything that would necessitate the universe measuring things infinitely precisely.
Stephen M. Barr
July 11, 2012
two replies
Dear Ageingcrofter,
I would agree with you that attempts to discuss mathematical subjects "verbally" (if by that is meant "natural languages" rather than mathematical symbols) can lead to distortion. This is especially
the case with subatomic physics, where the concepts needed are far removed from everyday experience. For example, the idea that the wavefunction is a "vector in a Hilbert space" involves a high level
of mathematical abstraction. On the other hand, it is impossible to learn either mathematics or physics without using natural language in addition to mathematical symbolism. One starts with ordinary
experience, and everything one learns must be somehow connected to ordinary experience if one is to grasp it. One cannot understand complex numbers, for instance, or develop intuition about them,
unless one first has some experience with ordinary counting numbers.
Dear jarruda,
If by retrocausality you mean effects coming before their causes, then this is not allowed in present theories of physics. Theories that "violate causality" are regarded as "pathological" and
But do you mean something else?
July 11, 2012
I'm curious as to your take on how the concept of retrocausality impacts your discussion. Very good article as well, thank you!
Stephen M. Barr
July 11, 2012
Reply to Josh's "yes ... but"
I think I know what is bothering you, Josh. As I said in reply to Wallace Forman (in the third paragraph), it all comes down to what the wavefunction of a system is. One would like to be able to
say that it is just a straightforward description of what is happening in the world, of the world as it really is, apart from what you or I know about it. That leads straight to the Many Worlds
picture, because the wavefunction typically contains descriptions of many alternative branches. In the traditional or Copenhagen interpretation, one has a more modest view of what the wavefunction
is: It is not simply "the world as it is", but rather it encodes what some observers know or are in a position to assert about the world. that is why heisenberg himself said that the mathematics of
quantum mechanics "represents no longer the behavior of elementary particles, but rather our knowledge of this behavior". And it is why Rudolf peierls said, "the quantum mechanical description is in
terms of knowledge."
That raises a very important question --- which, I think, is your question: What DOES describe "the world as it really is"? Even if the wavefunction does not describe it, there must be some
comprehensiove and complete and accurate description of physical reality --- call it the "God's eye view of things" (even thouigh I don't want to drag God back into the discussion).
In other words, what IS really going on when no one is looking? What if beings such as ourselves had never evolved? What about regions of the universe that no human or other sentient organism is
ever going to observe or make measurements of? What about what will be happening in the universe after all life has died out? Good questions! The wimpy answer is that science cannot speak about
things that cannot be observed, and what is going on in places that will never be observed is, by definition, something that cannot be observed! But that seems a pretty unsatisfactory answer. The
traditional Copehagen interpretation doesn't give an answer. I have an answer that satisfies me, and I give a very brief sketch of it in my reply to jrd261. Here I will only say that I think that
even in the context of the traditional interpretation of quantum mechanics there does exist an answer to the question "what is really going on in the world even when no observers are looking". In
other words, the traditional interpretation does NOT commit one to some form of subjectivism or Berkeleyan idealism, but can be consistent with a robust philosophical "realism". But this is a tricky
business, and probably beyond what can be discussed in such a forum.
Stephen M. Barr
July 11, 2012
Reply to Vishmehr24
Dear Vishmehr24
The branching of a wavefunction takes place, roughly speaking, when some microscopic system (e.g. a radioactive nucleus) interacts with a macroscopic system (e.g. a Geiger counter). The wavefunction
has various parts describing the nucleus decaying at different times. In these different parts of the wavefunction the Geiger counter is also doing different things (because it is affected by what
the nucleus does). Since a Geiger counter is made up of a huge number of
particles, a Geiger counter "doing different things" entails a huge number of particles behaving differently in the different parts of the wavefunction. That means that the different parts of the
wavefunction differ very greatly from each other, and therefore they "decohere" from each other, to use the technical jargon. That means that for all practical purposes, they no longer can affect
each other and are like different worlds.
In other words, the splitting apart of the branches happens whenever macroscopic objects are involved. The big question that distinguishes MWI from the traditional interpretation of quantum mechanics
is whether all the branches are equally real. In MWI, they are equally real. In the traditional interpretation, they are not. E.g. if the observer sees the nucleus decay at 1 PM (because the Geiger
counter clicked at 1 PM) then (according to the traditional view) that is what really happened and all the other branches of the wavefunction (for example the one where the nucleus
decayed at 5 PM) are just unrealized possibilities.
You are right that one is doing a great extrapolation in supposing that oen could (in principle) write down a wavefunction of the entire physical universe. Speaking just for myself, I see no problem
in making that extrapolation. Of course one is always making similar extrapolations when doing physics. When one assumes that nature obeys universal laws, so that every electron in a piece of metal
obeys the same Dirac equation, that is an extrapolation. (I think a reasonable and justified one. )
Stephen M. Barr
July 11, 2012
Reply to Wallace Forman
Dear Wallace,
First, the logical structure of the argument against materialism that I outlined in my essay is based on what happens when a measurement is made and how we are to understand the famous "collapse of
the wavefunction" that occurs when a measurement gives one definite result. The argument is to the effect that the mind of the measurer (or "observer") cannot be completely described by wavefunctions
governed by the laws of quantum physics. So, it is the mind of measurers that is in question, not the mind of God or gods. Let us therefore leave them out of the discussion.
Second, you say that we "might imagine a world with variables that look probabilistic to its inhabitants, but are not really undetermined." That is true, and it is exactly what happens in classical
physics. In classical physics, all measureable quantities have definite values at all times, but because we don't know most of those values, we must use probabilities. Many physicists, including
Einstein, have suggested that the same may be true in quantum mechanics, i.e. there are "hidden variables," whose values we don't know and that this is why probabilities seem to be needed in quantum
mechanics. The most successful attempt to construct a hidden variables theory was made by David Bohm. While such theories can reproduce many of the successes of standard quantum mechanics, they are
unable to deal with full-blown "quantum field theory." Most physicists, therefore, think the hidden variables approach is extremely unlikely to succeed.
Third, the structure of the argument I explained in the essay was not so much about the values that variables truly have, but rather about what a wavefunction means. Basically, a wavefunction is one
of two things: it is either (a) a straightforward description of the world as it is, or (b) it encodes what some observer knows or is in a position to assert about the world. If it is (a), one is led
perforce to the MWI. The reason is simple: The wavefunction contains many branches describing alternative histories of the world. So if the wavefunction is just "the world as it is", then such
histories are really all going on at once, and that is the essence of MWI. On the other hand, if (b) is correct, then "what some observer knows" --- and thus mind --- enters into how our fundamental
theories of physics. That is why Wigner wrote about 50 years ago
“While it may be premature to imagine that the present philosophy of quantum mechanics will remain a permanent feature of future physical theories, it will remain remarkable, in whatever way our
future concepts may develop, that the very study of the external world led to the conclusion that the content of the consciousness is an ultimate reality."
July 11, 2012
Limits of language?
A problem and a possibility. The problem: Jane’s relationship to the french exam is accidental to Jane – it is the relationship which is probable, not Jane. It is easy to verbalise this relationship.
In quantum physics probability is of the essence of the subject, a fact which mathematics reveals consistently and persistently. Verbalization of probability in this sense of essence, to use
Aristotelian language, seems impossible to adopt within the necessary constraints of grammar and syntax. Is it possible to verbalize quantum mechanics without gross distortion of its fundamentals?
The possibility: Panentheism ‘God in the world’ finds ‘in’ difficult. If probability is of the essence of all created things then probability is at the heart of ‘in’. A creator God might therefore be
said to be unchangeably committed to a probability inspired universe.
July 11, 2012
Measurement in MWI
Prof Barr,
How does a many-world wavefunction know when to branch?.
That is, the measurement is supposed to be external to the wavefunction but if everything is internal, what would measurement mean anyway?.
Also, quantum mechanics was initially formulated as a theory of interaction of microscopic systems interacting with measuring devices. Now the physicists define and use wavefunctions of the entire
universe. How is this extrapolation justified?. Or are they doing it as a matter of faith in atomism that all macroscopic objects are fully explainable as interactions of fundamental microscopic
In Thomism, the parts are only defined with respect to the whole and can not be fully understood
without reference to the whole. This whole-part loop is called the Formal Cause of the thing.
Perhaps this is what quantum mechanics paradoxes require?
Wallace Forman
July 10, 2012
Determined, Non MWI
Hi Dr. Barr,
Thank you for the interesting article, however I am afraid I am not convinced. Here is my objection:----------------------------------------If god observed all the quantum variables outside of time
and then disappeared, the universe would be determined, but still appear probabilistic to us as we observed the variables. If we can imagine such a probabilistic, determined universe as a possibly
existing thing, we can also imagine it existing without the need for determination by observation. If process Y will create a quantifiable object X with characteristics A and B, we can also just
imagine quantifiable object X with characteristics A and B, not necessarily created by process Y (determination by observation), a process which is itself fundamentally incapable of our observation.
(Wouldn't we prefer this under Occam's Razor?) This reasoning tells me I don't need to choose between MWI and determination by observation.----------------------------------------Since I composed the
above I see you have included some discussion about whether God can observe the variables in the comments section. What you say there seems more to be a discussion of theology than how we should
think about probability so let me offer just one more thought about my objection: It doesn't really matter for my point whether god exists, whether he can observe the variables, or determine them by
doing so. The point is that we can imagine a universe with variables that look probabilistic to its inhabitants, but are not really undetermined. If we can imagine such a universe, then we need not
choose between MWI and determination by observation. Moreover, if there is no way for us to test between the imagineably "determined" and untestable "determined, by observation," Occam's Razor tells
us to prefer "determined."
Joshua Weiner
July 10, 2012
Yes. . .but
There is something I still do not understand. My understanding is that until someone observes a quantum system, the wavefunction remains probabilistic. If God has already observed everything, because
he is out of time, then there would be no uncollapsed wavefunctions. If not, however, then there must still be quantum events that even God does not know the result of, because they are logically
unknowable. God can do anything that is logical, but not that which is by its nature illogical, correct? So I guess in response to your statement: "That is, he does not learn about things as a result
of their happening, but rather they happen because he knows them"; I would ask, doesn't this preclude the possibility of there being any QM events that remain probabilistic? If all events (including,
say, particle decays) happen because he knows they are going to happen (and where, and when, and how), then his very knowing would preclude any uncollapsed wavefunctions. . . and we would not have
made the observations that led to the formulation of QM. Probably I missed something here. . .
I guess where this my argument may go wrong is in your point that God does not "physically" interact with the physical world, but rather through his will. But in traditional theism, God can, if he
wills, cause a physical event such as a flood. Though the distal cause is his "will" (and it is difficult for us to understand quite what this means), the proximal cause is rainfall, which presumably
is not some special kind of supernatural precipitation but the same kind of physical rainfall we know (and this summer, wish for). So I don't quite understand why the action-at-a-distance of
observation by God (or we can call it willing) could not be possible; Even in the case of the flood, there must be some interface between God's will and a physical event. If it is not collapse of
wavefunctions, it is presumably something else. Am I way off base?
Stephen M. Barr
July 10, 2012
Reply to Joshua Weiner
Hi Josh,
Good questions. I don't think God (as understood in what is sometimes called "classical theism") can play the role of an "observer" in quantum mechanics, for several reasons.
In the traditional (or "Copenhagen") interpretation of quantum mechanics, as expounded by Wigner,
Peierls, and others, and explained in my essay, the collapse of the wavefunction corresponds to a change in the knowledge of the observer. But in classical theism, God is outside of time, and his
knowledge does not change.
Another way of looking at all this is that the wavefunction gives the probability correlations between the results of one set of measurements and the results of another (usually later) set of
measurements. For example, if I measure that an electron is located in a cubical box whose sides are of length L and that it has a certain energy, then that gives me an initial wavefunction that I
can use to calculate the relative chances of finding various values for the electron's position or momentum at a later time. In other words, in practice the wavefunction is a way of correlating the
of various measurements. But, in classical theism, God does not make measurements. His knowledge is not "a posteriori" but "a priori". That is, he does not learn about things as a result of their
happening, but rather they happen because he knows them, so speak.
The wavefunction encodes what some observer is in a position to assert about the physical world given some knowledge that he already possesses by virtue of other measurements or observations made by
him.. So the observer is, practicaly by definition, some being who acquires information by means of physicallly interacting with the rest of the universe. God, however, (according
to traditional theism) does not physically interact with things in the universe, as he is not a physical entity. Physical entities cause things to happen in the universe by means of physically
interacting with other entities. But God causes things in an utterly different way: by willing the reality or being of the whole universe. An analogy: a character in a novel has effects within the
novel's plot by means of acting within the plot in accordance with the internal rules of the novel. The author of the novel, on the other hand, causes things to happen as they do in the novel not by
being'an actor in the plot, but by simply by conceiving of the plot (and of the rules that givern it).
As far as free will goes, and possible connection to "quantum indeterminacy", I would suggest the relevant sections of my book "Modern Physics and Ancient Faith".
Joshua Weiner
July 10, 2012
God as observer
Excellent and succinct essay on the problem, Stephen! I believe I asked this naive question before but perhaps it is worthwhile to raise here again. Given the nature of quantum mechanics, many events
will remain probabilistic; that is, they will never have an observer, and thus the wavefunction will never collapse and there will be no "jump". While this obviously is outside the main question, in
that I am simply assuming an omniscient God, my question is: could God be the ultimate observer who can collapse any wave function? Would God's assumed omniscience preclude this (that is, would he
see all wavefunctions at once and thus there would be no need for him to "turn his attention" to any particular set of quantum events), or, philosophically might it remain a possibility? I suppose
one problem with this is that, while God could interact with the physical world "at a distance" in this way (which seems attractive in that one objection atheists have is how some kind of
supernatural being can possibly interact with the physical world), the result would not be determined by God but rather the wavefunction could collapse either way (if there is a way for the observer
to "choose" which direction the collapse goes, I've never heard of such). So God would be necessary, in this sense, for anything to happen (consistent with traditional theism), but he would still not
necessarily know the outcome (clearly inconsistent with traditional theism). I realize this is nebulous but I still find some aspect of the argument compelling, in that observation of quantum events
could be a way to explain God's continual interaction with our world in a way that is intellectually defensible. I would love to hear your opinion, and I apologize if you gave it previously but I
forgot what you said!
As to the main point of the essay I certainly agree that QM does NOT directly prove anything about God, but does prevent, pretty clearly, it seems to me, an purely materialistic (or deterministic)
worldview. It leaves open the door. The MWI is as ludicrous as any desperate attempt to square the circle is: sometimes one's instinctual resistance to the data leads one to propound ever-more
improbable explanations. It seems there was a similar resistance to the Big Bang, which still exists but has become fringe, as I understand.
An interesting question is the relationship of quantum uncertainty to events in the brain (or rather, as the very fact that they are there is trivial, the outward effects on how the mind works). I
don't think I've seen anything that isn't fairly silly on this, pace Penrose. It does seem to me that QM undermines our traditional understanding of free will as much as any purely materialistic
world view would. So working those issues out will be a key area--I'd love to hear any sources you know of on this topic that are worth reading.
Stephen M. Barr
July 10, 2012
jrd261,Your suggestions have merit
Dear jrd261,
What you are suggesting, namely solipsism, the idea that there is only one mind, is a perfectly consistent way of interpreting quantum mechanics. I quoted Eugene Wigner in my essay as saying, that
materialism is “logically consistent with present quantum mechanics." The full quote was actually, "Solipsism may be logically consistent with present quantum mechanics, but materialism is not." In
the solipsistic interpretation, there would still be "forks in the road": Each time the one observer (you) made an observation, the wavefunction would collapse, i.e. the universe would take one road
at the fork.
Your other suggestion --- all minds being a collective --- also can be made consistent. In particular one could take the view that whenever there is a branching of the wavefunction (which happens
when different parts of the wavefunction "decohere" from each other, in the technical jargon) all consciousness in the universe proceeds down just one branch. The wavefunction would continually
branch, exactly as MWI says, but there would never be a situation where the same observer existed in several conscious versions in distinct branches. In this picture, the wavefunction itself is
constantly branching, like train tracks; and what happens at the "collapse of the wavefunction" is not really any change in the wavefunction --- all the tracks are still there --- but rather all
consciousness proceeds down a single track, so to speak. (What I have just described is my own speculative view of quantum mechanics, for what it's worth.)
There are different ways that mind could be imagined to come into the story. But unless mind comes in somehow at a fundamental level, then everything is just matter governed by Schrodinger's
equation, and one is left with all parts of the wavefunction being equally real (even after decoherence). That is, one is left with MWI, and each observer existing in a huge number of equally real
July 10, 2012
Some Other Thoughts...
I enjoyed the read.
I'd first like to consider a very egocentric hybrid that can eliminate the need for MWI. Perhaps it is only "my" observation that can collapse the wavefunction and the rest of nature just follows
Schoedingers Equation. This would make "me" a sort of god of some closed system, but into necessarily God. I guess that is similar to MWI but with no forks in the road.
We could also treat humanity as a collective. If our minds are together one perfectly coupled wave collapsing detector than there need only be one world, and one mind... collective free will? That
would fit nicely with original sin and Christian dogma.
July 10, 2012
Some Other Thoughts...
I enjoyed the read.
I'd first like to consider a very egocentric hybrid that can eliminate the need for MWI. Perhaps it is only "my" observation that can collapse the wavefunction and the rest of nature just follows
Schoedingers Equation. This would make "me" a sort of god of some closed system, but into necessarily God. I guess that is similar to MWI but with no forks in the road.
We could also treat humanity as a collective. If our minds are together one perfectly coupled wave collapsing detector than there need only be one world, and one mind... collective free will? That
would fit nicely with original sin and Christian dogma.
|
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\(\qquad\qquad\qquad\qquad\qquad\qquad\qquad\Huge\mathsf{Basics!}\) \(\huge\mathsf{☞}\) \(\tt{Put\ your\ commands\ and\ messages\ inside\ this:}\) `\(message\)`. \(\qquad\tt{\ so\ that\ would\
look\ like:}\) \(message\). \(\huge\mathsf{☞}\) \(\tt{Put\ a\ tilde}\) `~` \(\tt{wherever\ you\ want\ a\ space.}\) \(\qquad\tt{so}\) `\(sentence~with~spaces\)` \(\tt{will\ give\ you:}\) \
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\(\qquad\qquad\qquad\qquad\qquad\qquad\qquad\Huge\mathsf{Sizes!}\) \(\large\mathsf{☯}\) \(\large\tt{To~ change~ sizes~ do}\) `\(\size{message}\)`. \(\large\mathsf{☯}\) \(\large\tt{So}\) `\(\large
{sacapuntas}\)` \(\tt\large{is}\) \(\large{sacapuntas}\). \(\large\mathsf{⇢}\) \(\tiny{tiny}\) \(\large\mathsf{⇢}\) \(\scriptsize{scriptsize}\) \(\large\mathsf{⇢}\) \(\small{small}\) \(\large\
mathsf{⇢}\) \(\normalsize{normalsize}\) \(\large\mathsf{⇢}\) \(\large{large}\) \(\large\mathsf{⇢}\) \(\Large{Large}\) \(\large\mathsf{⇢}\) \(\LARGE{LARGE}\) \(\large\mathsf{⇢}\) \(\huge{huge}\) \
(\large\mathsf{⇢}\) \(\Huge{Huge}\) \(\normalsize\tt{*Remember,~some~of~them~are~case~sensitive!}\)
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\(\qquad\qquad\qquad\qquad\qquad\qquad\qquad\Huge\mathsf{Colors!}\) \(\Large\mathsf{⚡}\) \(\large\tt{To~ make~ colors~ do}\) `\(\color{color}{message}\)`. \(\Large\mathsf{⚡}\) \(\large\tt{So}\)
`\(\color{blue}{sacapuntas}\)` \(\large\tt{would~ be}\) \(\color{blue}{sacapuntas}\). \(\Large\color{maroon}{\tt{Maroon}}\) \(\Large\color{brown}{\tt{Brown}}\) \(\Large\color{red}{\tt{Red}}\) \(\
Large\color{orangered}{\tt{OrangeRed}}\) \(\Large\color{salmon}{\tt{Salmon}}\) \(\Large\color{orange}{\tt{Orange}}\) \(\Large\color{goldenrod}{\tt{Goldenrod}}\) \(\Large\color{gold}{\tt{Gold}}\)
\(\Large\color{yellow}{\tt{Yellow}}\) \(\Large\color{greenyellow}{\tt{GreenYellow}}\) \(\Large\color{yellowgreen}{\tt{YellowGreen}}\) \(\Large\color{Olive}{\tt{Olive}}\) \(\Large\color{lime}{\tt
{Lime}}\) \(\Large\color{springgreen}{\tt{SpringGreen}}\) \(\Large\color{green}{\tt{Green}}\) \(\Large\color{forestgreen}{\tt{ForestGreen}}\) \(\Large\color{Seagreen}{\tt{SeaGreen}}\) \(\Large\
color{teal}{\tt{Teal}}\) \(\Large\color{turquoise}{\tt{Turquoise}}\) \(\Large\color{aqua}{\tt{Aqua~or~cyan}}\) \(\Large\color{aquamarine}{\tt{Aquamarine}}\) \(\Large\color{cadetblue}{\tt
{CadetBlue}}\) \(\Large\color{cornflowerblue}{\tt{CornflowerBlue}}\) \(\Large\color{royalblue}{\tt{RoyalBlue}}\) \(\Large\color{midnightblue}{\tt{MidnightBlue}}\) \(\Large\color{navy}{\tt{Navy}}
\) \(\Large\color{blue}{\tt{Blue}}\) \(\Large\color{purple}{\tt{Purple}}\) \(\Large\color{blueviolet}{\tt{BlueViolet}}\) \(\Large\color{darkorchid}{\tt{DarkOrchid}}\) \(\Large\color{Magenta}{\tt
{Magenta~or~fuchsia}}\) \(\Large\color{orchid}{\tt{Orchid}}\) \(\Large\color{plum}{\tt{Plum}}\) \(\Large\color{grey}{\tt{Grey}}\) \(\Large\color{tan}{\tt{Tan}}\) \(\large\mathsf{⚡}\) \(\large\tt
{To\ use\ both\ a\ size\ and\ a\ color\ it's}\) `\(\size\color{color}{message}\)`. \(\large\mathsf{⚡}\) \(\large\tt{So}\) `\(\LARGE\color{teal}{sacapuntas}\)` \(\large\tt{would\ output}\) \(\
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\(\qquad\qquad\qquad\qquad\qquad\qquad\qquad\Huge\mathsf{Fonts!}\) \(\tt{To\ change\ the\ font,\ put:}\) `\(\font{message}\)`. \(\tt{so}\) `\(\mathsf{sacapuntas}\)` \(\tt{will\ give\ you:}\) \(\
mathsf{sacapuntas}\). \(\huge\mathtt{Mathtt\ or\ tt}\) \(\huge\sf{Mathsf\ or\ sf}\) \(\huge\mathrm{Mathrm\ or\ rm}\) \(\huge\mathbf{Mathbf\ or\ bf}\) \(\huge\mathcal{Mathcal\ or\ cal}\) \(\huge\
mathbb{Mathbb}\) \(\Huge\frak{Mathfrak~or~frak}\) \(\huge\it{Mathit~or~it}\) \(\huge\mathscr{Mathscr\ or\ scr}\) \(\huge\boldsymbol{Boldsymbol}\) \(\huge\text{Text}\) \(\tt{To\ use\ a\ size,\
color,\ and\ font,\ type}\) `\(\size\color{color}{\font{message}}\)`. \(\tt{So}\) `\(\large\color{teal}{\mathsf{Sacapuntas}}\)` \(\tt{will~ look~ like}\) \(\large\color{teal}{\mathsf{Sacapuntas}}
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\(\qquad\qquad\qquad\qquad\qquad\qquad\qquad\Huge\mathsf{Symbols!}\) \(\large\sf{Check\ out\ this\ site\ for\ a\ list\ of\ symbols:}\) \(\Large{➱}\) http://detexify.kirelabs.org/symbols.html \(\
large\sf{or\ go\ here\ if\ you're\ looking\ for\ a\ specific\ symbol:}\) \(\Large{➱}\) http://detexify.kirelabs.org/classify.html
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\[\color{red}{\huge\mathtt{I}} \color{orange}{ \huge\sf{m}} \color {yellow}{ \huge\mathrm{p}} \color {green}{\huge\mathbf{r}} \color {blue}{ \huge\mathcal{e} } \color {indigo}{\huge\mathbb{s}} \
color{violet}{ \Huge\frak{s}} \color{cyan}{\huge\it{i}} \color{magneta}{ \huge\mathscr{v}} \color{purple}{ \huge\boldsymbol{e}} \Huge \color{blue}{\text{~}}\]
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how do u put different colors?
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\(\color{red}{\huge\tt{T}}\color{orange}{\huge\tt{r}}\color{gold}{\huge\tt{y}}\) \(\color{forestgreen}{\huge\tt{t}}\color{teal}{\huge\tt{h}}\color{blue}{\huge\tt{i}}\color{blueviolet}{\huge\tt
{s}}\) `\(\color{red}{\huge\tt{T}}\color{orange}{\huge\tt{r}}\color{gold}{\huge\tt{y}}\) \(\color{forestgreen}{\huge\tt{t}}\color{teal}{\huge\tt{h}}\color{blue}{\huge\tt{i}}\color{blueviolet}{\
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thank you!
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\(\color{red}{\huge\tt{G}}\color{orange}{\huge\tt{a}}\color{gold}{\huge\tt{b}}\) \(\color{forestgreen}{\huge\tt{y}}\color{teal}{\huge\tt{B}}\color{blue}{\huge\tt{o}}\color{blueviolet}{\huge\tt
{r}}\) \(\color{red}{\huge\tt{g}}\color{orange}{\huge\tt{e}}\color{gold}{\huge\tt{s}}\)
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Great work @kymber :)
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Another good one is the yfonts package. A friend of mine used it in his thesis to emphasize the chapters. The syntax goes like this: \yinipar{<letter of your choice>} , unfortunately it doesn't
work here.
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Thanks, mathslover. :D
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You're most welcome. I think most people wouldn't use it for science but actually i love it for giving my scripts some edge :)
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I like ur colors!
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\[Thank~you,~ I~ will~ now ~be ~able~ to~ write~ pretty~ papers!!\]
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@kymber You know what? You're epic.
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Thank you, Parth! :D
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Ahh...My student teaching others \(\Huge \color{red}{*}\)\(\huge \color{blue}{*}\)\(\LARGE \color{green}{*}\)\(\Large \color{yellow}{*}\)\(\large \color{orange}{*}\)\(\small \color{purple}{*}\)\
(\Tiny \color{pink}{*}\)\(\tiny \color{violet}{*}\)\(\Tiny \color{pink}{*}\)\(\small \color{purple}{*}\)\(\large \color{orange}{*}\)\(\Large \color{yellow}{*}\)\(\LARGE \color{green}{*}\)\(\huge
\color{blue}{*}\)\(\Huge \color{red}{*}\)\(\huge \color{blue}{*}\)\(\LARGE \color{green}{*}\)\(\Large \color{yellow}{*}\)\(\large \color{orange}{*}\)\(\small \color{purple}{*}\)\(\Tiny \color
{pink}{*}\)\(\tiny \color{violet}{*}\)\(\Tiny \color{pink}{*}\)\(\small \color{purple}{*}\)\(\large \color{orange}{*}\)\(\Large \color{yellow}{*}\)\(\LARGE \color{green}{*}\)\(\huge \color{blue}
{*}\)\(\Huge \color{red}{*}\)\(\huge \color{blue}{*}\)\(\LARGE \color{green}{*}\)\(\Large \color{yellow}{*}\)\(\large \color{orange}{*}\)\(\small \color{purple}{*}\)\(\Tiny \color{pink}{*}\)\(\
tiny \color{violet}{*}\)\(\Tiny \color{pink}{*}\)\(\small \color{purple}{*}\)\(\large \color{orange}{*}\)\(\Large \color{yellow}{*}\)\(\LARGE \color{green}{*}\)\(\huge \color{blue}{*}\)\(\Huge \
color{red}{*}\)\(\huge \color{blue}{*}\)\(\LARGE \color{green}{*}\)\(\Large \color{yellow}{*}\) \(\LARGE \color{red}{~~~~\:\:\:\:\:\:\mathbb Well\:\:\mathbb Done\:\:\mathbb Young~~\mathbb Jedi,~\
mathbb Well~~\mathbb Done.~~}\) \(\large \color{red}{~\:\:\:\:\:\:\:\:\:\:\:\: ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~-\mathbb Snuggie\mathbb Lad }\) \(\Huge \color{red}
{*}\)\(\huge \color{blue}{*}\)\(\LARGE \color{green}{*}\)\(\Large \color{yellow}{*}\)\(\large \color{orange}{*}\)\(\small \color{purple}{*}\)\(\Tiny \color{pink}{*}\)\(\tiny \color{violet}{*}\)\
(\Tiny \color{pink}{*}\)\(\small \color{purple}{*}\)\(\large \color{orange}{*}\)\(\Large \color{yellow}{*}\)\(\LARGE \color{green}{*}\)\(\huge \color{blue}{*}\)\(\Huge \color{red}{*}\)\(\huge \
color{blue}{*}\)\(\LARGE \color{green}{*}\)\(\Large \color{yellow}{*}\)\(\large \color{orange}{*}\)\(\small \color{purple}{*}\)\(\Tiny \color{pink}{*}\)\(\tiny \color{violet}{*}\)\(\Tiny \color
{pink}{*}\)\(\small \color{purple}{*}\)\(\large \color{orange}{*}\)\(\Large \color{yellow}{*}\)\(\LARGE \color{green}{*}\)\(\huge \color{blue}{*}\)\(\Huge \color{red}{*}\)\(\huge \color{blue}{*}
\)\(\LARGE \color{green}{*}\)\(\Large \color{yellow}{*}\)\(\large \color{orange}{*}\)\(\small \color{purple}{*}\)\(\Tiny \color{pink}{*}\)\(\tiny \color{violet}{*}\)\(\Tiny \color{pink}{*}\)\(\
small \color{purple}{*}\)\(\large \color{orange}{*}\)\(\Large \color{yellow}{*}\)\(\LARGE \color{green}{*}\)\(\huge \color{blue}{*}\)\(\Huge \color{red}{*}\)\(\huge \color{blue}{*}\)\(\LARGE \
color{green}{*}\)\(\Large \color{yellow}{*}\)
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where do latex works?
