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Results 1 - 10 of 39
- IEEE TRANSACTIONS ON INFORMATION THEORY , 2003
"... A new class of distances appropriate for measuring similarity relations between sequences, say one type of similarity per distance, is studied. We propose a new "normalized information
distance", based on the noncomputable notion of Kolmogorov complexity, and show that it is in this class and it min ..."
Cited by 192 (29 self)
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A new class of distances appropriate for measuring similarity relations between sequences, say one type of similarity per distance, is studied. We propose a new "normalized information distance",
based on the noncomputable notion of Kolmogorov complexity, and show that it is in this class and it minorizes every computable distance in the class (that is, it is universal in that it discovers
all computable similarities). We demonstrate that it is a metric and call it the similarity metric. This theory forms the foundation for a new practical tool. To evidence generality and robustness we
give two distinctive applications in widely divergent areas using standard compression programs like gzip and GenCompress. First, we compare whole mitochondrial genomes and infer their evolutionary
history. This results in a first completely automatic computed whole mitochondrial phylogeny tree. Secondly, we fully automatically compute the language tree of 52 different languages.
- in Advances in Minimum Description Length: Theory and Applications. 2005
"... ..."
- Trends in Cognitive Sciences , 2003
"... This article reviews research exploring the idea that simplicity does, indeed, drive a wide range of cognitive processes. We outline mathematical theory, computational results, and empirical
data underpinning this viewpoint. Key words: simplicity, Kolmogorov complexity, codes, learning, induction, B ..."
Cited by 48 (2 self)
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This article reviews research exploring the idea that simplicity does, indeed, drive a wide range of cognitive processes. We outline mathematical theory, computational results, and empirical data
underpinning this viewpoint. Key words: simplicity, Kolmogorov complexity, codes, learning, induction, Bayesian inference 30-word summary:This article outlines the proposal that many aspects of
cognition, from perception, to language acquisition, to high-level cognition involve finding patterns that provide the simplest explanation of available data. 3 The cognitive system finds patterns in
the data that it receives. Perception involves finding patterns in the external world, from sensory input. Language acquisition involves finding patterns in linguistic input, to determine the
structure of the language. High-level cognition involves finding patterns in information, to form categories, and to infer causal relations. Simplicity and the problem of induction A fundamental
puzzle is what we term the problem of induction: infinitely many patterns are compatible with any finite set of data (see Box 1). So, for example, an infinity of curves pass through any finite set of
points (Box 1a); an infinity of symbol sequences are compatible with any subsequence of symbols (Box 1b); infinitely many grammars are compatible with any finite set of observed sentences (Box 1c);
and infinitely many perceptual organizations can fit any specific visual input (Box 1d). What principle allows the cognitive system to solve the problem of induction, and choose appropriately from
these infinite sets of possibilities? Any such principle must meet two criteria: (i) it must solve the problem of induction successfully; (ii) it must explain empirical data in cognition. We argue
that the best approach to (i)...
- IEEE Trans. Inform. Theory
"... approach to statistics and model selection. Let data be finite binary strings and models be finite sets of binary strings. Consider model classes consisting of models of given maximal
(Kolmogorov) complexity. The “structure function ” of the given data expresses the relation between the complexity l ..."
Cited by 32 (14 self)
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approach to statistics and model selection. Let data be finite binary strings and models be finite sets of binary strings. Consider model classes consisting of models of given maximal (Kolmogorov)
complexity. The “structure function ” of the given data expresses the relation between the complexity level constraint on a model class and the least log-cardinality of a model in the class
containing the data. We show that the structure function determines all stochastic properties of the data: for every constrained model class it determines the individual best-fitting model in the
class irrespective of whether the “true ” model is in the model class considered or not. In this setting, this happens with certainty, rather than with high probability as is in the classical case.
We precisely quantify the goodness-of-fit of an individual model with respect to individual data. We show that—within the obvious constraints—every graph is realized by the structure function of some
data. We determine the (un)computability properties of the various functions contemplated and of the “algorithmic minimal sufficient statistic.” Index Terms— constrained minimum description length
(ML) constrained maximum likelihood (MDL) constrained best-fit model selection computability lossy compression minimal sufficient statistic non-probabilistic statistics Kolmogorov complexity,
Kolmogorov Structure function prediction sufficient statistic
, 2008
"... Inferring the causal structure that links n observables is usually basedupon detecting statistical dependences and choosing simple graphs that make the joint measure Markovian. Here we argue why
causal inference is also possible when only single observations are present. We develop a theory how to g ..."
Cited by 11 (11 self)
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Inferring the causal structure that links n observables is usually basedupon detecting statistical dependences and choosing simple graphs that make the joint measure Markovian. Here we argue why
causal inference is also possible when only single observations are present. We develop a theory how to generate causal graphs explaining similarities between single objects. To this end, we replace
the notion of conditional stochastic independence in the causal Markov condition with the vanishing of conditional algorithmic mutual information anddescribe the corresponding causal inference rules.
We explain why a consistent reformulation of causal inference in terms of algorithmic complexity implies a new inference principle that takes into account also the complexity of conditional
probability densities, making it possible to select among Markov equivalent causal graphs. This insight provides a theoretical foundation of a heuristic principle proposed in earlier work. We also
discuss how to replace Kolmogorov complexity with decidable complexity criteria. This can be seen as an algorithmic analog of replacing the empirically undecidable question of statistical
independence with practical independence tests that are based on implicit or explicit assumptions on the underlying distribution. email:
"... We compare the elementary theories of Shannon information and Kolmogorov complexity, the extent to which they have a common purpose, and where they are fundamentally different. We discuss and
relate the basic notions of both theories: Shannon entropy, Kolmogorov complexity, Shannon mutual informati ..."
Cited by 10 (2 self)
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We compare the elementary theories of Shannon information and Kolmogorov complexity, the extent to which they have a common purpose, and where they are fundamentally different. We discuss and relate
the basic notions of both theories: Shannon entropy, Kolmogorov complexity, Shannon mutual information and Kolmogorov (`algorithmic') mutual information. We explain how universal coding may be viewed
as a middle ground between the two theories. We consider Shannon's rate distortion theory, which quantifies useful (in a certain sense) information. We use the communication of information as our
guiding motif, and we explain how it relates to sequential question-answer sessions.
- In Proc. 43rd Symposium on Foundations of Computer Science , 2002
"... We vindicate, for the first time, the rightness of the original “structure function”, proposed by Kolmogorov in 1974, by showing that minimizing a two-part code consisting of a model subject to
(Kolmogorov) complexity constraints, together with a data-to-model code, produces a model of best fit (for ..."
Cited by 10 (0 self)
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We vindicate, for the first time, the rightness of the original “structure function”, proposed by Kolmogorov in 1974, by showing that minimizing a two-part code consisting of a model subject to
(Kolmogorov) complexity constraints, together with a data-to-model code, produces a model of best fit (for which the data is maximally “typical”). The method thus separates all possible model
information from the remaining accidental information. This result gives a foundation for MDL, and related methods, in model selection. Settlement of this long-standing question is the more
remarkable since the minimal randomness deficiency function (measuring maximal “typicality”) itself cannot be monotonically approximated, but the shortest two-part code can. We furthermore show that
both the structure function and the minimum randomness deficiency function can assume all shapes over their full domain (improving an independent unpublished result of Levin on the former function of
the early 70s, and extending a partial result of V’yugin on the latter function of the late 80s and also recent results on prediction loss measured by “snooping curves”). We give an explicit
realization of optimal two-part codes at all levels of model complexity. We determine the (un)computability properties of the various functions and “algorithmic sufficient statistic ” considered. In
our setting the models are finite sets, but the analysis is valid, up to logarithmic additive terms, for the model class of computable probability density functions, or the model class of total
recursive functions. 1
, 2006
"... ABSTRACT. By applying the minimality principle for model selection, one should seek the model that describes the data by a code of minimal length. Learning is viewed as data compression that
exploits the regularities or qualitative properties found in the data, in order to build a model containing t ..."
Cited by 8 (0 self)
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ABSTRACT. By applying the minimality principle for model selection, one should seek the model that describes the data by a code of minimal length. Learning is viewed as data compression that exploits
the regularities or qualitative properties found in the data, in order to build a model containing the meaningful information. The theory of causal modeling can be interpreted by this approach. The
regularities are the conditional independencies reducing a factorization and the v-structure regularities. In the absence of other regularities, a causal model is faithful and offers a minimal
description of a probability distribution. The causal interpretation of a faithful Bayesian network is motivated by the canonical representation it offers and faithfulness. A causal model decomposes
the distribution into independent atomic blocks and is able to explain all qualitative properties found in the data. The existence of faithful models depends on the additional regularities in the
data. Local structure of the conditional probability distributions allow further compression of the model. Interfering regularities, however, generate conditional independencies that do not follow
from the Markov condition. These regularities has to be incorporated into an augmented model for which the inference algorithms are adapted to take into account their influences. But for other
regularities, like patterns in a string, causality does not offer a modeling framework that leads to a minimal description. 1
"... Andrei Nikolaevich Kolmogorov was the foremost contributor to the mathematical and philosophical foundations of probability in the twentieth century, and his thinking on the topic is still
potent today. In this article we first review the three stages of Kolmogorov's work on the foundations of proba ..."
Cited by 7 (2 self)
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Andrei Nikolaevich Kolmogorov was the foremost contributor to the mathematical and philosophical foundations of probability in the twentieth century, and his thinking on the topic is still potent
today. In this article we first review the three stages of Kolmogorov's work on the foundations of probability: (1) his formulation of measure-theoretic probability, 1933, (2) his frequentist theory
of probability, 1963, and (3) his algorithmic theory of randomness, 1965--1987. We also discuss another approach to the foundations of probability, based on martingales, that Kolmogorov did not | {"url":"http://citeseerx.ist.psu.edu/showciting?cid=28553","timestamp":"2014-04-19T23:49:21Z","content_type":null,"content_length":"39422","record_id":"<urn:uuid:49856942-9607-42de-be20-e5021c1f51cd>","cc-path":"CC-MAIN-2014-15/segments/1397609537754.12/warc/CC-MAIN-20140416005217-00134-ip-10-147-4-33.ec2.internal.warc.gz"} |
Digital Signal Processing Using
Chapter 5. The Discrete Fourier
Gao Xinbo
School of E.E., Xidian Univ.
Review 1
The DTFT provides the frequency-domain (w)
representation for absolutely summable sequences.
The z-transform provides a generalized frequency-
domain (z) representation for arbitrary sequences.
Two features in common:
Defined for infinite-length sequences
Functions of continuous variable (w or z)
From the numerical computation viewpoint, these two
features are troublesome because one has to evaluate
infinite sums at uncountably infinite frequencies.
Review 2
To use Matlab, we have to truncate sequences and
then evaluate the expression at finitely many points.
The evaluation were obviously approximations to the
exact calculations.
In other words, the DTFT and the z-transform are not
numerically computable transform.
Introduction 1
Therefore we turn our attention to a numerically computable
It is obtained by sampling the DTFT transform in the
frequency domain (or the z-transform on the unit circle).
We develop this transform by analyzing periodic sequences.
From FT analysis we know that a periodic function can always
be represented by a linear combination of harmonically related
complex exponentials (which is form of sampling).
This give us the Discrete Fourier Series representation.
We extend the DFS to finite-duration sequences, which leads
to a new transform, called the Discrete Fourier Transform.
Introduction 2
The DFT avoids the two problems mentioned above
and is a numerically computable transform that is
suitable for computer implementation.
The numerical computation of the DFT for long
sequences is prohibitively time consuming.
Therefore several algorithms have been developed to
efficiently compute the DFT.
These are collectively called fast Fourier transform
(or FFT) algorithms.
The Discrete Fourier Series
Definition: Periodic sequence
~(n) ~(n kN), n, k
x x
N: the fundamental period of the sequences
From FT analysis we know that the periodic functions can be
synthesized as a linear combination of complex exponentials
whose frequencies are multiples (or harmonics) of the
fundamental frequency (2pi/N).
From the frequency-domain periodicity of the DTFT, we
conclude that there are a finite number of harmonics; the
frequencies are {2pi/N*k,k=0,1,…,N-1}.
The Discrete Fourier Series
A periodic sequence can be expressed as
N 1
~ ( n) 1 ~
X (k )e , n 0,1,
j 2N kn
N k 0
{ X ( K ), k 0,1, } are called the discrete Fourier series
coefficients, which are given by
N 1
~ ~(n)e j 2N kn , k 0,1,
X (k ) x
n 0
The discrete Fourier series representation of periodic sequences
The Discrete Fourier Series
X(k) is itself a (complex-valued) periodic
sequence with fundamental period equal to N.
j 2N
Let WN e
N 1
X (k ) DFS[ ~ (n)] ~ (n)WN
x x nk
n 0
N 1
~ (n) IDFS[ X (k )] 1
~ ~
X (k )WN
k 0
Example 5.1
Matlab Implementation
Matrix-vector multiplication:
Let x and X denote column vectors corresponding to the
primary periods of sequences x(n) and X(k), respectively.