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I'm certainly not your student, Snuggie. Not sure where you got that idea.
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\(\Huge \color{aqua}{\cal{LOL ~Kymber...}}\)
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I'm pretty sure @ParthKohli was the one who showed me LaTeX, actually... and @shruti LaTeX works in questions and responses on OpenStudy and you can download programs to use it on your computer
if you wanted. I don't know of any other sites that support it.
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I learned from snuggie ... And you i learned more fonts and colors from u .Snuggie taught me how to change colors and fonts and how to set it up
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oh...i thought you were one of mine...well...ok...
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does'nt i works on Other sites ? @Kymber
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@shruti , that may be because , other sites may have not installed these packages on their sites.
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btw, can you name those "Other sites" ?
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okay thanks! @mathslover
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You missed a few colors
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Are you guys still curious about what you can do? If you would like you can go to: http://latex-project.org/guides/usrguide.tex It is a rather dry read but you can learn a lot of commands to use
on this site!
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This one is not so dry but it is more recent. I don't know if the team has kept up with the most recent \(\LaTeX\) versions but if they have this will be helpful. this is one of the pages i refer
to a lot. http://latex-project.org/guides/fntguide.pdf
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\(\setlength{\unitlength}{0.75mm} \begin{picture}(60,40) \put(30,20){\vector(1,0){30}} \put(30,20){\vector(4,1){20}} \put(30,20){\vector(3,1){25}} \put(30,20){\vector(2,1){30}} \put(30,20){\
vector(1,2){10}} \thicklines \put(30,20){\vector(-4,1){30}} \put(30,20){\vector(-1,4){5}} \thinlines \put(30,20){\vector(-1,-1){5}} \put(30,20){\vector(-1,-4){5}} \end{picture}\)
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What colours did I miss, care to share?
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Try ones that I missed. They're probably not included in my tutorial here because they don't work.
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The ones with names are easier to remember
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Not if you understand how the Hexadecimal code works. If you understand the layout you can make any color you want with only swapping a number @kyber Names are for wimps when you've got millions
of codes and all you have to do is find the sequencing. When you learn that you can create any color you want with only the typing of a couple of memorized number letter mixtures that all follow
each other in a specific order that once you get it is dumbly easy.
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That's like..... a lot O_O
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@kymber why do you write "\(\color{red}{\huge\tt{T}}" emphazizing on the "\tt{T}}"?
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The double t and the double }
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double t = \tt = a font double } is to close the color tag, though it's unnecessary as you can write it as `\huge\tt\color{red}T`
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Ah ok much less confusing thanks thom :)
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You need to put your words in brackets if you want to write more. `\(\huge\tt\color{red}Tomatoes\)` will give you \(\huge\tt\color{red}Tomatoes\). So then you can do `\(\color{red}{\huge\tt
{Tomatoes}}\)` for \(\color{red}{\huge\tt{Tomatoes}}\). As far as I know. Parth taught me that.
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Yea my explanatin only works when you use 1 character. Like in this code: `\(\color{red}{\huge\tt{T}}\color{orange}{\huge\tt{r}}\color{gold}{\huge\tt{y}}\) \(\color{forestgreen}{\huge\tt{t}}\
color{teal}{\huge\tt{h}}\color{blue}{\huge\tt{i}}\color{blueviolet}{\huge\tt{s}}\)` You can reduce it to: `\(\huge\tt\color{red}T\color{orange}r\color{gold}y~\color{forestgreen}t\color{teal}h\
color{blue}i\color{blueviolet}s\)` \(\huge\tt\color{red}T\color{orange}r\color{gold}y~\color{forestgreen}t\color{teal}h\color{blue}i\color{blueviolet}s\)
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Math lesson 1: Do you know why there are 16 million possible color combinations? Answer: These colors are written in hexadecimal, where we use letters A-F and 0-9. Now A-F is like 6 letters and
0-9 is 10 different numbers. So there are 16 total characters in hex. Hey, now you have six letters in hex to make a color, like \(\rm \color{red}{FF0000}\) or \(\color{#C00}{C00000}\). So there
are \(16×16×16×16×16×16≈16,000,000\) colors.
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actualy it's 16,777,216 :P
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That's why I wrote \(\approx\)
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i know, but let's not further spoil the comments :P
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Thanks @kymber :)
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@kymber Very helpful. Thanks.
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Wow. I didn't see this earlier. So just to be clear, I can use hex code that I use in CSS in LATEX?
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Now what about RGBA?
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BTW really awesome tut @kymber
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is replying to Can someone tell me what button the professor is hitting...
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Quantum Algorithms and Complexity for Certain Continuous and Related Discrete Problems
Quantum Algorithms and Complexity for Certain Continuous and Related Discrete Problems
Permanent URL:
Part Number:
Department of Computer Science, Columbia University
Publisher Location:
New York
The thesis contains an analysis of two computational problems. The first problem is discrete quantum Boolean summation. This problem is a building block of quantum algorithms for many continuous
problems, such as integration, approximation, differential equations and path integration. The second problem is continuous multivariate Feynman-Kac path integration, which is a special case of
path integration. The quantum Boolean summation problem can be solved by the quantum summation (QS) algorithm of Brassard, HíŸyer, Mosca and Tapp, which approximates the arithmetic mean of a
Boolean function. We improve the error bound of Brassard et al. for the worst-probabilistic setting. Our error bound is sharp. We also present new sharp error bounds in the average-probabilistic
and worst-average settings. Our average-probabilistic error bounds prove the optimality of the QS algorithm for a certain choice of its parameters. The study of the worst-average error shows that
the QS algorithm is not optimal in this setting; we need to use a certain number of repetitions to regain its optimality. The multivariate Feynman-Kac path integration problem for smooth
multivariate functions suffers from the provable curse of dimensionality in the worst-case deterministic setting, i.e., the minimal number of function evaluations needed to compute an
approximation depends exponentially on the number of variables. We show that in both the randomized and quantum settings the curse of dimensionality is vanquished, i.e., the minimal number of
function evaluations and/or quantum queries required to compute an approximation depends only polynomially on the reciprocal of the desired accuracy and has a bound independent of the number of
variables. The exponents of these polynomials are 2 in the randomized setting and 1 in the quantum setting. These exponents can be lowered at the expense of the dependence on the number of
variables. Hence, the quantum setting yields exponential speedup over the worst-case deterministic setting, and quadratic speedup over the randomized setting.
Item views:
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MathGroup Archive: February 2011 [00145]
[Date Index] [Thread Index] [Author Index]
Re: Numerical equation solving
• To: mathgroup at smc.vnet.net
• Subject: [mg116203] Re: Numerical equation solving
• From: Gary Wardall <gwardall at gmail.com>
• Date: Sat, 5 Feb 2011 05:44:05 -0500 (EST)
• References: <iigpkp$21b$1@smc.vnet.net>
On Feb 4, 5:58 am, florian.mau... at schott.com wrote:
> Hi everybody,
> I have a quite challenging question about numerical equation solving with
> MATHEMATICA:
> f1 - f2 + ((f2 - f1) t)/t1 = (r1^2 rho vt^2)/(2 \[Pi] r2^4) (3/2 + l/(2
> r2) k/((2 rho vt)/(\[Pi] r2 eta))^(1/4))
> The equation given above is implicit when solving to variable vt. The
> variable vt itself is a differential operator (D[v,t]), so vt must be
> replaced with D[v,t]. As I am interested in v the solution has to be
> integrated with respect to t (i.e. Integrate[D[v,t],t,{0,t1}] or Integrate
> [D[v,t],t,{0,t1}]). Finally, I want to visualize in a contour plot the
> values of f1 and t1 for which the integrated equation fulfills a certain
> number q. All variables are positive real numbers.
> To maybe better explain my problem I have summarized all steps in one
> MATHEMATICA command (I know that the syntax is not correct):
> ContourPlot[q = NIntegrate[D[v, t]/.Solve[f1-f2 + ((f2 - f1) t)/t1 ==
> (r1^2 rho D[v, t]^2)/(2 \[Pi] r2^4) (3/2 + l/(2 r2) k/((2 rho D[v,
> t])/(\[Pi] r2 eta))^(1/4)), D[v, t]],t, {0, t1}], {t1, 0.1, a}, {f1, 0, b}]
> How can I solve solve this problem with MATHEMATICA? Thanks in advance for
> your support!
> Mr.Mason
There might be a solution. If you can change your equation to and ODE
or a system of ODE's.
Here is a much simpler example:
Given: Sin[x+y]+y=x
Solve for y.
The solution for y by the following method will be implicit.
Let yp be the first derivative of y with respect to x.
Solving for yp:
Now find a point on the graph of y, numerically or otherwise:
Note that:
So y(0)=0
Now solve the ODE:
yp=(1-Cos[x+y])/(Cos[x+y]+1) when y(0)=0
I hope this will work for you.
Good Luck
Gary Wardall
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Definitions for summationsəˈmeɪ ʃən
This page provides all possible meanings and translations of the word summation
Random House Webster's College Dictionary
sum•ma•tionsəˈmeɪ ʃən(n.)
1. the act or process of summing.
2. an aggregate or total.
3. a review or recapitulation of previously stated facts or statements, often with final conclusions drawn from them.
4. the final arguments of opposing attorneys before a case goes to the jury.
Category: Law
5. the arousal of nerve impulses by a rapid succession of sensory stimuli.
Category: Psychology
Origin of summation:
Princeton's WordNet
1. summation, summing up, rundown(noun)
a concluding summary (as in presenting a case before a law court)
2. summation(noun)
(physiology) the process whereby multiple stimuli can produce a response (in a muscle or nerve or other part) that one stimulus alone does not produce
3. sum, summation, sum total(noun)
the final aggregate
"the sum of all our troubles did not equal the misery they suffered"
4. summation, addition, plus(noun)
the arithmetic operation of summing; calculating the sum of two or more numbers
"the summation of four and three gives seven"; "four plus three equals seven"
1. summation(Noun)
A summarization.
2. summation(Noun)
: An adding up of a series of items.
Webster Dictionary
1. Summation(verb)
the act of summing, or forming a sum, or total amount; also, an aggregate
1. Summation
Summation is the operation of adding a sequence of numbers; the result is their sum or total. If numbers are added sequentially from left to right, any intermediate result is a partial sum,
prefix sum, or running total of the summation. The numbers to be summed may be integers, rational numbers, real numbers, or complex numbers. Besides numbers, other types of values can be added as
well: vectors, matrices, polynomials and, in general, elements of any additive group. For finite sequences of such elements, summation always produces a well-defined sum. Summation of an infinite
sequence of values is not always possible, and when a value can be given for an infinite summation, this involves more than just the addition operation, namely also the notion of a limit. Such
infinite summations are known as series. Another notion involving limits of finite sums is integration. The term summation has a special meaning related to extrapolation in the context of
divergent series. The summation of the sequence [1, 2, 4, 2] is an expression whose value is the sum of each of the members of the sequence. In the example, 1 + 2 + 4 + 2 = 9. Since addition is
associative the value does not depend on how the additions are grouped, for instance + and 1 + both have the value 9; therefore, parentheses are usually omitted in repeated additions. Addition is
also commutative, so permuting the terms of a finite sequence does not change its sum.
Find a translation for the summation definition in other languages:
Use the citation below to add this definition to your bibliography:
Are we missing a good definition for summation?
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Capitol College: Laplace and Fourier Analysis - Description
Laplace and Fourier Analysis
Definition of transform: Laplace transform of algebraic, exponential and trigonometric functions; basic theorems including shifting, initial and final-value theorems; unit-step, periodic and delta
functions; methods of inverting transforms; solutions of differential equations by transform methods. Fourier series and coefficients; expansion of functions in Fourier series; complex Fourier
coefficients; Parseval's Theorem; Fourier transform and its properties. Prerequisite: MA-340. (3-0-3)
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question regarding integer type
Author question regarding integer type
could you please look at my question,and let me know where I get wrong? I am greatly appreciate any tips or suggestions.
May 17, The question is asked to give the last value of i where a[i] output correctly.
Posts: 7 my solution: a[i+2] = 2^{i+2} - 1, and a[i+2] is a integer so that it is stored in 32 bits. 2^{i+2} - 1 <= 2^{32}, and I get i = 30.
but the correct answer is a = 31.
how to get a = 31?
Oct 27,
2005 int cannot go up to 2^{32}, it can only go up to 2^{31}-1. So if you need to go beyond that you must use long.
SCJP 1.4 - SCJP 6 - SCWCD 5 - OCEEJBD 6
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Hi Martha,
Sep 07, Just to expand a little on why that program works for a[31] = (2^31) - 1. Now, since the largest possible positive int value is (2^31) - 1, you're right that you would get numeric
2010 overflow if you tried to calculate it in that order: the (2^31) will overflow before you have a chance to subtract the 1. However, if you unwind the calculation that is being done in line
Posts: 98 12 of the program, what is actually being calculated is essentially (2^30) + (2^30 - 1). So the final step is an addition, and no overflow takes place.
[edit:] In looking back over your question, I realized that maybe you're confused about a different aspect of the problem. In the loop iteration when i = 31, it is true that the
calculation for a[i+2] is going to be wrong due to overflow (as will have been a[i+1]). But that doesn't matter, because what is being printed in that iteration is a[i], which was
calculated two iterations back and is correct.
subject: question regarding integer type
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viewing the second fundamental form as a tensor
up vote 2 down vote favorite
Dear all,
Thank you for your time reading this post. I am a student in computer science so this viewpoint of the second fundamental form may be interesting to you.
I would like to understand the second fundamental form of an affine (or projective) variety of dimension $m$ in affine (or projective) space $\mathbb{A}^n$ (or $\mathbb{P}V$). It is a bilinear form
from the tangent space to the normal space. So it is naturally identified as a three-way tensor.
My problem is that: is there any geometric meaning of the tensorial viewpoint? In particular, I would like to know if there is some geometric intuition for the tensor rank of this tensor.
Thank you.
Jimmy Qiao
p.s. The point is mainly to view the second fundamental form as a three−way tensor. Especially, will the tensor rank tell us something about the infinitesimal variation of the tangent space in the
neighborhood? There is some claim that I would like to see: for some point $p\in X$, if there is an affine space $S\subseteq X$ passing through $p$ (in some neighborhood of $p$) then the tensor rank
of $II_p$ is somehow bounded by the codimension of $S$. Thank you again.
The above claim is to generalize the following. Consider a hypersurface $H$. If there is an affine space $S\subseteq X$ passing through $p$ (in some neighborhood of $p$) then the rank of Hessian is
bounded by 2 times the codimension of $S$.
This may be a wild conjecture... But thank you all!
ag.algebraic-geometry dg.differential-geometry computer-science
add comment
2 Answers
active oldest votes
I believe the definition of the second fundamental form for a projective variety is explained very nicely in
up vote 1 down vote accepted Griffiths, Phillip; Harris, Joseph Algebraic geometry and local differential geometry. Ann. Sci. École Norm. Sup. (4) 12 (1979), no. 3, 355–452.
add comment
The geometric meaning you are looking for is related to the so called Gauss map.
The Gauss map for a smooth projective variety $X \subset \mathbb{P}^n$ is the rational map
$\mathcal{G} \colon X \to \mathbb{G}(k, n)$
sending the point $p \in X$ to its projective tangent plane $\mathbb{T}_p(X)$.
Such a Gauss map $\mathcal{G}$ induces a differential map on (affine) tangent spaces as follows:
$(d\mathcal{G})_p \colon T_p(X) \to \textrm{Hom}(\mathbb{T}_p(X), K^{n+1}/\mathbb{T}_p(X))$.
Every homomorphism $\phi \colon \mathbb{T}_p(X) \to K^{n+1}/\mathbb{T}_p(X)$ in the image of the Gauss map has $p$ in its kernel, so $(d\mathcal{G})_p$ induces a map
$(d\mathcal{G})_p \colon T_p(X) \to \textrm{Hom}(\mathbb{T}_p(X)/p, K^{n+1}/\mathbb{T}_p(X))$.
Using the identifications
$T_p(X)=\textrm{Hom}(p, \mathbb{T}_p(X)/p)$,
$N_p(X)=T_p(\mathbb{P}^n)/T_p(X)=\textrm{Hom}(p, K^{n+1}/\mathbb{T}_p(X))$
up vote 6 one sees that this is equivalent to a map
down vote
$(d\mathcal{G})_p \colon T_p(X) \to \textrm{Hom}(T_p(X), N_p(X))$, i.e. to a bilinear map
$(d\mathcal{G})_p \colon T_p(X) \otimes T_p(X) \to N_p(X)$ inducing
$(d\mathcal{G})_p \colon S^2 T_p(X) \to N_p(X)$
which is exactly the second fundamental form $II_p$.
So, roughly speaking, $II_p$ measures the "variation" of the (projective) tangent space of $X$ in a neighborhood of the point $p$.
You can find more details in the book of Joe Harris "Algebraic geometry: a first course", especially Chapter 17.
EDIT The following result is due to Landsberg (see the paper "On second fundamental forms of projective varieties", Inventiones Mathematicae 117) and deals with the rank of the second
fundamental form of a variety containing a linear space. The original statement is more general and also includes the case where $X$ is singular.
THEOREM Let $X^n \subset \mathbb{P}^{n+a}$ be a smooth variety not contained in a hyperplane. Let $L \subset X$ be a $r$-plane. Then for general $p \in L$ we have
$\dim II_p(\underline{L}, T_p(X)) \geq \min \{r,a\}$,
where $\underline{L}$ denotes the tangent directions to $L$ and $II_p(\underline{L}, T_p(X)) = \textrm{Image} \; II_p|_{L \times T_p(X)}$
Thank you. But I think the point is mainly to $\emph{view the second fundamental form as a three-way tensor}$. Especially, will the tensor rank tell us something about the infinitesimal
variation of the tangent space in the neighborhood? There is some claim that I would like to see: for some point $p\in X$, if there is an affine space $S\subseteq X$ passing through $p$
(in some neighborhood of $p$) then the tensor rank of $II_p$ is somehow bounded by the tensor rank. Thank you again. – Jimmy Feb 17 '11 at 13:45
Sorry, the formatting is not good. Please refer to the original post (modified). – Jimmy Feb 17 '11 at 14:05
I added in the answer a theorem of Landsberg that seems related to what you are looking for – Francesco Polizzi Feb 18 '11 at 9:47
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Not the answer you're looking for? Browse other questions tagged ag.algebraic-geometry dg.differential-geometry computer-science or ask your own question.
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Arrays, pointers, trouble..
08-05-2010 #1
Registered User
Join Date
Aug 2010
Arrays, pointers, trouble..
Hi all!
I am trying to write a Matlab mex-file, and I find the c-coding part a bit challenging. I can get the code below to compile, but it doesn't do what is expected:
#include <math.h>
#include "mex.h"
/*Subroutine to calculate the Fuzzy Sample Entropy*/
void FSE(double *cop, int M, double r, double c, mwSize N, double *FuzzSampEnt) {
mwIndex m, i, ii, j, jj, k, l;
double expo, d, *seg_i, *seg_j, count, sumCim, *Cim, CIM[2];
expo=(log(log(pow(2, c)))/log(r));
for (m=M;m<M+2;m++) {
seg_i=mxCalloc(m, sizeof(double));
seg_j=mxCalloc(m, sizeof(double));
Cim=mxCalloc(N-m, sizeof(double));
for (i=0; i<N-m; i++) {
for (ii=0; ii<m; ii++){
for (j=0; j<N-m; j++) {
for (jj=0; jj<m; jj++) {
/*Finding the maximum absolut value of seg_i-seg_j*/
for (k=0; k<m; k++) {
if (d < fabs(seg_i[k]-seg_j[k])) {
/*if d=0, self match*/
if (d>0) {
count+=exp(-(pow(d, expo/c)));
if (m==M){
else if (m==M+1){
*FuzzSampEnt=seg_j[k];//-log(CIM[1]/ CIM[0]);
/*the gateway function*/
void mexFunction(int nlhs, mxArray *plhs[ ],
int nrhs, const mxArray *prhs[ ]) {
int M;
double *cop, *FuzzSampEnt;
double c, r;
mwSize N;
if(nrhs!=4) {
mexErrMsgTxt("Four inputs required (cop,M,r,c).");
/*Assign pointers*/
plhs[0] = mxCreateDoubleMatrix(1, 1, mxREAL);
FuzzSampEnt = mxGetPr(plhs[0]);
/*Call the subroutine */
FSE(cop, M, r, c, N, FuzzSampEnt);
The trouble seems to be here: seg_i[ii]=cop[i+ii]; cop[i+ii] seems to be what it should be, but I can't seem to get it's value to seg_i[ii]. Also I'm not sure if it is necessary to use pointers
here at all. I know that seg_i and seg_j are either M or M+1 in length and Cim is either N-M or N-M-1 in length.
The idea would be to make the code as fast as possible, as fast as possible.. So how would you deal with the seg_i, seg_j and Cim? Can you see what I'm doing wrong in here?
Thank You!
The trouble seems to be here: seg_i[ii]=cop[i+ii]; cop[i+ii] seems to be what it should be, but I can't seem to get it's value to seg_i[ii].
What exactly do you mean by this?
C + C++ Compiler: MinGW port of GCC
Version Control System: Bazaar
Look up a C++ Reference and learn How To Ask Questions The Smart Way
Well, probably the real problem is earlier than in this point, but until this point I get the stuff out what I want. I have changed the ending of the code, the output parameter FuzzSampEnt to see
what it gives out at what point (I'm sure there is a better way). So it would seem that for example FuzzSampEnt=expo; or FuzzSampEnt=cop[ii+i]; give out what they are supposed to, but
*FuzzSampEnt=seg_i[k], gives out something funny like -9*10^303. The true output, -log(CIM[1]/ CIM[0]), is commented, but using it the function gives out NaN.
08-05-2010 #2
08-05-2010 #3
Registered User
Join Date
Aug 2010
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Nash Equilibrium Proof Sketch
A selection of articles related to nash equilibrium proof sketch.
Original articles from our library related to the Nash Equilibrium Proof Sketch. See Table of Contents for further available material (downloadable resources) on Nash Equilibrium Proof Sketch.
Nash Equilibrium Proof Sketch is described in multiple online sources, as addition to our editors' articles, see section below for printable documents, Nash Equilibrium Proof Sketch books and related
Suggested Pdf Resources
Suggested Web Resources
Great care has been taken to prepare the information on this page. Elements of the content come from factual and lexical knowledge databases, realmagick.com library and third-party sources. We
appreciate your suggestions and comments on further improvements of the site.
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User Benjamin Steinberg
bio website sci.ccny.cuny.edu/~benjamin
location New York City
age 41
visits member for 2 years, 10 months
seen Apr 16 at 13:33
stats profile views 42
I am an algebraist interested in a broad range of areas. I've worked on semigroups, geometric group theory, algebraic combinatorics, representation theory, self-similar groups (aka automaton groups),
profinite groups and random walks on semigroups and groups. I've been particularly interested in interactions between these areas and Computer Science and am fond of algorithmic questions. I've also
dabbled with operator algebras associated to etale groupoids and inverse semigroups. Currently, I am interested in applications of finite semigroup theory to finite state Markov chains.
My blog is http://bensteinberg.wordpress.com/author/bsteinbg/
MathOverflow 12,893 rep 22880
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231 Votes Cast
all time by type month
186 up 79 question 3
45 down 152 answer
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The Novikov Conjecture: Geometry and Algebra
In his 1951 review of Chevalley’s Introduction to the Theory of Algebraic Functions of One Variable (which carried a price-tag of $4.00 in those halcyon times) Weil famously observed: “Here is
algebra with a vengeance; algebraic austerity could go no further.” Noting that one should only steal from the best, we might paraphrase Weil and observe in regard to the book under review, The
Novikov Conjecture: Geometry and Algebra, by Matthias Kreck and Wolfgang Lück, that here is algebraic topology with a vengeance! And it’s perhaps proper to note that here, too, we encounter no small
measure of algebraic austerity: a geometry book with no pictures. Of course this isn’t really a criticism per se, given that Kreck and Lück generally work in dimension at least five; besides,
trumping everything with algebra is a virtuous act.
But naturally it’s not just algebra. The Novikov conjecture, as well as the other, related, conjectures the book is concerned with (e.g. the Borel conjecture, the Baum-Connes conjecture) are truly
geometrical and topological assertions. Already in the Introduction the authors place their emphasis on questions surrounding the classification of manifolds, playing off the Poincaré conjecture
(which may presently acquire theorem-status, if Perelman is right). Regarding the specific focus of what lies ahead Kreck and Lück note that “[t]he Borel Conjecture, which is closely related to the
Novikov Conjecture, implies that the fundamental group determines the homeomorphism type of an aspherical closed manifold.” Thus, the prevailing context is that of topological invariants (e.g.
homotopy, (co)homology, characteristic classes).
Well, then, what does the Novikov conjecture assert? The answer to this question is a bit involved: G is any discrete group, BG its classifying space, M a closed, oriented, smooth manifold, and u: M
→ BG. Let L(M) be the L-class of M, so that we get that L(M) is a sum of homogeneous polynomials in rational Pontjagin classes. If x is an element of the direct product of the non-negative
dimensional cohomology groups of BG taking values in Q, define the signature sign(x, M, u) := , where <.,.> is the Kronecker product, [M] is M’s fundamental class in the dim(M)-dimensional homology
group of M taking values in Z, and otherwise we’re dealing with the intersection pairing. The Novikov conjecture says that this signature is a homotopy invariant for every x.
By contrast, the Borel conjecture is a lot easier to state. To wit: given a pair of closed, aspherical manifolds, M, N. Then, first, each homotopy equivalence M ® N is homotopic to a homeomorphism;
and, second, M and N are homeomorphic if and only if they have isomorphic fundamental groups. Kreck and Lück go on to tantalize the reader by observing that e. g. if u is a homotopy equivalence then
Borel ⇒ Novikov, and that a certain instance of Borel implies nothing less than the 3-dimensional Poincaré conjecture. Exciting stuff.
Now for the question of the proper audience for this book. The propaganda on the book-cover warns that “[t]he prerequisites consist of a solid knowledge of the basics of manifolds, vector bundles,
(co-)homology, and characteristic classes.” And we might couple this somewhat understated appraisal to the observation that the book springs from the authors’ Oberwolfach seminar on the indicated
material, evidently involving an audience of initiates if not aficionados.
Thus, The Novikov Conjecture: Geometry and Algebra is not for the timid, the dabbler, or the dilettante. It is serious business and deals with a wealth of interesting and deep material from modern
algebraic topology and differential geometry; here is a short sampling of topics: bordism, the signature, the Whitehead group, Whitehead torsion, s-cobordism, surgery, Poincaré duality, equivariant
homology. Twelve chapters are taken care of by Kreck, eleven by Lück, one by Lück and Varisco. There are 71 exercises at the end of the book, with solution hints, and, apparently for good measure, a
copy of the Oberwolfach schedule for the whole affair. The reader, if he is prepared to work hard (and is well-prepared to begin with), will learn a lot of wonderful contemporary mathematics by
following the path Kreck and Lück lay out.
Michael Berg is Professor of Mathematics at Loyola Marymount University in California.
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Approaching Proof in a Community of Mathematical Practice
Abstract (Summary)
This thesis aims to describe how students encounter proof in a community of mathematical practice at a mathematics department and how they are drawn to share mathematicians’ views and knowledge of
proof. Considering the department as a community of practice where the joint enterprise is learning mathematics in a broad sense made it possible to perceive the newcomers as active participants in
the practice. The combination of a socio-cultural perspective, Lave and Wenger’s and Wenger’s social practice theories and theories about proof offers a fresh framework in understanding and
describing the diversity of the culture involving such a complex notion as proof. Proof is examined from historical, philosophical and didactical points of view and considered as reification and as
an artefact from a socio-cultural perspective. The metaphor of transparency of artefacts that refers to the intricate dilemma about how much to focus on different aspects of proof at a meta-level and
how much to work with proof without focusing on it, both from teacher and student perspectives, is a fundamental aspect in the data analysis. The data consists of transcripts of interviews with
mathematicians and students as well as survey responses of university entrants, protocols of observations of lectures, textbooks and other instructional material. Both qualitative and quantitative
methods were applied in the data analysis. A theoretical model with three different teaching styles with respect to proof could be constructed from the data. The students related positively to proof
when they entered the practice. Though the mathematicians had no explicit intention of dealing so much with proof in the basic course, students felt that they were confronted with proof from the very
beginning of their studies. Proof was there as a mysterious artefact and a lot of aspects of proof remained invisible as experienced by students when they struggled to find out what proof is and to
understand its role and meaning in the practice. The first oral examination in proof seems to be significant in drawing students to the practice of proof.
Bibliographical Information:
School:Stockholms universitet
School Location:Sweden
Source Type:Doctoral Dissertation
Keywords:MATHEMATICS; proof; university mathematics; community of practice; participation; reification
Date of Publication:01/01/2006
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Baker's Game
Baker's Game
FreeCell type. 1 deck. No redeal.
Move all cards to the foundations.
Quick Description
Like FreeCell, but the piles build down by suit.
All cards are dealt at the start of the game. To compensate for this there are 4 free cells which can hold any - and just one - card.
Cards may only be moved onto cards of the same suit.
The number of cards you can move as a sequence is restricted by the number of free cells - the number of free cells required is the same as if you would make an equivalent sequence of moves with
single cards. (As a shortcut, the computer also considers the number of free piles so that you can move even more cards as one single sequence.)
Baker's Game is named after the mathematician C.L. Baker and was first published in Martin Gardner's June 1968 Mathematical Games column in Scientific American.
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Topic: Queries about Species
Replies: 4 Last Post: Jan 13, 2003 10:41 AM
Messages: [ Previous | Next ]
Re: Queries about Species
Posted: Jan 6, 2003 11:37 AM
Chris Hillman kindly pointed out to me that
(1 - x^2) ^-1/2 = cosh(arctanh x)
To turn this into a species, note that cosh corresponds to being a
set of even cardinality, and arctanh corresponds to being a set of odd
cardinality up to cyclic order. Then from what Baez tells us in
Week 190, to compose species we look for an even numbered
set of odd numbered sets, the latter taken up to cyclic order.
This spits out 9 answers when a 4 element set is fed to it as it should.