Then X W x x 1 W * X
N N
1 W 1 WN N 1)
WN WN 0 k , n N 1
kn N
( N 1) ( N 1) 2
1 WN WN
DFS Matrix
dfs1 function(row vect.for dfs)
Function [Xk] = dfs (xn, N)
% Xk and xn are column(not row) vectors
if nargin < 2 N = length(xn)
n = 0: N-1; k = 0: N-1;
WN = exp(-j*2*pi/N);
kn = k’*n;
WNkn = WN.^kn;
Xk = WNkn * xn
Ex5.2 DFS of square wave seq.
L, k 0, N ,2 N ,
X (k ) j ( L 1) / N sin(kL / N )
sin(k / N )
1. Envelope like the sinc function;
2. Zeros occur at N/L (reciprocal of duty cycle);
3. Relation of N to density of freq. Samples;
Relation to the z-transform
Nonzero, 0 n N 1 N 1
x(n) X ( z ) x(n) z n
0, elsewhere n 0
~(n), 0 n N 1 ~
x N 1
X (k ) x(n) e
j 2N k n
0, elsewhere n 0
X (k ) X ( z ) | j 2 k
z e N
The DFS X(k) represents N evenly spaced samples of the z-
transform X(z) around the unit circle.
Relation to the DTFT
N 1 N 1
X (e ) x ( n )e
jw jwn
~ (n)e jwn
n 0 n 0
X (k ) X (e jw ) |w 2 k
Let w1 , and wk k kw1
N N
X (k ) X (e jwk ) X (e jkw1 )
The DFS is obtained by evenly sampling the DTFT at w1 intervals.
The interval w1 is the sampling interval in the frequency domain. It is called
frequency resolution because it tells us how close are the frequency samples.
Sampling and construction in the
X (k ) X ( z ) | j 2 k
, k 0,1,2,
DFS & z- z e N
x ( m)e x(m)WNkm
j 2N km
m m
~ ( n) 1 ~ N 1
1 N 1 km
IDFS x X (k )WN x(m)WN WN kn
N k 0 m
Nk 0
1 N 1 k ( n m )
x ( m ) WN x(m) (n m rN )
m N k 0 m r
x(m) (n m rN ) x(n rN )
m r r
When we sample X(z) on the unit circle, we
obtain a periodic sequence in the time domain.
This sequence is a linear combination of the
original x(n) and its infinite replicas, each
shifted by multiples of N or –N.
If x(n)=0 for n<0 and n>=N, then there will be
no overlap or aliasing in the time domain.
x(n) ~(n) for 0 n N 1
~(n) R (n) ~(n)1 0 n N 1
x ( n) x x
0 else
RN(n) is called a rectangular window of length N.
THEOREM1: Frequency Sampling
If x(n) is time-limited (finite duration) to [0,N-1], then N
samples of X(z) on the unit circle determine X(z) for all z.
Reconstruction Formula
Let x(n) be time-limited to [0,N-1]. Then from Theorem 1 we should be able to
recover the z-transform X(z) using its samples X~(k).
~ ( n) z n 1
N 1 N 1 N 1 N 1
X ( z ) x ( n) z x
N X (k )WN kn z n
n 0 n 0 n 0 k 0
1 N 1 ~ N 1 kn n 1 N 1 ~ N 1 k 1 n
X (k ) WN z X (k ) WN z
N k 0 n 0 N k 0 n 0
1 N 1
~ 1 WN kN z N
X (k ) 1 W k z 1 WN-kN=1
N k 0 N
N N 1 ~
1 z X (k )
X ( z)
1 W k z 1
k 0 N
The DTFT Interpolation Formula
1 e jwN N 1
X (k ) N 1
~ 1 e jwN
X (e )
1 e j 2k / N e jw X (k )
N 1 e j 2k / N e jw
k 0 k 0
sin wN jw N21 An interpolation polynomial
( w) 2
N sin 2
N 1 This is the DTFT interpolation
X (e ) X (k )w 2Nk
jw ~
formula to reconstruct X(ejw) from
k 0 its samples X~(k)
Since (0) 1 , we have that X(ej2pik/N)=X~(k), which
means that the interpolation is exact at sampling points.
The Discrete Fourier Transform
The discrete Fourier series provided us a mechanism
for numerically computing the discrete-time Fourier
It also alert us to a potential problem of aliasing in the
time domain.
Mathematics dictates that the sampling of the discrete-
time Fourier transform result in a periodic sequences
But most of the signals in practice are not periodic.
They are likely to be of finite duration.
The Discrete Fourier Transform
Theoretically, we can take care of this problem by
defining a periodic signal whose primary shape is that
of the finite duration signal and then using the DFS on
this periodic signal.
Practically, we define a new transform called the
Discrete Fourier Transform (DFT), which is the
primary period of the DFS.
This DFT is the ultimate numerically computable
Fourier transform for arbitrary finite duration
The Discrete Fourier Transform
First we define a finite-duration sequence x(n) that has N
samples over 0<=n<=N as an N-point sequence
~ ( n)
x x(n rN ) ~(n) x(n mod N ) x((n))N
The compact relationships between x(n) and x~(n) are
~(n) x((n)) ( Periodic extension)
x N
x(n) ~(n) R (n) (Window operation)
x N
The function rem(n,N) can be used to implement our
modulo-N operation.
The Discrete Fourier Transform
The Discrete Fourier Transform of an N-point sequence is
given by ~
X (k ) 0 k N 1 ~
X (k ) DFT[ x(n)] X ( k ) RN ( n )
0 else
N 1
X (k ) x(n)WN , 0 k N 1
n 0
Note that the DFT X(k) is also an N-point sequence, that is, it is not
defined outside of 0<=n<=N-1.
DFT X(k) is the primary interval of X~(k).
N 1
~ ( n) R ( n) 1
x(n) IDFT[ X (k )] x N
X (k )WN kn ,0 n N 1
k 0
Matlab Implementation
X WN x
1 *
x WN X
1 W 1 WN N 1)
WN WN 0 k , n N 1
kn N
( N 1) 2
1 WN N 1)
Examples 5.6 5.7 Comments
Zero-padding is an operation in which more zeros are appended
to the original sequence. The resulting longer DFT provides
closely spaced samples of the discrete-times Fourier transform
of the original sequence.
The zero-padding gives us a high-density spectrum and
provides a better displayed version for plotting. But it does not
give us a high-resolution spectrum because no new information
is added to the signal; only additional zeros are added in the
To get high-resolution spectrum, one has to obtain more data
from the experiment or observations.
Properties of the DFT
1. Linearity: DFT[ax1(n)+bx2(n)]=aDFT[x1(n)]+bDFT[x2(n)]
N3=max(N1,N2): N3-point DFT
2. Circular folding:
x(0) k 0
x( N k ) 1 k N 1
X (0) k 0
DFT[ x((n))N ] X ((k ))N
X ( N k ) 1 k N 1
Matlab: x=x(mod(-n,N)+1)
Properties of the DFT
3. Conjugation: DFT [ x * ( n)] X * (( k )) N
4. Symmetry properties for real sequences:
Let x(n) be a real-valued N-point sequence
X (k ) X * (( k )) N
Re[ X (k )] Re[ X ((k )) N ] : circular even sequence
Im[X (k )] Im[X ((N k )) N ] : circular odd sequence
| X (k ) || X ((k )) N |
X (k ) X ((k )) N
Circular symmetry
Periodic conjugate symmetry
About 50% savings in computation as well as in storage.
X(0) is a real number: the DC frequency
X(N/2)(N is even) is also real-valued: Nyquist component
Circular-even and circular-odd components:
xec (n) [ x(n) x((n)) N ] DFT[ xec (n)] Re[ X (k )] Re[ X ((k ))N ]
xoc (n) [ x(n) x((n)) N ] DFT[ xoc (n)] Im[X (k )] Im[X ((k ))N ]
The real-valued signals Function, p143
5. Circular shift of a sequence
DFT [ x(( n m)) N RN (n)] WN X (k )
6. Circular shift in the frequency domain
DFT [WN ln x(n)] X (( k l )) N RN (k )
7. Circular convolution**
N 1
x1 (n) x2 (n) x1 (m) x2 ((n m))N , 0 n N 1
m 0
DFT [ x1 (n) x2 (n)] X 1 (k ) X 2 (k )
8. Multiplication: DFT[ x1 (n) x2 (n)] X 1 (k ) X 2 (k )
9. Parseval’s relation:
N 1 N 1
E x | x ( n) | 2
| X ( k ) |2
n 0 N k 0
Energy spectrum | X ( k ) |2
Power spectrum X (k )
Linear convolution using the DFT
In general, the circular convolution is an aliased version of the
linear convolution.
If we make both x1(n) and x2(n) N=N1+N2-1 point sequences by
padding an appropriate number of zeros, then the circular
convolution is identical to the linear convolution.
N 1
x4 (n) x1 (n) x2 (n) x1 (k ) x2 ((n k ))N RN (n)
m 0
N 1
x1 (k ) x2 (n k rN ) RN (n)
m 0 r
N1 1
x1 (k ) x2 (n k rN ) RN (n) x3 (n rN ) RN (n)
r m 0 r
Error Analysis
When N=max(N1,N2) is chosen for circular convolution, then
the first (M-1) samples are in error, where M=min(N1,N2).
e(n) x4 x3 x3 (n rN ) RN (n) x3 (n)
x3 (n rN ) RN (n) N max(N1 , N 2 )
r 0
e(n) [ x3 (n N ) x3 (n N )]RN (n)
x3 (n N ) 0 n N 1
X3(n) is also causal
Block Convolution
Segment the infinite-length input sequence into smaller sections (or blocks),
process each section using the DFT, and finally assemble the output sequence
from the outputs of each section. This procedure is called a block convolution
Let us assume that the sequence x(n) is sectioned into N-point sequence and
that the impulse response of the filter is an M-point sequence, where M<N.
We partition x(n) into sections, each overlapping with the previous one by
exactly (M-1) samples, save at last (N-M+1) output samples, and finally
concatenate these outputs into sequence.
To correct for the first (M-1) samples in the first output block, we set the first
(M-1) samples in the first input blocks to zero.
Matlab Implementation
Function [y]=ovrlpsav(x,h,N)
Lenx = length(x); M=length(h); M1=M-1; L=N-M1;
H = [h zeros(1,N-M)];
X = [zeros(1,M1), x, zeros(1,N-1);
K = floor((Lenx+M1-1)/L);
For k=0:K, xk = x(k*L+1:k*L+N);
Y(k+1,: ) = circonvt(xk,h,N); end
The Fast Fourier Transform
Although the DFT is computable transform, the
straightforward implementation is very inefficient,
especially when the sequence length N is large.
In 1965, Cooley and Tukey showed the a procedure to
substantially reduce the amount of computations
involved in the DFT.
This led to the explosion of applications of the DFT.
All these efficient algorithms are collectively known
as fast Fourier transform (FFT) algorithms.
The FFT
Using the Matrix-vector multiplication to
implement DFT:
X=WNx (WN: N*N, x: 1*N, X: 1*N)
takes N×N multiplications and (N-1)×N
additions of complex number.
Number of complex mult. CN=O(N2)
A complex multiplication requires 4 real
multiplications and 2 real additions.
Goal of an Efficient computation
The total number of computations should be linear
rather than quadratic with respect to N.
Most of the computations can be eliminated using the
symmetry and periodicity properties
k ( n N ) (k N )n
W kn
N W N W
WN N / 2 WN
kn kn
If N=2^10, CN=will reduce to 1/100 times.
Decimation-in-time: DIT-FFT, decimation-in-frequency: DIF-FFT
4-point DFT→FFT example
X (k ) x(n)W4nk , 0 k 3; W4 e j 2 / 4 j
n 0
Efficient W40 W44 1; W41 W49 j
Approach W42 W46 1; W43 j
X(0) = x(0)+x(2) + x(1)+x(3) = g1 + g2
X(1) = x(0)-x(2) – j(x(1)-x(3)) = h1 - jh2
X(2) = x(0)+x(2) - x(1)+x(3) = g1 - g2
X(3) = x(0)-x(2) + j(x(1)-x(3)) = h1 + jh2
It requires only 2 complex multiplications.
Signal flowgraph
A 4-point DFT→FFT example
X (0) X (1) g1 g 2 1 1 g1 g 2
X (2) X (3) h jh *W2 1 j . * h h *W2
1 1 x(0) x(1)
. *W2 * x(2) x(3) *W2
1 j
W2 0*0 W2 0*1 1 1
where W2
W2 W2
1*0 1*1 1 1
so W2 * A or A *W2 no multiplic ation needed
Divide-and-combine approach
To reduce the DFT computation’s quadratic dependence on N,
one must choose a composite number N=LM since L2+M2<<N2
for large N.
Now divide the sequence into M smaller sequences of length L,
take M smaller L-point DFTs, and combine these into a large
DFT using L smaller M-point DFTs. This is the essence of the
divide-and-combine approach.
n Ml m, 0 l L 1, 0 m M 1
k p Lq , 0 p L 1, 0 q M 1
Divide-and-combine approach
M 1 L 1
X ( p, q ) x(l , m)WN Ml m )( p Lq )
m 0 l 0
M 1
mp L 1
WN x(l , m)WNMlp WNLmq
m 0 l 0
M 1 mq
mp L 1 lp
Twiddle factor WN x(l , m)WL WM
m 0 l
L int DFT
M po int DFT
Three-step procedure: P155
Divide-and-combine approach
The total number of complex multiplications
for this approach can now be given by
This procedure can be further repeat if M or L
are composite numbers.
When N=Rv, then such algorithms are called
radix-R FFT algorithms.
A 8-point DFT→FFT example
1. two 4-point DFT for m=1,2
1 pm 3 mq
X ( p, q) W8 W4 x(l , m) W2
l 0
m 0
W4 00 W4 01 W4 02 W4 03 x(0,0) x(0,1)
3 W410 W411 W412 W413 x(1,0) x(1,1)
F ( p, m) *
l 0 W4 20 W4 21 W4 22 W4 23 x(2,0) x(2,1)
30 31 32 33 x(3,0) x(3,1)
W4 W4 W4 W4
2. G ( p, m) W8 pm
F ( p, m)
is a 4 2 matrix dot mult. of 8 cplx multns
3. 2 poi nt DFTX with
( p, q ) pm
F ( p, m) W2 mq
m 0
G (0,0) G (0,1)
G (1,0) G (1,1) W 00 W 01
* 2
G (2,0) G (2,1) W 10 W 11
G (3,0) G (3,1)
Number of multiplications
A 4-point DFT is divided into two 2-point
DFTs, with one intermedium matrix mult.
number of multiplications=
4×4cplx→ 2 ×1+ 1 ×4 cplx 16 →6
A 8-point DFT is divided into two 4-point
DFTs, with one intermedium matrix mult.
8×8→2 ×6 + 2×4 64 →20
For 16-point DFT:
16×16→2 ×20 + 2×8 256 →56
In general the reduction of mult.
If N=M*L
N-pt DFT divided into
M times L-pt DFT +
Intermediate matrix transform +
L times M-pt DFT
Radix-2 FFT Algorithms
Let N=2v; then we choose M=2 and L=N/2 and divide x(n) into
two N/2-point sequence.
This procedure can be repeated again and again. At each stage
the sequences are decimated and the smaller DFTs combined.
This decimation ands after v stages when we have N one-point
sequences, which are also one-point DFTs.
The resulting procedure is called the decimation-in-time FFT
(DIF-FFT) algorithm, for which the total number of complex
multiplications is: CN=Nv= N*log2N;
using additional symmetries: CN=Nv= N/2*log2N
Signal flowgraph in Figure 5.19
function y=mditfft(x)
% 本程序对输入序列 x 实现时域抽取快速傅立叶变换DIT-FFT基2算法
m=nextpow2(x); n=2^m; % 求x的长度对应的2的最低幂次m
if length(x)<n
x=[x,zeros(1,n-length(x))]; % 若x的长度不为2的幂次，补零到2的整数幂
nxd=bin2dec(fliplr(dec2bin([1:n]-1,m)))+1; % 求1:2^m数列的倒序
y=x(nxd); % 将x倒序排列得y的初始值
for mm=1:m % 将DFT作m次基2分解,从左到右，对每次分解作DFT运算
le=2^mm;u=1; % 旋转因子u初始化为w^0=1
w=exp(-i*2*pi/le); % 设定本次分解的基本DFT因子w=exp(-i*2*pi/le)
for j=1:le/2 % 本次跨越间隔内的各次蝶形运算
for k=j:le:n % 本次蝶形运算的跨越间隔为le=2^mm
kp=k+le/2; % 确定蝶形运算的对应单元下标
t=y(kp)*u; % 蝶形运算的乘积项
y(kp)=y(k)-t; % 蝶形运算
y(k)=y(k)+t; % 蝶形运算
u=u*w; % 修改旋转因子,多乘一个基本DFT因子w
Decimation-in-frequency FFT
In an alternate approach we choose L=2, M=N/2 and
follow the steps in (5.49).
We can get the decimation-frequency FFT (DIF-FFT)
Its signal flowgraph is a transposed structure of the
DIT-FFT structure.
Its computational complexity is also equal to CN=Nv=
Matlab Implementation
Function: X = fft(x,N)
If length(x)<N, x is padded with zeros.
If the argument N is omitted, N=length(x)
If x is matrix, fft computes the N-point DFT of each
column of x
It is written in machine languag and not use the
Matlab command. Therefore, it executes very fast.
It is written as a mixed-radix algorithm.
N=2v; N=prime number, it is reduced to the raw DFT.
Fast Convolutions
Use the circular convolution to implement the linear
convolution and the FFT to implement the circular
The resulting algorithm is called a fast convolution algorithm.
If we choose N=2v and implement the radix-2 FFT, then the
algorithm is called a high-speed convolution.
If x1(n) is N1-point, x2(n) is N2-point, then
log 2 ( N1 N 2 1)
N 2
Compare the linear convolution and the high-speed conv.
High-speed Block Convolution
Overlap-and-save method
We can now replace the DFT by the radix-2
FFT algorithm to obtain a high-speed overlap-
and-save algorithm.
Ex5.21 MATLAB verification of DFT time
versus data length N;
Ex5.22 Convolution-time comparison of
using FFT versus DFT;
Fast block-convolution using FFT;
Textbook: pp.116~172
Chinese reference book: pp.35~37,
pp.68~82 ,pp.97~109
1: p5.1b,d; 选ch.1.3a
2: p5.3; p5.6; 选p5.9
3: p5.15;5.18a,d;
4: p5.23;p5.29; | {"url":"http://www.docstoc.com/docs/36552941/Digital-Signal-Processing-Using","timestamp":"2014-04-24T07:21:10Z","content_type":null,"content_length":"84293","record_id":"<urn:uuid:ddafc613-f930-4e3a-9f42-09e7d51e7e40>","cc-path":"CC-MAIN-2014-15/segments/1398223205375.6/warc/CC-MAIN-20140423032005-00287-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Summary of Analysis
Next: Evaluation of Implementation Up: Conclusion Previous: Conclusion
Chapter 7 provides an analysis of some of the algorithms and the performance of the implementation. We can draw a number of conclusions from this.
Firstly, operations may be performed using either the dyadic or signed binary representations. Algorithms performing the same operation using different representations have different behaviours. In
general the signed binary algorithms are more complex and have greater lookahead requirements, but on the other hand perform better than the corresponding dyadic digit operations which generally
suffer if the size of the dyadic digits involved swell.
Secondly, computing certain expressions exactly (eg. the iterated logistic map) necessarily involves examining many more digits of input and performing a significantly greater number of manipulations
than would normally be performed with floating point arithmetic.
Thirdly, the order in which operations are performed can greatly affect the lookahead required. Rearranging the same expression can significantly reduce the computation time of complex expressions.
Lastly, the present implementation is slow when compared to the floating point arithmetic packages commonly used, even when those operations are performed to a high precision in a package such as
Maple. This is in part be due to the fact that the algorithms themselves are more complex than the floating point operations one might otherwise use (c.f. Chapter 7, especially section 7.1.5).
However the fact that the implementation uses a functional language and most floating point arithmetic is either written using an imperative language, assembler, or embedded in hardware makes it
unreasonable to make a direct comparison.
Next: Evaluation of Implementation Up: Conclusion Previous: Conclusion Martin Escardo | {"url":"http://www.dcs.ed.ac.uk/home/mhe/plume/node133.html","timestamp":"2014-04-18T18:10:51Z","content_type":null,"content_length":"4836","record_id":"<urn:uuid:4bfdb1b7-1262-4a5e-94c2-22fca17f51d7>","cc-path":"CC-MAIN-2014-15/segments/1397609535095.7/warc/CC-MAIN-20140416005215-00043-ip-10-147-4-33.ec2.internal.warc.gz"} |
This Article
Bibliographic References
Add to:
Counting All Possible Ancestral Configurations of Sample Sequences in Population Genetics
July-September 2006 (vol. 3 no. 3)
pp. 239-251
ASCII Text x
Yun S. Song, Rune Lyngs?, Jotun Hein, "Counting All Possible Ancestral Configurations of Sample Sequences in Population Genetics," IEEE/ACM Transactions on Computational Biology and Bioinformatics,
vol. 3, no. 3, pp. 239-251, July-September, 2006.
BibTex x
@article{ 10.1109/TCBB.2006.31,
author = {Yun S. Song and Rune Lyngs? and Jotun Hein},
title = {Counting All Possible Ancestral Configurations of Sample Sequences in Population Genetics},
journal ={IEEE/ACM Transactions on Computational Biology and Bioinformatics},
volume = {3},
number = {3},
issn = {1545-5963},
year = {2006},
pages = {239-251},
doi = {http://doi.ieeecomputersociety.org/10.1109/TCBB.2006.31},
publisher = {IEEE Computer Society},
address = {Los Alamitos, CA, USA},
RefWorks Procite/RefMan/Endnote x
TY - JOUR
JO - IEEE/ACM Transactions on Computational Biology and Bioinformatics
TI - Counting All Possible Ancestral Configurations of Sample Sequences in Population Genetics
IS - 3
SN - 1545-5963
EPD - 239-251
A1 - Yun S. Song,
A1 - Rune Lyngs?,
A1 - Jotun Hein,
PY - 2006
KW - Ancestral configurations
KW - coalescent
KW - recombination
KW - contingency table
KW - enumeration.
VL - 3
JA - IEEE/ACM Transactions on Computational Biology and Bioinformatics
ER -
Given a set D of input sequences, a genealogy for D can be constructed backward in time using such evolutionary events as mutation, coalescent, and recombination. An ancestral configuration (AC) can
be regarded as the multiset of all sequences present at a particular point in time in a possible genealogy for D. The complexity of computing the likelihood of observing D depends heavily on the
total number of distinct ACs of D and, therefore, it is of interest to estimate that number. For D consisting of binary sequences of finite length, we consider the problem of enumerating exactly all
distinct ACs. We assume that the root sequence type is known and that the mutation process is governed by the infinite-sites model. When there is no recombination, we construct a general method of
obtaining closed-form formulas for the total number of ACs. The enumeration problem becomes much more complicated when recombination is involved. In that case, we devise a method of enumeration based
on counting contingency tables and construct a dynamic programming algorithm for the approach. Last, we describe a method of counting the number of ACs that can appear in genealogies with less than
or equal to a given number R of recombinations. Of particular interest is the case in which R is close to the minimum number of recombinations for D.
[1] S.N. Ethier and R.C. Griffiths, “The Infinitely-Many-Sites Model as a Measure Valued Diffusion,” Annals of Probability, vol. 15, pp. 515-545, 1987.
[2] S.N. Ethier and R.C. Griffiths, “On the Two-Locus Sampling Distribution,” J. Math. Biology, vol. 29, pp. 131-159, 1990.
[3] P. Fearnhead and P. Donnelly, “Estimating Recombination Rates from Population Genetic Data,” Genetics, vol. 159, pp. 1299-1318, 2001.
[4] R.C. Griffiths, “Genealogical-Tree Probabilities in the Infinitely-Many-Site Mode,” J. Math. Biology, vol. 27, pp. 667-680, 1989.
[5] R.C. Griffiths and P. Marjoram, “Ancestral Inference from Samples of DNA Sequences with Recombination,” J. Computational Biology, vol. 3, pp. 479-502, 1996.
[6] R.C. Griffiths and S. Tavaré, “Ancestral Inference in Population Genetics,” Statistics in Science, vol. 9, pp. 307-319, 1994.
[7] R.C. Griffiths and S. Tavaré, “Simulating Probability Distributions in the Coalescent,” Theoretical Population Biology, vol. 46, pp. 131-159, 1994.
[8] D. Gusfield, “Efficient Algorithms for Inferring Evolutionary Trees,” Networks, vol. 21, pp. 19-28, 1991.
[9] J.F.C. Kingman, “The Coalescent,” Stochastic Processing Applications, vol. 13, pp. 235-248, 1982.
[10] J.F.C. Kingman, “On the Genealogy of Large Populations,” J. Applied Probability, vol. 19A, pp. 27-43, 1982.
[11] M.K. Kuhner, J. Yamato, and J. Felsenstein, “Estimating Effective Population Size and Mutation Rate from Sequence Data Using Metropolis-Hastings Sampling,” Genetics, vol. 140, pp. 1421-1430,
[12] M.K. Kuhner, J. Yamato, and J. Felsenstein, “Maximum Likelihood Estimation of Recombination Rates from Population Data,” Genetics, vol. 156, pp. 1393-1401, 2000.
[13] J. De Loera, R. Hemmecke, J. Tauzer, and R. Yoshida, “Effective Lattice Point Counting in Rational Convex Polytopes,” J. Symbolic Computation, vol. 38, pp. 1273-1302, 2004.
[14] R. Lyngsø, Y.S. Song, and J. Hein, “Minimum Recombination Histories by Branch and Bound,” Proc. 2005 Workshop Algorithms in Bioinformatics, pp. 239-250, 2005.
[15] K.L. Simonsen and G.A. Churchill, “A Markov Chain Model of Coalescence with Recombination,” Theoretical Population Biology, vol. 52, pp. 43-59, 1997.
[16] M. Stephens and P. Donnelly, “Inference in Molecular Population Genetics,” J. Royal Statistical Soc. Series B, vol. 62, pp. 605-655, 2000.