After integrating this species to arcsin, do we find a case of the elusive
categorification of pi?
"John Baez" <baez@galaxy.ucr.edu> wrote in message
> I'll tackle the easiest one now and attempt the harder ones
> later, but I sure hope other people try too.
> >X/(1 - e^X) looks like a simple composition of species - pick out a one
> >element set and arrange a set of sets whose union is the remainder - yet
> >it can't be that simple to get at the Bernoulli numbers. I guess lots of
> >unwanted empty sets appear in the union.
> I don't see what you're worrying about here, but I presume
> it's related to the naive "0/0" you get when you evaluate
> this expression at X= 0.
What I was driving at is as follows:
X/(1 - e^X) as a species is constructed by multiplying X with the composite
of Permuations (i.e., 1/(1 - X)) acting on Set (i.e., e^X). Following your
rules in Week 190, let's give it a 2-element set {a, b}. First split this
into a one-element set and its complement. For the complement, find an
ordered set of unordered sets whose union is that complement.
So, if the X picks up {a}, the 1/(1 - e^X) will start listing:
<{b}>, <phi, {b}>, <{b}, phi>, <phi, phi, {b}>, ...
[phi the empty set]
Clearly an infinite number of such things, but all is not lost.
Decategorified the number concerned gives us:
1 + 2 + 3 + 4 +..., or zeta(-1), which we know to be -1/12.
So the coefficient of (X^2/2!) in X/(1 - e^X) is twice this, i.e., -1/6,
minus the 2nd Bernouilli number, as one would hope.
Cartier does this kind of thing (without species) in his fascinating
Mathemagics (A Tribute to L. Euler and R. Feynman)
He presents this style as complementary to Hilbert-Bourbaki.
Will species bring them together?
> I've been meaning to think about this ever since I read Connes'
> comments on Bernoulli numbers in this book:
As for how this relates to Connes' comments, I'll leave that to others.
David Corfield
Date Subject Author
1/5/03 David Corfield
1/6/03 Re: Queries about Species John Baez
1/6/03 Re: Queries about Species David Corfield
1/8/03 Re: Queries about Species John Baez
1/13/03 Re: Queries about Species ayatollah potassium
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A persistent store for values of arbitrary types. Variant for the ST monad.
The Vault type in this module is strict in the keys but lazy in the values.
lookup :: Key s a -> Vault s -> Maybe aSource
insert :: Key s a -> a -> Vault s -> Vault sSource
Insert a value for a given key. Overwrites any previous value.
adjust :: (a -> a) -> Key s a -> Vault s -> Vault sSource
Adjust the value for a given key if it's present in the vault.
delete :: Key s a -> Vault s -> Vault sSource
union :: Vault s -> Vault s -> Vault sSource
lock :: Key s a -> a -> Locker sSource
unlock :: Key s a -> Locker s -> Maybe aSource
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# Find the kth largest or smallest number
Four kinds of method to "Select the kth largest or smallest element"
==> to make is O(n)
A simpler formulation of a worst-case O(n) algorithm, is as follows:
Specialized partial sorting algorithms based on mergesort and quicksort.
ONE: The simplest is the quicksort variation: there is no need to recursively sort partitions which only contain elements that would fall after the kth place in the end. Thus, if the pivot falls in
position k or later, we recur only on the left partition:
function quicksortFirstK(list, left, right, k)
if right > left
select pivotIndex between left and right
pivotNewIndex := partition(list, left, right, pivotIndex)
quicksortFirstK(list, left, pivotNewIndex-1, k)
if pivotNewIndex < k
quicksortFirstK(list, pivotNewIndex+1, right, k)
The resulting algorithm requires an expected time of only O(n + klogk), and is quite efficient in practice, especially if we substitute selection sort when k becomes small relative to n. However, the
worst-case time complexity is still very bad, in the case of a bad pivot selection. Pivot selection along the lines of the worst-case linear time selection algorithm could be used to get better
worst-case performance.
TWO: Even better is if we don't require those k items to be themselves sorted. Losing that requirement means we can ignore all partitions that fall entirely before or after the kth place. We recur
only into the partition that actually contains the kth element itself.
function quickfindFirstK(list, left, right, k)
if right > left
select pivotIndex between left and right
pivotNewIndex := partition(list, left, right, pivotIndex)
if pivotNewIndex > k // new condition
quickfindFirstK(list, left, pivotNewIndex-1, k)
if pivotNewIndex < k
quickfindFirstK(list, pivotNewIndex+1, right, k)
The resulting algorithm requires an expected time of only O(n), which is the best such an algorithm can hope for.
Others :
Another simple method is to add each element of the list into an ordered set data structure, such as a
self-balancing binary search tree
, with at most
elements. Whenever the data structure has more than
elements, we remove the largest element, which can be done in O(log
) time. Each insertion operation also takes O(log
) time, resulting in O(
) time overall.
It is possible to transform the list into a
in Θ(
) time, and then traverse the heap using a modified
Breadth-first search
algorithm that places the elements in a
Priority Queue
(instead of the ordinary queue that is normally used in a BFS), and terminate the scan after traversing exactly k elements. As the queue size remains O(
) throughout the traversal, it would require O(
) time to complete, leading to a time bound of O(
) on this algorithm.
We can achieve an O(log
) time solution using
skip lists
. Skip lists are sorted data structures that allow insertion, deletion and indexed retrieval in O(log
) time. Thus, for any given percentile, we can insert a new element into (and possibly delete an old element from) the list in O(log
), calculate the corresponding index(es) and finally access the percentile value in O(log
) time. See, for example, this Python-based implementation for calculating
running median
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CONTINGENCY THEORY - Professor Ahmed PPT
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Source : http://www.educ.uidaho.edu/sportpsych/leadership/Contingency%20Theories.pptx
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Source : https://facultystaff.richmond.edu/~choyt/supportingdocs/lecture9_contingency.ppt
Presentation Summary : Leadership Chapter 6 - Contingency Theory Northouse, 5th edition Overview Contingency Theory Approach Perspective Leadership Styles Situational Variables Research ...
Source : http://www.lssu.edu/faculty/lschmitigal/documents/06_PowerPoint.rev.ppt
Chapter 4: SYSTEMS THEORY - University of Kentucky PPT
Presentation Summary : Chapter 4: SYSTEMS THEORY Provides a general analytical framework (perspective) for viewing an organization. Systems Theory Synergy Interdependence Interconnections ...
Source : http://www.uky.edu/%7Edrlane/orgcomm/325ch04.ppt
Presentation Summary : Leadership Theories Andrea Reger Theories Trait Approach Skills Approach Style Approach Situational Approach Contingency Theory Path-Goal Theory Leader Member ...
Source : http://leadership.okstate.edu/sites/leadership.okstate.edu/files/files/docs/contingency.pathgoaltheories.ppt
Charismatic Leadership - SIUE PPT
Presentation Summary : Chapter 8 Contingency Theories of Effective Leadership LPC Contingency Model The Path-Goal Theory of Leadership Leadership Substitutes Theory Subordinate ...
Source : http://www.siue.edu/~marlove/MBA533/Powerpoint533/Charismatic_Leadership.ppt
Continguency Approach - FreeQuality PPT
Presentation Summary : Therefore contingency theory applies to promote a structures approach to quality An attentiveness to the fact that there is no one optimal management system that ...
Source : http://www.freequality.org/documents/Training/Classes%20Fall%202002/Contingency_approach.ppt
Presentation Summary : Chapter 3 Contingency Approaches Chapter Objectives Understand how leadership is often contingent on people and situations. Apply Fiedler’s contingency model to key ...
Source : http://www.business.unr.edu/faculty/simmonsb/mgt423/ch03.ppt
Presentation Summary : According to Fiedler’s Contingency Theory, a score greater than 76 indicates a relationship orientation, and a score of less than 62 indicates a task orientation.
Source : http://www.drluisortiz.com/PPT/CHAP03PP.PPT
Approaches to reviewing leadership literature - Homepages at WMU PPT
Presentation Summary : Contingency Theory of Leadership Contingency theory of leadership assumes that there is no one best way to lead. Effective leadership depends on the leader’s and ...
Source : http://homepages.wmich.edu/%7Eshen/teaching/edld609/10thcontingency.ppt
SOCW 5307: Class 3: Theory - Western Michigan University PPT
Presentation Summary : Title: SOCW 5307: Class 3: Theory Author: ssw Last modified by: Fritz Created Date: 5/28/1995 4:26:58 PM Document presentation format: On-screen Show
Source : http://homepages.wmich.edu/~macdonal/Addl.%20Files/Organizations%20and%20Systems/theory_systems_contingency%201.ppt
Chapter 14 - PowerPoint Presentation - Murray State ... PPT
Presentation Summary : Fiedler’s Contingency Theory of Leadership House’s Path-Goal Theory A contingency model of leadership proposing that effective leaders can motivate subordinates ...
Source : http://campus.murraystate.edu/academic/faculty/roger.schoenfeldt/FA07/PowerPoint/Chapter14.ppt
Presentation Summary : Leadership Chapter 6 - Contingency Theory * Contingency Theory Approach Description Contingency theory is a leader-match theory (Fiedler & Chemers, 1974) Tries to ...
Source : http://www.leadmore.org/NWCOR/Content/Lectures/MGT%20395%20Chap%206-%20Contingency%20Theory%20Web%20Based.PPT
Presentation Summary : Contingency Theory Approach Description. Contingency theory is a . leader-match. theory (Fiedler & Chemers, 1974) Tries to match leaders to appropriate situations
Source : http://people.uncw.edu/nottinghamj/documents/Ch6%20Contingency.pptx
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Title page for ETD etd-04282000-13520019
A fundamental task of algebraists is to classify algebraic structures. For example, the classification of finite groups has been widely studied and has benefited from the use of computational
tools. Advances in computer power have allowed researchers to attack problems never possible before.
In this dissertation, algorithms for noncommutative algebra, when ab is not necessarily equal to ba, are examined with practical implementations in mind. Different encodings of associative
algebras and modules are also considered. To effectively analyze these algorithms and encodings, the encoding neutral analysis framework is introduced. This framework builds on the ideas used in
the arithmetic complexity framework defined by Winograd. Results in this dissertation fall into three categories: analysis of algorithms, experimental results, and novel algorithms.
Known algorithms for calculating the Jacobson radical and Wedderburn decomposition of associative algebras are reviewed and analyzed. The algorithms are compared experimentally and a
recommendation for algorithms to use in computer algebra systems is given based on the results.
A new algorithm for constructing the Drinfel'd double of finite dimensional Hopf algebras is presented. The performance of the algorithm is analyzed and experiments are performed to demonstrate
its practicality. The performance of the algorithm is elaborated upon for the special case of group algebras and shown to be very efficient.
The MeatAxe algorithm for determining whether a module contains a proper submodule is reviewed. Implementation issues for the MeatAxe in an encoding neutral environment are discussed. A new
algorithm for constructing endomorphism rings of modules defined over path algebras is presented. This algorithm is shown to have better performance than previously used algorithms.
Finally, a linear time algorithm, to determine whether a quotient of a path algebra, with a known Gröbner basis, is finite or infinite dimensional is described. This algorithm is based on the
Aho-Corasick pattern matching automata. The resulting automata is used to efficiently determine the dimension of the algebra, enumerate a basis for the algebra, and reduce elements to normal
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Open and Closed sets
November 27th 2007, 07:34 AM #1
Oct 2007
Open and Closed sets
The set B is given by
B = { n/n+1 | n = a natural number } as a subset of the Real numbers. Is this set
i) Open
ii) Closed
Provide justification.
a) What does it mean to be an open set in the reals?
b) What does it mean to be an closed set in the reals?
The set is not open. Note, $1/2 \in B$ but if we choose $\epsilon$ small enough then $(1/2-\epsilon,1/2+\epsilon)$ does not lie in $B$, take for example $\epsilon = .1$.
The set is not closed. We need to show that its complement is not open. Let $B^*$ be its complement. Since $1ot \in B$ it means $1\in B^*$ but not interval $(1-\epsilon,1+\epsilon)$ lies
completely outside $B$ because $n/(n+1)$ converges to $1$.
Last edited by ThePerfectHacker; November 27th 2007 at 08:34 AM.
Hey, thanks for that. Could you help me out with interior points. The next part of the question asks to state (if any) which are the non-interior points.
So all points in B are non-interior points of B, I see now why B is not open.
If B* is the complement of B, does this mean that all points in B* are non-interior points of B*?
ah right, thanks.
November 27th 2007, 08:12 AM #2
November 27th 2007, 08:15 AM #3
Global Moderator
Nov 2005
New York City
November 27th 2007, 08:39 AM #4
Oct 2007
November 27th 2007, 08:47 AM #5
Global Moderator
Nov 2005
New York City
November 27th 2007, 08:51 AM #6
Oct 2007
November 27th 2007, 08:58 AM #7
Global Moderator
Nov 2005
New York City
November 27th 2007, 09:00 AM #8
Oct 2007
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Pre-Cal 40S
So Long
I'm so glad we've had this time together,
Just to have a laugh or learn some math,
Seems we've just got started and before you know it,
Comes the time we have to say, "So Long!"
So long everybody! Watch this space in the fall for pointers to new blogs for each of my classes. Some of the students from this class will be mentoring the students in the new classes. If anyone
else (former students or other mathophile readers of this blog) is interested in being a mentor
email me
and let me know.
Farewell, Auf Wiedersehen, Adieu, and all those good bye things. ;-)
The exam is over and we did a little survey in class. The results are below; 20 students participated. We also discussed the future use of blogs and wikis in this course. The consensus seems to be
that the next class should have their own blog but there should be a link back to this one for the new students to see what the old ones did. We also talked about having students from this class
volunteer to be mentors for students in next year's classes. I've got a few volunteers signed up already. If you'd like to mentor one or more of my classes (grade 10, 11 or 12) next year please email
me or leave a comment below this post.
Without any further ado, here are the results of our class's survey. Please share your thoughts by commenting (anonymously if you wish) below .....
How prepared were you to write this exam? (Average score out of 100)
How much effort did you put into preparing for this exam? (Average score out of 100)
How good a job did your teacher do preparing you for this exam? (Average score out of 100)
Did you have enough preparation using your calculator?
Yes 85% No 15%
Did you have enough preparation without using your calculator?
Yes 65% No 30% Middle 5%
Was I too hard or too easy on you??
Easy 5% Hard 0% Middle 95%
What was the best learning experience you had in this class?
┃group work (14) │review sites ┃
┃math dictionary (8) │quizzes ┃
┃online quizzes (3) │humour ┃
┃lectures (3) │clearly detailed lessons ┃
┃Go For Gold assignment │showing how math affects our daily lives and history ┃
┃pre-tests │using stories to help us understand ┃
┃this blog (6) │ ┃
What was the worst learning experience you had in this class?
┃quizzes │too long getting back tests ┃
┃online quiz (3) │none (6) ┃
┃class time too short │sun shining on the white board ┃
┃learning too fast │group work ┃
┃this blog (5) │web assignments ┃
┃not always going over homework │tests ┃
┃homework (2) │exercises that have answers but not solutions ┃
What suggestions can you share for next year?
┃no quiz for marks on blog │project work (3) │give make-up tests ┃
┃fewer jokes and stories │more online quizzes │make blogging optional ┃
┃more group work (4) │tell more jokes │give take home tests ┃
┃an extra class every 2^nd day │homework should count for marks │don't use the same exercise books ┃
┃keep blogging (7) │students should do more work at the board │more class participation ┃
┃more time on binomial theorem │give tests back ASAP │keep using the same teaching strategies ┃
┃go over selected homework exercises in class │have a mentor for this class │ ┃
It's interesting to compare the items that were considered both the worst and best learning experiences. Also, take a look at the list of worst learning experiences compared to suggestions for next
year. Help me do a better job next year by commenting on what you see here ....
Here is where you can login to the online quiz site to try your hand at the review quiz we made as a class.
There's lots more you can do to get ready for Tuesday!
Take this quiz on Probabilities of Compound Events.
Take this quiz on Conditional Probability.
Take this quiz on The Binomial Theorem and Probability.
Take this test on Combinatorics and Probability.
Take Harry Potter's quiz on Combinatorics. Try not to run into any walls. ;-)
Try this Probability quiz.
take your own customized quiz on a variety of topics we've studied. You can create customized review quizzes by choosing up to 15 questions from: Algebra, Trigonometry, Exponents and Logarithms,
Geometry, Problem Solving and Graphs (Transformations). Create as many quizzes as you like and practice, Practice, PRACTICE!!! When you start each quiz a timer counts down from 50 minutes -- that's
how you prepare for a timed test -- you take LOTS of timed tests! You can probably skip the Geometry questions and not all of the Problem Solving and Algebra applies to our course, but the kind of
thinking you need to do for these questions WILL DEFINITELY help you gear up for the exam!
Remember, luck has nothing to do with it! It's all about doing your best! How well you do is a direct function of the effort you put into preparing!!!
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Java program error
Join Date
May 2011
Rep Power
i want to use at least 2 methods
i want to write a program that inputs 3 integers, which are sides of a triangle
and determines the type of triangle it is:
not valid sides of a triangle
If it is a valid triangle print the type of triangle it is, its area and perimeter.
the program will then repeat for another set of sides, until a 0 is entered for the 1st side of the triangle (the other 2 sides should not be read in for this case)
This is what i got so far:
Java Code:
import java.util.*;
public class BookExample{
public static void main(String[] args)
{int a, b, c, p;
System.out.println("Enter the length of side \"a\" of the triangle to begin:");
Scanner in=new Scanner(System.in);
p = a+b+c;
public static void getTriType()
{System.out.println("According to the given sides these are invalid sides of a Triangle");
else if(a==b&&a==c&&b==c)
{System.out.print("According to the given sides, since a,b and c are equal ");
System.out.print("the triangle is an Equilateral triangle.");
else if(a==b||a==c||b==a||b==c||c==a||c==b)
{System.out.print("According to the given sides, since two sides are equal ");
System.out.print("the triangle is an Isosceles triangle.");
{System.out.print("According to the given sides, since a,b and c are not equal ");
System.out.print("the triangle is a Scalene triangle.");
public static double getArea(int a,int b,int c)
{double s = 0.5 * (a + b + c);
double area = Math.sqrt(s*(s-a)*(s-b)*(s-c));
I know i'm still missing to output the area and perimeter but i'm trying to get rid of my error(s) first. My first error is "BookExample.java:16: getArea(int,int,int) in BookExample cannot be
applied to ()" i know what it has to do with i just can't figure out how to fix the problem. Thanks for the help.
Join Date
Jan 2011
Rep Power
You need to make the parameters of the function definition the same as the parameters in the function call within your main.
getArea(); is not the same as
public static double getArea(int a, int b, int c)
when you call getArea from main, make sure you include the values that you inputted from the console.
Join Date
May 2011
Rep Power
You need to make the parameters of the function definition the same as the parameters in the function call within your main.
getArea(); is not the same as
public static double getArea(int a, int b, int c)
when you call getArea from main, make sure you include the values that you inputted from the console.
so what your saying is when i call getArea from main i would need to call it as is getArea(int a,int b,int c); ?
Join Date
Jan 2011
Rep Power
Yep. Your function call needs to match some function definition, somewhere.
In this problem, you want to pass the values that you received from the console (Your user) to the getArea() function so you can calculate the area of the user's triangle.
Join Date
May 2011
Rep Power
Thanks. I knew more errors were going to follow up right behind that one. Now i'm getting something like "BookExample.java:16: '.class' expected
getArea(int a,int b,int c);"
Join Date
May 2011
Rep Power
I see what your talking about i changed it and just called the variables instead and did the same to my other function which began getting an error that said something like 'can't find a'
etc...(a,b,c); now my error is completely different.
Java Code:
import java.util.*;
public class BookExample{
public static void main(String[] args)
{int a, b, c, p;
System.out.println("Enter the length of side \"a\" of the triangle to begin:");
Scanner in=new Scanner(System.in);
p = a+b+c;
public static void getTriType(int a, int b, int c)
{System.out.println("According to the given sides these are invalid sides of a Triangle");
else if(a==b&&a==c&&b==c)
{System.out.print("According to the given sides, since a,b and c are equal ");
System.out.print("the triangle is an Equilateral triangle.");
else if(a==b||a==c||b==a||b==c||c==a||c==b)
{System.out.print("According to the given sides, since two sides are equal ");
System.out.print("the triangle is an Isosceles triangle.");
{System.out.print("According to the given sides, since a,b and c are not equal ");
System.out.print("the triangle is a Scalene triangle.");
public static double getArea(int a,int b,int c)
{double s = 0.5 * (a + b + c);
double area = Math.sqrt(s*(s-a)*(s-b)*(s-c));
my new error is "BookExample.java:41: missing return statement
Join Date
May 2011
Rep Power
Join Date
Jan 2011
Rep Power
What does that error message suggest to you? I think it is telling you that there is supposed to be a return statement but it is missing. Do you have any methods that SHOULD have a return
statement but DON'T?
Join Date
May 2011
Rep Power
Being that i'm new to java i'm actually lost right now.
Java Code:
public static double getArea(int a,int b,int c)
This is your getArea method. It's return type is double. This means you are promising that the method will return a double. Are you?
Join Date
May 2011
Rep Power
no. I have a question being that in that method i have s and area as a type double does it matter if i change the type double to void?
Java Code:
public static void getArea(int a,int b,int c)
Last edited by Taszk; 06-06-2011 at 05:43 AM.
Join Date
Jan 2011
Rep Power
The method is named "getArea". You don't want to change it to void or it won't "get" anything.
You want that method to return a value of type double which is the area of the triangle.
Returning a Value from a Method (The Java™ Tutorials > Learning the Java Language > Classes and Objects)
Join Date
May 2011
Rep Power
Thanks a lot to the both of you i got the errors out the way now to try and finish it up. Hopefully i won't run into anymore mistakes but if i do i hope i could comeback.
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How Misinformed are Tea Party Protesters About Tax Policy?
March 25, 2010
By Drew Conway
For those of you used to reading about international relations, I apologize for the following brief foray into American politics. It appears that the American Enterprise Institute and David Frum have
decided to (abruptly) part ways. Before David left, however, he and his team of interns provided some interesting statistical insight into the Tea Party movement, as he writes:
Over the course of our survey, FrumForum interviewed approximately 60 people of the estimated 300-500 protesters assembled on Capitol Hill to protest the healthcare bill currently before the
House*. We asked them questions about their perception of current taxation rates and the economy.
*To be sure, this survey lacked a control group and a statistically significant sample, but based on our estimates, we surveyed between 11% and 19% of the protesters on Capitol Hill. It is
perhaps more valid to treat the results as that of a focus group, and a general contribution to the understanding of the Tea Party movement.
Frum asserts that the survey, “shows that Tea Partiers tend to be more financially pessimistic than average Americans, and perceive the United States’ tax burden to be significantly higher than it
actually is.” Given the small number of respondents (as noted in the highlighted footnote above), it is difficult to get a sense of the actual distribution of beliefs within the population of Tea
Partiers. We, can, however bootstrap (simulate) these distribution by making parametric assumption, and then more accurately test the assertion of Frum.
A brief caveat: as Frum has pointed out, these data were by no means scientifically collected, and therefore any results generated will be biased in whatever direction the collection pointed them. In
the follow experiments, I will be treating these data as though they were legitimate, despite the dangers of doing so. That said, we will model the results of these questions using the Gamma
distribution and approximate the shape (k) and scale ($theta$) parameters using descriptive statistice from the data gathered in the survey.
To generate the approximate values for k and $theta$ I will use the mean and standard deviation values provided for each of the survey questions. From the definition of the Gamma distribution, the
mean is equal to $ktheta$ and the variance is equal to $ktheta^2$. By substituting the standard deviation of the survey results for the variance we can easily approximate values of these parameters.
To do so, I use sympy, an open source symbolic mathematics Python package, which among other things simply makes my life easier.
Once these values have been calculated it is very easy to generate simulated distributions. I use the following R code to do so:
Next, we will visualize these distributions with ggplot2, noting where the actual values for these questions fall on the distributions using a vertical red line.
With the simulated distributions it is a bit easier draw conclusions as to the level of “misinformation” within the population of the Tea Party movement. My initial reaction is that while the actual
values for both questions are clearly in the tails of these distributions, they are not so far in these tails as to make them extreme. In fact, for the federal income tax question the actual value is
well within one standard deviation of the mean. Perhaps the Tea Partiers are not as misinformed as many would be presumed from their rhetoric. At the same time, however, these are not necessarily
difficult questions, so any deviation from the actual value could be viewed negatively.
What are your thoughts?
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2. The Barber's Paradox
"One of themselves, even a prophet of their own, said, the Cretians are alway liars, evil beasts, slow bellies. This testimony is true."
Titus 1:12-14 (King James Version)
A paradox is a statement or group of statements that lead to a logical self-contradiction. For example,
• The next bulleted statement is a lie.
• The previous bulleted statement is true.
There is a barber who lives on an island. The barber shaves all those men who live on the island who do not shave themselves, and only those men.
Does the barber shave himself?
If the barber shaves himself then he is a man on the island who shaves himself hence he, the barber, does not shave himself. If the barber does not shave himself then he is a man on the island who
does not shave himself hence he, the barber, shaves him(self).
This is not actually a paradox.
Consider the proposition Shave(x,y) which is true if x shaves y and false if x does not shave y. We can restate the proposition as:
x y (Shave(x,y) Shave(y,y))
There exists an x such that for every y, x shaves y iff y does not shave y.
Suppose this proposition were true. Let b be the x whose existance is hypothesized. thus
y (Shave(b,y) Shave(y,y))
Since this holds for all y it holds for yb. So
Shave(b,b) Shave(b,b)
Hence one may hypothesize that the proposition is false. It is worth checking that this hypothesis does not also lead to a contradiction.
(x y (Shave(x,y) Shave(y,y)))
x(( y (Shave(x,y) Shave(y,y)))
x y(Shave(x,y) Shave(y,y))
Which is no problem since if for any x if we chose yx it is certainly the case that
(Shave(x,x) Shave(x,x))
For any meaning of Shave(x,x)!!!!!!
Of course there is no problem if the barber ferries in from the mainland. In particular, he is not a member of the "set" of people on the island referred to in the second sentence of the proposition.
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Exponential Model
Exponential model is associated with the name of Thomas Robert Malthus (1766-1834) who first realized that any species can potentially increase in numbers according to a geometric series. For
example, if a species has non-overlapping populations (e.g., annual plants), and each organism produces R offspring, then, population numbers N in generations t=0,1,2,... is equal to:
When t is large, then this equation can be approximated by an exponential function:
There are 3 possible model outcomes:
1. Population exponentially declines (r < 0)
2. Population exponentially increases (r > 0)
3. Population does not change (r = 0)
Parameter r is called:
• Malthusian parameter
• Intrinsic rate of increase
• Instantaneous rate of natural increase
• Population growth rate
"Instantaneous rate of natural increase" and "Population growth rate" are generic terms because they do not imply any relationship to population density. It is better to use the term "Intrinsic rate
of increase" for parameter r in the logistic model rather than in the exponential model because in the logistic model, r equals to the population growth rate at very low density (no environmental
Assumptions of Exponential Model:
1. Continuous reproduction (e.g., no seasonality)
2. All organisms are identical (e.g., no age structure)
3. Environment is constant in space and time (e.g., resources are unlimited)
However, exponential model is robust; it gives reasonable precision even if these conditions do not met. Organisms may differ in their age, survival, and mortality. But the population consists of a
large number of organisms, and thus their birth and death rates are averaged.
Parameter r in the exponential model can be interpreted as a difference between the birth (reproduction) rate and the death rate:
where b is the birth rate and m is the death rate. Birth rate is the number of offspring organisms produced per one existing organism in the population per unit time. Death rate is the probability of
dying per one organism. The rate of population growth (r) is equal to birth rate (b) minus death rate (m).
Applications of the exponential model
• microbiology (growth of bacteria),
• conservation biology (restoration of disturbed populations),
• insect rearing (prediction of yield),
• plant or insect quarantine (population growth of introduced species),
• fishery (prediction of fish dynamics).
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The smallest composite number, the second smallest square number, the first non-Fibonacci number, the smallest Smith number, and the smallest number that can be written as the sum of two prime
numbers. Four is the number of dimensions that make up spacetime (three of space and one of time). It is the most number of colors needed to color any map so that no two neighboring areas are the
same color (see four-color problem). There are four cardinal points on the compass, four Riders of the Apocalypse, and four Gospels.
Related category
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Is the following invariant of rooted trees a complete invariant?
up vote 11 down vote favorite
Recall that rooted trees may be generated by starting with a trivial rooted tree (just a vertex), along with the operations of grafting a number of trees (identify their roots) and adding a new
vertex to the tree to be a new minimum element. We will call this second operation "leafing".
Now let us define an invariant of rooted trees. If $T$ is a rooted tree, we will denote $P_T(z)$ to be the associated polynomial.
If the number of edges of $T$ is zero, then $P_T(z)=1$.
If $T'$ is the leafing of $T$, then $P_{T'}(z)=(z+1)P(z)+1$.
If $T$ is the grafting of $T_i, i=1\ldots n$, then $P_T(z)=P_{T_1}(z)P_{T_2}(z)\ldots P_{T_n}(z)$.
This polynomial is an isomorphism invariant of rooted trees. My question is
If $P_T=P_{T'}$, are the rooted trees, $T,T"$ isomorphic? If these trees are not isomorphic, what is the smallest counterexample? Any references to this invariant would be appreciated.
graph-theory enumerative-combinatorics trees
Interesting question. You might compare what you are doing with the paper "On distinguishing trees by their chromatic symmetric functions" by Jeremy Martin, Matthew Morin and Jennifer Wagner,
1 though they are dealing with unrooted trees. I wonder if your polynomial is somehow a specialization of the subtree polynomial $S_T$, which they prove is not as strong (for purpose of
distinguishing trees) as the chromatic symmetric function. It looks like it is probably an open question whether the chromatic symmetric function can distinguish trees -- or at least it was when
this paper was written. – Patricia Hersh Sep 23 '12 at 15:23
1 Hope you don't mind the new tag I just added. People in enumerative combinatorics definitely work on questions of this flavor. – Patricia Hersh Sep 23 '12 at 16:07
Thank you for the addition of the tag. It could only help the question. – Spice the Bird Sep 23 '12 at 19:58
I went back and calculated your above polynomial $P_T$ for the two trees given in Figure 2 of the paper I mention above which the subtree polynomial $S_T$ also mentioned above couldn't
distinguish. Your polynomial did distinguish these. To save time, I actually just compared a few evaluations of the two polynomials, which was enough to see this. I didn't look at how your
polynomial relates to the chromatic symmetric functions though. Any chance that when your root has a single child, your polynomial is irreducible over ${\bf Z}[z]$? Or is that too much to hope
for? – Patricia Hersh Sep 24 '12 at 13:19
1 The linear tree with n nodes (n+1 vertices, one of them the root) has polynomial $((z+1)^{(n+1)}−1)/(z)$. So no – Spice the Bird Sep 24 '12 at 16:52
show 14 more comments
5 Answers
active oldest votes
Chaudhary and Gordon ("Tutte polynomials for trees," J. Graph Theory 15, no. 3 (1991), 317-331) construct a couple of invariants that look very similar to yours. They prove that
these invariants do in fact determine a rooted tree up to isomorphism.