[17] R.H. Ward, B.L. Frazier, K. Dew, and S. Pääbo, “Extensive Mitochondria Diversity within a Single Amerindian Tribe,” Proc. Nat'l Academy Science, vol. 88, pp. 8720-8724, 1991.
Index Terms:
Ancestral configurations, coalescent, recombination, contingency table, enumeration.
Yun S. Song, Rune Lyngs?, Jotun Hein, "Counting All Possible Ancestral Configurations of Sample Sequences in Population Genetics," IEEE/ACM Transactions on Computational Biology and Bioinformatics,
vol. 3, no. 3, pp. 239-251, July-Sept. 2006, doi:10.1109/TCBB.2006.31
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GEOL3650: Energy: A Geological Perspective: Laboratory: Visualizing the Earth in Three Dimensions - Geologic Maps, Cross-sections & Block Diagrams
Intro | Tabular Bodies | Definition | Map Symbols | Special Symbols | Measuring
On a map, strike and dip are represented by a T-shaped symbol (Fig. 1). The long bar of the T is parallel to the strike and the short bar of the T indicates the dip direction. Since dip is always
measured at right angles to strike, the strike and dip components are always perpendicular to each other. The dip angle is indicated by placing the angle (in degrees) next to the strike and dip
symbol. In contrast, the strike orientation is not explicitly recorded by the strike and dip symbol.
Fig. 1: Strike and dip symbol on a map.
To determine the strike of a feature on a geologic map, use a straight edge to lightly draw in pencil a line parallel the map's north direction that intersects the strike line of the strike and dip
symbol then xtend the strike line. The strike of the feature is the angle between these two lines and can be measured with a protractor.
Intro | Tabular Bodies | Definition | Map Symbols | Special Symbols | Measuring
[ Lab Listing | Geologic Maps ]
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Antievolution.org - Antievolution.org Discussion Board -Topic::Evolutionary Computation
Posts: 529
Joined: Sep. 2008
(Permalink) Posted: June 11 2009,14:05
I haven't run the program, but perusing the description of the algorithm, it seem to me that this section that describes how fitness is assigned is the part that determines the ultimate behavior of
the model. They clearly have built in an asymmetry in the fitness of beneficial vs deleterious mutations, and their justifications of the asymmetry smell fishy to me, but IANAB. (bolding mine)
To provide users of Mendel even more flexibility in specifying the fitness effect distribution, we have chosen to use a
form of the Weibull function [12] that is a generalization of the more usual exponential function. Our function, expressed
by eq. (3.1), maps a random number x, drawn from a set of uniformly distributed random numbers, to a fitness effect d(x)
for a given random mutation.
d(x) = (dsf) exp(?ax^gamma), 0 < x < 1. (3.1)
Here (dsf) is the scale factor which is equal to the extreme value which d(x) assumes when x = 0. We allow this scale
factor to have two separate values, one for deleterious mutations and the other for favorable ones. These scale factors are meaningful relative to the initial fitness value assumed for the population
before we introduce new mutations. In Mendel we assume this initial fitness value to be 1.0. For deleterious mutations, since lethal mutations exist, we choose dsf del = ?1. For favorable mutations,
we allow the user to specify the (positive) scale factor dsf fav. Normally, this would be a small value (e.g., 0.01 to 0.1), since it is only in very special situations that a single beneficial
mutation would
have a very large effect.
The parameters a and gamma, both positive real numbers, determine the shape of the fitness effect distribution. We applythe same values of a and gamma to both favorable and deleterious mutations. The
parameter a determines the minimum absolute values for d(x), realized when x = 1. We choose to make the minimum absolute value of d(x) the inverse of the haploid genome size G (measured in number of
nucleotides) by choosing a = loge(G). For example, for the human genome, G = 3 × 109, which means that for the case of deleterious mutations, d(1) = ?1/G = ?3 × 10?10. For large genomes,
this minimum value is essentially 0. For organisms with smaller genomes such as yeast, which has a value for G on
the order of 107, the minimum absolute effect is larger. This is consistent with the expectation that each nucleotide in a smaller genome on average plays a greater relative role in the organism’s
The second parameter gamma, can be viewed as ontrolling the fraction of mutations that have a large absolute fitness
effect. Instead of specifying gamma directly, we select two quantities that are more intuitive and together define gamma. The first is theta, a threshold value that defines a “high-impact mutation”.
The second is q, the fraction of mutations that exceed this threshold in their effect. For example, a user can first define a high-impact mutation as one that results in 10% or more change in fitness
(theta = 0.1) relative to the scale factor and then specify that 0.001 of all mutations (q = 0.001) be in this category. Inside the code the value of is computed that satisfies these requirements. We
reiterate that Mendel uses the same value for gamma, and thus the same values for theta and q, for both favorable and deleterious mutations. Figure 3.1 shows the effect of the parameter q on the
shape of the distribution of fitness effect. Note that for each of the cases displayed the large majority of mutations are nearly neutral, that is, they have very small effects. Since a utation’s
effect on fitness can be measured experimentally only if it is sufficiently large, our strategy for parameterizing the fitness effect distribution in terms of high-impact situtations provides a means
for the Mendel user to relate the numerical model input more directly to available data regarding the actual measurable frequencies of mutations in a given biological context.
Part of the justification for asymmetry is that some mutations are lethal, meaning that individual has zero probability of reproducing. OK, but the maximum fitness benefit of a beneficial mutation
is "a very small number like 0.001", which is then subject to "heritability factor", typically 0.2, and other probabilities that severely limit its ability to propagate.
To make matters worse, for some unjustified reason, the same distribution for beneficial and deleterious is used, after severely skewing the results with the above.
Again, IANOB, but it seems to me that a single beneficial mutation can, in many situations like disease resistance, blonde hair, big boobs, etc, virtually guarantee mating success, just like a
deleterious mutation can be reproductively lethal.
I can see easily how the skewed treatment of beneficial vs deleterious mutations could virtually guarantee "genetic entropy", as evidenced by monotonically decreasing population fitness caused by
accumulation of deleterious mutational load.
ETA source. link is above
Sanford, J., Baumgardner, J., Gibson, P., Brewer, W., & ReMine, W.
(2007a). Mendel’s Accountant: A biologically realistic forward-time population genetics program. Scalable Computing: Practice and Experience 8(2), 147–165.
The majority of the stupid is invincible and guaranteed for all time. The terror of their tyranny is alleviated by their lack of consistency. -A. Einstein (H/T, JAD)
If evolution is true, you could not know that it's true because your brain is nothing but chemicals. ?Think about that. -K. Hovind | {"url":"http://www.antievolution.org/cgi-bin/ikonboard/ikonboard.cgi?s=50a399b6a4631568;act=ST;f=14;t=6034;st=30","timestamp":"2014-04-16T15:25:01Z","content_type":null,"content_length":"174275","record_id":"<urn:uuid:f2f90592-1094-47aa-a797-b2fd1abff2c1>","cc-path":"CC-MAIN-2014-15/segments/1397609523429.20/warc/CC-MAIN-20140416005203-00161-ip-10-147-4-33.ec2.internal.warc.gz"} |
Higher-Dimensional Algebra VI: Lie 2-Algebras
John C. Baez and Alissa S. Crans
The theory of Lie algebras can be categorified starting from a new notion of `2-vector space', which we define as an internal category in Vect. There is a 2-category 2Vect having these 2-vector
spaces as objects, `linear functors' as morphisms and `linear natural transformations' as 2-morphisms. We define a `semistrict Lie 2-algebra' to be a 2-vector space L equipped with a skew-symmetric
bilinear functor [ . , . ] : L x L -> L satisfying the Jacobi identity up to a completely antisymmetric trilinear natural transformation called the `Jacobiator', which in turn must satisfy a certain
law of its own. This law is closely related to the Zamolodchikov tetrahedron equation, and indeed we prove that any semistrict Lie 2-algebra gives a solution of this equation, just as any Lie algebra
gives a solution of the Yang--Baxter equation. We construct a 2-category of semistrict Lie 2-algebras and prove that it is 2-equivalent to the 2-category of 2-term L_\infty-algebras in the sense of
Stasheff. We also study strict and skeletal Lie 2-algebras, obtaining the former from strict Lie 2-groups and using the latter to classify Lie 2-algebras in terms of 3rd cohomology classes in Lie
algebra cohomology. This classification allows us to construct for any finite-dimensional Lie algebra g a canonical 1-parameter family of Lie 2-algebras g_h which reduces to g at h = 0. These are
closely related to the 2-groups G_h constructed in a companion paper.
Keywords: Lie 2-algebra, L_\infty-algebra, Lie algebra cohomology
2000 MSC: 17B37,17B81,17B856,55U15,81R50
Theory and Applications of Categories, Vol. 12, 2004, No. 15, pp 492-528.
Revised 2010-02-25. Original version at
TAC Home | {"url":"http://www.tac.mta.ca/tac/volumes/12/15/12-15abs.html","timestamp":"2014-04-20T20:55:39Z","content_type":null,"content_length":"3301","record_id":"<urn:uuid:b789fac5-a4ac-4e58-8696-e591c94c6af8>","cc-path":"CC-MAIN-2014-15/segments/1397609539230.18/warc/CC-MAIN-20140416005219-00510-ip-10-147-4-33.ec2.internal.warc.gz"} |
"Building a patio" problem, must write an equation
September 11th 2009, 05:06 PM #1
Sep 2009
Duluth, California
Here's a problem a friend and I have been trying to figure out. Maybe one of you will be able to help us?
Here it goes:
"Someone is planning to build a patio along the back wall of her house which is 32 feet long. The patio will be rectangular in shape and will fit against the full length of the back wall (so one
side of the patio will be 32 feet long).
Let's assume the patio tiles are each 1ft x 1ft.
If this person has 256 tiles to work with, how far out from the wall will the patio extend?
Pretend you are the patio builder and do these tasks:
* Choose the variable you are going to use and state what it represents.
* Write an equation that represents the problem.
* Then solve the problem and equation.
Also ...
Benito is also going to build a patio but his patio does not have to fit exactly against a wall. In fact all that Benito has decided is that the patio should be rectangular in shape and should
use all of the 144 tiles he has available. His tiles are the same size as the problem above.
Find as many possibilities you can for the dimensions of Benito's patio."
Any tips, suggestions, etc on how to solve this two are appreciated.
Ok, let's assume the following variables!
L= (The distance of the back wall = the length of the patio) = 32 feet (this is told to us)
W = how far the patio extends from the wall (what we're trying to find)
A = The total area of the patio tiles
Clearly, if the patio tiles are 1ft by 1ft big and there's 256 of these, then the total area (A) will be 256 feet squared, correct?
(1 patio tile = 1ft squared. Hence, 256 = 256 ft squared)
Therefore, A = 256 feet squared.
By using the Area formula for a rectangle (which happens to contain all of our variables) we can solve the problem:
256 = 32 * W
Solving for W:
256/32 = W
Therefore, W (what we're trying to find) = 8 feet
Hello Sarah-
Benito is also going to build a patio but his patio does not have to fit exactly against a wall. In fact all that Benito has decided is that the patio should be rectangular in shape and should
use all of the 144 tiles he has available. His tiles are the same size as the problem above.
Find as many possibilities you can for the dimensions of Benito's patio."
Any tips, suggestions, etc on how to solve this two are appreciated.
When you're building a rectangular patio, you can work out the number of tiles you'll need by multiplying the length of the patio by its breadth. For example, if the length is 10 feet and the
breadth 6 feet, then the area of the patio is 10 x 6 = 60 square feet. So if each tile is 1 ft by 1 ft, that will use 60 tiles.
So, if you have 144 of these tiles, you can make any shape rectangle you like provided its length multiplied by its breadth is 144. I'll start you off:
1 ft x 144 ft (Well, it's possible, but it's a very long thin patio!)
2 ft x 72 ft
3 ft x 48 ft
... and so on. Can you find all the other possible shapes now? (Bear in mind that each number you choose will have to divide exactly into 144. So it's no good trying to make a patio measuring 5
feet along one side, is it?)
I think so.... wouldn't the only other possible combination be 4 * 36? If not, please correct me.
Most certainly not,
This is just a matter of being proficient with your times tables. You're looking for two factors of 144 that are both whole integers (that is, not fractions nor decimals).
There are quite a number of combinations!
6 * 24
8 * 18
12 * 12
16 * 9
Can you think of anymore?
September 11th 2009, 05:18 PM #2
Junior Member
Sep 2009
September 11th 2009, 10:14 PM #3
September 12th 2009, 12:56 PM #4
Sep 2009
Duluth, California
September 12th 2009, 05:03 PM #5
Junior Member
Sep 2009 | {"url":"http://mathhelpforum.com/algebra/101742-building-patio-problem-must-write-equation.html","timestamp":"2014-04-21T06:00:53Z","content_type":null,"content_length":"44506","record_id":"<urn:uuid:f6da8dae-8543-489f-89f1-58f3faa65531>","cc-path":"CC-MAIN-2014-15/segments/1397609539493.17/warc/CC-MAIN-20140416005219-00520-ip-10-147-4-33.ec2.internal.warc.gz"} |
HPL_dlaswp04N copy rows of U in A and replace them with columns of W.