Update: I think the answer to your original question is no.
The relevant invariant from the Chaudhary-Gordon paper is what they call $f_p(T;t,z)$. This is a polynomial in two variables $t,z$ that satisfies the recurrence $$ f_p(L(T);t,z) =
up vote 14 down t(z+1)f(T) + 1 - tz,$$ $$ f_p(T_1*\cdots*T_r;t,z) = f(T_1)\cdots f(T_r)$$
vote accepted
where $L$ means leafing and $*$ means grafting. (These are Prop 4(b) and and Prop 5 in Chaudhary-Gordon.) If I'm doing the algebra right, your invariant is $P_T(z) = f_p(T;z+1,0).$
Chaudhary and Gordon give an example of two rooted trees on 8 vertices with the same values of $f_p(T;t,z)$. The edge sets could be labeled as 01,12,24,13,35,56,57 and
01,12,13,34,35,56,67, with 0 the root vertex in both cases. (Probably a good idea to confirm this if you have code to compute your invariant quickly.)
Did somebody vote this down? My calculations indicate the Jeremy is absolutely right. The polynomials as modified by Owen are equal and are $z^7 + 3z^6 + 4z^5 + 4z^4 + 3z^3 + 2z^
2 +z + 1$. I am very glad to have this answer, and I believe it absolutely deserves to be upvoted. – Todd Trimble♦ Sep 26 '12 at 20:51
"the Jeremy" --> that Jeremy – Todd Trimble♦ Sep 26 '12 at 20:53
Great! This excellent answer absolutely settles the question, and provides a truly complete invariant. – Owen Biesel Sep 26 '12 at 21:40
Very happy to help! – Jeremy Martin Sep 26 '12 at 21:44
Their is a smaller set of trees that also works. Just chop off the first root to get 12,24,13,35,56,57 and 12,13,34,35,56,67. – Spice the Bird Sep 27 '12 at 19:58
add comment
This is not a complete answer, but there is a nice description of the information in $P_T$ which may prove useful to someone else.
First of all, I will define a slightly different polynomial $\tilde P_Z(T)$: Grafting works the same way, but if $T'$ is the leafing of $T$, then I define $\tilde P_{T'}(z)= z\tilde P_T(z)
+1$. It's an easy proof by recursion that $\tilde P_T(z) = P_T(z-1)$, so this new polynomial determines $T$ just as well or poorly as yours.
By "node" of $T$, I mean a vertex of $T$ other than its root, and by "subtree" $T'$ of $T$, I mean a subgraph of $T$, such that for every node of $T$ included in $T'$, the node's parent and
the edge to it are also included in $T'$. [Edit: These are non-standard uses of those words.] Then the coefficient of $z^n$ in $\tilde P_T(z)$ is the number of $n$-node subtrees of $T$. This
is because, for $n>0$, choosing an $n$-node subtree of the leafing of $T$ is the same as choosing an $(n-1)$-node subtree of $T$, and for any $n$, choosing an $n$-node subtree of the grafting
of $T$ and $T'$ is the same is choosing a $k$-node subtree of $T$ and an $(n-k)$-node subtree of $T'$ for some $k$ between $0$ and $n$.
Some consequences include:
• If $T$ has $n$ nodes (vertices other than the root), then the highest-order term of $\tilde P_T(z)$ is $z^n$.
• The coefficient of $z$ in $\tilde P_T(z)$ is the degree of the root of $T$.
• If $T$ has $a$ nodes at distance $1$ from the root, and $b$ nodes at distance $2$, then the coefficient of $z^2$ in $\tilde P_T(z)$ is ${a \choose 2}+ b$.
• If $T$ has a total of $n$ nodes, then the coefficient of $z^{n-1}$ is the number of leaves of $T$ (nodes with degree 1).
Hence if $\tilde P_T(z)=\tilde P_{T'}(z)$, then $T$ and $T'$ have the same numbers of vertices and leaves, their roots have the same degrees, and they have the same total number of vertices
at distance $2$ from the root. It seems that more should be true, but I haven't proven any more.
Edit: I've now proved the following result: if $T$ and $T'$ are graphs whose nodes are distance at most $2$ from the root, and such that $\tilde P_T(z) = \tilde P_{T'}(z)$, then $T\cong T'$.
Proof: A rooted tree of depth at most $2$ corresponds to a sequence of natural numbers $b_1, b_2, \ldots, b_a$, where $a$ is the number of children of the root, and $b_i$ is the number of
up vote children of the $i$th child of the root. Then $$ \tilde P_{T}(z) = \prod_{i=1}^a (z(z+1)^{b_i} + 1)$$ $$ = \prod_{i=1}^a \left(1+{b_i\choose 0}z + {b_i\choose 1}z^2 + \ldots + {b_i\choose k}z
10 down ^{k+1} + \ldots + {b_i\choose b_i-1}z^{b_i} + {b_i\choose b_i}z^{b_i+1}\right) $$ I show that $\tilde P_T(z)$ determines the $b_i$ up to reordering, and hence $T$ up to isomorphism.
First, note that knowing the $b_i$ up to reordering is the same as knowing the elementary symmetric polynomials in the $b_i$, because they are the solutions of $\prod_{i=1}^a(x-b_i)=0$. Or
equivalently, by Newton's identities, that information is contained in the sums $\sum_{i=1}^a b_i^k$ for all $k\geq 0$. In turn, knowing the $\sum_{i=1}^a b_i^k$ for $k$ up to $n$ is the same
as knowing the $\sum_{i=1}{b_i\choose k}$ for $k$ up to $n$, through simple linear identities relating the two sets of data.
Now I show that $\tilde P_T(z)$ does determine each $\sum_{i=1}^a {b_i\choose k}$ for $k \geq 0$, by induction on $k$. Suppose we know that $\tilde P_T(z)$ determines $\sum_{i=1}^a {b_i\
choose k}$ for $0\leq k < n$, and now consider the coefficient of $z^{n+1}$. There is a contribution from each partition $n+1 = \lambda_1+\lambda_2+\ldots+\lambda_m$ of $n+1$, given by $$\
sum_{i_1,\ldots,i_m\text{ distinct}}\left(\prod_{j=1}^m {b_i\choose \lambda_i-1}z^{\lambda_i}\right)=\left(\sum_{i_1,\ldots,i_m\text{ distinct}}\prod_{j=1}^m {b_i\choose \lambda_i-1}\right)z^
{n+1}.$$ Considering the term in parentheses on the right-hand side as a polynomial in the $b_i$, note that it is symmetric in the $b_i$ and has degree $\sum_{j=1}^m(\lambda_j-1) = (n+1)-m<
n$ if $m>1$. Hence such contributions are expressible in terms of the $\sum_{i=1}^a {b_i\choose k}$ for $0\leq k < n$, and so can be deduced from $\tilde P_T(z)$ by the induction hypothesis,
unless the partition is simply $n+1=(n+1)$, in which case the resulting term is $\sum_{i=1}^a{b_i\choose n}z^{n+1}$. Therefore the coefficient of $z^{n+1}$ in $\tilde P_T(z)$ differs
predictably from $\sum_{i=1}^a {b_i\choose n}$, so the latter is deducible from $\tilde P_T(z)$ as well.
Knowing the $\sum_{i=1}^a {b_i\choose k}$ for all $k$, we can work backwards: first we inductively deduce the $\sum_{i=1}^a b_i^k$, from which Newton's identities tell us the values of the
elementary symmetric polynomials evaluated at the $b_i$. Then we recover the simplified form of $\prod_{i=1}^a (x-b_i)$, and the $b_i$ are its roots.
For example: If $\tilde P_T(z) = z^4 + 3z^3 + 3z^2 + 2z + 1$ and we know $T$ has no nodes of distance more than $2$ from the root, then we can recover $T$ as follows. The coefficient of $z$
is $a=2$, so we are trying to find $b_1$ and $b_2$ such that $$P_T(z) = (z(z+1)^{b_1} + 1)(z(z+1)^{b_2} + 1).$$ The coefficient of $z^2$ is ${a\choose 2} + (b_1+b_2) = 1 + (b_1+b_2) = 3$, so
$b_1+b_2 = 2$. And the coefficient of $z^3$ is ${a\choose 3} + \sum_{i\neq j} b_i + \sum_i {b_i\choose 2} = (0) + (b_1+b_2) + \left({b_1\choose 2} + {b_2\choose 2}\right) = 2 + {b_1\choose 2}
+ {b_2\choose 2} = 3$, so ${b_1\choose 2} + {b_2\choose 2} = 1$. Hence $\frac{b_1^2-b_1}{2} + \frac{b_2^2-b_2}{2} = \frac{(b_1^2 + b_2^2) - (b_1 + b_2)}{2} = \frac{(b_1^2 + b_2^2) - 2}{2} =
1$, so $b_1^2 + b_2^2 = 4$. Therefore $b_1b_2 = \frac{(b_1 + b_2)^2 - (b_1^2 + b_2^2)}{2} = \frac{2^2 - 4}{2} = 0$, so $b_1$ and $b_2$ are the roots of $$x^2 - (b_1+b_2)x + (b_1b_2) = x^2 -
2x.$$ Therefore $b_1$ and $b_2$ are $0$ and $2$, up to reordering, so $T$ is the tree whose root has $a=2$ children, one of which has $0$ children, and the other of which has $2$.
Thank you for your answer. Is their a reference for your invariant? I am wondering about complete invariants. The absence of a discussion of that seems to indicate that this might be an
open question? Also, are you sure that your polynomial is mine shifted by one? Mine shifted by one is $zP(z-1)+1$. maybe I am missing something silly? – Spice the Bird Sep 22 '12 at 22:50
2 I just made mine $\tilde P(z) = P(z-1)$, so the recurrence relation for mine is $\tilde P'(z) = P'(z-1) = z P(z-1) + 1 = z\tilde P(z) +1$, which is simpler. I don't know if it's in the
literature, but it's slightly easier to read of some of the graph's information from my polynomial than yours. I've tried to clarify in the edit. – Owen Biesel Sep 23 '12 at 4:53
Well, I realized that I was being silly. Again thank you very much. – Spice the Bird Sep 23 '12 at 6:14
add comment
Since there seems to be some confusion in the comments below Richard Stanley's answer, and maybe also some discrepancy in terminology between Owen's answer and Richard's, I will record what
I think is going on.
Vertices of rooted trees can be ordered by $x \lt y$ if $x$ is a descendant of $y$. The notion of subtree used by Owen looks as though he means upward closed subsets, since his subtrees
include the root of the original tree (I apologize to Owen if that's not what he meant, although I think it is because that seems to be consistent with his remark on coefficients).
But that's not the usual notion of subtree, which according to Wikipedia is a (principal) downward closed subset of the original tree (i.e., if $y$ belongs to the subtree and $x$ is a
descendant of $y$, then $x$ belongs to the subtree). Under that notion, Richard's answer made a lot more sense. Let me describe what I think the isomorphism types of his examples are using
ZF sets. Let $a, b, c, d$ be ur-elements, and order sets by the transitive closure of the membership relation (so that $x \in y$ implies $x \lt y$). Then I can guess one of his trees looks
$$\{ \{a, b, c\}, \{ \{ d \} \} \}$$
up vote 6
down vote which has four one-node subtrees $a, b, c, d$, one two-node subtree $\{d\}$, one three-node subtree $\{\{ d \} \}$, one four-node subtree $\{a, b, c\}$, and one eight-node subtree which is
the original tree (N.B. here, $n$-node subtree means there are $n$ vertices, including its root.) The other of his trees looks like
$$\{ \{a, \{ b \} \}, \{c, d \} \}$$
which has four one-node subtrees $a, b, c, d$, one two-node subtree $\{ b \}$, one three-node subtree $\{c, d \}$, one four-node subtree $\{a, \{ b \} \}$, and one eight-node subtree which
is the original tree. (If those were not the isomorphism types he had in mind, then again I apologize.)
I don't believe however that these two trees have the same polynomial. If I did my arithmetic correctly, I believe they have different constant coefficients (one is 35 and the other is 36),
using the original definition of the polynomial, not Owen's modification.
Thank you Todd. I suspected that we were all talking about different things. – Spice the Bird Sep 24 '12 at 15:38
Yes, thank you Todd. I didn't realize that (a) "subtree" is already standard terminology and (b) it doesn't mean what I thought it should mean. I'll clarify. – Owen Biesel Sep 24 '12 at
add comment
The number of different values taken by the polynomial is given by
1, 1, 2, 4, 9, 20, 47, 112, 274, 679, 1717, ...
Comparing with the sequence A000081 given by
up vote 3 down vote
1, 1, 2, 4, 9, 20, 48, 115, 286, 719, 1842, ...
one can easily see that this is not at all a complete invariant.
add comment
My feeling is (or was) that it should not be an isomorphism invariant. But my attempt to find a counterexample suggests that perhaps it is (or that my quick and dirty programming had an
error). Consider rooted trees where no vertex has out-degree greater than $2$. It is pretty quick work to generate them and find their polynomials (for a while). The number for depth
up vote (vertices in longest direct path) $2,3,4,5$ receptively are $2, 7, 56, 2212. $ Up to that far, the polynomials are unique. This seems to be (equal to) the number of rooted 3 trees suggesting
2 down that the next number is $2595782.$ That was too big for my quick Maple program.
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Not the answer you're looking for? Browse other questions tagged graph-theory enumerative-combinatorics trees or ask your own question.
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Mathematica, how to construct a tridiagonal matrix?
August 20th 2010, 09:14 PM #1
Junior Member
Oct 2009
Mathematica, how to construct a tridiagonal matrix?
Hello I want to view the entries of a block tridiagonal matrix. The matrix is defined by
$u_{i,j-1}+u_{i-1,j}+4 u_{i,j}+u_{i+1,j}+u_{i,j+1}=h^2f_{i,j}$
$1\leq i \leq 3$ and $1\leq i \leq 4$
I want to examine the structure of this matrix. Obviously writing this out by hand is a laborious task so I was wondering if mathematica can help me.
I tried
Table[Subscript[u, i, j - 1] + Subscript[u, i - 1, j] +
4 Subscript[u, i, j] + Subscript[u, i + 1, j] + Subscript[u, i,
j + 1] - h^2 Subscript[f, i, j], {i, 3}, {j, 4}]
But this doesn't quite output what I want. Can somebody tell me how I can construct this matrix more explicitly in Mathematica?
That looks like a discretization of Poisson's equation. Am I right? If so, shouldn't the $4u_{i,j}$ be negative?
I can't say I know how to visualize the matrix in Mathematica, but here's a visualization of it.
Thanks, thats correct, it is the discretisation of the Poisson PDE, I was really hoping mathematica could help me in trying to visualise these systems of equations and matrices. Doing it by hand
is such a laborious task.
Try this:
Table[Subscript[u, i, j - 1] + Subscript[u, i - 1, j] -
4 Subscript[u, i, j] + Subscript[u, i + 1, j] + Subscript[u, i,
j + 1] - h^2 Subscript[f, i, j], {i, 3}, {j, 4}]//MatrixForm
Last edited by Ackbeet; August 23rd 2010 at 02:38 AM. Reason: Minus Sign.
August 21st 2010, 04:51 AM #2
August 21st 2010, 10:05 PM #3
Junior Member
Oct 2009
August 23rd 2010, 02:38 AM #4
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Extracting particles from a determinantal point process
up vote 8 down vote favorite
Consider $N$ real random particles $x_1,\cdots, x_N$ distributed according to a density $\rho(x_1,\ldots,x_N)$ with respect to the Lebesgue measure on $\mathbb R^N$, which is assumed to be invariant
under permutations : $$ \rho(x_{\sigma(1)},\ldots,x_{\sigma(N)})=\rho(x_1,\ldots,x_N),\qquad \sigma\in\mathfrak S_N. $$ We moreover assume the particles to interact as a determinantal point process
with a kernel $K:\mathbb R \times \mathbb{R}\rightarrow\mathbb R$ which is, seen as a kernel operator on $L^2(dx)$, a projection operator.
This means roughly that the density distribution is given by
$$ \frac{1}{N!}\det\Big[ K(x_i,x_j) \Big] \prod_{i=1}^N dx_i. $$
Questions :
If we order the particles $x_1<\cdots< x_N$ and then extract a new system of particles $\{x_i\}_{i\in I}$, where $I\subset \{1,\ldots,N\}$, do we keep a determinantal structure (once forgetting the
ordering on the $x_i$'s, $i\in I$ )?
And if Yes, what would be the new kernel ?
This question is motivated from the following particular case : Once the particles ordered, namely $x_1<\ldots< x_N$, it is well known that one has the Fredholm determinant representation : $$ \
mathbb P(x_N\leq s)=\det(I-K)_{L^2(s,+\infty)}, \qquad x_1,\ldots,x_N\in\mathbb R. $$ I'm looking for a similar formula involving the operator $K$ for $ \mathbb P(x_k\leq s)$ when $1\leq k < N.$
1 For example, we take an $N \times N$ random Hermitian matrix, the eigenvalues $\lambda_1, \dots, \lambda_N$ form a determinantal process. We can put these real numbers in order $-2 \sqrt{N} \
approx \lambda'_1 < \dots < \lambda'_N \approx 2 \sqrt{N}$. Do the $\lambda_{2k}$ form a determinantal process? I don't know. – john mangual Jul 27 '12 at 1:51
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The sum of the lengths of any two sides of a triangle must - JustAnswer
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The sum of the lengths of any two sides of a triangle must be greater than the length of the third side. For example, the numbers 3, 4, and 5 can form a triangle because 3+4 > 5, 4+5 > 3, and 5+3 >
4. In contrast, the numbers 1, 2, and 5 cannot form a triangle because 1+2 < 5. Thus, if you are given any three integers, you can determine whether they could possibly form a triangle or not by
applying this general principle
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Programming Assignment 2 i for got the second part and yes i will pay more
The sum of the lengths of any two sides of a triangle must be greater than the length of the third side. For example, the numbers 3, 4, and 5 can form a triangle because 3+4 > 5, 4+5 > 3, and 5+3 >
4. In contrast, the numbers 1, 2, and 5 cannot form a triangle because 1+2 < 5. Thus, if you are given any three integers, you can determine whether they could possibly form a triangle or not by
applying this general principle.
Write a JavaScript program that allows a user to input three integers using text boxes in a form. (Hint: You need to use the built-in parseInt function to convert the input strings to integers.) Test
the three integers to determine if they can be formed into a triangle using the rule given above. Also test if the resulting triangle would be a right triangle using the Pythagorean theorem, namely
that the square of the hypotenuse (the longest side) equals the sum of squares of the other two sides. Display an alert box to inform the user whether their integers can form a triangle or a right
triangle (tell them which), or if the integers cannot form a triangle. Continue accepting sets of three integers and testing them until the user decides to quit.
The firt part of the progarm was good i just need it to out put a triangle whe the answer
Programming Assignment 4Create an html document that includes a JavaScript program that creates a new constructor function named Automobile in the document head. Include at least five properties in
the object definition, such as make, model, color, engine, seats, and so forth. Then assign the values of your car to each of the Automobile properties. Print the properties to the screen. i will pay
$20.00 for help on this one
if you make one just like the last one thats fine html
I have 0 Balance in my account i will load it tommorrow
and you were right the program can be html but have some javascript in the program
you dont have to sumit the program until i pay tomorrow
Write a JavaScript program that allows a user to input three integers using text boxes in a form. (Hint: You need to use the built-in parseInt function to convert the input strings to integers.) Test
the three integers to determine if they can be formed into a triangle using the rule given above. Also test if the resulting triangle would be a right triangle using the Pythagorean theorem, namely
that the square of the hypotenuse (the longest side) equals the sum of squares of the other two sides. Display an alert box to inform the user whether their integers can form a triangle or a right
triangle (tell them which), or if the integers cannot form a triangle. Continue accepting sets of three integers and testing them until the user decides to quit.
MS in IT.Several years of programming experience in Java C++ C C# Python VB Javascript HTML
Expert in C, C++, Java, DOT NET, Python, HTML, Javascript, Design.
Good knowledge of OOP principles. 3+ years of programming experience with Java and C++. Sun Certified Java Programmer 5.0.
Several years of intensive programming and application development experience in various platforms.
Bachelor of computer science, 5+ years experience in software development, software company owner
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Re: TeXForm, OutputForm questions ?
[Date Index] [Thread Index] [Author Index]
Re: TeXForm, OutputForm questions ?
Your query has been forwarded to me by the
comp.soft-sys.math.mathematica moderator.
> To: comp-soft-sys-math-mathematica@moderators
> Message-ID: <34E97FB6.460C374C@ico.unil.ch>
> Date: Tue, 17 Feb 1998 13:16:54 +0100
> From: Alexej Jerschow <Alexej.Jerschow@ico.unil.ch>
To: mathgroup@smc.vnet.net
> Subject: [mg11118] TeXForm, OutputForm questions ?
> 1) TeXForm creates matrices where the columns are separated by \& rather
> than &, so I have to change that every time I include text in the LaTeX
> file. Can this be switched ?
> 2) Matrices are created with the\matrix command, this does not seem to
> suit amstex or amsmath package. Simple way to fix this ?
> 3) Possibility to create LaTeX tabulars with formulae as entries ? (so
> that I don't have to include $ pairs for every entry myself)
The behavior of TeXForm under Mathematica 2.x and 3.0 are significanly
different. It is not clear from the behaviors you describe what version
you might be using. Could you give me a more precise description of what
you do to generate the TeX form of matrices and tables? If you are using
Mathematica 3.0, I may be able to provide you with some guidance on how to
get the best output possible by taking advantage of the front end's
typesetting mechanisms.
> These questions may be rather related to Mathematica output form:
> 4) Simple way to represent exp ans exp rather than e^{...}.
(* First, disable spell warning message because exp is similar to Exp *)
(* Temporarily disable protection on changes to the Power function *)
(* Use Format to tell TeXForm how to format expression. See Mathematica
book, Section 2.8.16 *)
Format[Power[E,x_],TeXForm]:= exp[x]
(* Restore protection on Power *)
(* Reenable spelling warnings *)
(* Now when we apply TeXForm... *)
E^(x^2+y^2) // TeXForm
(* we get exp instead *)
\exp({x^2} + {y^2})
> 5) represent factors like \frac{1}{\sqrt{2}} as 2^{-1/2}, possible ?
Similar line of reasoning, except this time we have to turn to
Mathematica 3.0's box typesetting mechanism:
RowBox[{"-", "1"}], "/", "2"}]]
When we evaluate:
1/(x^2 + y^2)^(1/2) // TeXForm
We get:
> 6) represent \cos(\frac{\beta}{2})^2 rather as \cos^2\frac{\beta}{2},
> possible ?
Similar to question (5)
Format[Power[Cos[x_],2],TeXForm]:= RowBox[{
SuperscriptBox["Cos", "2"], "(", MakeBoxes[x], ")"}]
Evaluation of:
Cos[beta/2]^2 //TeXForm
{{\cos }^2}\Big(\frac{\beta}{2}\Big)
P.J. Hinton
Mathematica Programming Group paulh@wolfram.com Wolfram
Research, Inc. http://www.wolfram.com/~paulh/
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; Let d(n) be defined as the sum of proper divisors of n (numbers less than n
; which divide evenly into n).
; If d(a) = b and d(b) = a, where a ≠ b, then a and b are an amicable pair and
; each of a and b are called amicable numbers.
; For example, the proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55
; and 110; therefore d(220) = 284. The proper divisors of 284 are 1, 2, 4, 71
; and 142; so d(284) = 220.
; Evaluate the sum of all the amicable numbers under 10000.
; Answer: 31626
(defn factors [n]
(let [max (inc (int (Math/sqrt n)))]
(loop [fs [] i 2]
(if (= i max)
(set (cons 1 fs)) ; Unique items
(if (zero? (mod n i))
(recur (concat fs [i (/ n i)]) (inc i))
(recur fs (inc i)))))))
(defn d [n]
(apply + (factors n)))
(defn amicable? [n]
(let [i (d n)]
(and (not (= n i)) (= (d i) n))))
(prn (apply + (filter amicable? (range 1 10001))))
Tip: Filter by directory path e.g. /media app.js to search for public/media/app.js.
Tip: Use camelCasing e.g. ProjME to search for ProjectModifiedEvent.java.
Tip: Filter by extension type e.g. /repo .js to search for all .js files in the /repo directory.
Tip: Separate your search with spaces e.g. /ssh pom.xml to search for src/ssh/pom.xml.
Tip: Use ↑ and ↓ arrow keys to navigate and return to view the file.
Tip: You can also navigate files with Ctrl+j (next) and Ctrl+k (previous) and view the file with Ctrl+o.
Tip: You can also navigate files with Alt+j (next) and Alt+k (previous) and view the file with Alt+o.
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Mathematical writing
, 1994
"... Your success as a scientist will in part be measured by the quality of your research publications in high-quality journals and conference proceedings. Of the three classical rhetorical
techniques, it is logos, rather than pathos or ethos, which is most commonly associated with scientific publication ..."
Cited by 19 (1 self)
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Your success as a scientist will in part be measured by the quality of your research publications in high-quality journals and conference proceedings. Of the three classical rhetorical techniques, it
is logos, rather than pathos or ethos, which is most commonly associated with scientific publications. In the mathematical sciences the paradigm for publication is to describe the mathematical proofs
of propositions in sufficient detail to allow duplication by interested readers. Quality control is achieved by a system of peer review commonly referred to as refereeing. This guide is an attempt to
distill the experience of the theoretical computer science community on the subject of refereeing into a convenient form which can be easily distributed to students and other inexperienced referees.
Although aimed primarily at theoretical computer scientists, it contains advice which maybe relevant to other mathematical sciences. It may also be of some use to new authors who are unfamiliar with
the peer review process. However, it must be understood that this is not a guide on how to write papers. Authors who are interested in improving their writing skills can consult the "Further Reading"
section. The main part of this guide is divided into nine sections. The first section describes the
, 1998
"... Students are expected to have reading knowledge of PASCAL or C, and to be able to follow explanations of recursive algorithms. ..."
- IN THIRD INTERNATIONAL WORKSHOP ON COMPUTATIONAL SEMANTICS (IWCS-3 , 1999
"... Knowledge is essential for understanding discourse. Generally, this has to be common sense knowledge and therefore, discourse understanding is hard. For the understanding of textbook proofs,
however, only a limited quantity of knowledge is necessary. In addition, we have gained something very essent ..."
Cited by 1 (1 self)
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Knowledge is essential for understanding discourse. Generally, this has to be common sense knowledge and therefore, discourse understanding is hard. For the understanding of textbook proofs, however,
only a limited quantity of knowledge is necessary. In addition, we have gained something very essential: inference. A prerequisite for parsing textbook proofs is to being able to parse formulae that
occur in these proofs. Parsing formulae alone in the empty context is trivial. But within the context of textbook proofs the task soon gets complex. Several kinds of references from the text to parts
or sets of terms and formulae have to be handled. We describe some of the linguistic phenomena that occur in mathematical texts. The focus is on our treatment of term reference which is embedded in
the DRT.
, 1992
"... is a much greater percentage of what I am supposed to be doing in life ..."
- Int. Workshop on FirstOrder Theorem Proving (FTP'98), Technical Report E1852-GS-981 , 1998
"... . Our long-range goal is to implement a program for the machine verification of textbook proofs. We study the task from both the linguistics and deduction perspective and give an in-depth
analysis for a sample textbook proof. A three phase model for proof understanding is developed: parsing, str ..."
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. Our long-range goal is to implement a program for the machine verification of textbook proofs. We study the task from both the linguistics and deduction perspective and give an in-depth analysis
for a sample textbook proof. A three phase model for proof understanding is developed: parsing, structuring and refining. It shows that the combined application of techniques from both NLP and AR is
quite successful. Moreover, it allows to uncover interesting insights that might initiate progress in both AI disciplines. Keywords: automated reasoning, natural language processing, discourse
analysis 1 Introduction In [12], John McCarthy notes that "Checking mathematical proofs is potentially one of the most interesting and useful applications of automatic computers". In the first half
of the 1960s, one of his students, namely Paul Abrahams, implemented a Lisp program for the machine verification of mathematical proofs [1]. The program, named Proofchecker, "was primarily directed
"... We discuss the kind of writing that's appropriate in a paper submitted to the math department to complete Phase Two of MIT's writing requirement. First, we review the general purpose of the
requirement and the specific way of completing it for the math department. ..."
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We discuss the kind of writing that's appropriate in a paper submitted to the math department to complete Phase Two of MIT's writing requirement. First, we review the general purpose of the
requirement and the specific way of completing it for the math department.
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|
What is the difference between variance and Mean Square Error(MSE)?
Variance measures the dispersion of an estimator around its mean, whereas the mean square error measures the dispersion around the true value of the parameter being estimated. If the estimator is
unbiased then both are identical.
$V(T) = E[T-E(T)]^2$
$MSE = E(T-\mu)^2=E[T-E(T) +E(T)-\mu]^2 = V(T) +[Bias(T)]^2$
If $T$ is Unbiased for $\mu$ then $E(T) = \mu$, so $[Bias(T)]=0$. Hence $V(T) = MSE(T)$.
Hope you understand this...
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...She went through college at an accelerated pace of 3 years instead of 4, while maintaining her HOPE scholarship. She even studied abroad in Ireland during those three years! She's been
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[Numpy-discussion] Numpy Advanced Indexing Question
Robert Kern robert.kern@gmail....
Thu Jul 17 03:39:46 CDT 2008
On Thu, Jul 17, 2008 at 03:16, Stéfan van der Walt <stefan@sun.ac.za> wrote:
> Hi Robert
> 2008/7/17 Robert Kern <robert.kern@gmail.com>:
>> In [42]: smallcube = cube[idx_i,idx_j,idx_k]
> Fantastic -- a good way to warm up the brain-circuit in the morning!
> Is there an easy-to-remember rule that predicts the output shape of
> the operation above? I'm trying to imaging how the output would
> change if I altered the dimensions of idx_i or idx_j, but it's hard.
Like I said, they all get broadcasted against each other. The final
output is the shape of the broadcasted index arrays and takes values
found by iterating in parallel over those broadcasted index arrays.
> It looks like you can do all sorts of interesting things by
> manipulation the indices. For example, if I take
> In [137]: x = np.arange(12).reshape((3,4))
> I can produce either
> In [138]: x[np.array([[0,1]]), np.array([[1, 2]])]
> Out[138]: array([[1, 6]])
> or
> In [140]: x[np.array([[0],[1]]), np.array([[1], [2]])]
> Out[140]:
> array([[1],
> [6]])
> and even
> In [141]: x[np.array([[0],[1]]), np.array([[1, 2]])]
> Out[141]:
> array([[1, 2],
> [5, 6]])
> or its transpose
> In [143]: x[np.array([[0,1]]), np.array([[1], [2]])]
> Out[143]:
> array([[1, 5],
> [2, 6]])
> Is it possible to separate the indexing in order to understand it
> better? My thinking was
> cube_i = cube[idx_i,:,:].squeeze()
> cube_j = cube_i[:,idx_j,:].squeeze()
> cube_k = cube_j[:,:,idx_k].squeeze()
> Not sure what would happen if the original array had single dimensions, though.
You'd have a problem.
So the way fancy indexing interacts with slices is a bit tricky, and
this is why we couldn't use the nicer syntax of cube[:,:,idx_k]. All
axes with fancy indices are collected together. Their index arrays are
broadcasted and iterated over. *For each iterate*, all of the slices
are collected, and those sliced axes are *added* to the output array.
If you had used fancy indexing on all of the axes, then the iterate
would be a scalar value pulled from the original array. If you mix
fancy indexing and slices, the iterate is the *array* formed by the
remaining slices.
So if idx_k is shaped (ni,nj,3), for example, cube[:,:,idx_k] will
have the shape (ni,nj,ni,nj,3). So
Is that clear, or am I obfuscating the subject more?
> Back to the original problem:
> In [127]: idx_i.shape
> Out[127]: (10, 1, 1)
> In [128]: idx_j.shape
> Out[128]: (1, 15, 1)
> In [129]: idx_k.shape
> Out[129]: (10, 15, 7)
> For the constant slice case, I guess idx_k also have been (1, 1, 7)?
> The construction of the cube could probably be done using only
> cube.flat = np.arange(nk)
Yes, but only due to a weird feature of assigning to .flat. If the RHS
is too short, it gets repeated. Since the last axis is contiguous,
repeating arange(nk) happily coincides with the desired result of
cube[i,j] == arange(nk) for all i,j. This won't check the size,
though. If I give it cube.flat=np.arange(nk+1), it will repeat that
array just fine, although it doesn't line up.
cube[:,:,:]=np.arange(nk), on the other hand broadcasts the RHS to the
shape of cube, then does the assignment. If the RHS cannot be
broadcasted to the right shape (in this case because it is not the
same length as the final axis of the LHS), an error is raised. I find
the reuse of the broadcasting concept to be more memorable, and robust
over the (mostly) ad hoc use of plain repetition with .flat.
Robert Kern
"I have come to believe that the whole world is an enigma, a harmless
enigma that is made terrible by our own mad attempt to interpret it as
though it had an underlying truth."
-- Umberto Eco
More information about the Numpy-discussion mailing list
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Palmetto, GA Trigonometry Tutor
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11 Subjects: including trigonometry, biology, algebra 1, algebra 2
...She went through college at an accelerated pace of 3 years instead of 4, while maintaining her HOPE scholarship. She even studied abroad in Ireland during those three years! She's been
tutoring for over 5 years in many different environments that include one-on-one tutoring in-person and online, as well as tutoring in a group environment.
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...I have my own 6 year old working on the 2nd grade level in mathematics. I have passed the Elementary Math qualifying test. I am a huge fan of the game and I relate basketball to mathematics in
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...I love working with children and exploring new ways to help them learn when conventional methods of learning do not meet their needs. I believe every child is intelligent beyond measure. They
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Problem Solving
July 19th 2011, 06:43 PM
Problem Solving
Find the letter that is the 118th entry in the following sequence. Explain how you determined your answer.
I know the answer is Y, but I can't figure out why.
July 19th 2011, 07:43 PM
Re: Problem Solving
The $n$th letter is Y if and only if $n \equiv 1 {\text{ mod }} 3$
July 19th 2011, 07:53 PM
Re: Problem Solving
I appreciate your reply, but I don't know what any of that means. I'm awful at math. I think the instructor is just looking for a basic answer on how to figure it out.
July 19th 2011, 07:56 PM
Re: Problem Solving
Well maybe you can say its the first letter, then every third letter.
So the pattern goes, 1, 4, 7, 10, ...
July 19th 2011, 07:59 PM
Re: Problem Solving
Yeah that's what I had, but I didn't know if it made sense or if it was even an answer at all. Thanks again.
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How to store transformation result? [Archive] - OpenGL Discussion and Help Forums
10-11-2006, 08:58 PM
Hi, I am new to OpenGL hope someone can help me with following problem.
Is it possible to store a transformation result? ie: If I apply glScalef(2,2,0) to a square struct (with 4 points in the struct) and if I want to save the transformed result of the points back to the
square variable structure I have. How should I do that?
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Stepper Motor
05-24-2011 08:23 PM
well, it is a positioning control system, and the reason I trying to find a transfer function is because I need to look at the step response of the control system and get the performance parameters
such as rise time, settling time etc... So this positioning control system is more like digital control system where you use a Z transform--so that's the reason I need a transfer function in S domain
and convert that into z domain
05-25-2011 09:55 AM
05-25-2011 02:12 PM
05-25-2011 02:27 PM
05-25-2011 02:34 PM
05-25-2011 03:15 PM
07-01-2011 04:57 PM
07-06-2011 04:14 PM
07-08-2011 07:29 PM
07-08-2011 07:38 PM
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Financial Ratio Analysis
Financial ratio analysis – relating line items from an organisation’s financial statements to assess the organisation’s financial status or performance at a point in time; or by indicating a trend
over a time series; or by comparison with another similar organisation.
Ratios may be calculated slightly differently by different analysts, the important thing is whether the particular ratio makes sense and actually addresses the underlying economic issue of interest
(eg short- or long-term solvency, asset management, profitability and market value).
Following below (alphabetically) are a descriptions of a number of common financial ratios. And some free downloads!
Accounts payable turnover. A measure of efficiency or activity.
Accounts payable turnover = cost of goods sold / average accounts payable.
Accounts payable turnover = credit purchases / average accounts payable.
Accounts Payable period. A specific measure of working capital management.
Accounts payable period = (average accounts payable / purchases) x 365.
Accounts receivable turnover. A measure of efficiency or activity.
Accounts receivable turnover = credit sales / average accounts receivable.
Acid test. See quick ratio.
Activity ratio. A measure of efficiency or activity. For example, see working capital turnover, inventory turnover, collection period, fixed asset turnover, asset turnover, net asset turnover,
receivables turnover and accounts payable turnover. Download our template for some of these activity ratios.
Adjusted accounting measures. See Return on Invested Capital (ROIC), Economic Profit (EP) and Market Value Added (MVA).
Altman Z-Score. Developed by Edward I Altman, the Z-Score is a quick way to assess the solvency of an organisation – it can indicate current and potential financial distress. It uses 5 financial
ratios based on 8 variables from the organisation’s financial statements. A Z-Score of less than 1.8 indicates a high probability of financial failure. Whereas a Z-Score of greater than 3.0 indicates
a low probability of financial distress. Download our template for the Altman Z-Score. Download Predicting Financial Distress of Companies: revisiting the Z-Score and ZETA Models , a paper by Edward
I Altman (July, 2000). Download our template for the Z-Score.
Asset turnover. A measure of overall asset management.
Asset turnover = sales / assets.
Note: This maybe current, non-current or total assets and either beginning, ending or average value.
Cash coverage. A measure of long-term solvency.
Cash coverage = (EBIT + depreciation) / interest.
Cash ratio. A measure of short-term solvency or liquidity.
Cash ratio = cash / current liabilities.
Cash ratio = (cash + marketable securities) / current liabilities.
Collection period. A specific measure of working capital management.
Collection period = accounts receivable / (annual sales / 360).
Creditors turnover. See accounts payable turnover.
Current asset turnover. A measure of efficiency or activity.
Current asset turnover = sales / average current assets.
Current ratio. A measure of short-term solvency or liquidity.
Current ratio = current assets / current liabilities.
Days in cash operating cycle. A measure of efficiency or activity.
Days in cash operating cycle = days to collect + days in inventory – days in payables.
Days in inventory. A measure of efficiency or activity.
Days in inventory = 365 / inventory turnover.
Days in payables. A measure of efficiency or activity.
Days in payables = 365 / accounts payable turnover.
Days sales in inventory. A specific measure of working capital management.
Days sales in inventory = 365 / inventory turnover.
Days sales in receivables. A specific measure of working capital management.
Days sales in receivables = 365 / receivables turnover.
Days sales outstanding. See days sales in receivables.
Days to collect. A measure of efficiency or activity.
Days to collect = 365 / accounts receivable turnover.
Debt to equity ratio. A measure of long-term solvency.
Debt to equity ratio = (total assets – shareholders’ equity) / shareholders’ equity.
Debtors turnover. See receivables turnover.
Discount to growth. For listed companies, a comparative measure of value relative to potential for growth. For example: Establish the ratio of the company’s share price to cash flow, compare that
with the sustainable growth rate of cash flow per share.
Dividend payout ratio. A measure of cash dividends paid to income.
Dividend payout ratio = cash dividends paid / net income.
Dividend retention ratio. See retention ratio.
Dividend yield.
Dividend yield = dividend per share / current market price per share.
Dividend yield = profit after interest and tax / total dividend.
du Pont analysis. See Return on Equity (ROE) decomposition and Return on Assets (ROA) decomposition. Download our template for calculating the du Pont financial ratios.
Earnings per share. A measure of market value.
Earnings per share = net income – preferred stock dividends / weighted average number of shares outstanding.
EBIT. Earnings Before Interest and Taxes.
EBITDA. Earnings Before Interest, Taxes, Depreciation and Abnormals.
EBIT margin. A ‘before interest and taxes’ measure of the price premium that the organisation’s products or services can commend in the marketplace and the efficiency of the organisation’s
procurement, production, sales and distribution processes.
EBIT margin = EBIT / sales.
EBITDA margin. A ‘before interest, taxes, depreciation and abnormals’ measure of the price premium that the organisation’s products or services can commend in the marketplace and the efficiency of
the organisation’s procurement, production, sales and distribution processes.
EBITDA margin = EBITDA / sales.
Economic Profit (EP). A measure of economic value created over time period.
EP = invested capital x (ROIC – WACC). Where WACC is the Weighted Average Cost of Capital.
Economic Value Added (EVA). EVA is measure of whether a company is earning better than its cost of capital.
EVA is calculated from reported earnings, specific accounting adjustments and after deducting the cost of capital. It is a sophisticated but complex measure requiring specialist expertise. Also see
Market Value Added (MVA) and Future Growth Value (FGV). EVA was devised by Stern Stewart & Co.
Equity multiplier. A measure of long-term solvency.
Equity multiplier = total assets / shareholders’ equity.
Equity multiplier = 1 + debt to equity ratio.
Fixed asset turnover. A measure of efficiency or activity.
Fixed asset turnover = sales / average PP&E (net).
Fixed asset turnover = sales / net fixed assets.
Future Growth Value (FGV). FGV is a measure of the market’s expectation of EVA growth. Also see Economic Value Added (EVA) and Market Value Added (MVA). EVA was devised by Stern Stewart & Co.
FGV = MVA – EVA.
Gearing ratios. See debt to equity ratio, cash coverage and total coverage.
Gross profit margin. A measure of the price premium that the organisation’s products or services can commend in the marketplace and the efficiency of the organisation’s procurement and production
Gross profit margin = (sales – cost of goods sold) / sales.
Historical accounting measures. See profitability ratios, activity ratios, liquidity ratios, capital gearing ratios and shareholders and investors ratios.
Interest rate cover. See cash coverage.
Inventory turnover. A specific measure of working capital management.
Inventory turnover = cost of goods sold / inventory.
Note: This maybe beginning, ending or average value for inventory.
Investors ratios. See earnings per share, price earnings ratio and dividend yield.
Leverage. This is a measure of gearing, the relative value of the organisation’s debt to shareholders’ equity.
Leverage = assets / shareholders’ equity.
Liquidity ratios. See current ratio, quick ratio and no credit period.
Pages: 1 2
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Riemannian Hausdorff distance between two conjugacy classes in a compact Lie group
up vote 1 down vote favorite
I am interested in the distance between two conjugacy classes in a group like $SO(n)$. However let's consider $U(n)$ for simplicity. My conjecture is that the Hausdorff distance between the conjugacy
classes of $\Lambda_1$ and $\Lambda_2$, both of which are diagonal matrices, is given by the distance of the eigenvalues $(\lambda_1, \ldots, \lambda_n)$ and $(\lambda^{(2)}_1, \ldots, \lambda^{(2)}
_n)$ as elements in the homogeneous space $\mathbb{T}^n / S_n$, where the action of the symmetric group $S_n$ on $\mathbb{T}^n$ is by permutation of the coordinates. Thus by translation invariance,
the closest element to $X$ in the conjugacy class of $Y$ is any one that can be simultaneously diagonalized with $X$. However I don't know how to prove this plausible statement.
lie-groups dg.differential-geometry mg.metric-geometry
add comment
1 Answer
active oldest votes
Diagonalization gives a map $f\colon U(n)\to \mathbb{T}^n/S_n$. The map $f$ is also the projection to the orbit-space of of the $U(n)$-action by conjugacy on $U(n)$. Hence $f$ is a
up vote 3 down submetry; i.e., $f(B_r(M))=B_r(f(M))$ for any matrix $M$. Hence your statement follows.
Do people really say submetry? – Mariano Suárez-Alvarez♦ Jan 4 '12 at 6:16
arxiv.org/abs/0909.4650 – Anton Petrunin Jan 4 '12 at 6:21
I guess your claim that $f$ is a submetry can be proved by Lipschitzness of the exponential map at every point on $U(n)$, which in turn can be proved by Baker-Hausdorff lemma. Is
this how you think about it? – John Jiang Jan 4 '12 at 6:24
1 $DEITY, now Lipschitzness!... :P – Mariano Suárez-Alvarez♦ Jan 4 '12 at 6:26
+1 for appeal to deity. – John Jiang Jan 4 '12 at 6:58
show 2 more comments
Not the answer you're looking for? Browse other questions tagged lie-groups dg.differential-geometry mg.metric-geometry or ask your own question.
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Math Forum Discussions
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Views expressed in these public forums are not endorsed by Drexel University or The Math Forum.
Topic: Algebra I Sample Regents?????
Replies: 2 Last Post: Jun 17, 2013 10:03 PM
Messages: [ Previous | Next ] Topics: [ Previous | Next ]
Re: Algebra I Sample Regents?????
Posted: Jun 17, 2013 8:40 PM
These are 8 sample items here...hoping to get more by September!! Pls post any other info ;)
Date Subject Author
6/17/13 Algebra I Sample Regents????? Robin Schwartz
6/17/13 Re: Algebra I Sample Regents????? Robin Schwartz
6/17/13 Re: Algebra I Sample Regents????? Herminia
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Projectile motion via a spring (well, actually, rubber bands...)
Okay, here's the deal. 2nd day back at college and I pretty much have forgot most everything Physics related already
Had a very easy assignment today in one of my engineering classes to take two dowels and a bunch of rubber bands and build something that could launch/throw a tennis ball from atop a terrace towards
a point on the ground below (goal to get as close as possible, if not a bulls-eye). So, you say, that's elementary stuff, what's the problem? Here ya go:
I have:
-the mass of the ball
-the distance it traveled in 4 different trials
-the times it traveled
-the actual distance from the terrace to the target (x)
-the height of the terrace (y)
I need to write and use equations to:
1) predict where the ball will land (assuming we hadn't actually launched it)
2) determine the effective spring constant for the system (in this case, two strings of rubber bands)
3) in addition to, I guess, finding the initial velocity using what info. I have
So, while this is, in essence, so easy it hurts, I'm brain dead right now and just need to know how to get/do #1,2,3 based on what I already have (ball mass,travel/height distances, and times).
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15-819K Logic Programming
Fall 2006
Frank Pfenning
TuTh 10:30-11:50
WeH 4623
12 units
Logic programming is a paradigm where computation arises from proof search in a logic according to a fixed, predictable strategy. It thereby unifies logical specification and implementation in a way
that is quite different from functional or imperative programming. This course provides a thorough, modern introduction to logic programming. It consists of a traditional lecture component and a
project component. The lecture component introduces the basic concepts and techniques of logic programming followed by successive refinement towards more efficient implementations or extensions to
richer logical concepts. We plan to cover a variety of logics and operational interpretations. The project component will be one or several projects related to logic programming.
Prerequisites: For undergraduates, 15-317 Constructive Logic or 15-312 Foundations of Programming Languages. No prerequisites for graduate students.
What's New?
Class Material
Course Information
Lectures TuTh 10:30-11:50, WeH 4623
Office Hours Wed 11:00-12:00, WeH 8117
Notes There is no textbook, but notes on Logic Programming and papers will be handed out
Credit 12 units
Grading 30% Homework, 15% Midterm, 55% Project
Homework Weekly homework is assigned each Thursday and due the following Thursday.
Midterm Date Thu Oct 19, in class.
Closed book.
Project Project topic or topics will be selected after the midterm.
Topics Horn logic, intuitionistic logic, linear logic,
unification, constraints, backward chaining, backtracking,
forward chaining, saturation,
proof terms, higher-order patterns, concurrency,
resource management, logical compilation.
Time permitting: deontic, epistemic, or other modal logics,
reasoning about logic programs.
Mailing List lp-course@cs [.cmu.edu]
Home http://www.cs.cmu.edu/~fp/courses/lp/
[ Home | Schedule | Assignments | Projects | Handouts | Software | Resources ]
Frank Pfenning
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PI-ATLAS LHC Day 2010
The Large Hadron Collider (LHC), the world's most powerful particle accelerator, promises to unveil a new spectrum of particles and unravel the mystery of electroweak symmetry breaking. PI-ATLAS LHC
Days are an ongoing series of meetings between researchers working on LHC-related physics in southwestern Ontario. The goal of these meetings is to share new ideas and encourage interaction in our
common fields of interest through a series of informal talks followed by discussions. This meeting will gather experimental and theoretical particle physicists focused on early new physics
opportunities during the first run of the LHC.
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Greg Astfalk, Editor
This volume conveniently brings together updated versions of 30 articles that originally appeared in SIAM News from 1990 to 1995. The objective of the column from which the articles are taken is to
present applications that have been successfully treated on advanced architecture computers. Astfalk edits this popular series of articles in SIAM's flagship publication, SIAM News. Algorithmic
issues addressed are those which have found general use in building parallel codes for solving problems. In addition to updates that reflect advances and changes in the field of applications on
advanced architecture computers, Astfalk has added an index and introductory comments to each article, making this book cohesive and interesting to practitioners and researchers alike.
Although some of the computers used in older articles are no longer available and some of the languages are rarely used, what is valuable in these articles is the diversity of areas to which advanced
computing can be applied. Perhaps more valuable than the diversity are the successful methods used to achieve parallelism. This may be used by others for similar applications. The applications and
technology discussed cover many disciplines, with significant overlap. Articles deal with molecular dynamics, optimization/math programming, finance/economics, geometry, and partial differential
Contributing Authors; Preface; Chapter 1: Massively Parallel Algorithms for Electronic Structure Calculations in Quantum Chemistry, Andrew L. Sargent, Jan Almlöf, and Martin W. Feyereisen; Chapter 2:
Massively Parallel Lattice QCD Calculations, Clive F. Baillie; Chapter 3: Parallel Weiner Integral Methods for Elliptic BVPs: A Tale of Two Architectures, Michael Mascagni; Chapter 4: A Parallel
Genetic Algorithm Applied to the Mapping Problem, El-Ghazali Talbi and Pierre Bessiere; Chapter 5: Supercomputers in Seismology: Determining 3-D Earth Structure, Robert J. Geller; Chapter 6: Advanced
Architectures: Current and Future, Greg Astfalk; Chapter 7: Large-Scale Molecular Dynamics on MPPs: Part I, David M. Beazley and Peter S. Lomdahl; Chapter 8: Large-Scale Molecular Dynamics on MPPs:
Part II, David M. Beazley and Peter S. Lomdahl; Chapter 9: Symbolic and Parallel Computation in Celestial Mechanics, Liam M. Healy; Chapter 10: Parallel Methods for Systems of Ordinary Differential
Equations, Kevin Burrage; Chapter 11: Parallelizing Computational Geometry: First Steps, Isabel Beichl and Francis Sullivan; Chapter 12: Parallel Inverse Iteration for Eigenvalue and Singular Value
Decompositions, Richard J. Hanson and Glenn R. Luecke; Chapter 13: Parallel Branch-and-Bound Methods for Mixed Integer Programming, Jonathan Eckstein; Chapter 14: Optimal Scheduling Results for
Parallel Computing, Håkan Lennerstad and Lars Lundberg; Chapter 15: Airline Crew Scheduling: Supercomputers and Algorithms, Jeremey Schneider and Theresa Hull Wise; Chapter 16: Parallel Molecular
Dynamics on a Torus Network, Klaas Esselink and Peter A.J. Hilbers; Chapter 17: A Review of Numerous Parallel Multigrid Methods, Craig C. Douglas; Chapter 18: Shared-Memory Emulation Enables
Billion-Atom Molecular Dynamics Simulation, Eduardo F. D'Azevedo, Charles H. Romine, and David W. Walker; Chapter 19: Probing the Playability of Violins by Supercomputer, Robert T. Schumacher and Jim
Woodhouse; Chapter 20: Ray Tracing with Network Linda, Rob Bjornson, Craig Kolb, and Andrew H. Sherman; Chapter 21: A SIMD Algorithm for Intersecting hree-Dimensional Polyhedra, David Strip and
Michael Karasick; Chapter 22: Numerical Simulation of Laminar Diffusion Flames, Craig C. Douglas, Alexandre Ern, and Mitchell D. Smooke; Chapter 23: Applications of Algebraic Topology to Concurrent
Computation, Maurice Herlihy and Nir Shavit; Chapter 24: Parallel Computation of Economic Equilibria, Anna Nagurney; Chapter 25: Solving Nonlinear Integer Programs with a Subgradient Approach on
Parallel Computers, Robert Bixby, John Dennis, and Zhijun Wu; Chapter 26: Parallelizing FDTD Methods for Solving Electromagnetic Scattering Problems, Sandy Nguyen, Brian Zook, and Xiaodong Zhang;
Chapter 27: Using a Workstation Cluster for Physical Mapping of Chromosomes, Steven W. White and David C. Torney; Chapter 28: Massively Parallel Computations in Finance, Stavros A. Zenios; Chapter
29: Robust Optimization on PC Supercomputers, John M. Mulvey; Chapter 30: History Matching of Multiphase Reservoir Models on Hypercubes, Jianping Zhu; Appendix: A Retrospective on the "Applications
on Advanced Architecture Computers" SIAM News Column, Greg Astfalk; Index.
Royalties from the sale of this book are contributed to the SIAM Student Travel fund.
1996 / xviii + 359 pages / Softcover / ISBN-13: 978-0-898713-68-8 / ISBN-10: 0-89871-368-4 /
List Price $66.00 / SIAM Member Price $46.20 / Order Code SE03
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PT Final Exam Review
A quantity that is fully specified by measuring its size is called a:
Tension, compression, bending and twisting are kinds of mechanical:
A resultant force has both magnitude and:
The measure of gravitational pull is:
Newtons or kilogram weight
The SI units of mechanical force are:
45 lb×ft
A 30 lb. force applied to the end of a 1.5 ft wrench produces a torque of:
The formula for torque (T) is:
2 lbs
Dave and Deborah push a cart with a force of 5 lb. and 7 lb. respectively, in the opposite direction. The magnitude of the resultant is:
15 lbs
Bill and Mary push a cart with a force of 6 lb. and 9 lb. respectively, in the same direction. The magnitude of the resultant is:
15 lbs
Kevin and Sandy push a cart with a force of 7 lb and 8 lb. respectively, in the same direction. The magnitude of the resultant is:
30 lb×ft
A 20 lb. force applied to the end of a 1.5 ft. wrench produces a torque of:
T=F x L
Torque is a measure of what kind of force?
36 lb×ft
A 20 lb. force applied to the end of a 1.8 ft. wrench produces a torque of:
T=F x L
Pneumatic or hydraulic
All fluid systems can be classified as either:
What is a gas or a liquid that conforms to the shape of its container?
A hydraulic system uses:
Specific gravity
In fluid systems the quantity that has no units is:
Absolute pressure
Total pressure above zero pressure is:
14.7 lb/in²
Sea level air pressure is:
The basic reference for density is:
Mass divided by volume
Density is equal to:
Density times volume
Mass is equal to:
Specific gravity
A hydrometer measures:
A unit of pressure is:
Weight density time height
Pressure equals:
An SI unit of pressure is:
A battery
The most common source of DC voltage is:
A term that pertains to DC voltage is:
In a circuit, the battery would be the:
Control element
In a circuit, the switch would be the:
In a circuit, the lamp would be the:
Which of the following does NOT refer or pertain to batteries?
18 volts
Three 6-volt batteries connected in series would produce a voltage of:
15 volts
A 9-volt and a 6-volt battery connected in series would produce:
24 volts
Two 9-volt and one 6-volt batteries connected in series would produce:
3 volts per lamp
A series circuit containing one 6-volt battery and two identical lamps would produce a voltmeter reading of:
6 volts per lamp
A parallel circuit containing one 6-volt battery and two lamps would produce a voltmeter reading of:
Voltage difference is measured with a/an:
DC Volts
To measure the strength of a battery, set the multimeter function to:
Warmer regions to cooler regions
Heat energy moves from:
Temperature depends on the presence or absence of:
Dry ice (-196°C)
The substance with the LEAST energy of molecular motion would be:
Which of the following is a form of energy?
68 ⁰F
Room temperature would be approximately:
20⁰F is equal to:
T ⁰C = 5/9 (T ⁰F- 32)
100C⁰ equals:
108⁰F equals how many degrees Celsius?
T ⁰C=5/9(T ⁰F-32)
108⁰C equals how many degrees Fahrenheit?
T⁰F = 9/5(T⁰C)+32
The object remains in place
Which of the following is NOT an effect of mechanical work on an object?
1000 ft×lbs
A vertical force raises a 200 lb. Barbell 5 ft. The amount of work done is:
W=F x d
How much work is being done by a person holding a 60 pound suitcase 18 inches above the floor?
Radius of the circle in which the force is applied.
The prime mover in an angular mechanic system is called torque which is determined by multiplying the force applied, by the lever arm. The lever arm is equal to the:
Angle moved through
Work done by a torque is equal to the torque applied times the:
The length of the radius of a circle measured on the circumference is one:
37680 joules
How much work is done while an electric motor drives a pump through 40 revolutions, given the motor delivers 150 N×m of torque?
W=T x ∅
10.4 meters
520 N×m of work is done when a 50 Newton force is moved a distance. How far is the load moved?
W=F x d
2070 in×lb
A wrench with an 18 inch handle turns through an angle of 2.3 radians while a force of 50 lbs. is applied to the end of the handle. How much work is done?
T=F x L
W=T x ∅
Two hundred Newtons of force are required to lift a mass 1.2 meters above the floor. How many Newton.meters of work are done?
W=F x d
9.0 × 10⁴ N×m
A loaded cart is pulled a distance of 100 meters by an electric truck that exerts 900 Newtons of force to pull the cart at a constant speed. If the 100 meters is covered in 12 seconds, how much work
is done?
W=F x d
900 ft×lbs
A vertical force raises a 150 lb. box 6 ft. The amount of work done is:
W=F x d
37 Newtons of force are required to lift a mass 3.5 meters above the floor. How many Newton.meters of work is done?
W=F x d
The hydraulic brake system in most cars can be described as a closed fuel system because the fluid is:
Multiplying .7854 x d²
The cross sectional area of a cylinder can be found by:
Pressure times change in volume
In fluid systems, the work equation may take two forms: work equals pressure difference times fluid volume moved, or work equals:
It is a way to distinguish between torque and work units
In the English system of units, why is torque expressed in lb×ft and work in ft×lb
.045 N×m
A pressure of 200 N/m² is applied to a pneumatic cylinder that has a piston with a 3 cm diameter and a 0.32 m stroke. How much fluid work can be done in one piston stroke?
V=.7854d² (l)
Weight Density
The work required to lift water from a lake to a tank above the lake is given by the formula W=∆P×V. To find the pressure difference (∆P), you must know the height (h) of the lift as well as the:
A box weighing 200 pounds is pushed 8 inches on to a conveyor belt by the shaft on a hydraulic cylinder. The cylinder has a 3 inch diameter and a pressure of 50lb/in2 applied to move the piston and
shaft. How efficient is the system?
W=F x d
W= ∆pV
V = .7584d²(l)
% Eff = W₀/W₁
A = 7854d²
4 × 10⁵ N×m
A pump lifts 2 cublic meters (m³) of water through a pressure difference of 2 × 10⁵N/m². How much fluid work is done by the pump?
W= ∆PV
4071.5 cm³
An engine has eight cylinders each has a bore (diameter) of 9 centimeters and a stroke of 8 centimeters. What is the engines's displacement in cubic centimeters?
V = .7584d²(l)
500 joules/cylinder
In an engine where each cylinder is ten centimeters in diameter and has a stroke of twelve centimeters, how much work can be done by each cylinder if combustion produces 530.56 x 10³ N/m² of pressure
in each cylinder when the spark plug fires?