#include "hpl.h"
void HPL_dlaswp04N( const int M0, const int M1, const int N, double * U, const int LDU, double * A, const int LDA, const double * W0, const double * W, const int LDW, const int * LINDXA, const int *
LINDXAU );
HPL_dlaswp04N copies M0 rows of U into A and replaces those rows of U with columns of W. In addition M1 - M0 columns of W are copied into rows of U.
M0 (local input) const int
On entry, M0 specifies the number of rows of U that should be
copied into A and replaced by columns of W. M0 must be at
least zero.
M1 (local input) const int
On entry, M1 specifies the number of columns of W that should
be copied into rows of U. M1 must be at least zero.
N (local input) const int
On entry, N specifies the length of the rows of U that should
be copied into A. N must be at least zero.
U (local input/output) double *
On entry, U points to an array of dimension (LDU,N). This
array contains the rows that are to be copied into A.
LDU (local input) const int
On entry, LDU specifies the leading dimension of the array U.
LDU must be at least MAX(1,M1).
A (local output) double *
On entry, A points to an array of dimension (LDA,N). On exit,
the rows of this array specified by LINDXA are replaced by
rows of U indicated by LINDXAU.
LDA (local input) const int
On entry, LDA specifies the leading dimension of the array A.
LDA must be at least MAX(1,M0).
W0 (local input) const double *
On entry, W0 is an array of size (M-1)*LDW+1, that contains
the destination offset in U where the columns of W should be
W (local input) const double *
On entry, W is an array of size (LDW,M0+M1), that contains
data to be copied into U. For i in [M0..M0+M1), the entries
W(:,i) are copied into the row W0(i*LDW) of U.
LDW (local input) const int
On entry, LDW specifies the leading dimension of the array W.
LDW must be at least MAX(1,N+1).
LINDXA (local input) const int *
On entry, LINDXA is an array of dimension M0 containing the
local row indexes A into which rows of U are copied.
LINDXAU (local input) const int *
On entry, LINDXAU is an array of dimension M0 that contains
the local row indexes of U that should be copied into A and
replaced by the columns of W.
See Also
HPL_dlaswp00N, HPL_dlaswp10N, HPL_dlaswp01N, HPL_dlaswp01T, HPL_dlaswp02N, HPL_dlaswp03N, HPL_dlaswp03T, HPL_dlaswp04N, HPL_dlaswp04T, HPL_dlaswp05N, HPL_dlaswp05T, HPL_dlaswp06N, HPL_dlaswp06T. | {"url":"http://www.netlib.org/benchmark/hpl/HPL_dlaswp04N.html","timestamp":"2014-04-17T04:20:24Z","content_type":null,"content_length":"4465","record_id":"<urn:uuid:12dc404d-df22-47b0-a8b2-89c419e43c3d>","cc-path":"CC-MAIN-2014-15/segments/1397609526252.40/warc/CC-MAIN-20140416005206-00155-ip-10-147-4-33.ec2.internal.warc.gz"} |
SPORTS >>Cabot skirts by Lions
SPORTS >>Cabot skirts by Lions
IN SHORT: The Cabot Panthers had a tough time with Searcy, but got past the winless team to remain undefeated as they head into conference play in the brand new 7A-East Conference against rival
By JOHNNY RAY LAKE
Special to the Leader
For the second week in a row, the Cabot Panthers overcame a second half deficit to come away with a victory on the road as Cabot defeated the Searcy Lions 23-16. Vince Aguilar’s second touchdown of
the game with 4:06 left in the fourth quarter proved to be the deciding points in the game.
The fourth quarter seemed to be dominated by the Panthers as they controlled both the clock and the line of scrimmage.
With Aguilar gaining 61 of his 167 rushing yards in the final 12 minutes, Cabot held the ball for more than seven minutes, while limiting the Lions to just less than five minutes and only two
possessions to end the game. An interception of Searcy quarterback Justin Rowden by senior Johnnie Stone with 2:55 left ended the Lions hopes of tying the score.
Cabot broke a 16-16 tie on the first play of the fourth quarter on a 24-yard field goal by Alex Tripp. On the next possession, facing fourth and one from the Cabot 46, Rowden snuck up the middle for
two yards and a first down. But on the very next play, Rowden’s pass under pressure was intercepted by Ethan Coffee, which led to the go-ahead score for the Panthers.
Cabot got on the board first with an impressive ten play 65-yard march that ended with a 28-yard touchdown pass from Corey Wade to Josh Clem on fourth down and five. Searcy responded with their own
ten play drive that led to a 32-yard field goal by Ryan Wilbourn. On the drive, Rowden ran three straight draw plays for 13, 14, and 16 yards. But a holding penalty on brought back a 15-yard
touchdown run and Searcy settled for the field goal.
After exchanging punts heading through the second quarter, Cabot took over at their own 34 with 9:35 left in the first half. 13 plays later, Aguilar busted out from the middle of the field down the
right sideline for a 27-yard score with 3:48 to go until halftime.
Matt Cramblett broke through to block the extra point, so the score was 13-3 in favor of the Panthers.
Using the sidelines and their timeouts, the Lions went 65 yards in under four minutes to take the lead on a Rowden quarterback sneak with 12 seconds left in the first half. The key play on the 17-
play drive was converting on fourth and one on the Cabot 19-yard line. Rather than take the field goal attempt, Adam Robertson ran up the middle for a two-yard gain.
Cabot led Searcy 13-10 at the break.
The Lions took the ball on their own 34 to start the third quarter. On fourth and two on the Panthers’ 41, Searcy failed to convert as Robinson this time was stuffed short of the first down marker.
The ensuing Panthers drive ended in a fumble, and after taking over on their own 37, Searcy faced once again fourth and short in Panther territory. Pressure up the middle forced Rowden to scramble to
his right, where he found a wide open Nick Evans on the 6 who then went in for the score.
Cabot returned the favor and blocked the extra point. Cabot then forced the two fourth quarter interceptions to end the Lions’ final two drives.
Aguilar carried the ball 31 times for 167-yards and two scores, with 96 of those coming in the second half as the Panthers offensive line began to wear down the Lions defense. Both teams combined to
convert five of six fourth down plays, with Cabot going two for two, and Searcy converting three of four. Both teams scored six of their points on fourth down.
When asked how important it was that his team responded quickly to the Searcy scores, Cabot head coach Mike Malham said ”Yes, that and the fact we didn’t turn the ball over in the fourth were the key
factors in the game. “Searcy is always a tough place to go and win on the road, he added.
The Panthers will travel to archrival Conway next week to open up play in the 7A-East Conference. | {"url":"http://www.arkansasleader.com/2006/09/sports-cabot-skirts-by-lions.html","timestamp":"2014-04-18T08:41:47Z","content_type":null,"content_length":"21411","record_id":"<urn:uuid:312720a4-26c3-42fe-8d3c-8ac987290704>","cc-path":"CC-MAIN-2014-15/segments/1398223204388.12/warc/CC-MAIN-20140423032004-00301-ip-10-147-4-33.ec2.internal.warc.gz"} |
Schur's lemma
Schur's lemma
Two related basic facts in representation theory bear the name of Schur: one general about categories of modules, and another specific to the case of complex numbers. If $G$ is a group, a $G$-module
is any $k$-module with an action of $G$, where $k$ is a fixed commutative unital ground ring.
1. Given a group $G$ and a linear map $\phi : M\to N$ between two irreducible (= simple) $G$-modules (linear maps intertwining the actions) then $\phi$ is either the zero morphism or an isomorphism.
It follows that the endomorphisms of simple irreducible $G$-module form a division ring.
2. Set $M=N$, and suppose further that the ground ring $k$ is an algebraically closed field; then $\phi$ is a multiple $\lambda I$ of the identity operator. In other words, the nontrivial
automorphisms of simple modules, a priori possible by (1), are ruled out over algebraically closed fields.
Part (1) is essentially category-theoretic and can be generalized in many ways, for example, by replacing $G$ by some $k$-algebra and taking the representations compatible with the action of $k$;
more generally, given an abelian category, the endomorphism ring of a simple object is a division ring. Schur’s lemma is one of the basic facts of representation theory. For (2), if the endomorphism
rings of all objects in an abelian category are finite-dimensional over an algebraically closed ground field $k$ (as is the case for group representations), then the endomorphism ring of a simple
object is $k$ itself.
Revised on October 17, 2009 20:12:57 by
Toby Bartels | {"url":"http://ncatlab.org/nlab/show/Schur's+lemma","timestamp":"2014-04-20T19:00:42Z","content_type":null,"content_length":"17384","record_id":"<urn:uuid:9fc42c26-e0eb-4870-85bf-ecf6f569c8a1>","cc-path":"CC-MAIN-2014-15/segments/1398223203841.5/warc/CC-MAIN-20140423032003-00447-ip-10-147-4-33.ec2.internal.warc.gz"} |
Collapsing of Riemannian manifolds with a group action
up vote 4 down vote favorite
Let $M$ be a complete Riemannian manifold with bounded sectional curvature and $G$ a compact connected Lie group acts smoothly on $M$. Consider the fixed point set $F$, it is of course a submanifold
of $M$ by the slice theorem. Let $\{F_i\}$ be the connected components of $F$. Then for each $i$, is there a sequence of Riemannian manifolds $\{M_j\},j\in\mathbb{N}$ with $M_0=M$ such that $\{M_j\}$
collapses to $F_i$ while keeping their sectional curvatures uniformly bounded?
If in general such a sequence does not exist, how about the case $G=T$? Here $T$ is a finite-dimensional torus.
riemannian-geometry mg.metric-geometry transformation-groups
1 Motivation? Could you point us to a good definition of collapsing'? Is it true that each Euclidean space collapses' to a point? [If not, take T = S^1 acting in standard linear fashion on the plane
to get a counterexample. – Richard Montgomery Sep 19 '12 at 17:58
1 Collapsing in the Gromov-Hausdorff sense. – Acky Sep 19 '12 at 18:05
2 Do you require that all $M_j$ are diffeomorphic to $M$? If not, why not take $M_j=F_i\times $(small circle)? – Sergei Ivanov Sep 19 '12 at 18:16
3 "...Consider the fixed point set F, it is of course a submanifold of M by the slice theorem". Note that it is really simpler than that; in geodesic coordinates at a point p of F, the fixed point
set is locally the linear subspace left fixed by the linearized action at p. – Dick Palais Sep 19 '12 at 18:22
1 I assume you give $F$ the induced metric from $M$. If the answer to your question is yes, then the curvature of $F$ is bounded from below, so in effect you hope that the fixed point set of any
smooth compact group action has curvature bounded below. Why would that be true? – Igor Belegradek Sep 19 '12 at 18:25
show 5 more comments
1 Answer
active oldest votes
As it was noted in the comments you probably wanted to say that the action is isometric and $M_n$ is diffeomorphic to $M$ for all $n$. (Otherwise the question has no
In this case answer is NO. Consider $\mathbb S^1$ action on $\mathbb S^3$ with fixed point set $\mathbb S^1$ and note that simply connected spaces can not GH-converge to $\
up vote 2 down vote mathbb S^1$.
For the second part, it seems that you may only get a torus as the fixed set (?). In this case the answer is obviously YES.
add comment
Not the answer you're looking for? Browse other questions tagged riemannian-geometry mg.metric-geometry transformation-groups or ask your own question. | {"url":"http://mathoverflow.net/questions/107597/collapsing-of-riemannian-manifolds-with-a-group-action","timestamp":"2014-04-20T06:35:10Z","content_type":null,"content_length":"57943","record_id":"<urn:uuid:687553f9-a8ca-4f57-af25-33b9e87b7820>","cc-path":"CC-MAIN-2014-15/segments/1397609538022.19/warc/CC-MAIN-20140416005218-00172-ip-10-147-4-33.ec2.internal.warc.gz"} |
Sébastien Van Bellegem, CORE
Monday, 13 February 2012, 12:00 - 13:30
Sébastien Van Bellegem, CORE
Functional linear instrumental regression
In an increasing number of empirical studies, the dimensionality mea- sured e.g. as the number of variables in a model, can be very large. Two instances of large dimensional models are the linear
regression with a large number of covariates and the estimation of a regression function with many instrumental variables. We will recall these examples in the talk by means of some examples. We also
recall why classical least square or IV estimators behaves poorly in such large dimensional regression problems. An appropriate setting to analyze high dimensional problems is provided by a
functional linear model, in which the covariates are a vector in Rp for large p (p can tend to infinity). More generally we consider that covariates belongs to some Hilbert space. We also consider
the case where covariates are endogenous and assume the existence of instrumental variables (that are functional as well). In this talk we show that estimating the regression function is a linear
ill-posed inverse problem, with a known but data-dependent operator. Our main contribution is to analyse the rate of convergence of the Tikhonov regularized estimator, when we premultiply the problem
by an instrument dependent operator. This extends the technology of Generalized Method of Moments to functional (GMM) to functional data. We then discuss the optimal choice of the premultiplication
operator and propose an extension of the notion of “weak instrument” to this nonparametric framework. The performance of the resulting nonparametric estimator is also studied through simulations.