V = .7854d²(l)
W= ∆pV
62.4 lb/ft³
160 lb/ft² of pressire is required to lift a certain liquid 25 feet from a tank car to a storage tank. What is the weight density of the liquid?
∆p= pwh
15 ft×lb
A hydraulic cylinder rod is moved 1 ft. by pressure of 60 lb/ft². What is the amount of fluid work done if 0.25 ft³ of oil is displaced?
W = p(∆V)
.253 N×m
A pressure of 300 N/m² is applied to a pneumatic cylinder that has a piston with a 5 cm diameter and a 0.43 m stroke. How much fluid work can be done in one piston stroke?
V = .7854d² (l)
6.25 × 10¹⁸ electrons
The coulomb is a unit of electrical charge that is made up of:
Amount of charge moved/second
Current is defined as:
The force like quantity in the equation for electrical work is:
The rate of movement of electrical charge is measured in:
5000 joules
5,760 joules of electric work is done on an electric motor that is connected to an overhead crane that lifts steel "I" beams. each beam weighs 2,000 Newtons and is lifted a height of 2.5 meters, how
much work is done by the motor while lifting on the beam? W=F x d
An electric motor does 800 Nm of useful work while it uses 1000 joules of electrical energy. The efficiency of the motor is:
%Eff.= W₀/W₁ x 100
15 coulombs
A 12-volt battery does 180 volt coulombs of work. How much charge has been moved?
One coulomb of charge times one bolt is equal to one:
50 coulombs
A battery-powered windshield wiper motor is activated for 10 seconds and draws a current of 5 amps at a voltage of 12 volt. The total amount of charge moved is:
I = q/t
6 volts
Forty eight thousand joules of work is done in an electrical circuit when 8,000 coulombs of electrical charge is moved. What voltage difference causes the charge to move?
W = Vq
750000 joules
An electric motor that is connected to an overhead crane lifts stacks of reinforcement rods. Each stack of rods weighs 3,000 Newtons. How much work is done by the motor if it lifts 5 stacks of rods
50 meters?
E=F x d
An electric motor does 317 Nm of useful work while it use 400 joules of electrical energy. The efficiency of the motor is approximately:
%Eff.= W₀/W₁ x 100
8.18 volts
27 thousand joules of work is done in an electrical circuit when 3,300 coulombs of electrical charge is moved. When voltage difference causes the charge to move?
W= Vq
A negative sign in front of an acceleration would indicate the object is:
Measuring acceleration
In the English system of units, ft/sec² indicates a unit for:
Buckshot fired from a gun
Which of the following is associated with linear rate?
The shaft of an electric drill
Which of the following is associated with a rotational rate?
Direction and time
Angular rate is expressed in which terms?
Meters per second
When using the international system of units (SI), the mechanical rate is measured in:
A jet airplane accelerating down a runway
Which of the following has a linear rate?
9.8 meters per second squared
In the SI system, the acceleration of gravity is:
Number of items produced per unit of time.
Production rate is the:
A deceleration
Negative acceleration is also referred to as:
Revolutions per minute
The term RPM refers to:
Rate = Displacement-like Quantity / Elapsed Time
The general unifying equation for expressing rate is:
V = I/t
A letter definition, or equation, for linear speed is:
Appear to stand still
A wheel is turning at a rate of 240 RPM. A strobe light is set at 240 FPM. Which one best describes what will happen? The wheel will:
2 feet per second
A box placed on a conveyor belt, moving at a constant speed, travels 240ft in 2 minutes. What is the speed of the box and the conveyor in feet per second?
V = I/t
50 miles per hour
A bus changes speed frequently. The total distance it travels is 850 miles. The total driving time is 17 hours. What is the average speed of the bus?
V = I/t
-.08 rads/second²
A brake is applied to a flywheel for 5 seconds, reducing the angular velocity of the flywheel from 2.2 rads/second to 1.8 rads/second. What is the angular acceleration of the flywheel?
a = (Vf - Vi)/t
Magnitude and direction
A vector must have:
Fluid-flow rate units such as "kg per min" or "gm per sec" are units of:
A unit of measure, such as "gal" or "liter" is a unit of:
A "volume or Mass"-flow rate
A rate associated with a fluid system:
The flow rate is inversely proportional to the distance traveled
When gasses are compressed, storage tanks should be located as close to the point of use as possible because:
A supertanker loaded with 1.5 million gallons of crude oil is to unload at an offshore discharge buoy. The ship pump can discharge the oil in 30 hours. What is the volume flow rate of crude oil in
gallons per minute?
Qv = V/t
2.5 gallons per hour
IF a leak in a faucet causes a water loss of 120 gallons in two days, what is the volume flow rate per hour? Qv = V/t
3 gal/hour
If a leak in a water line causes the loss of 144 gallons in 2 days, what is the volume-flow rate of the leak?
Qv = V/t
920 gallons
A bath tub faucet drips 200 gallons in ten days. How many gallons will have been lost in 46 days? Qv = V/t
5 cubic inches/sec
A weir gage, with a cross-sectional area of 10 square inches at a height of 3 inches at the weir is being used to measure liquid flow. If the float takes 20 seconds to move 10 inches. what is the
volume flow rate in the weir?
V = Al
Qv = V/t
2 gallons per second
If a factory produces 120 gallons of hydrogen peroxide per minute, how many gallons of hydrogen peroxide are produced per second Qv = V/t
A supertanker loaded with 2 million gallons of crude oil run aground and begins to leak oil. The leak causes the ship to lose 15,873 gallons per hour. How many gallons of oil will be spilled during a
24 hour period? Qv = V/t
A toilet leaks 1 gallon of water every 20 minutes. Approximately how many gallons will be lost in a month? Qv = V/t
One coulomb per second
One ampere equals:
.001 amperes
One milliampere of electric current equals:
1 × 10⁻⁶ amperes
One microampere equals:
½ ampere
500 milliamperes equals:
D.C. voltage
A DMM is often used to measure:
60 coulombs
Find the charge that flows through an electric iron in 10 seconds if the iron is rated at 6 amperes. I = q/t
6.25 × 10¹⁸
A generator produces a current of one ampere. How many electrons are induced in the coils in one second?
4950 hertz
A function generator changes frequency from 50 to 5000 Hz. What is ∆f?
Voltage waveform
An oscilloscope can be used to measure the period of a:
The frequency of a voltage can be measured with an:
Voltage increments
The vertical volts per division control of an oscilloscope is used to adjust the:
420 coulombs
Find the charge that flows through an electric toaster in 1 minute if the toaster is rated at 7 amperes.
I = q/t
One calorie
The amount of heat required to raise one gram of water one Celsius degree is:
Heat energy
A calorie is a unit of:
Latent heat
When a substance (such as water) goes from a solid to a liquid state without a change in temperature, the heat added during the transformation is called:
One calorie per gram Celsius degree
The specific heat of water is:
A Cromel and Constantan Thermocouple used to measure:
Celsius degrees
In the SI system, temperature difference is expressed in:
Constantan and chromel
Type 'E' thermocouples are made from:
4000 Btus per hour
A toaster element produces 2000 Btus of heat energy in 30 minutes. What is the heat-flow rate?
Dewar flask
A device used to contain and store heat energy is called a:
Heat-flow rate is represented by:
300000 Btus
A heater is rated a 25,000 Btu/hr. What is the heat energy generated in 12 hours in a 144 square foot room?
QH = H/t
10 Btus/Fahrenheit degree
The heat capacity of 10 pounds of water is :
Zero degrees Celsius
When using type "E" thermocouples in laboratory experiments, what reference temperature is used to construct the temperature chart?
Drag force
For an object moving through a fluid, streamlining is a way to reduce:
Kinetic friction
Rolling friction is a type of:
Which of the following shapes is most streamlined?
15 lb.
A wooden crate that weighs 50 lb. is pilled across a wooden floor at a constant speed. The coefficient of friction for wood on wood is 0.3. What is the frictional force necessary to keep the crate
F = µN
It takes a fore of 20 lb. to move a 50 lb. wooded crate across a stone floor. What is the coefficient of friction for wood on stone? F = mN
What is the SI unit that represents force?
2800 lb.
The weight of a truck is 4000 lb. and is parked on a wet concrete loading ramp. Another truck bumps into it and slides it our of position. What force was necessary to overcome friction? Static
friction of rubber on wet concrete is 0.7.
F = µN
66.67 N
A plastic box weighing 100 N will start to slide on a metal truck bed when a push of 50 lb. is applied. What is the coefficient of static friction for plastic on metal? F = µN
Kinetic friction
Static friction is greater than:
66.67 N
It takes 20 Newtons of force to move a wood desk across a wood floor. The coefficient of kinetic friction for wood on wood is 0.3. How much does the desk weigh.
F = µN
16.67 N/(m/sec)
A drag force of 200 N is experienced by a boat moving through water at a speed of 12 m/sec. What is the drag resistance.
RD = F / V
0.12 N/(m/sec)
What is the drag resistance of a rocket with a drag force measured at 6 Newton when traveling at 50 m/sec?
RD = F / V
The English unit for drag resistance is:
Lengthen the pipe
A way to increase resistance in a pipe is to:
Heat the fluid
A way to decrease resistance in a pipe is to:
Fluid pressure in English units is usually measured in:
Motor oil
Which of the following has the greatest viscosity?
0.2 gal/min
What is the fluid flow rate when the pressure drop (∆p) is 5 lb/in² and the fluid resistance (Rf) is 25 lb/in² / gal/min
RF = P/Qv
∆p = Rf x Qv
Given: Rf = ∆p / Qv
Find: ∆p
Qv = ∆p / Rf
Given: Rf = ∆p/Qv
Find: Qv
0.36 (lb/in²) / (gal/min)
Water flowing through a pipe at 550 gal/min has a pressure difference of 200 lb/in². What is the fluid resistance?
Rf = P/Qv
Which of the following instruments does not measure fluid pressure?
1100 lb/in²
The resistance (Rf) in a fuel-transfer pipeline is 2 (lb/in²)/(gal/min). Fuel moves through the pipe at a volume flow rate of 550 gal/min. What is the pressure difference along the length of the
Rf = P/Qv
0.2 gal/min
What is the approximate fluid flow rate when the pressure drop (∆p) is 7 lb/in² and the fluid resistance (Rf) is 33 lb/in² / gal/min?
Rf = P/Qv
0.53 (lb/in²) / (gal/min)
Water flowing through a pipe at 330 gal/min has a pressure difference of 175 lb/in². What is the fluid resistance?
Rf = P/Qv
Electrical resistance occurs because electrons moving through a wire bump into:
Tolerance band
The fourth band of a 4 band resister is the:
To find the total resistance in a series circuit the individual resistances must be:
Copper wire and solder are each classified as:
6 volts
A current of 0.25 amperes flows through a 24 ohm resister. The voltage drop across the resistor is: V=I x R
6 ohms
∆ Vtot = 60 V.
R₁ = 3 Ω.
R₂ = 3 Ω
Rtot = :
10 amps
∆ Vtot = 60 V.
R₁ = 3 Ω.
R₂ = 3 Ω
Itot = :
10 amps
∆ Vtot = 60 V.
R₁ = 3 Ω.
R₂ = 3 Ω
I₁ = :
30 volts
∆ Vtot = 60 V.
R₁ = 3 Ω.
R₂ = 3 Ω
∆V₁ = :
10 amps
∆ Vtot = 60 V.
R₁ = 3 Ω.
R₂ = 3 Ω
I₂ = :
30 volts
∆ Vtot = 60 V.
R₁ = 3 Ω.
R₂ = 3 Ω
DV2 = :
A current of 0.03 amperes flows through a 100 ohm resistor. The voltage drop across the resistor is: V=I x R
384.6 ft³
3.6 x 10⁶ ft lb of work is required to mobe a volume of water from a lake to a tank 150 ft above the lake. How much water is pumped while doing this much work?
pw = 62.4lbs/ft³ (for water)
∆p = pw x h
W = ∆pV
Thermal flow rate in SI is usually measured in:
Thermal resistance in English units may be expressed as:
Thermal conductivity
Which of the following indicates how well material conducts heat energy?
A defining equation for thermal resistance is:
0.25 F°/Btu/hr
The temperature difference (∆T) between the inside and outside of an electric oven is 300 F°. The heating element must supply 1200 Btu/hr of heat energy to the oven to overcome the heat-flow rate out
of the oven. Find the thermal resistance of the oven walls. Rt = ∆T / Qh
Rearrange the formula, RT = l / kA, to solve for l.
20 kcal/hr
A firefighter's suit has a thermal resistance of 50 C°/kcal/hr. When exposed to a fire the temperature difference from outside to inside the suit may be 1000 C°. What is the heat-flow rate through
the suit at 1000 C°?
RT = ∆T / Qh
307.69 kcal/hr
A house has a thermal resistance of 0.13 C°/kcal/hr. What is the buidings heat-flow rate when the temperature difference fro minside to outside is 40C°? RT = ∆T / Qh
109.37 C°/kcal/hr
The temperatire inside a house is 24°. The temperature outside is 4° A furnace supplies 2625 Kilocalories to the house every hour to maintain a constant temperature of 24°. What is the overall
thermal resistance of the house?
Rt = ∆T / Qh
400 Btu/hr
A hot water heater runs at 140°F in a room with a temperature of 80°F. The insulation of the heater has a termal resistance of 0.15 F°/Btu/hr. What is the heat flow rate?
RT = ∆T/Qh
.23 F°/Btu/hr
The temperature difference (∆T) between the inside and outside of an electric oven is 350 F°. The heating element must supply 1500 Btu/hr of heat energy to the oven to overcome the heat-flow rate out
of the oven. Find the thermal resistance of the oven walls. RT = ∆T/Qh
200 kcal/hr
A house has a thermal resistance of 0.15 C°/kcal/hr. What is the building's heat-flow rate when the temperature difference from inside to outside is 30C°? Rt = ∆T/Qh
Gravitational potential energy
What aspect of an ovject increases as its height increases?
Kinetic energy invloves the mass of an object and its:
An avalanche in progress is an example of what type of energy?
20000000 ft×lbs
What's the potential energy of a cloud at 10,000 feet if the condensed moisture in the cloud has a total weight of 2,000 pounds? Ep = wh
4 pounds per inch
A spring i compressed 4 inches by a force of 16 pounds. What's the spring constant (k) for this spring? K = F/d
In the equation Ep = ½ kd², what does d represent?
2940000 N×m
The density of water is 1000 kg/m³. The mass of a volume of water is equal to the density of the water times its volume. How much potential energy is stored in 75 m³ of water that is raised to a
height of 4 meters?
Ep = wh
W = m × g
5000 ft×lb
What is the gravitational potential energy stored in a box that weighs 1000 lbs. and located 5 ft above the floor? Ep = wh
112500 ft×lb
Water is stored in a storage tank of a factory fire-sprinkler system. The tank is located 75 feet above the factory floor. How much potential energy is released when 1500 lb. of water is released
into the system. Ep = wh
343200 ft×lb
A water pump is rated at 4 hp and pumps 5.5 cubic feet of water per minute. The pump lifts the water 100 feet into a tank. How much potential energy is added to the water after 10 minutes of pumping?
Ep = wh
2500000000 ft×lb
5000000 lbs of water are stored behind a dam that is 500 feet above a town. How much potential energy would be released if the dam collapsed? Ep = wh
2.4 pounds per inch
A spring is compressed 5 inches by a force of 12 pounds. What's the spring constant (k) for this spring? K = F/d
Angular speed
In the formula, Ek = ½Iw², the w represents the:
Linear kinetic energy
Objects moving in a straight line contain:
Moment of intertia
When an automobile tire is unbalanced, what needs to be adjusted?
Rotational kinetic energy
Objects moving in a curved path contain:
2141 N×m
How much kinetic energy can be stored in a flywheel that has a radius of 0.25 meters, a mass of 50 kg, and is rotating at 500 rpm?
I = ½ mr²
Ek = ½w²
25 kg×m²
Calculate the moment of intertia for a disk type flywheel. The flywheel has a mass of 200 kg and a radius of 0.5 meters.
I = ½ mr²
34256.5 J
A flywheel has a moment of inertia of 25 kg×m² and is rotating at 500 rpm (52.35 rad/sec). How much rotational kinetic energy is stored in the flywheel?
Ek = ½ Iw²
1000000 J
A box of cargo was pushed out of an open airplane and its parachute failed to open. The box weighed 1,000 N and the air plane was at an altitude of 1,000 meters when the cargo box was pushed out of
the plane. How much kinetic energy did the box have at the instant it hit the ground?
Ep = wh
480000 J
Steam under high pressure flows through a nozzle and is forced through a turbine. The steam has a density of 60 kg/m³ and flows with a speed of 20 m/sec. In one second, 40 m³ of steam strike the
turbine blades. How much kinetic energy does the steam have?
Ek = ½ mv²
m = pV
878.9 ft×lb
Water under high pressure strikes the blades of a turbine in a dam. During each second, 1,000 lb. of water moving at 7.5 ft/sec strike the turbine blades. How much kinetic energy does the water have?
Coils of wire wrapped around a conducting core
An inductor consists of:
Color code of the wire in the coil
Inductance for a given coil DOES NOT depend on the:
Age of the plates
The capacitance of a capacitor DOES NOT depend on the:
Filter circuits
Capacitors and inductors are used in circuits called:
Ep = ½ LI²
What is the correct equation for calculating the potential energy stored in an inductor?
Ep = ½ C(V)²
Which of the following equations is the correct equation for calculating the potential energy stored in a capacitor?
0.0625 J
Find the energy stored in a capacitor rated at 50 µF (50 × 10⁻⁶ F) where the voltage difference across the plates is 50 volts.
Ep = ½ C(V²)
45 J
An inductor has an inductance of 25 henries and it draws 20 amps of current. How much energy is stored in the inductor?
Ep = ½ LI²
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Re: st: plot population distribution over sample histogram
Notice: On March 31, it was announced that Statalist is moving from an email list to a forum. The old list will shut down at the end of May, and its replacement, statalist.org is already up and
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Re: st: plot population distribution over sample histogram
From Nick Cox <njcoxstata@gmail.com>
To statalist@hsphsun2.harvard.edu
Subject Re: st: plot population distribution over sample histogram
Date Thu, 2 Jun 2011 21:07:05 +0100
Some technique
. sysuse auto
(1978 Automobile Data)
. histogram mpg , addplot(function normalden(x,26,10), ra(10 42))
But -qnorm- is a better method in my view.
On Thu, Jun 2, 2011 at 8:11 PM, jacques detiege <pies.stata@gmail.com> wrote:
> I would like to plot the normal curve for my population (mean=80,
> SD=10) over a histogram for a set of sample scores.
> I have only been able to produce the normal distribution of my sample
> data over the histogram.
> Using Stata, how would I produce a graph of both the normal curve for
> the population and the histogram for the sample data?
* For searches and help try:
* http://www.stata.com/help.cgi?search
* http://www.stata.com/support/statalist/faq
* http://www.ats.ucla.edu/stat/stata/
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Welcome to the finance homework help subject page. Let's take a look at the topics covered by a typical course in finance:
• Value
• Risk
• Capital Budgeting
• Financing Decisions and Market Efficiency
• Dividend Policy and Capital Structure
• Options
• Debt Financing
• Risk Management
• Financial Planning and Short-term Financial Management
• Mergers, Corporate Control, and Governance
That just about covers it - now let's have some fun.
The first topic in the list above is Value, which refers to present value and future value problems. You may have seen the following two equations already, the simple derivations of which can be
found in any good finance text:
The first equation is the future value formula for compound interest, where A = the future value of the account, P = the present value of the account, often called the principle, r = the interest
rate as a decimal, n = the number of compounding periods each year, and t = the time in years. The second equation is the future value formula for an annuity, where m = the amount of the periodic
payment. As a reminder, an annuity is any sequence of payments into or out of an interest-bearing account.
A couple of useful finance notes before we continue with this homework help session. The first equation can easily be solved for P, in which case it becomes a present value formula for compound
interest. With that, you can calculate how much you would have to invest now (present value) in order for your account to be worth a specified amount at some time in the future (future value) with a
given interest rate. The second equation can easily be solved for m, after which you can calculate, for example, the size of a monthly payment you would need to make in order to have a certain
amount of money at some point in the future. When solved for m, the second equation is called the sinking fund formula.
The question we wish to consider now is: How would we calculate the present value of an annuity? Notice that the second equation has no P in it, so we cannot simply solve for P as we did in the
first equation. In a sense, we will consider both equations together by thinking about the following situation. Suppose Nick was setting up an interest bearing account at his local bank while his
friend Pete was setting up an annuity with his stock broker. Nick's account would be governed by the first equation, while Pete's account would be governed by the second.
Let's assume that we want to have Nick and Pete set up their accounts with the same interest rate (r) and compounding (n), and that after the same amount of time (t), we want the accounts to have
the same value (A). Notice that the two variables not mentioned in this scenario so far are the amount Nick will put into his bank account initially (P), his present value, and the size of the
periodic payment of Pete's annuity (m). Now, let's choose the size of Pete's annuity payment (m) and let the annuity run its course. Is it not true that we could play around with the size of Pete's
initial bank deposit (P) until the two accounts had the same value at the end of this experiment? When this condition is met, we can say that Pete's initial deposit (P), which is a present value,
correlates with the present value of Nick's annuity, since it gives the same future value under identical conditions.
This tells us how to calculate the present value of an annuity using the two formulas above. Since the future values will be the same (in our experiment), set the A of one equal to the A of the
other. That means we can set the right side of the first equation equal to the right side of the other to obtain:
All we have to do now is solve the equation for P, and with a little algebraic manipulation, we will have the formula for the present value of an annuity:
If you solve the equation above for m, you will be able to calculate the size of a monthly payment when you know the size of a loan (present value) and the other required parameters. When solved for
m, this equation is called the payout annuity formula and is discussed in the blog article Landlords and Logarithms. By the way, if you are unable to do the algebra to obtain the various forms of
these equations, you are weak in algebra and need to do some skill building in this area.
Now let's see how useful the last equation is. Suppose you decided you could afford $250 per month as a car payment and the best interest rate you could get was 13% over a four-year car loan period.
What is the most expensive car you could buy? This is the same as asking what is the largest loan I can afford, and anytime you are trying to determine a loan amount it is a present value problem.
(Remember that because of interest you will end up paying much more than the face value of the loan, and that larger amount is the future value of the loan.) If you plug in everything you know to
the formula for the present value of an annuity, you find out you can afford a car that costs $9,319. That is a pretty useful calculation for just about anyone.
Let's do one more. Suppose you decide you can afford $1,575 each month as a mortgage payment and interest rates are 10.5% on a 30-year fixed mortgage. How expensive a home can you afford? Once
again, you are looking to determine the size of the loan, and that is a present value problem. After plugging in, you find out you can buy a house for $172,180. Congratulations - you now have a
house and a car!
Don't be fooled into thinking that finance is a dull subject isolated from math and the sciences. A quick visit to books on Amazon.com and Google will soon dispel that notion.
To fulfill our mission of educating students, our online tutoring centers are standing by 24/7, ready to assist students who need extra finance homework help and tutoring.
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Definitions for option 6 (part 2 of 2)
From: Andy Seaborne <andy.seaborne@epimorphics.com> Date: Fri, 30 Mar 2012 10:32:04 +0100 Message-ID: <4F757D94.8000203@epimorphics.com> To: SPARQL Working Group <public-rdf-dawg@w3.org>
== Option 6
The path operators are:
6.A: /, |, ! as there are in 2LC.
6.B: *, +, ? are non-counting
No DISTINCT, No {} forms: {n}, {n,m}, {n,}, {,m}
== Definitions
Of all those, the critical one is *. Given that, the machinery used to
define it and the existing transformation rules, the major item is the
definition of * (ZeroOrMorePath/distinct) here. The description below
is not completely formal.
An arbitrary length path P = (X (path)* Y) is all solutions from
X to Y by repeated use of path. It collects the set of nodes visited by
repeated applying path.
ZeroOrMorePath includes X.
OneOrMorePath starts after one application of path.
First, we define a function that returns the set of end points reached
by using a path expression N times. Then we use that to define path* as
the bindings caused by any number of applications of the path.
== Auxiliary definition: PathPoints
The "path points" of a path expression P and start point X (an RDF term)
is the set of nodes reached by exactly N repeated applications of P.
These two definitions are equivalent:
Definition 1 (recursive):
PathPoints(X, P, 0) =
{ X }
PathPoints(X, P, N) =
{ y | N>0 & Exists z in PathPoints(X, P, N-1) such that z P y }
Definition 2 (fill in the dots):
PathPoints(X, P, 0) = { X }
PathPoints(X, P, N) =
SELECT DISTINCT ?V
WHERE { X P ?V1 . # N patterns with path P in them.
?V1 P ?V2 .
?V(N-1) P ?V
They are equivalent because they define sets. If you directly
implemented like this, definition 1 is likely faster as the size
increases. It reduces intermediate results to distinct as it goes along
because it used the set answer to N-1. Definition 2 leaves all that to
end unless the optimizer is smart enough.
== Example
:x :p :y .
:y :q :z .
path: :p/:q
PathPoints(:x, :p/:q, 1) = { :z }
and not :y because :p/:q does have :y as an endpoint.
== Main Definitions
** Definition: ZeroOrMorePath
This defines ZeroOrMorePath to be the set of nodes reachable by any
number of applications of a path expression.
Let X be an RDF Node, ?V a variable and P a path expression.
ZeroOrMorePath(X, P, ?V) =
{ ?V=y | y in PathPoints(X, P, N) for some N >= 0 }
Now the case of two variables for end points: consider all nodes in the
Let V1 and V2 be variables:
ZeroOrMorePath(?V1, P, ?V2) =
{ (?V1=z,?V2=y) | z in nodes(G) and y in ZeroOrMorePath(z, P, ?V2) }
and for a path from a variable to a fixed node:
ZeroOrMorePath(V, P, X)
= ZeroOrMorePath(X, reverse P, V)
= { ?V=z | z in nodes(G) and exists y in ZeroOrMorePath(z, P, y) }
and for a path between two fixed ends:
ZeroOrMorePath(X, P, Y)
= { {} } if ZeroOrMorePath(X, P, ?V) contains ?V=y
= { } if ZeroOrMorePath(X, P, ?V) does not contains ?V=y
Notes: More identities:
ZeroOrMorePath(x, P, V) =
Projection of ZeroOrMorePath(V1, P, V2) of first of pair to a set
= { a | (a,b) in ZeroOrMorePath(V1, P, V2) }
ZeroOrMorePath(V, P, z) =
Projection of ZeroOrMorePath(V1, P, V2) of second of pair to a set
= { b | (a,b) in ZeroOrMorePath(V1, P, V2) }
** Definition: OneOrMorePath
OneOrMorePath(X, P, ?V) =
{ ?V=y | y in PathPoints(X, P, N) for some N >= 1 }
etc etc
== Non-normative text to help implementers
We can also provide a (now informative) function to calculate the
reachable points from x, some term. This the result of PathPoints for
any N. It's the same as the function ALP in 2LC with the last line
removed, and using the result set as the set of nodes visited.
# R is the overall result and the visited set
ALP_1(x:term, path, R:set of RDF terms) =
if ( x in R ) return
add x to R
X = eval(x,path)
For n:term in X
ALP_1(n, path, R)
The result is in R.
For comparison, here is function ALP from 2LC:
ALP(x:term, path, R:multiset of RDF terms , V:set of RDF terms) =
if ( x in V ) return
add x to V
add x to R
X = eval(x,path)
For n:term in X
ALP(n, path, R, V)
remove x from V
Received on Friday, 30 March 2012 09:32:36 GMT
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New Energy Watch: 12-Volt DC Basics
Recent Comments
DC means Direct Current. It's the kind of electrical current produced by batteries. 12-volt DC batteries are the kind found in cars, trucks, RVs, and boats.
AC means Alternating Current, the kind of electrical current found in wall sockets in buildings.
In buildings, we can replace any amount of AC electricity, from a small fraction to 100 percent, with energy collected by solar panels and stored in 12-volt DC batteries. Direct current from the
batteries can be changed to AC by means of an inverter, or it can be used directly to run equipment designed for DC; for example, a car vacuum that plugs into a cigarette lighter socket.
On this website we talk about very basic 12-volt concepts and systems. If you would like to replace a bit of your AC requirement with energy derived from the sun -- get your toe in the water for a
minimal investment -- this will be a good place to start. If you'd like to get involved with solar energy in a serious way, try www.findsolar.com.
Now, on to more basics. Note that these are utterly rudimentary definitions. There's much more to know, but you don't need to know much more in order to get involved.
Voltage/Volts = the amount of potential energy available to push electrical current. Think of it as pressure in a water system. For a battery-powered system, think of a water tower with a big tank on
top. It's drained by gravity, and the way water flows out of it depends on the diameter of the pipe, the length of the pipe, and the weight of the water in the tank at any given time.
Amperage/Amps = the flow of electrical current through conductors like wires. Think of it as the amount of water current flowing through a pipe.
Wattage/Watts = the amount of energy expended, or used. Think of it as the water that fills a glass (a few watts) or a swimming pool (lots of watts).
Ampere-hours (Ah) = the current in amperes multiplied by the amount of time it flows. Deep-cycle batteries, the kind used for solar-power storage, have ampere-hour capacity ratings that give a
general idea of how many amps can be drawn from the battery for how long. In a perfect world, a battery rated for 90 amp-hours would be able to give you 90 amps for one hour, 45 amps for two hours,
one amp for 90 hours, and so on. In reality, you can and would use only a portion of those amp-hours before the battery should be charged again.
Ohm = a measure of resistance in a wire or other conductor. Resistance is determined both by the wire's length and its thickness, or gauge. The thicker the wire, the more easily current will flow
through it. In a
simple 12-volt solar panel arrangement, resistance can exist between the solar panel and the battery, and between the battery and whatever it's supplying power to. Resistance always creates heat, and
the greater the resistance, the more heat. Try to put too much current through too small a wire, and you can create enough resistance to start melting things and causing fires. This can happen even
in a simple 12-volt system, so always use common sense and generous wire gauges. See 12-Volt Safety for more.
Now, here are some of the easiest equations you'll ever have to use:
• Volts x Amps = Watts
• Watts / Amps = Volts
• Watts / Volts = Amps
If you read the owner's manual for any electrical appliance, or the stamped electrical infomation on the appliance itself, you can usually discover what it needs for energy input and how much energy
it uses. Then it's a matter of arithmetic to find out if your 12-volt system can handle the task. Bear in mind, though, that sometimes there's more to the story than simple volts, amps, watts, and
ohms. For example, an inexpensive automotive-grade DC-to-AC inverter can be used for a myriad of tasks that are electrically simple, like running an incandescent light bulb, but some appliances that
are sensitive to interference, like TVs, often need to be run through a better, more expensive inverter.
For starters, we'd recommend one of the simpler inverters. A 400-watt model from an auto-parts store or online retailer will cost about $35. If you find that you need a more sophisticated inverter
later, you won't have too much of an investment in the simpler ones. See our Inverters section for more detailed information.
~ Doug Logan, New Energy Watch
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Figure 5
Resolution: standard / high
Figure 5.
3D-BLAST performance with E values on the protein query set SCOP-516. (a) The relationship between 3D-BLAST E values and the root mean square deviation (rmsd) values of aligned residues. The average
rmsd values with E value below e^-10, e^-15, e^-20, and e^-25 are 3.57 Å, 2.85 Å, 2.37 Å and 2.25 Å, respectively, based on 22,415 protein structures randomly selected from the 516 returned lists.
(b) The relationship between E values and the percentages of true (black) and false (gray) function assignment. The correct percentages of the superfamily assignments with E values below e^-10, e^
-15, e^-20 and e^-25, are 95.26%, 97.67%, 99.31%, and 99.75%, respectively. The coverage values of the function assignment are 98.06% (<e^-10), 91.47% (<e^-15), 83.72% (<e^-20), and 76.74 (<e^-25).
Tung et al. Genome Biology 2007 8:R31 doi:10.1186/gb-2007-8-3-r31
Download authors' original image
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Inductors in Series and in Parallel
In this article, we will go over how inductors add in series and how they add in parallel.
We will go over the mathematical formulas for calculating series and parallel inductance so that we can compute the total inductance values of actual circuits.
Inductors in Series
Inductors in series are inductors that are placed back-to-back.
Below is a circuit where 3 inductors are placed in series.
You can see the inducotrs are in series because they are back-to-back against each other. The best way to think of a series circuit is that if current flows through the circuit, the current can only
take one path. You can see in the above circuit that if current flowed through it, it could only take one path.
Formula for Adding Inductors in Series
The formula to calculate the total series inductance of a circuit is:
So to calculate the total inductance of the circuit above, the total inductance, LT would be:
So using the above formula, the total inductance is 60H.
Note- When inductors are in series, as the formula shows, they simply add together. Thus, the total inductance of a series circuit will always be greater than any of the individual inductor values.
Inductors in Parallel
Inductors in parallel are inductors that are connected side-by-side in different branches of a circuit.
Below is a circuit where 3 inductors are in parallel:
You can see that the inductors are in parallel because they are all on their own separate branches in the circuit. The best way to think about parallel circuits is by thinking of the path that
current can take. When current is travelling through a parallel circuit, the current can take various paths through the circuits, such as to go through any of the branches containing the inductors.
In series, this is not the case. Current can only take one path.
Formula for Adding Inductors in Parallel
The formula to calculate the total parallel inductance is:
So to calculate the total inductance of the circuit above, the total inductance, LT would be:
So using the above formula, the total inductance is 5.45Ω.
Note- When inductors are in parallel, the total inductance value is always less than the smallest inductor of the circuit. In other words, when inductors are in parallel, the total inductance
shrinks. It's always less than any of the values of the inductors.
Inductor Circuit in Series and In Parallel
We'll now do an inductor circuit in which inductors are both in series and in parallel in the same circuit.
Below is a circuit which has inductors in both series and parallel:
So how do we add them to find the total inductance value?
First, we can start by finding the resistance of the resistors in series. In the first branch, containing the 20H and 40H inductors, the series resistance is 60H. And in the second branch, containing
the 30H and 60H inductors, the series inductance is 90H. Now in total, the circuit has 3 inductances in parallel, 10H, 60H, and 90h. Now, we plug these 3 values into the parallel inductance formula
and we get a total inductance of 7.83H.
If you want to test the above series and parallel connections out practically, get 1mH inductor or whatever inductors you have, but let them be of the same value. In this example, I'll stick with 2
1mH inductors. Take the inductors and place them in series. Now take a multimeter and place the multimeter in the inductance setting (if available) and place the probes over the 2 inductors You
should read just about 2mH, which is double the value of both inductors. This proves that inductors add when connected in series. Now place the inductors in parallel. Take the multimeter probes and
place one end on one side of a inductor (either one) and place the other probe on the other side of that inductor. You should now read about 0.5mH, or half the value, because inductance decreases in
parallel. This is a practical, real-life test you can do to show how inductors add.
If you have a circuit where you would like to calculate the series or parallel inductance, check out our online Parallel and Series Inductor Calculator. This allows you to calculate the total
inductance value of circuits.
Related Resources
Inductance Calculator
Inductor Energy Calculator
Inductor Impedance Calculator
Ferrite Core Inductor Calculator
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Calculating inferential statistics for a single proportion from the Course SPSS Statistics Essential Training
Start learning with our library of video tutorials taught by experts. Get started
In this course, author Barton Poulson takes a practical, visual, and non-mathematical approach to the basics of statistical concepts and data analysis in SPSS, the statistical package for business,
government, research, and academic organization. From importing spreadsheets to creating regression models to exporting presentation graphics, this course covers all the basics, with an emphasis on
clarity, interpretation, communicability, and application.
For many people, when they think of statistics, they think of inferential statistics, and not always fondly. Of course, there is much more to statistics and data analysis than the calculation of
probability values, and this should be evident by the amount of time we spent so far on graphics and descriptive statistics. However, the ability to go beyond the data at hand and make inferences
about a larger group of people--hence the name inferential statistics--is one of the great beauties of analysis. In this set of movies, I want to start with the simplest kinds of inferential
statistics, those for one variable at a time.
There are few different procedures that we'll cover, such as confidence intervals and hypothesis tests, for scale variables and proportions, as well as the distribution of a single categorical
variable. But let's start with what is probably the simplest and most familiar, the confidence interval and hypothesis test for a single proportion. For this example, I'm going to be using the
GSS.sav data set. That stands for General Social Survey. And it has one variable on the end here that I think is interesting. If I scroll to the end, I have a variable here that's called ReadBook,
and what it means is whether the person says that they've read a novel, a poem, or a play in last year.
We might be interested in the percentage of people who say that they have read one, whether that is significantly higher then, for example 50% and what the confidence interval for that might be, like
you would get from a political poll where they say 73% of respondents plus or minus 3% who are in favor of a particular candidate. To do this, I'm going to many use one of SPSS's more interesting
features. It's called nonparametric tests, and I get to it by going to the Analyze menu, down to Nonparametric Tests.
It's called nonparametric because we're not using parameters like means and standard deviations. Then I come over to One Sample. And here it will do a lot of things automatically, but I'm going to be
a little bit selective and customize it to actually make things simpler for right now. The first thing I'm going to do is I'm going to come here to Fields, and that really means variables. And right
now it's putting in nearly every variable. It would test for equality of distribution on categorical variables, and it would also test for scale variables, whether they are normally distributed like
a bell curve.
I don't want to do all of that, so what I'm going to do is I'm going to take all of these variables, I'm going to put them back into the original field. The only test variable that I want is this
one: Read Novel, Poem, or Play. So I'll double-click to move that over. Then I go to the Settings tab to choose exactly what test it is that I want to do. Now I'm going to do Customized tests here,
and I'm going to choose Compare the observed binary probability--binary means two answers: yes or no--to the hypothesized value with what's called the binomial tests.
And click on Options, and what it's going to do is it's going to do a hypothesis test to see if the proportion of people who say they've read a novel, poem, or play in the last year is statistically
significantly different from a hypothesized proportion, which right now I'll leave at 50%. I can also get what's called the confidence interval. That's like the plus or minus 3% in a political poll.
Now sometimes you can use conventional statistics, but right here SPSS is doing a very nice thing and it's letting me use what's called an exact statistic. In this case, it's called the Clopper-
Pearson for the confidence interval.
We don't need to go into any details except to say this would be a good choice. So I'm just going to click on that and I'm going to come down and press OK, and then I'm going to press Run. Now the
output for this looks little different from what we've had so far, because it's a table with colors and shading in it. Also, it's not showing me everything right now. This is actually what's called a
model viewer. Now right now, all it's telling me is that the proportion of people who say they've read a novel, poem, or play in the last year is significantly different from 50%. It's not telling me
what's the actual proportion was or how far away it is, but I can get that through going onto the Model Viewer.
I'll double-click here and it brings up the Model Viewer. I'll maximize that window. And what I have here is the output that I saw on the other page. It tells me that the proportion of people who say
they've read one of these is not 50%. It's significantly different from 50%. In fact, what I can do is I can come over here and the hypothesized, that 50%, is this blue bar right here. But what I
really have is an observed 71% of the people say that they've read a novel, poem, or play in the last year. That's out of 349 people, and this tells me that that is significantly different from 0.
To get the confidence interval, I need to do one other thing. I come back over to this left pane and I go down to where it says View. Right now we're looking at the Hypothesis Summary. If I click on
that, I can get the Confidence Interval Summary. It's a slightly different table here, and it tells me how it calculated the confidence interval by using the Clopper-Pearson. It tells me what the
Parameter was, the probability that a person read a novel, a poem, or play in the last year. It tells me that the proportion of people who said yes, because they put ones instead of zeroed, is 71%.
That corresponds to what I have over here.
The yes is the 71%. The confidence interval at the 95% confidence interval, which is the most common, is from 66% to 76%. And what this means is that while in my sample of 349 people 71% may have
said they've read these, in the population of those 349 people came from, the true value could be somewhere between 66% and 76%. This is like the plus or minus 5% that you would get from a political
poll. So the new nonparametric tests in SPSS is actually a very flexible procedure that can perform an entire range of tests all on its own.
It's also the easiest way to get confidence intervals and hypothesis tests for a single proportion. We'll come back to this procedure in another movie on testing nominal variables with multiple
categories, but for now this should give you a good start on dealing with inferential statistics for dichotomous variables in SPSS. In the next movie, we'll look at common tests for scale variables.
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10. Develop A Logic Circuit Whose Output Ishigh ... | Chegg.com
10. Develop a logic circuit whose output ishigh when the input has had three
consecutive ones to the input. Overlappingthrees are allowed. (e.g.,
x=011110110 then the output z=000110000assuming starting state was 0).
• Develop astate diagram and table for the circuit.
• Develop a logic circuit toimplement this circuit.
Can someone explain what the question is asking forplease?
Electrical Engineering
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The De Morgan Forum
Tag Archives: calculators
Press release from DfE and discussion of underlying statistics in fullfact.org. A table from fullfact.org
based on Trends in International Mathematics and Science Study (TIMSS):
From a post by Alex Knapp on Forbes blog:
QAMA [...] is a calculator designed to reverse the last several decades of education by actually improving students’ intuitive understanding and appreciation of math skills. It does this in a
deceptively simple way.
The name itself gives the method away – QAMA stands for “Quick Approximate Mental Arithmetic” (and in Hebrew, it means “How much?”). As with most calculators, to solve a problem with a QAMA, you
first do what you’d do with a regular calculator: type in the problem. But rather than just give you the answer right away, QAMA asks you for one more step: you have to estimate the answer. If
your estimation demonstrates that you understand the math, the calculator will give you the precise answer. If your estimation isn’t close, then you have to try again before you get the precise
Quick – what’s the square root of 2? What do you mean you don’t have a calculator? Well, you can start guessing, right? So let’s work this through. You know you have an upper bound – it has to be
less than 1.5, because 1.5 x 1.5 is 2.25. And it has to be more than 1, because 1 x 1 is just 1. But 2.25 is pretty close, right? So what if you guess 1.4? Well, then you’d be pretty close. 1.4
x 1.4 is 1.96, and the square root of 2 is about 1.414.
But did you notice something? Without your calculator, you had to estimate. In order to estimate, you had to think about and engage with the math behind exponents and square roots. Which means,
hopefully, that you came out of that first paragraph with a bit better understanding of math.
That’s the theory behind QAMA, which is a calculator designed to reverse the last several decades of education by actually improving students’ intuitive understanding and appreciation of math
skills. It does this in a deceptively simple way.
The name itself gives the method away – QAMA stands for “Quick Approximate Mental Arithmetic” (and in Hebrew, it means “How much?”). As with most calculators, to solve a problem with a QAMA, you
first do what you’d do with a regular calculator: type in the problem. But rather than just give you the answer right away, QAMA asks you for one more step: you have to estimate the answer. If
your estimation demonstrates that you understand the math, the calculator will give you the precise answer. If your estimation isn’t close, then you have to try again before you get the precise
How close is close? Well, that depends on the calculation. If you put in 5×6, you have to estimate 30 – the calculator expects you to know your multiplication tables. For exponent problems, if
you have an integer – say something like 23^2, the tolerance is such that you still have to be pretty close, but there’s a wider berth for say, 23^2.1, because non-integer exponents are a tougher
Developing the calculator to have these different tolerances for different types of calculations was the key challenge for QAMA inventor Ilan Samson.
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To Make Algebra Fun, Rethink The Problem
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Behind the Mics
Production Director & host of All This Jazz
7:00 am
Sat December 24, 2011
To Make Algebra Fun, Rethink The Problem
By editor
Originally published on Sat December 24, 2011 7:25 am
SCOTT SIMON, HOST:
Schools across the country are on break this week, meaning that millions of students don't have to think even about algebra - or are they just missing the algebra that's all around them? We're joined
now by our Math Guy, Keith Devlin of Stanford University, who joins us this week from member station KJAU in Boulder, Colorado. Keith, thanks so much for being with us.
KEITH DEVLIN, BYLINE: Hi, Scott. Nice to be here.
SIMON: Well, nice to join us electronically. I bet you're pretty happy you're in Boulder for the holidays though, aren't you?
DEVLIN: That's right. I came out here for a white Christmas and right now it looks as though we're getting one.
SIMON: Oh, all right. Now, I think a lot of people think of algebra as kind of arithmetic with letters. You say that's wrong.
DEVLIN: That's right. Arithmetic with letters is still arithmetic. Algebra's is actually a very different way of thinking. With arithmetic, you actually take some numbers and you calculate a new
number. With algebra, you think about numbers. You end up naming an unknown number - you called it X or Y - and then you reason, not arithmetically so much as logically. So, algebra is really logical
thinking about numbers as opposed to arithmetic, which is calculating with numbers. And, in fact, it's because people, I think, when kids don't realize it's a different way of thinking that they find
algebra was so hard 'cause they try to solve algebra problems using arithmetic and it doesn't work.
SIMON: But why was algebra invented? And I recognize it may not have been invented by just one person but what need in the world was being addressed when algebra was created?
DEVLIN: What we think of as algebra - and it goes back into the ancient Babylonians and the Greeks - but it was really in the Muslim empire in the Ninth and 10th centuries where it really developed.
In fact, there was one particular individual called Al-Khwarizmi who wrote a textbook in the Ninth century, which essentially lays out modern algebra. It wasn't symbolic. He wrote it out entirely in
words because he was actually telling stories about how to reason with number. It was trying to describe to people how to think logically about numbers in this method called algebra. But that book
that he wrote not only established modern algebra but it also gave us its name. The name algebra is an Arabic term, al-jabr. He also wrote a book on arithmetic, how to do calculations with numbers
and that's why we have this word algorithm. You take al-Khwarizmi and it becomes algorithm for rules, procedures for doing numerical calculations.
SIMON: And was this done to assist this is the trading for which the Mid-East was the center of the world then.
DEVLIN: Oh, yes. Yeah, in fact, one of the things I think that's typical of the work done in mathematics in the ninth, 10th, 11th century in the Arabic-speaking world, is they were not interested in
theorizing. They were very practical people and they developed algebra to be more efficient traders, more efficient commercial people, to be better engineers.
In fact, Baghdad and in the ninth century is the equivalent of Silicon Valley in the 20th century and the 21st century. That was the center of new technologies for many centuries as a result of
developing that algebra.
SIMON: How do package this story for students these days?
DEVLIN: It's very difficult actually because the revolutions, both in arithmetic and then later algebra, was so pervasive and so fundamental that once people got used to it, what had been regarded as
this brilliant breakthrough was regarded as dull and routine. We see it today with technologies.
To me, the iPad, the iPhone, those are incredible breakthroughs. But to anybody under the age of 15 or 16, that's just part of life. It's no big deal for kids when they have these things, and that's
a measure of how fundamental a major change is.
SIMON: Keith, do you think schools sometimes make a mistake in the way they try and convey algebra to students?
DEVLIN: Oh, I think they make a terrible mistake. First of all, students typically will ask this question: what is a useful for? And if a child is asking that, then it has been introduced in the
wrong way. I think it will be much more effective to begin with a bit of the history. Why did people develop this stuff? Algebra didn't come and it wasn't just sort of discovered hidden in a box
somewhere. People invented to solve real-life problems.
Now, in the ninth century people literally had to solve those problems themselves with a paper and pencil. Today, we literally don't have to use algebra because we have devices in our pockets; we
have spreadsheets in our computers that implement the algebra for us. But we are actually better users of technologies when we have some idea of what goes on underneath the hood.
SIMON: Keith, I hope you'll have a wonderful holiday.
DEVLIN: And the same to you, Scott.
SIMON: Our Math Guy, Keith Devlin speaking with us from Boulder, Colorado this week. Transcript provided by NPR, Copyright NPR.
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A wrinkle in space-time: Math shows how shockwaves could crinkle space
July 19th, 2012 in Other Sciences / Mathematics
Mathematicians at UC Davis have come up with a new way to crinkle up the fabric of space-time -- at least in theory.
"We show that space-time cannot be locally flat at a point where two shock waves collide," said Blake Temple, professor of mathematics at UC Davis. "This is a new kind of singularity in general
The results are reported in two papers by Temple with graduate students Moritz Reintjes and Zeke Vogler, respectively, both published in the journal Proceedings of the Royal Society A.
Einstein's theory of general relativity explains gravity as a curvature in space-time. But the theory starts from the assumption that any local patch of space-time looks flat, Temple said.
A singularity is a patch of space-time that cannot be made to look flat in any coordinate system, Temple said. One example of a singularity is inside a black hole, where the curvature of space
becomes extreme.
Temple and his collaborators study the mathematics of how shockwaves in a perfect fluid can affect the curvature of space-time in general relativity. In earlier work, Temple and collaborator Joel
Smoller, Lamberto Cesari professor of mathematics at the University of Michigan, produced a model for the biggest shockwave of all, created from the Big Bang when the universe burst into existence.
A shockwave creates an abrupt change, or discontinuity, in the pressure and density of a fluid, and this creates a jump in the curvature. But it has been known since the 1960s that the jump in
curvature created by a single shock wave is not enough to rule out the locally flat nature of space-time.
Vogler's doctoral work used mathematics to simulate two shockwaves colliding, while Reintjes followed up with an analysis of the equations that describe what happens when shockwaves cross. He found
this created a new type of singularity, which he dubbed a "regularity singularity."
What is surprising is that something as mild as interacting waves could create something as extreme as a space-time singularity, Temple said.
Temple and his colleagues are investigating whether the steep gradients in the space-time fabric at a regularity singularity could create any effects that are measurable in the real world. For
example, they wonder whether they might produce gravity waves, Temple said. General relativity predicts that these are produced, for example, by the collision of massive objects like black holes, but
they have not yet been observed in nature. Regularity singularities could also be formed within stars as shockwaves pass within them, the researchers theorize.
Reintjes, now a postdoctoral scholar at the University of Regensburg, Germany presented the work at the International Congress on Hyperbolic Problems in Padua, in June.
Provided by UC Davis
"A wrinkle in space-time: Math shows how shockwaves could crinkle space." July 19th, 2012. http://phys.org/news/2012-07-wrinkle-space-time-math-shockwaves-crinkle.html
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I. Unit I - Introduction (Organizing Data)
A. Data Displays( Organizing Data)
1. Bar Graphs
2. Circle (Pie) Graphs
a. fractions to decimals to percents
b. construct converting to degrees
3. Stem and Leaf
4. Box and Whisker Plot
a. five-number summary
b. quartiles
c. median
B. Data Patterns
1. Notes & Worksheet -Proportional People
a. Class Data Tables I & II with worksheet discussion
b. Ratios and proportions
c. Scatter Plots with armspans
d. Coroner's Report
Discuss proportions in class, assign for homework.
In groups, share dimensions and draw the body.
C. Quiz
1. Bar Graphs
2. Circle (pie) Graphs
3. Stem and Leaf
4. Box and Whisker Plot
D. Matrices
1. Organizing Data in Matrix form
2. Adding Matrices
3. Scalar Multiplication
II. Unit 2 - Related Variables
A. Related Variables Notes: #1
1. Models and Graphs (Scatter Plots)
2. Variables
3. Domain and Range (Realistic Boundaries)
4. Independent and Dependent
B. Related Variables: Notes #2
1. Working with pairs of Data (If the Shoe Fits)
2. Positive /Negative Associations shown on scatter plots
3. WS#2 Scatter plots of data
C. Related Variables: Notes & WS#3 (Variable/Constant)
D. Related Variables: Notes & WS#4 (Describing Change)
1. Inversely related
2. Directly related
E. Test-Related Variables
III. Unit 3 - Linear Functions
A. Part I - Functional Relationships
1. Funct. Relations Notes #1: WS#1& #2
Situations to tables and graphs, domain and range
2. Funct. Relations Notes #2: WS#3a,b
Relationships as Equations
Tables to Equations
3. Funct. Relations Review: WS #4a,b
Review of situations to tables and graphs.
4. Funct. Relations Notes #3
Function Notations & Solving Equations
5. Funct. Relations Test - Part I
Situations, tables and graphs.
B. Part II - Linear Function Family
1. Linear Function Exploration (Translations )
2. Linear Function Toolkit
C. Part III - Linear Models
1. Linear Model Worksheets
(Four Faces of a Function)
2. Sums of Angles - Data collecting Activity
IV. Unit 4 - Linear Systems
1. Linear System Model Problems (solve by graphing)
2. Linear Systems solve by substitution
V. Unit 5 - Absolute Value
1. Situations and Graphs
2. Exploring the Family of Absolute Value Functions
3. Equations to Graphs and Tables
4. Graphs to Equations
5. Function Notation and Intersection fo Graphs
6. Applications of Absolute Functions
7. Solving Absolute Value Functions Graphically
8. Solving absolute value equations. (Algebraically)
9. Absolute Value Function Toolkit
VI. Unit 6 - Quadratic Functions
1. Situations and Graphs
2. Graphing Calculator Exploration
3. Equations to Tables and Graphs
4. Tables to Equations
5. Graphs to Equations
6. Function Notation and Intersections of Graphs
7. Finding Equations from the x-intercepts (Zeros or Roots).
8. Multiplying lines and Multiplying Binomials
9. W Finding Area and Dimensions of Rectangles
10. Find the x-intercepts Algebraically by Quadratic Formula.
11. Find the x-intercepts by completing the square method
12. Find the vertex by completing the square method
13. Quadratic Function Toolkit
14. Vertical Motion Notes and Examples
15. Vertical Motion Model Problems
VII. Unit 7 - Square Root Functions
1. WS#1 Situations and Graphs
2. WS#2 Graphing Calculator Exploration
3. WS#3 Equations to Tables and Graphs
4. WS#4 Tables to Equations
5. WS#5 Graphs to Equations
6. WS#6 Function Notation and Intersections of Graphs
7. WS#7 Applications and Models
Email: Marjorie Ader at mader@k12.colostate.edu
Return to Project Page
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A cork of density 0.15 g/cm^3 floats in a bucket of water with 10 cm^3 of its volume above the surface of water.Find the mass of the cork.
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do u know how to do this?
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\[d=\frac{m}{V}\] Therefore, \[m=dV\] Hope this helps, somehow. I'm unsure.
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how?can u solve it?
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help tomas......
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i think you need to use Archimedus law or something
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ya so now?
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also \[F_a>mg\] if that matters lol
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i know but then i am stuck at hw to find the missin volume..............
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u hv the same problem?
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i posted fr u
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jacob help man!
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wait sachin
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I think you have to use Archimedes Principle and Density- Mass relationship. Still I am not getting the equation.
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i know but how????
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how is d=m/g
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Sorry, i was saying whether we can use density = mass/volume?
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yeah we can use anythin
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Then it is simple. Mass= volume X density, so Mass= 10 * 0.15= 1.5 gm (Still u check the answer)
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AADARSH!did u see my diagram and read the question again ....
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answer is wrong
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Saw it, but confused about the equation
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volume is 10+some x
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I will consult and say u tomorrow
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\[\rho=0.15\,g/cm^3\quad,\quad \rho_w=1\,g/cm^3\quad,\quad V\uparrow=10\,cm^3\]\[\rho Vg=\rho_w V\downarrow g\quad\Rightarrow\quad \frac{V}{V\downarrow}=\frac{V\uparrow+V\downarrow}{V\downarrow}=
rho\rho_w}{\rho_w-\rho}V\uparrow\]\[m=\frac{0.15}{0.85}\cdot 10\,g=1.765\,g\]
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nikvist, this a good answer. I understand your notation. But perhaps for the others could you explain in words what you've done here? Thanks.
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|dw:1326987424332:dw|For practical purposes water is incompressible so the submerged part would displace an amount of water equal to its own volume. so same mass that is what nikivist has done
...equate the mass of cork as a whole to mass of water displaced then manipulations gives us everything written in terms of density which we know(data in the question)
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nikvist could u explain ure answer in words or JamesJ could u do it fr him?
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hmmm..... that is wat i did.............
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Please explain in simple words. Are we to use that Density formula or not?
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I WILL TELL EVERYTHING FROM FIRST by archimedes principle, the wt of cork=wt of water displaced(buoyant force) this is because the body is floating upward forces=downward then wt of cork=
volume*density*g wt of water displaced=volume(V)*density oif water*g write volume of cork as volume of upper immersed portion+vol of downward immersed portion(V) REMEMBER IT IS V that we need to
calculate do it in simple terms after this |dw:1327073074048:dw|
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knowingV it is easy to calculate the mass of that part of cork+mass of upper part of cork...........got it adarsh?
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u need to know here that by archimedes principle,buoyant force=wt of water displaced in this case as the cork is floating the wt mg equals the buoyant force so to to keep the cork at rest
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Yeah, that's right.
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here is yours ans attachment below hope it will help you
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is replying to Can someone tell me what button the professor is hitting...
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Here's the question you clicked on:
Factorise X^2 - y^2 - 2x + 2y
• one year ago
• one year ago
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Did you try factoring out the x and y? If it helps, try rearranging the terms like so: \[x^2 - 2x - y^2 + 2y\] and try grouping/factoring the x out and the y out
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Actually i tried that and it doesnt work... grouping doesnt work... Completing the square doesnt work...
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x^2 - y^2 - 2x + 2y (x^2 - y^2) + ( - 2x + 2y) (x - y)(x + y) + ( - 2x + 2y) (x - y)(x + y) - 2( x - y) See where to go from here?
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@jim_thompson5910 started you off great. There is still something in common you could factor out to further simplify it. Think about it
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I think he left it out on purpose
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Thanks you so much jim.. i know how to complete it now. Thanks
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is replying to Can someone tell me what button the professor is hitting...
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Reeb Foliation
leaves of the foliation, and all foliations are assumed to be at least of class. C2.
Reeb Foliation is described in multiple online sources, as addition to our editors' articles, see section below for printable documents, Reeb Foliation books and related discussion.
Suggested Pdf Resources
Suggested News Resources
We had concluded before that the in-spiraling trajectories must form something like a “Reeb foliation” on the way to the unmoving horizon. He suddenly realized that an “anchored rotating Reeb
foliation” is the answer.
Suggested Web Resources
Great care has been taken to prepare the information on this page. Elements of the content come from factual and lexical knowledge databases, realmagick.com library and third-party sources. We
appreciate your suggestions and comments on further improvements of the site.
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Properties of the Regression Line
Regression is concerned with the study of relationship among variables. The aim of regression (or regression analysis) is to make models for prediction and for making other inferences. Two variables
or more than two variables may be treated by regression. Regression line usually written as
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Dynamics Control of the Complex Systems via Nondifferentiability
Journal of Applied Mathematics
Volume 2013 (2013), Article ID 137056, 12 pages
Research Article
Dynamics Control of the Complex Systems via Nondifferentiability
^1Faculty of Materials Science and Engineering, “Gheorghe Asachi” Technical University of Iaşi, 41 D. Mangeron Boulevard, 700050 Iaşi, Romania
^2Faculty of Civil Engineering and Building Services, “Gheorghe Asachi” Technical University of Iaşi, 1 D. Mangeron Boulevard, 700050 Iaşi, Romania
^3Faculty of Textile, Leather Engineering and Industrial Management, “Gheorghe Asachi” Technical University of Iaşi, Dimitrie Mangeron 29, 700050 Iaşi, Romania
^4Institute of Macromolecular Chemistry Petru Poni Iaşi, Aleea Grigore Ghica Voda, No. 41A, 700487 Iaşi, Romania
^5Physics Department, Faculty of Machine Manufacturing and Industrial Management, “Gheorghe Asachi” Technical University of Iaşi, Professor Dr. Docent Dimitrie Mangeron Road, No. 59A, 700050 Iaşi,
^6Lasers, Atoms and Molecules Physics Laboratory, University of Science and Technology, Villeneuve d’Ascq, 59655 Lille, France
Received 29 May 2013; Accepted 12 July 2013
Academic Editor: Zhiwei Gao
Copyright © 2013 Carmen Nejneru et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any
medium, provided the original work is properly cited.
A new topic in the analyses of complex systems dynamics, considering that the movements of complex system entities take place on continuum but nondifferentiable curves, is proposed. In this way, some
properties of complex systems (barotropic-type behaviour, self-similarity behaviour, chaoticity through turbulence and stochasticization, etc.) are controlled through nondifferentiability of motion
curves. These behaviours can simulate the standard properties of the complex systems (emergence, self-organization, adaptability, etc.).
1. Introduction
Complex systems are very large interdisciplinary research topics that have been intensively studied, particularly since the 1980s, by means of a combination of basic theory, derived especially from
physics and computer simulation. Such kind of systems is composed of many interacting elemental units that were called “agents.” Examples of complex systems can be found in human societies, the
brain, internet, ecosystems, biological evolution, stock markets, economies, and many others [1–4].