Location: ULB R42.2.113
Contact: Claude Adan, This e-mail address is being protected from spam bots, you need JavaScript enabled to view it | {"url":"http://www.ecares.org/index.php?option=com_events&task=view_detail&agid=639&year=2012&month=02&day=13&Itemid=149&catids=39","timestamp":"2014-04-20T15:50:42Z","content_type":null,"content_length":"30344","record_id":"<urn:uuid:501f290e-63af-4a7d-8890-c518255180a6>","cc-path":"CC-MAIN-2014-15/segments/1397609538824.34/warc/CC-MAIN-20140416005218-00070-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Power Scaling for Spatial Modulation with Limited Feedback
International Journal of Antennas and Propagation
Volume 2013 (2013), Article ID 718482, 5 pages
Research Article
Power Scaling for Spatial Modulation with Limited Feedback
National Key Laboratory of Science and Technology on Communications, University of Electronic Science and Technology of China, Chengdu 610054, China
Received 8 February 2013; Revised 6 May 2013; Accepted 8 May 2013
Academic Editor: Feifei Gao
Copyright © 2013 Yue Xiao 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.
Spatial modulation (SM) is a recently developed multiple-input multiple-output (MIMO) technique which offers a new tradeoff between spatial diversity and spectrum efficiency, by introducing the
indices of transmit antennas as a means of information modulation. Due to the special structure of SM-MIMO, in the receiver, maximum likelihood (ML) detector can be combined with low complexity. For
further improving the system performance with limited feedback, in this paper, a novel power scaling spatial modulation (PS-SM) scheme is proposed. The main idea is based on the introduction of
scaling factor (SF) for weighting the modulated symbols on each transmit antenna of SM, so as to enlarge the minimal Euclidean distance of modulated constellations and improve the system performance.
Simulation results show that the proposed PS-SM outperforms the conventional adaptive spatial modulation (ASM) with the same feedback amount and similar computational complexity.
1. Introduction
Spatial modulation (SM) [1] is a recently proposed multiple-input multiple-output (MIMO) transmission technique, in which the index of the transmit antenna is utilized for information modulation in
the spatial domain. The main advantages of SM-MIMO can be described as follows. Firstly, only one transmit antenna is active on each time slot so that strict time synchronization is saved. Secondly,
SM-MIMO can be used for any number of receive antennas so that it adapts to downlink communications on an unbalanced MIMO channels in which the number of transmit antennas is much larger than that of
receive antennas. Finally, SM-MIMO can offer a new balance between spectrum efficiency and spatial diversity compared to the conventional MIMO techniques [2, 3].
For detecting the SM-MIMO signals at the receiver side, maximum likelihood (ML) algorithm was suggested in [4] to achieve the optimal performance, which also gives a performance bound of the SM-MIMO
system. For controlling the computational complexity of ML detection while approaching the performance bound, in current research, a series of near-ML detectors has been proposed [5–7] to make a
tradeoff between complexity and performance.
Traditional SM-MIMO cannot exceed the ML performance bound. In this case, for further improving the system performance, in recent research, limited feedback was considered for link adapted SM-MIMO.
However, these techniques have their disadvantages. For example, adaptive SM (ASM) was firstly proposed in [8] to select the modulation styles according to the channel information. In ASM, the
transmission rate of each data frame is not fixed, which is not good for system design. And error propagation may happen in low-SNR domain due to the wrong demodulation for any single data symbol.
Furthermore, a combination of adaptive spatial modulation and antenna section scheme is proposed in [9] to enhance the performance. However, it inherits the drawbacks of ASM, and the introduction of
additional antennas will introduce extra implementation cost and channel estimation complexity.
This paper considers another aspect as adaptive power scaling for SM-MIMO with limited feedback. A novel power scaling spatial modulation (PS-SM) scheme is proposed, based on the introduction of
scaling factor (SF) for weighting the modulated symbols and enlarging the minimum Euclidean distance of the transmit signals. As a result the proposed method can effectively improve the system
performance. We will show that the proposed PS-SM outperforms the conventional adaptive spatial modulation (ASM) with the same feedback. In this case, PS-SM can be an alternative scheme to ASM
schemes by overcoming their disadvantages. Furthermore, the enlarging of the minimum distance could lead to an improvement of equivalent transmit power. In this paper, a power attenuation factor is
introduced to keep the minimal transmit power for improving the energy efficiency.
The rest of the paper is organized as follows. In Section 2, the basic structure of SM-MIMO and main idea of adaptive schemes are summarized. The proposed PS-SM scheme is described in Section 3. In
Section 4, bit-error rate (BER) performance of PS-SM and conventional schemes are disclosed by simulation results. Finally, conclusions are given in Section 5.
2. Conventional SM and Adaptive Scheme
Assume an SM-MIMO [1] with transmit and receive antennas. The information bits vector , with length , can be divided into two parts as and ( is the size of QAM constellation) bits, as . Then the bit
information vector is mapped into transmit vector with length , in which there is only one nonzero element , where is the index of transmit antennas with and is the index of QAM constellation with .
Let be the MIMO channel matrix. Then the receive vector , with length , can be written as
where represents the th column of and is AWGN noise with variance of .
For optimal ML detector [4], the estimated SM-MIMO symbol is given by
where denotes the estimated transmit symbol vector, is the set of all possible transmit symbols, and represents the Frobenius norm of the vector.
The performance bound of ML detection is decided by the minimal Euclidean distance , defined as [8]
In general, the minimal Euclidean distance is determined by the transmit vector and channel matrix for conventional SM. In ASM [8, 9], the modulation styles for each antenna can be selected to offer
extra freedom for generating a larger . However, ASM schemes suffer from a nonconstant data rate when different antennas employ different modulation styles. For instance, for an SM-MIMO with 2
transmit antennas, if the first and second antennas separately select BPSK and QPSK, the data rate may be variable between 2 and 3 bits/slot. Furthermore, in the receiver side, if the antenna index
is estimated incorrectly, the number of the received bits will differ from that of original transmission, which will cause consecutive errors.
3. Power Scaling Spatial Modulation
For avoiding the drawback of ASM schemes, in this paper, a novel power scaling spatial modulation scheme is proposed. The system block is shown in Figure 1. Firstly, the information bits are divided
into two parts to modulate the digital constellations and the active transmit antenna index. Then at each time instant one active transmit antenna is selected with a scaling factor (SF) , for
weighting the modulated symbols, where and is the total number of candidate SF groups. In this case, the transmit SM symbol can be expressed as . For normalizing the transmit power, let . According
to (1) and (3), in PS-SM, the minimal Euclidean distance is a function of , given and further computed as with
where denotes the real part of the input and are the inner products of the columns of the channel matrix and can be reused for reducing the calculation complexity. Comparing (4) to the processing of
[8], it is shown that the proposed PS-SM scheme has similar computational complexity as ASM.
From every possible set of SF candidates as , there is one minimal Euclidean distance . Then the optimal value of , which is defined as the maximal value of , is expressed as
The only limitations for the selection of SF are given as and . For example, we target a PS-SM-MIMO transmission system with the scaling factors obeying uniform distribution, which is generated in
Table 1, as . With limited feedback, the proposed PS-SM algorithm presets the same candidate SF sets at both transceiver sides. And the selected index of the optimal candidate will be delivered to
the transmitter.
Figure 2 gives the complementary cumulative distribution functions (CCDF) of the minimum Euclidean distance for a PS-SM-MIMO and conventional SM-MIMO. It is shown that with the introduction of
scaling factor, the minimum Euclidean distance can be enlarged effectively. Moreover, with the increase in feedback amount, the increase in minimum Euclidean distance is more considerable. In this
case, the system performance will be effectively improved. The following simulation results will show the benefit of PS-SM to system performance. In this case, the proposed PS-SM is an alternative
scheme by overcoming the disadvantage of traditional ASM.
4. Simulation Results
Computer simulation is performed for comparing the system performance of conventional SM, ASM, and PS-SM. We consider an SM-MIMO system with 2 and 4 transmit antennas and QPSK modulation under
Rayleigh flat fading channel, so the transmission data rate is 3 and 4 bits per symbol, respectively. The number of receive antennas is 2. Assume that the channel information of Rayleigh flat fading
channel is perfectly estimated at the receiver side.
Figure 3 gives the system performance of SM-MIMO system. Both ASM, and PS-SM schemes are with 5 candidates for a fair comparison. Firstly, with limited feedback, both ASM and PS-SM outperform
conventional SM. For example, at BER of 10^−3, there is a 1.7dB SNR gap between ASM and conventional SM. Furthermore, with the same amount of candidate adaptive sets, the proposed PS-SM outperforms
ASM. More specifically, at BER of 10^−3, 0.9dB SNR gain can be achieved by the proposed PS-SM scheme.
Figure 4 shows the comparison of system performance of PS-SM with different amounts of feedback as 1, 2, and 3 bits in which 2, 4, and 8 candidate SF sets are assumed, respectively. The system is
with transceiver configuration and QPSK modulation. It is shown that the performance of PS-SM improves with the increase in the number of feedback bits which is related to the number of factor
candidates. At a BER of 10^−3, there are 2.5dB, 3.8dB, and 4.4dB performance gains in PS-SM with 1-bit, 2-bit, and 3-bit feedback, respectively. In this case, a feedback of 2 bits gives a better
tradeoff between feedback amount and system performance.
Since PS-SM can effectively enlarge the minimum Euclidean distance, the equivalent transmit power is also enlarged. In this part, we consider a tradeoff between system performance and
energy-efficient minimum power transmission (Figure 5). A power attenuation factor is introduced to normalize the enlarged minimum Euclidean distance to original SM. In this case, we keep the same
system performance to traditional SM while minimizing the transmit power for improving the energy efficiency. A PS-SM with different amounts of feedback as 1, 2, and 3 bits in which 2, 4, and 8
candidate SF sets are assumed, respectively. The system is with transceiver configuration and QPSK modulation. A PS-SM with different amounts of feedbacks as 1, 2, and 3bits are assumed, with 2, 4,
and 8 candidates SF sets respectively. Furthermore, energy saving of 0.4dB, 1.5dB, and 2.2dB is achieved with feedback amounts of 1, 2, and 3 bits, respectively.
5. Conclusions
For overcoming the disadvantages of original ASM schemes, a novel power scaling spatial modulation scheme was proposed as an alternative adaptive SM-MIMO scheme with limited feedback. In PS-SM,
multiple candidate scaling factor sets are introduced for weighting the modulated symbols on each transmit antenna, so as to enlarge the minimal Euclidean distance and improve the system performance.
Simulation results showed that the proposed PS-SM outperforms ASM schemes with the same feedback amount on similar computational complexity.
This work was supported in part by the National Science Foundation of China under Grant no. 61101101, the National High-Tech R&D Program of China (“863” Project under Grant no. 2011AA01A105), and the
Foundation Project of National Key Laboratory of Science and Technology on Communications under Grant 9140C020404110C0201.
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2. J. Fu, C. Hou, W. Xiang, L. Yan, and Y. Hou, “Generalised spatial modulation with multiple active transmit antennas,” in Proceedings of the IEEE Globecom Workshops (GC '10), pp. 839–844, December
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and Networking in China (CHINACOM '11), pp. 102–107, August 2011. View at Publisher · View at Google Scholar · View at Scopus
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6. C. Xu, S. Sugiura, S. Ng, and L. Hanzo, “Spatial modulation and space-time shift keying: optimal performance at a reduced detection complexity,” IEEE Transactions on Communications, vol. 61, no.
1, pp. 206–216, 2013.
7. P. Yang, Y. Xiao, L. Li, Q. Tang, and S. Li, “An improved matched-filter based detection algorithm for space-time shift keying systems,” IEEE Signal Processing Letters, vol. 19, no. 5, pp.
271–274, 2012. View at Publisher · View at Google Scholar · View at Scopus
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at Google Scholar · View at Scopus
9. P. Yang, Y. Xiao, L. Li, Q. Tang, Y. Yu, and S. Li, “Link adaptation for spatial modulation with limited feedback,” IEEE Transactions on Vehicular Technology, vol. 61, no. 8, pp. 3808–3813, 2012. | {"url":"http://www.hindawi.com/journals/ijap/2013/718482/","timestamp":"2014-04-16T12:18:14Z","content_type":null,"content_length":"138960","record_id":"<urn:uuid:8d46d32c-d602-4ae0-b7cd-f77d3ed11b8d>","cc-path":"CC-MAIN-2014-15/segments/1397609523265.25/warc/CC-MAIN-20140416005203-00164-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Buoyancy Force Question
October 21st 2008, 12:43 AM #1
Mar 2008
Buoyancy Force Question
I am looking to investigate the buoyancy force of a cuboid. The dimensions of the cuboid are 30cm x 20cm x 15 cm. When the cuboid is placed in the water (the face 30cm x 20cm faces the water),
4cm of the cuboid will be submerged. If I push the cuboid so that the top face is 15mm below the surface of the water, I will have pushed the cuboid (15 - 4) + 1.5 = 12.5 cm. Therefore, using
some notes that I have I have worked out the buoyancy force to be:
F = p*A*y*g
where p = pressure of the water (998.2 kg/m^3, pressure of tap water at 20oC), A is the cross sectional area to face the water (0.06 m), y is the overall displacement (0.125 m) and g = 9.8m/s,
acceleration due to gravity. If I work this out, I get: 73.37 Newtons.
First things first, is this correct? Please note that this is NOT homework, just a problem that I am looking into. I have looked at some notes where I was placing a bottle filled with sand in
water, and then pushing it to a new depth. We used F = p*A*y*g to work out the buoyancy force in that case, however I dont believe that we considered the case if the bottle was pushed so far into
the water that it was fully submerged (as in this case). Therefore, I think that the equation is correct so far as I push the cuboid down to a depth of 11cm so that the top face of the cuboid
will be level with the top of the water, however I am unsure if I can carry the above reasoning through to when the object is fully submerged in water. I then want to consider pushing the cuboid
so that the top face is 15cm and then 30cm deep, if I know that I can apply the above principle at 15mm then I know that I can extend it to these two cases.
What I am really confused with (the reason i dont think that this works past 11cm) is that Archimedes said the Buoyancy Force is equal to the mass of the water displaced. However, when fully
submerged, surely there wont be any more water displaced if we push the cuboid deeper and deeper?!?
Any help on this would be greatly appreciated!
Cheers in advance,
I am looking to investigate the buoyancy force of a cuboid. The dimensions of the cuboid are 30cm x 20cm x 15 cm. When the cuboid is placed in the water (the face 30cm x 20cm faces the water),
4cm of the cuboid will be submerged. If I push the cuboid so that the top face is 15mm below the surface of the water, I will have pushed the cuboid (15 - 4) + 1.5 = 12.5 cm. Therefore, using
some notes that I have I have worked out the buoyancy force to be:
F = p*A*y*g
where p = pressure of the water (998.2 kg/m^3, pressure of tap water at 20oC), A is the cross sectional area to face the water (0.06 m), y is the overall displacement (0.125 m) and g = 9.8m/s,
acceleration due to gravity. If I work this out, I get: 73.37 Newtons.
What you have written as p is usualy $\rho$ and is the density of the water (as you should be able to see from the units)
The boyancy force is equal (and opposite if we are being carefull about direction) to the weight of the displaced water. In this case the cuboid is completley submerged so the boyancy force is:
$F=\rho A h = \rho Vg$
where $h$ is the dimension of the cuboid normal to a face of area $A$.
What you had is only applicable if the cuboid is not completly submerged.
What I am really confused with (the reason i dont think that this works past 11cm) is that Archimedes said the Buoyancy Force is equal to the mass of the water displaced. However, when fully
submerged, surely there wont be any more water displaced if we push the cuboid deeper and deeper?!?
That is not what Archimedes principle is, it is that the boyancy force is equal to the weight of the fluid displaced.
October 31st 2008, 08:03 AM #2
Grand Panjandrum
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Interpolated estimation of markov source parameters from sparse data,” Pattern Recognit. Practice
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- COMPUTATIONAL LINGUISTICS , 1996
"... The concept of maximum entropy can be traced back along multiple threads to Biblical times. Only recently, however, have computers become powerful enough to permit the widescale application of
this concept to real world problems in statistical estimation and pattern recognition. In this paper we des ..."
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The concept of maximum entropy can be traced back along multiple threads to Biblical times. Only recently, however, have computers become powerful enough to permit the widescale application of this
concept to real world problems in statistical estimation and pattern recognition. In this paper we describe a method for statistical modeling based on maximum entropy. We present a maximum-likelihood
approach for automatically constructing maximum entropy models and describe how to implement this approach efficiently, using as examples several problems in natural language processing.
, 1998
"... We present an extensive empirical comparison of several smoothing techniques in the domain of language modeling, including those described by Jelinek and Mercer (1980), Katz (1987), and Church
and Gale (1991). We investigate for the first time how factors such as training data size, corpus (e.g., Br ..."
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We present an extensive empirical comparison of several smoothing techniques in the domain of language modeling, including those described by Jelinek and Mercer (1980), Katz (1987), and Church and
Gale (1991). We investigate for the first time how factors such as training data size, corpus (e.g., Brown versus Wall Street Journal), and n-gram order (bigram versus trigram) affect the relative
performance of these methods, which we measure through the cross-entropy of test data. In addition, we introduce two novel smoothing techniques, one a variation of Jelinek-Mercer smoothing and one a
very simple linear interpolation technique, both of which outperform existing methods. 1
- Computational Linguistics , 1992
"... We address the problem of predicting a word from previous words in a sample of text. In particular we discuss n-gram models based on calsses of words. We also discuss several statistical
algoirthms for assigning words to classes based on the frequency of their co-occurrence with other words. We find ..."
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We address the problem of predicting a word from previous words in a sample of text. In particular we discuss n-gram models based on calsses of words. We also discuss several statistical algoirthms
for assigning words to classes based on the frequency of their co-occurrence with other words. We find that we are able to extract classes that have the flavor of either syntactically based groupings
or semantically based groupings, depending on the nature of the underlying statistics.
- IEEE Transactions on Acoustics, Speech and Signal Processing , 1987
"... Abstract-The description of a novel type of rn-gram language model is given. The model offers, via a nonlinear recursive procedure, a com-putation and space efficient solution to the problem of
estimating prob-abilities from sparse data. This solution compares favorably to other proposed methods. Wh ..."
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Abstract-The description of a novel type of rn-gram language model is given. The model offers, via a nonlinear recursive procedure, a com-putation and space efficient solution to the problem of
estimating prob-abilities from sparse data. This solution compares favorably to other proposed methods. While the method has been developed for and suc-cessfully implemented in the IBM Real Time
Speech Recognizers, its generality makes it applicable in other areas where the problem of es-timating probabilities from sparse data arises. Sparseness of data is an inherent property of any real
text, and it is a problem that one always encounters while collecting fre-quency statistics on words and word sequences (m-grams) from a text of finite size. This means that even for a very large
data col-lection, the maximum likelihood estimation method does not allow Turing’s estimate PT for a probability of a word (m-gram) which occurred in the sample r times is r* PT = where r We call a
procedure of replacing a count r with a modified count r ’ “discounting ” and a ratio rt/r a discount coefficient dr. When r ’ = r*, we have Turing’s discounting. Let us denote the m-gram wl, *.., w,
as wy and the number of times it occurred in the sample text as c(wT). Then the maxi-mum likelihood estimate is
- COMPUTATIONAL LINGUISTICS , 1990
"... In this paper, we present a statistical approach to machine translation. We describe the application of our approach to translation from French to English and give preliminary results. ..."
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In this paper, we present a statistical approach to machine translation. We describe the application of our approach to translation from French to English and give preliminary results.
- IN PROCEEDINGS OF THE THIRD CONFERENCE ON APPLIED NATURAL LANGUAGE PROCESSING , 1992
"... We present an implementation of a part-of-speech tagger based on a hidden Markov model. The methodology enables robust and accurate tagging with few resource requirements. Only a lexicon and
some unlabeled training text are required. Accuracy exceeds 96%. We describe implementation strategies and op ..."
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We present an implementation of a part-of-speech tagger based on a hidden Markov model. The methodology enables robust and accurate tagging with few resource requirements. Only a lexicon and some
unlabeled training text are required. Accuracy exceeds 96%. We describe implementation strategies and optimizations which result in high-speed operation. Three applications for tagging are described:
phrase recognition; word sense disambiguation; and grammatical function assignment.
- Readings in Speech Recognition , 1990
"... In the case of a trlgr~m language model, the proba-bility of the next word conditioned on the previous two words is estimated from a large corpus of text. The re-sulting static trigram language
model (STLM) has fixed probabilities that are independent of the document being dictated. To improve the l ..."
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In the case of a trlgr~m language model, the proba-bility of the next word conditioned on the previous two words is estimated from a large corpus of text. The re-sulting static trigram language model
(STLM) has fixed probabilities that are independent of the document being dictated. To improve the language mode] (LM), one can adapt the probabilities of the trigram language model to match the
current document more closely. The partially dictated document provides significant clues about what words ~re more likely to be used next. Of many meth-ods that can be used to adapt the LM, we
describe in this paper a simple model based on the trigram frequencies es-timated from the partially dictated document. We call this model ~ cache trigram language model (CTLM) since we are c~chlng
the recent history of words. We have found that the CTLM red,aces the perplexity of a dictated doc-ument by 23%. The error rate of a 20,000-word isolated word recognizer decreases by about 5 % at the
beginning of a document and by about 24 % after a few hundred words.
, 1991
"... Parameter sharing plays an important role in statistical modeling since training data are usually limited. On the one hand, we would like to use models that are as detailed as possible. On the
other hand, with models too detailed, we can no longer reliably estimate the parameters. Triphone generaliz ..."
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Parameter sharing plays an important role in statistical modeling since training data are usually limited. On the one hand, we would like to use models that are as detailed as possible. On the other
hand, with models too detailed, we can no longer reliably estimate the parameters. Triphone generalization may force two models to be merged together when only parts of the model output distributions
are similar, while the rest of the output distributions are different. This problem can be avoided if clustering is carried out at the distribution level. In this paper, a shared-distribution model
is proposed to replace generalized triphone models for speaker-independent continuous speech recognition. Here, output distributions in the hidden Markov model are shared with each other if they
exhibit acoustic similarity. In addition to detailed representation, it also gives us the freedom to use a large number of states for each phonetic model. Although an increase in the number of states
will inc...
- Computer, Speech and Language , 1996
"... An adaptive statistical languagemodel is described, which successfullyintegrates long distancelinguistic information with other knowledge sources. Most existing statistical language models
exploit only the immediate history of a text. To extract information from further back in the document's histor ..."
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An adaptive statistical languagemodel is described, which successfullyintegrates long distancelinguistic information with other knowledge sources. Most existing statistical language models exploit
only the immediate history of a text. To extract information from further back in the document's history, we propose and use trigger pairs as the basic information bearing elements. This allows the
model to adapt its expectations to the topic of discourse. Next, statistical evidence from multiple sources must be combined. Traditionally, linear interpolation and its variants have been used, but
these are shown here to be seriously deficient. Instead, we apply the principle of Maximum Entropy (ME). Each information source gives rise to a set of constraints, to be imposed on the combined
estimate. The intersection of these constraints is the set of probability functions which are consistent with all the information sources. The function with the highest entropy within that set is the
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simple graph is a tree?
May 7th 2009, 07:32 PM #1
May 2009
simple graph is a tree?
Prove that a simple graph is a tree if and only if it is connected but the deletion of any of its edges produces a graph that is not connected.
Can someone here please help me with this proof? Thanks in advance ...
This really depends on the definition of a tree you were given.
But for example if you were given the definition that a tree is a connected graph without any cycles, it follows straight from the definition that the graph is connected, and if you remove a path
between a and b, then no path exists between a and b, because otherwise in the original situation there would exist two paths, and thus a cycle.
The same argument works for other definitions like a tree is a connected graph with each vertice connected by only ONE path etc.
May 9th 2009, 07:45 AM #2
Junior Member
May 2009
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How to handle secondary keys in partitioning?
Re: How to handle secondary keys in partitioning?
Date: May 05, 2010 02:20AM
OK in this case it depends on how many partitions and how many matching rows there are for 'SELECT * FROM tp WHERE sender_id=1 ORDER BY date'.
Since it is partitioned by recipient_id pruning will not kick in.
Lets say you have N partitions and M matching rows (sender_id = 1).
This means that it has to do a index ordered read from each partition, and merge sort all N rows. And while there is still matching rows a new index ordered read would be done, until all matching
rows are found.
So if (M < N) i.e. more partitions than matching rows, it will need to do more index lookups (and the sum of all steps in the b-tree index will be higher).
But if (M > N) i.e. more matching rows than partitions, the number of index reads will be the same (but the sum of the b-tree depth will be slightly higher, and there will be an additional sorting
So there is an overhead of index search in partitioning, since the indexes are partitioned just like the data, but this overhead is higher per row if there are more partitions than matching rows and
shrinks per row as the matching row number increases.
And the cost for sorting is also added in case of partitioning.
The type of performance degradation is hard to estimate without benchmarking, it depends on how many partitions and how many matching rows are found, as well as used engine and if a covering index
can be used etc.
What I would suggest is that you benchmark your load on both partitioned and non partitioned tables. And if possible, add the results to this topic!
(Note that this is only index reads we discuss, the cost for index writes should be lower for partitioned tables, since the b-tree depth is lower per partition than it would be for a non partitioned
If you usually query recipient_id and sender_id independently, I would suggest you to try to also subpartition your data, perhaps like:
PARTITION BY LIST (recipient_id % 5)
SUBPARTITION BY HASH (sender_id) PARTITIONS 5
(PARTITION pREC_0 VALUES IN (0),
PARTITION pREC_1 VALUES IN (1),
PARTITION pREC_2 VALUES IN (2),
PARTITION pREC_3 VALUES IN (3),
PARTITION pREC_4 VALUES IN (4))
That way you have partitioned on both recipient_id and sender_id equally, and have a total of 25 partitions. (you could also use RANGE instead of LIST, but I used LIST to get an easy way to equally
distribute both columns.) | {"url":"http://forums.mysql.com/read.php?106,366049,366217","timestamp":"2014-04-21T07:20:52Z","content_type":null,"content_length":"22887","record_id":"<urn:uuid:b3c451be-27a0-4077-8178-c8be7307fee7>","cc-path":"CC-MAIN-2014-15/segments/1397609539665.16/warc/CC-MAIN-20140416005219-00352-ip-10-147-4-33.ec2.internal.warc.gz"} |
Vol. 6, No. 1, September 2011
Title: Integrals Involving I-Function
Authors: U.K. Saha, L.K. Arora and B.K. Dutta
Abstract: In this paper, we have presented certain integrals involving product of the I-function with exponential function, Gauss's hypergeometric function and Fox's H-function. The results derived
here are basic in nature and may include a number of known and new results as particular cases.