The way in which such a system manifests cannot be predicted only by the behaviour of individual elements or by adding their behavior but is determined by the manner in which the elements relate to
influence global behaviour. Among the most significant properties of complex systems are emergence, self-organization, adaptability, and so forth [5–9].
The emergence of a complex system can be represented by that state of the whole which does not include the sum of its elements.
Self-organization is another characteristic of complex systems. Examples of organization that manifests in nature are found from cells to organisms, ecosystems, and also planets, stars, and galaxies
Another characteristic of complex systems is adaptability. This occurs when the system changes in response to some information. Continuously oscillating between equilibrium and disorder, complex
systems are not “rigid.” Any modification induced at microlevel generates a series of fluctuations, exploring new states of self-organization [5–9].
An example of a complex system is represented by polymers. Their forms present a multitude of organizations starting from simple, linear chains of identical structural units to very complex chains
consisting of sequences of amino acids that form the building blocks of living things. Probably one of the most intriguing complex systems in nature is DNA, which creates cells by means of a simple
but very elegant language. It is responsible for the remarkable way in which individual cells organize into complex systems like organs, and these organs form even more complex systems like
organisms. The study of complex systems can offer a glimpse about the realistic dynamics of polymers, solving difficult problems as protein folding [6–10].
Correspondingly, the theoretical models that describe the complex systems dynamics become sophisticated and ambiguous too [6–10]. However the situation can be standardized taking into account that
the complexity of interaction process imposes various temporal resolution scales, and the pattern evolution imposes different degrees of freedom.
In order to develop a new theoretical model we must admit that the complex systems that display chaotic behaviour are known to acquire self-similarity (space-time structures seem to appear) in
association with strong fluctuations at all possible space-time scales [6–8]. Then, for temporal scales that are large with respect to the inverse of the highest Lyapunov exponent, the deterministic
trajectories are replaced by a collection of potential trajectories and the concept of definite positions by that of probability density [11–14]. One of the most interesting examples is the
collisions processes in plasma discharge as a complex system, where the dynamics of the particles can be described by Levi-type movements [15]: between two successive collisions, the particle
trajectory is a straight line, with the trajectory becoming nondifferentiable in the impact point, which implies that there are left and right derivatives in this point. We note that the
Brownian-type motion is a particular case of Levi-type motion [16].
Since the nondifferentiability appears as a universal property of the complex systems, it is necessary to construct a nondifferentiable physics. In such conjecture, by considering that the complexity
of the interactions processes is replaced by nondifferentiability, it is no longer necessary to use the whole classical “arsenal” of quantities from the standard physics (differentiable physics) [12–
14, 17, 18].
This topic was developed in [12–14, 19–23] using the scale relativity theory (SRT) [12–14]. In the framework of SRT we assume that the movements of complex system entities take place on continuous
but nondifferentiable curves (fractal curves) so that all physical phenomena involved in the dynamics depend not only on the space-time coordinates but also on the space-time scales resolution. From
such a perspective, the physical quantities that describe the dynamics of complex systems may be considered fractal functions [12–14]. Moreover, the entities of the complex system may be reduced to
and identified with their own trajectories, so that the complex system will behave as a special interactionless “fluid” by means of its geodesics in a nondifferentiable (fractal) space.
In the present paper, we propose a new topic to analyse the complex systems dynamics using SRT. Considering that the entities of the complex system are moving on continuous but nondifferentiable
curves, we show that the control of different behaviours of these systems implies nondifferentiability.
2. Geodesics Equations
Considering that the dynamics of the complex system entities take place on continuous but nondifferentiable curves, that is, fractal curves (e.g., the Koch curve, the Peano curve, or the Weierstrass
curve [11–14, 17, 18]), they are given by the fractal operator (for details see Appendices A and B): where is the complex velocity, is the differentiable and resolution scale independent velocity, is
the nondifferentiable and resolution scale dependent velocity, is the convective term, is the dissipative term, is the scale resolution identified here with by substitution principle [12–14], is the
fractal dimension of the movement curves, is the reference length scale, is the reference time scale, and is the Nottale coefficient specific to fractal-nonfractal transition [12–14]. In the case of
fractal dimension , we can use any definition (the Hausdorff-Besicovitch fractal dimension, the Kolmogorov fractal dimension, etc. [11]), but once such definition is accepted, it has to be constant
over the entire complex system dynamics analysis.
Applying the fractal operator (1) to the complex velocity (2) and accepting a generalized inertial principle (a generalization of Nottale’s principle of scale covariance [12–14]), we obtain the
geodesics in the form of the Navier-Stokes-type equations:
Equation (4) shows that at any point of a fractal path, local acceleration, , convection, , and dissipation, , is in equilibrium. According to [2–6], the complex system can be assimilated to a
“rheological” fractal fluid. The “rheology” of the fractal fluid gives hysteretic properties to the complex system (the complex system has a hysteresis cycle, memory, etc.).
Since the movement of the complex system’s entities lacks interaction, we practically make use of self-convection and self-dissipation type mechanisms.
3. Fractal Hydrodynamics Model
For irrotational motionwe can choose of the form or explicitly, with ,where is an amplitude and a phase. The function by means of ln defines the velocity scalar potential
By substituting ((7a)–(7c)) in (4) and separating the real and imaginary parts, up to an arbitrary phase factor which may be at zero by a suitable choice of the phase , we obtain with being the
fractal potential and the rest mass of the complex system’s entity. The first equation (9) is the momentum conservation law, the second equation (10) is the density conservation law, and they define
together the fractal hydrodynamics (FH) model. We note that, for Peano-type movements in fractal dimension and supposing that fractal potential is assimilated with the pressure, the fractal
hydrodynamic model becomes the standard hydrodynamic (for details see [19–23]).
The fractal potential (11) comes from the nondifferentiability and must be considered as a kinetic term and not as a potential one. Moreover, the fractal potential (11) can generate a viscosity
stress tensor type [22, 23] of which divergence is equal to the usual force density associated with
4. Barotropic-Type Behaviours of the Complex Systems via Nondifferentiability
For a barotropic-type behaviour of the complex system [24], with being the critical velocity (for details see [24]); the FH equations become
In the following, using (15a) and (15b) in a plane symmetry, we analyze the complex fluid dynamics. The presence of an external field is specified by adequate initial and boundary conditions (e.g.,
spatial-temporal Gaussian). In this situation, let us introduce the normalized coordinateswhere , , and are critical parameters of the complex fluid (in [19, 25–27] the complex fluid is identified
with a laser produced plasma; in such a context is the plasma pulsation, is the density of plasma at thermodynamic equilibrium, is the inverse of the Debye length, and is the ion acoustic speed).
Then (15a) and (15b) become
For the numerical integration we will impose the initial conditionsas well as the boundary conditions
By using the finite differences method [28], the system ((17a)–(17c)) with the initial conditions ((18a)–(18e)) and the boundary ones ((19a)–(19g)) was numerically resolved.
In Figures 1(a), 1(b), and 1(c), three-dimensional dependences of the normalized density , normalized velocities, and , on the normalized coordinates, and , are given for the normalized time . Also
in Figures 1(d), 1(e), and 1(f) the two-dimensional contours of the normalized density , and normalized velocities, and , are given for the same normalized time. The followings result in (i) the
generation of two complex fluid structures; (ii) the symmetry of the normalized velocity, , with respect to symmetry axis of the spatial-temporal Gaussian; and (iii) vertices at the complex fluid
periphery for the normalized velocity field, . These results are in agreement with the experimental data on the behaviours of a laser produced aluminium plasma [19, 26, 27, 29–33].
Starting from these numerical solutions we can build the normalized current density See Figures 2(a)–2(c).
It results in the splitting of the complex fluid into two structures. By assimilating the complex fluid with a laser produced aluminium plasma, the previous theoretical results are in agreement with
the experimental data [32, 33].
5. Self-Similarity Behaviour of the Complex Systems via Nondifferentiability
Neglecting both convection and dispersion (4) takes the form or, by separating the resolution scales, for the differentiable scale and for fractal scale. The velocity fields are totally separated,
first applying (22) and (23) to the Δ operator, that is, then, substituting in the transformed equations the dissipative terms by using (22) and (23). It results in the Kirchhoff-type equations
For one-dimensional case, the previous equations with the substitutionstake the unitary form
On (27), we impose for “clamping” conditions at
and for boundary conditions at
These four boundary conditions in associated with the two initial conditionsimply a unique solution to (27); see Figure 3.
Owing to the scaling , we seek a solution of (27) in the form where the self-similarity variable is
The boundary conditions for are derived from those for :
Substituting this self-similar form of into (27) yields the following equation for the self-similar solution :
Imposing that matches the initial condition for large implies that as shown with the help of an integral of motion. This last condition, combined with the previous ones, yields a unique self-similar
solution to (34): where we have introduced the Fresnel sine integral, , also arising in diffraction theory.
Equation (36) describes a self-similar solution. This reflects the dispersive nature of (27), see Figure 4.
In these conditions the phase associated with the velocity scalar potential is presented in Figure 5.
The states density associated with the same velocity scalar potential has the same behaviour as the one presented in Figure 5. It results in a self-similarity behaviour.
The theoretical oscillations (obtained by means of self-elimination of the dissipative terms between differentiable and fractal scales) can explain the experimental ionic oscillations of the current
density that take place in a laser produced aluminium plasma [19, 26, 27, 29]. In Figure 6 we give by comparison the experimental curve (red curve, which reflects the ionic oscillations of the
current density with time that take place in a laser produced aluminium plasma at a laser power of 40mJ/pulse) and the theoretical curve (blue curve). We note that in Figure 6 we used the normalized
It results in a good agreement (correlation factor 0.81) between theoretical model and the experimental data.
6. Chaoticity through Turbulence and Stochasticization via Nondifferentiability
Through the fractal velocities field, , the specific fractal potential is a measure of nondifferentiability of the complex system particle trajectories, that is, of their chaoticity.
Since the position vector of the particle is assimilated with a stochastic process of Wiener type (for details see [11–13], is not only the scalar potential of a complex speed (through ) in the frame
of fractal hydrodynamics but also density of probability (through ) in the frame of a Schrödinger-type theory.
It results in the equivalence between the formalism of the fractal hydrodynamics and the one of the Schrödinger-type equation. Moreover, the chaoticity, either through turbulence in the fractal
hydrodynamics approach or through stochasticization in the Schrödinger-type approach, is generated only by the nondifferentiability of the movement trajectories in a fractal space. In this way the
nondifferentiability becomes a “control parameter” of the complex system dynamics.
6.1. Full and Fractional Speed Scalar Potential Revivals in the Infinite Square Well: Various Criteria of Evolution to Chaos
6.1.1. Infinite Square Well System
Let us consider that the external perturbation applied to complex system simulates, in our opinion, one-dimension square well system. After solving the time-dependent Schrödinger-type equation
according to the method described in [23] we obtain the discrete eigenvalues and the eigenfunctions where is the well’s width and is the rest mass of the fractal fluid particle.
6.1.2. Time Scales
Some time scales of a speed potential evolution are contained in the coefficients of the Taylor series of the quantized energy levels around the main energy (see method from [34]) where often the
zero of energy is shifted to remove the term. Regrouping the infinite square well energies (17a)–(17c) in this form gives And comparing (39) and (40) we relate
We note that the time scale does not depend on the mean energy level . This will provide us with a “universal” time scale for describing speed potential evolution that does not depend on the particle
average energy.
6.1.3. Time Evolution
We write the particle’s time speed scalar potential in the infinite square well as
We expand this speed scalar potential using the energy eigenstate basis with
Using the time scale , the time evolution in the energy eigenstate basis was found from Schrödinger-type equation to be
6.1.4. Simulation of the Evolution to Chaos Criteria
Now, the full and fractional revivals formalism may be applied. Full and fractional revivals of a speed scalar potential in the infinite square well occur when a speed scalar potential evolves in
time to a state that can be described as a collection of spatially distributed subspeed scalar potentials that each closely reproduces the initial speed scalar potential shape; see for details [34].
Therefore, the full and fractional revivals of a speed scalar potential in the infinite square well impliy either or for any time and , , and integers. In any of the situations above, either for or
for , we can introduce Reynolds-type criterions wherehave the usual signification from fluid mechanics [24]. Up to the values the fractal fluids become turbulent. Then through and it is formally
simulated through the criterion of evolution to chaos via Feigenbaum scenario (cascade of period doubling bifurcations), while through with and the criterion of evolution to chaos via a cascade of
subharmonic bifurcations.
We admit that in any of these two situations mentioned above the fractal velocity (7c) is null, since ; meanwhile the differential velocity (7b) is not zero, since the phase is not constant, with the
increase of the systems phase incoherence being associated with the increase in turbulence of fractal fluid.
We note that in the standard model (Landau’s scenario [24, 35]) the Fourier spectrum is always discrete and cannot approximate a continuum spectrum than that in case of a large number of frequencies
that will generate an unlimited number of spectral components as a result of their beats which appear thanks to the presence of nonlinearities in the system. Yet, considering standard model, the flow
can never be truly chaotic, because, in case of multiple periodic functions, correlations tend to be not null but having an oscillating character. Therefore, Landau’s scenario can describe transition
towards chaotic behaviour only in a system with an infinite number of degrees of freedom, such as a fluid. In our case, because the Reynolds numbers (48) present scale dependencies (49c), when for ,
the fractals physical values that describe the dynamics of the system are no longer defined. So, in this approximation, a simulation of a system with an infinite number of degrees of freedom is used.
We note that the results from the present paragraph permit the chaoticity and self-structuring control by means of nondifferentiability. Thus the experimental results on the formation, dynamics, and
evolution towards chaos of complex space charge structures that emerge in front of a positively biased electrode immersed in a quiescent plasma from [20] are in agreement with our previous
theoretical results.
7. Conclusions
In the present paper, we propose a new topic in the control of complex systems dynamics using the nondifferentiability of complex system movement curves. This topic was developed through the scale
relativity approach. Considering the dynamics of complex system entities that take place on fractal curves, we show that the control of different behaviours of these systems implies
nondifferentiability. The main conclusions of the present paper are as follows: (i) the geodesics equations in the form of a Navier-Stokes-type equation are obtained. It results in the rheologic
properties of the complex system which implies memory and so forth; (ii) considering that the fractal fluid flows are irrotational, fractal hydrodynamics equations are obtained. These equations are
formed from the density and momentum conservations laws, wih the presence of nondifferentiability being induced by the fractal potential; (iii) a barotropic-type behaviour of the complex system is
numerically simulated using FH equations. In this way we highlight the self-multiplication mechanism of the substructures that constitute the complex system; (iv) the implications of self-similarity
in the dynamics of the complex system are presented; (v) the chaoticity through turbulence and stochaticization via nondifferentiability are obtained; (vi) these previous behaviours can simulate the
standard properties of the complex system.
In this way, standard properties of complex systems such as emergence, self-organisation, and adaptability are controlled through nondifferentiability of motion curves of the sub-systems that compose
the complex system. We note that general aspects of the dynamics control of the complex system are described in [36], while the concrete cases of this control are presented in [37–39].
A. Consequences of Nondifferentiability
The nondifferentiability implies the following [12–14, 19–22].
(i) A continuous and a nondifferentiable curve (or almost nowhere differentiable) is explicitly scale dependent. This means that its length tends to infinity, when the scale interval tends to zero.
Therefore, a continuous and nondifferentiable space is fractal, in the general meaning given by Mandelbrot to this concept [11].
(ii) Physical quantities will be expressed through fractal functions, namely, through functions that are dependent both on spatial and temporal coordinates and on resolution scale. The invariance of
the physical quantities in relation with the resolution scale generates special types of transformations, called resolution scale transformations. Particularly, the differentiality of the generalized
spatial coordinates, , takes the form where is the classical differential element and is a differential fractal one.
(iii) There is infinity of fractal curves (geodesics) relating to any couple of points (or starting from any point) and applied for any scale. The phenomenon can be easily understood at the level of
fractal surfaces, which, in their turn, can be described in terms of fractal distribution of conic points of positive and negative infinite curvatures. As a consequence, we have replaced velocity on
a particular geodesic by fractal velocity field of the whole infinite ensemble of geodesics. This representation is similar to that of fluid mechanics where the motion of the fluid is described in
terms of its velocity field, density, and pressure. We will, indeed, recover the fundamental equations of fluid mechanics (Euler and continuity equations), but we will write them in terms of a
density of probability (as defined by the set of geodesics) instead of a density of matter and adding an additional term of quantum pressure (the expression of fractal geometry).
(iv) The local differential time invariance is broken, so the time-derivative of the fractal field can be written twofold:
Both definitions are equivalent in the differentiable case . In the nondifferentiable situation, these definitions are no longer valid, since limits are not defined anymore. Fractal theory defines
physics in relationship with the function behaviour during the “zoom” operation on the time resolution , which is here identified with the differential element (substitution principle), which is
considered an independent variable. The standard field is therefore replaced by fractal field , explicitly dependent on time resolution interval, whose derivative is not defined at the unnoticeable
limit . As a consequence, this leads to the two derivatives of the fractal field as explicit functions of the two variables and
Notation “+” corresponds to the forward process, while “−” to the backward one.
Let us particularize (A.3) for the spatial coordinates. results in
Since, according to [12–14, 19–22], we can write from (A.4) by averaging, it results in
(v) The differential fractal part satisfies the fractal equation: where are some constant coefficients, is the time differential, is the reference time scale, and is a constant fractal dimension. We
note that the use of any Kolmogorov or Hausdorff definition [12–14, 19–22] can be accepted for fractal dimension, but once a certain definition is admitted, it should be used until the end of
analyzed dynamics.
(vi) The local differential time reflection invariance is recovered by combining the two derivatives, and , in the complex operator:
Applying this operator to the “position vector,” a complex velocity yields with
The real part, , of the complex velocity represents the standard classical velocity, which does not depend on resolution, while the imaginary part, , is a new quantity coming from fractality.
B. Covariant Derivative
Let us now assume that curves describing movements of complex systems (continuous but nondifferentiable) are immersed in a 3-dimensional space and that of components is the position vector of a point
on the curve. Let us also consider the fractal field and expand its total differential up to the second order:
Relations (B.1) are valid in any point both for the spatial manifold and for the points on the fractal curve selected in relations (B.1). Hence, the forward and backward average values of these
relations take the form
The following aspects should be mentioned: the mean value of function and its derivatives coincide with themselves, and the differentials and are independent; therefore, the average of their products
coincides with the product of averages. Consequently, (B.2) becomes or more, using (A.4) and (A.6),
Even if the average value of the fractal coordinate is null (see (A.6)), for higher order of fractal coordinate average, the situation can still be different. Let us focus on the averages . If ,
these averages are zero due to the independence of and . So, using (A.7), we can write
Then, (B.4) may be written as follows:
If we divide by and neglect the terms containing differential factors, (B.6) is reduced to
These relations also allow us to define the operators:
Under these circumstances, let us calculate . Taking into account (B.8), (A.8), and (A.9), we will obtain
This relation also allows us to define the fractal operator:
Particularly, by choosing where is the reference length scale, the fractal operator (B.10) (covariant derivative) takes the usual form:
1. B.-Y. Yaneer, Dynamics of Complex Systems, The Advanced Book Program, Addison-Wesley, Reading, Mass, USA, 1997.
2. S. V. Buldyrev, R. Parshani, G. Paul, H. E. Stanley, and S. Havlin, “Catastrophic cascade of failures in interdependent networks,” Nature, vol. 464, no. 7291, pp. 1025–1028, 2010. View at
Publisher · View at Google Scholar · View at Scopus
3. K. Kaneko, Complex Systems: Chaos and Beyond, A Constructive Approach with Applications in Life Sciences, 2001.
4. C. Gros, Complex and Adaptative Dynamical Systems: A Primer, 2011.
5. R. Badii and A. Politi, Complexity: Hierarchical Structure and Scaling in Physics, Cambridge University Press, Cambridge, UK, 1997.
6. M. Mitchell, Complexity: A Guided Tour, Oxford University Press, Oxford, UK, 2009.
7. G. W. Flake, The Computational Beauty of Nature, The MIT Press, Cambridge, Mass, USA, 1998.
8. A. T. Winfree, The Geometry of Biological Time, Springer, New York, NY, USA, 2nd edition, 2000.
9. P. Davies, The New Physics, Cambridge University Press, 1992.
10. C. H. Bennett, “How to define complexity in physics, and why,” in Complexity, Entropy, and the Physics of Information, W. H. Zurek, Ed., pp. 443–454, Addison-Wesley, Reading, Mass, USA, 1990.
11. Mandelbrot and B. Benoit, The Fractal Geometry of Nature, W. H. Freeman, New York, NY, USA, 1983.
12. L. Nottale, Fractal Space-Time and Microphysics: Towards a Theory of Scale Relativity, World Scientific, Singapore, 1993.
13. L. Nottale, Scale Relativity and Fractal Space-Time—A New Approach To Unifying Relativity and Quantum Mechanics, Imperial College Press, London, UK, 2011.
14. L. Nottale, “Fractals and the quantum theory of spacetime,” International Journal of Modern Physics A, vol. 4, pp. 5047–5117, 1989.
15. G. M. Viswanathan, F. Bartumeus, S. V. Buldyrev et al., “Lévy flight random searches in biological phenomena,” Physica A, vol. 314, no. 1–4, pp. 208–213, 2002. View at Publisher · View at Google
Scholar · View at Zentralblatt MATH · View at Scopus
16. M. F. Shlesinger, J. Klafter, and G. Zumofen, “Above, below and beyond Brownian motion,” American Journal of Physics, vol. 67, no. 12, pp. 1253–1259, 1999. View at Publisher · View at Google
Scholar · View at Scopus
17. M. S. El Naschie and O. E. Rossler, “Quantum mechanics and chaotic fractals,” Chaos, Solitons and Fractals, vol. 4, no. 3, pp. 307–309, 1994. View at Publisher · View at Google Scholar · View at
18. P. Weibel, G. Ord, and O. E. Rosler, Space Time Physics and Fractility, Springer, New York, NY, USA, 2005.
19. S. Gurlui, M. Agop, P. Nica, M. Ziskind, and C. Focşa, “Experimental and theoretical investigations of transitory phenomena in high-fluence laser ablation plasma,” Physical Review E, vol. 78,
Article ID 026405, 2008.
20. M. Agop, D. G. Dimitriu, O. Niculescu, E. Poll, and V. Radu, “Experimental and theoretical evidence for the chaotic dynamics of complex structures,” Physica Scripta, vol. 87, no. 4, Article ID
045501, 2013. View at Publisher · View at Google Scholar
21. M. Agop, N. Forna, I. Casian-Botez, and C. Bejenariu, “New theoretical approach of the physical processes in nanostructures,” Journal of Computational and Theoretical Nanoscience, vol. 5, no. 4,
pp. 483–489, 2008. View at Scopus
22. I. Casian-Botez, M. Agop, P. Nica, V. Paun, and G. V. Munceleanu, “Conductive and convective types behaviors at nano-time scales,” Journal of Computational and Theoretical Nanoscience, vol. 7,
no. 11, pp. 2271–2280, 2010. View at Publisher · View at Google Scholar · View at Scopus
23. M. Agop, P. E. Nica, S. Gurlui, C. Focşa, V. P. Păun, and M. Colotin, “Implications of an extended fractal hydrodynamic model,” The European Physical Journal D, vol. 56, no. 3, pp. 405–419, 2010.
View at Publisher · View at Google Scholar
24. E. M. Lifshiëtìs and L. D. Landau, Fluid Mechanics, Pergamon Press, 1987.
25. G. V. Munceleanu, V.-P. Paun, I. Casian-Botez, and M. Agop, “The microscopic-macroscopic scale transformation through a chaos scenario in the fractal space-time theory,” International Journal of
Bifurcation and Chaos, vol. 21, no. 2, pp. 603–618, 2011. View at Publisher · View at Google Scholar · View at Scopus
26. P. Nica, M. Agop, S. Gurlui, C. Bejinariu, and C. Focsa, “Characterization of aluminum laser produced plasma by target current measurements,” Japanese Journal of Applied Physics, vol. 51, Article
ID 106102, 10 pages, 2012. View at Publisher · View at Google Scholar
27. P. Nica, P. Vizureanu, M. Agop et al., “Experimental and theoretical aspects of aluminum expanding laser plasma,” Japanese Journal of Applied Physics, vol. 48, no. 6, Article ID 066001, 2009.
View at Publisher · View at Google Scholar · View at Scopus
28. O. C. Zienkievicz and R. L. Taylor, The Finite Element Method, McGraw-Hill, New York, NY, USA, 1991.
29. C. Ursu, O. G. Pompilian, S. Gurlui et al., “Al[2]O[3] ceramics under high-fluence irradiation: plasma plume dynamics through space- and time-resolved optical emission spectroscopy,” Applied
Physics A, vol. 101, no. 1, pp. 153–159, 2010. View at Publisher · View at Google Scholar · View at Scopus
30. O. G. Pompilian, S. Gurlui, P. Nemec, V. Nazabal, M. Ziskind, and C. Focşa, “Plasma diagnostics in pulsed laser deposition of GaLaS chalcogenides,” Applied Surface Science, vol. 278, pp. 352–356,
31. C. Focsa, P. Nemec, M. Ziskind, C. Ursu, S. Gurlui, and V. Nazabal, “Laser ablation of As[x]Se[100-x] chalcogenide glasses: plume investigations,” Applied Surface Science, vol. 255, no. 10, pp.
5307–5311, 2009. View at Publisher · View at Google Scholar · View at Scopus
32. C. Ursu, S. Gurlui, C. Focsa, and G. Popa, “Space- and time-resolved optical diagnosis for the study of laser ablation plasma dynamics,” Nuclear Instruments and Methods in Physics Research B,
vol. 267, no. 2, pp. 446–450, 2009. View at Publisher · View at Google Scholar · View at Scopus
33. S. Gurlui and C. Focşa, “Laser ablation transient plasma structures expansion in vacuum,” IEEE Transactions on Plasma Science, vol. 39, no. 11, pp. 2820–2821, 2011. View at Publisher · View at
Google Scholar · View at Scopus
34. D. L. Aronstein and C. R. Stround Jr., “Fractional wave-function revivals in the infinite square well,” Physical Review A, vol. 55, no. 6, pp. 1050–2947, 1997. View at Publisher · View at Google
35. C. P. Cristescu, Dinamici Neliniare Si Haos, Fundamente Teoretice Si Aplicatii, Academiei Romane, Bucuresti, Romania, 2008.
36. J. Zhang and W. Tang, “Optimal control for a class of chaotic systems,” Journal of Applied Mathematics, vol. 2012, Article ID 859542, 20 pages, 2012. View at Publisher · View at Google Scholar
37. J. B. Polly and J. M. McDonough, “Application of the poor Man's navier-stokes equations to real-time control of fluid flow,” Journal of Applied Mathematics, vol. 2012, Article ID 746752, 18
pages, 2012. View at Publisher · View at Google Scholar
38. E. S. Bacaita, C. Bejinariu, B. Zoltan, et al., “Nonlinearities in drug release process from polymeric microparticles: long-time-scale behaviour,” Journal of Applied Mathematics, vol. 2012,
Article ID 653720, 26 pages, 2012. View at Publisher · View at Google Scholar
39. K. Yuan, J. Cao, and S. Fei, “Synchronization of coupled networks with mixed delays by intermittent control,” Journal of Applied Mathematics, vol. 2012, Article ID 927609, 13 pages, 2012. View at
Publisher · View at Google Scholar
|
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Direct Products and Sums of Modules - Notation - 2nd Post
August 9th 2013, 06:57 PM
Direct Products and Sums of Modules - Notation - 2nd Post
This post follows my previous post on John Dauns book "Modules and Rings"
My issue is understanding the notation on Section 1-2, subsection 1-2.1 (see attachment).
Dauns is dealing with the product $\Pi \{ M_i | i \in I \} \equiv \Pi M_i$ and states in (ii) - see attachement
Alternatively, the product can be viewed as consisting of all strings or sets
$x = \{ x_i | i \in I \} \equiv (x_i)_{i \in I} \equiv (x_i) \equiv ( \_ \_ \_ \ , x_i , \_ \_ \_ ) , x_i \in M_i;$ i-th
I am not sure of the meaning of the above set of equivalences. Can someone briefly elaborate ... preferably with a simple example
If we take the case of I = {1,2,3} and consider the product $M_1 \times M_2 \times M_3$ then does Dauns notation mean
$x = (x_1, x_2, x_3)$ where order in the triple matters (mind you if it does what are we to make of the statement $x = \{ x_i | i \in I \}$
Can someone confirm that $x = (x_1, x_2, x_3)$ is a correct interpretation of Dauns notation?
================================================== =============================================
Dauns then goes on to define the direct sum as follows:
The direct sum $\oplus \{ M_i | i \in I \} \equiv \oplus M_i$ is defined as the submodule $\oplus M_i \subseteq \Pi M_i$ consisting of those elements $x = (x_i) \in \Pi M_i$ having at most a
finite number of non-zero coordinates or components. Sometimes $\oplus M_i , \Pi M_i$ are called the external direct sum and the external direct product respectively.
================================================== ==============================================
Can someone point out the difference between $\oplus M_i , \Pi M_i$ in the case of the example involving $M_1, M_2, M_3$ - I cannot really see the difference! For example, what elements exactly
are in $\Pi M_i$ that are not in $\oplus M_i$
I would be grateful if someone can clarify these issues.
August 9th 2013, 08:33 PM
Re: Direct Products and Sums of Modules - Notation - 2nd Post
Hi Bernhard,
If the index set I is finite, then certainly the direct product and direct sum are exactly the same. Only for I infinite is the direct sum a proper subset of the direct product.
August 9th 2013, 09:51 PM
Re: Direct Products and Sums of Modules - Notation - 2nd Post
Thanks johng
Are you implicitly answering the following question in the affirmative?
"Can someone confirm that $x = (x_1, x_2, x_3)$ is a correct interpretation of Dauns notation?"
August 10th 2013, 06:42 AM
Re: Direct Products and Sums of Modules - Notation - 2nd Post
Hi Peter,
See the attachment.
Attachment 28970
August 10th 2013, 03:23 PM
Re: Direct Products and Sums of Modules - Notation - 2nd Post
Thanks Johng ... Most helpful post!
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