PP. 1-14
Title: Solution for Boundary Value Problem of non-integer order in L^2- space
Author: Azhaar H. Sallo
Abstract: In this paper, we shall prove the existence and uniqueness of a square - integrable solution in L^2-space, for the boundary value problem of non- integer order which has the form:
^cD^α[x] y(x) = f(x, y(x)), 1<α ≤2
y(a) = y[a], y(b) = y[b]
Where ^cD^α is the Caputo fractional derivative and a, b are positive constants with b not equal a. The contraction mapping principle has been used in establishing our main results.
PP. 15-24
Title: Combined Effect of MHD and Radiation on Unsteady Transient Free Convection Flow between Two Long Vertical Parallel Plates with Constant Temperature and Mass Diffusion
Authors: U. S.Rajput and P. K. Sahu
Abstract: The paper studies combined effects of MHD and radiation on unsteady transient free convection flow of a viscous, incompressible, electrically conducting and radiating fluid between two long
vertical parallel plates with constant temperature and mass diffusion, under the assumption that the induced magnetic field is negligible. TheLaplacetransform method has been used to find the
solutions for the velocity, temperature and concentration profiles. The velocity, temperature, concentration and skin-friction are studied for different parameters like Prandtl number, Schmidt
number, magnetic parameter, buoyancy ratio parameter and time.
PP. 25-39
Title: On the Solution of Fractional Kinetic Equation
Authors: B.K. Dutta, L.K. Arora and J. Borah
Abstract: In this paper, the solution of a class of fractional Kinetic equation involving generalized I-function has been discussed. Special cases involving the I-function, H-function, generalized
M-series, generalized Mittag-Leffer functions are also discussed. Results obtained are related to recent investigations of possible astrophysical solutions of the solar neutrino problem.
PP. 40-48
Title: A Parametric Study on Multi-Objective Integer Quadratic Programming Problems under Uncertainty
Author: Osama E. Emam
Abstract: This paper presents a parametric study on multi-objective integer quadratic programming problem under uncertainty. The proposed procedure presents a quadratic multi-objective integer
programming problem with a stochastic parameters in the right hand sides, and all constraints occurs under certain probability. We consider all random variables are normally distributed. We shall be
essentially concerned with three basic notions: the set of feasible parameters; the solvability set and the stability set of the first kind (SSK1). An algorithm to clarify the developed theory as
well as an illustrative example is presented.
PP. 49-60
Title: The n- Dimensional Generalized Weyl Fractional Calculus Containing to n-Dimensional $\bar{H}$-Transforms
Authors: V. B. L. Chaurasia and RaviShanker Dubey
Abstract: The object of this paper is to establish a relation between the n-dimensional $\bar{H}$-transform involving the Weyl type n-dimensional Saigo operator of fractional integration.
PP. 61-72
Title: An Optimal Distributed Control for Age-Dependent Population Diffusion System
Authors: Jun Fu, Renzhao Chen and Xuezhang Hou
Abstract: The optimal distributed control problem for age-dependent population diffusion system governed by integral partial differential equations is investigated in this paper. As new results, the
existence and uniqueness of the optimal distributed control are proposed and proved, a necessary and suffcient conditions for the control to be optimal are obtained, and the optimality system
consisting of integro-partial differential equations and variational inequalities are constructed in which the optimal controls can be determined. The applications of penalty shifting method for
infinite dimensional systems to approximate solutions of control problems for the population system are researched. An approximation program is structured, and the convergence of the approximating
sequences in appropriate Hilbert spaces is derived. The results in this paper may significantly provide theoretical reference for the practical research of the control problem in population systems.
PP. 73-85 | {"url":"http://www.emis.de/journals/GMN/volumes/vol_6_no_1_september_2011.html","timestamp":"2014-04-20T11:03:28Z","content_type":null,"content_length":"23896","record_id":"<urn:uuid:f7e0a502-ae94-4b7d-ab95-5ddb351952eb>","cc-path":"CC-MAIN-2014-15/segments/1397609538423.10/warc/CC-MAIN-20140416005218-00428-ip-10-147-4-33.ec2.internal.warc.gz"} |
Particle in 3D Rectangular Rigid Box
From ICT-Wiki
6.2 Particle in 3D Rectangular Rigid Box
6.2.1 Schrödinger Equation in Cartesian Co-ordinates
Consider a particle of mass $m\,$ and energy $E\,$ constrained to move in a three-dimensional rectangular potential well having sides of length equal to $L_x,\, L_y,\, L_z$ parallel to the $x,\, y,\,
z$ -axes respectively. Suppose there is no force acting on the particle in the box. The appropriate potential is $V(x,\,y,\,z) = 0$ for its position at a point $(x,\, y,\, z)$ given by
$0< \, x \, < \, L_x$
$0< \, y \, < \, L_y$
$0< \, z \, < \, L_z$
and $V(x,\,y,\,z) = \infty$ outside the box
Fig. Particle in 3-D rectangular box
The total kinetic energy $T\,$ is given by
$T=-\frac{\hbar^2}{2m}\left(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}\right)=-\frac{\hbar^2}{2m}\triangledown^2\,...........$ ---------------(1)
$\triangledown^2\,$ is called as the Laplacian operator
$-\frac{\hbar^2}{2m}\triangledown^2 \psi=E \psi$
For the motion of the particle inside the box the Schrödinger's time independent equation:
$-\frac{\hbar^2}{2m}\left(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}\right)\psi(x,\,y,\,z)=E \psi(x,\,y,\,z)$ ---------------(2)
It is a partial differential equation. It is assumed that the function $\psi\,$ can be written as a product of three functions, $X,\, Y$ and $Z\,$ each of which depends on only one of the coordinates
$x,\, y,\, z$ respectively. The equation can be solved by the technique of separation of variables.
$\psi=X(x).\,Y(y).\,Z(z)$ ---------------(3)
The Hamiltonian is the sum of three independent one dimensional Hamiltonians in totally separate variables
$H= H_x+H_y+H_z\,$.---------------(4)
$[H_x+H_y+H_z]\psi(x,\,y,\,z)=E \psi(x,\,y,\,z)$ ---------------(5)
Substituting for $\psi\,$ the Schrödinger's equation in the separated variables is
$X^{''}YZ+XY^{''}Z+XY^{''}Z+\frac{2mE}{\hbar^2}XYZ=0$ ---------------(6)
The terms $X^{''},\, Y^{''},\, Z^{''}$ are ordinary derivatives instead of the partial derivatives as each of the functions $X,\, Y,\, Z$, is a function of one variable only. Dividing the equation by
Each term of this equation depends on a different variable $x,\, y$, or $z\,$ and the three variables $x,\, y$, and $z\,$ are independent. The term $k^2=\frac{2mE}{\hbar^2}$ is a constant for a
particular value of $T\,$ kinetic energy. Velocity of the particle is a vector quantity. Velocity and corresponding kinetic energy can be resolved in three components along three co-ordinate axes $x,
\, y,\, z$.
$E \,= E_x+E_y+E_z$ ---------------(8)
The only way for the equation to remain valid for all of $x,\, y$, and $z\,$ in the interval is for each term to be constant. Therefore with separation of variables, the Schrödinger's equation
separates into three independent equations in $x,\, y,\, z$ respectively as follows
$\frac{-\hbar^2}{2m}\left(\frac{\partial^2 X(x)}{\partial x^2}\right) \,=E_x X \,(x)$
$\frac{-\hbar^2}{2m}\left(\frac{\partial^2 Y(y)}{\partial y^2}\right) \,=E_y Y \,(y)$
$\frac{-\hbar^2}{2m}\left(\frac{\partial^2 Z(z)}{\partial z^2}\right) \,=E_z Z \,(z)$ ---------------(9)
These differential equations have the solution:
$X(x) \,= A_x \,sin \,(k_x x)+B_x \,cos\, (k_x x) \,\,\,\,E_x =\frac{\hbar^2 k_x^2}{2m}$ ---------------(10a)
$Y(y) \,= A_y \,sin \,(k_y y)+B_y \,cos\, (k_y y) \,\,\,\,E_y =\frac{\hbar^2 k_y^2}{2m}$ ---------------(10b)
$Z(z) \,= A_z \,sin \,(k_z z) +B_z \,cos\, (k_z z) \,\,\,\,E_z =\frac{\hbar^2 k_z^2}{2m}$ ---------------(10c)
Applying the boundary conditions
$\psi(x=0)=0\rArr\, B_x=0 \,\,\,\,\,\,\,\,\psi(x=L_x)=0\rArr \, k_x L_x=n_x \pi$ ---------------(10a)
$\psi(y=0)=0\rArr \,B_y=0 \,\,\,\,\,\,\,\,\psi(y=L_y)=0\rArr \, k_y L_y=n_y \pi$ ---------------(10b)
$\psi(z=0)=0\rArr \,B_z=0 \,\,\,\,\,\,\,\,\psi(z=L_z)=0\rArr \, k_z L_z=n_z \pi$ ---------------(11c)
The resulting x-components of eigenfunctions are of the form
$\Psi_n=\sqrt{\frac{2}{L}}\,sin\,\left(\frac{n\pi x}{L}\right) \,\,\,\,\mathbf{n}=1,\,2,\,3,\,4....$
The solution for $y,\, z$ components have the same form. The total normalized eigenfunctions for the motion of the particle in the box are given by
$\psi(x,\,y,\,z)=\sqrt{\frac{8}{L_x L_y L_z}}sin(k_x x)\,sin\, (k_y y)\,sin(k_z z)$
$k_x=\frac{\pi n_x}{L_x} ,\, k_y=\frac{\pi n_y}{L_y} ,\, k_z=\frac{\pi n_z}{L_z}$ ---------------(12)
$n_x= 1,\, 2,\, 3,\, 4\, ....$
$n_y= 1,\, 2,\, 3,\, 4 \,....$
$n_z= 1,\, 2,\, 3,\, 4\, ....$ ---------------(13)
The eigenfunctions are zero outside the box. The eigenvalues of energy $E\,$ are given by
$E=\frac{\hbar^2(k_x^2+k_y^2+k_z^2)}{2m}=\frac{\hbar^2\pi^2}{2m}\left(\frac{n_x^2}{L_x^2}+\frac{n_y^2}{L_y^2}+\frac{n_z^2}{L_z^2}\right)$ ---------------(14)
These eigenvalues of the energy $E\,$ are called as the energy levels of the particle. They form quantized or discrete energy spectrum.
$\ast$ The integers $n_x,\, n_y,\, n_z$ are called as the quantum numbers. They are required to specify each stationary state. If the sign of the quantum numbers is changed, then there is no change
in the energy and in the wave function except for the minus sign. Hence all the stationary states are given by the positive integral values of $n_x,\, n_y,\, n_z$.
$\ast$ The value of any of the quantum numbers $n_x,\, n_y,\, n_z$ cannot be zero. If any one is taken as zero then $\psi (x,\,y,\,z) = 0$. It implies that the particle does not exist in the box
$\ast$ The lowest possible energy is obtained for $n_{x,} =1,\, n_{y,}=1,\, n_z =1$. It is called as the ground state energy and it depends on the values of $L_x,\, L_y,\, L_z$
The wave function corresponding to ground state energy is called as ground state wave function and denoted by $\Psi_{111}\,$
$\ast$ The excited state energy levels $E_{nx,\,ny,\,nz}$ are obtained by substituting different positive integer values for $n_x,\,n_y,\,n_z$ and the corresponding eigen functions $\Psi_{nx,\,ny,
\,nz}$ are labeled by these quantum numbers | {"url":"http://ictwiki.iitk.ernet.in/wiki/index.php/Particle_in_3D_Rectangular_Rigid_Box","timestamp":"2014-04-17T15:48:05Z","content_type":null,"content_length":"25730","record_id":"<urn:uuid:d55a6610-196d-45f1-bb3a-d5b4b17b3a76>","cc-path":"CC-MAIN-2014-15/segments/1397609530136.5/warc/CC-MAIN-20140416005210-00617-ip-10-147-4-33.ec2.internal.warc.gz"} |
word problem
April 21st 2011, 05:53 AM
word problem
cant figure this out at all.
basically there are 5 cds that come in a cereal box randomly. what are the chances of getting all 5 cds if you buy 12 boxes?
can someone point me in the right direction?
April 21st 2011, 12:11 PM
You need more information, i.e. the probability of getting a cd
April 21st 2011, 02:27 PM
This is an inclusion/exclusion question.
I deleted one reply because the notation was too complicated.
Lets say the CD's are $a,b,c,d,e$.
Let X be the event that at least one disk x is present in a sample of 12.
Now we want to show http://quicklatex.com/cache3/ql_96c0...d8f7d1f_l3.png
That is http://quicklatex.com/cache3/ql_9162...289b3a6_l3.png
Note that http://quicklatex.com/cache3/ql_807c...5c6672e_l3.png
April 22nd 2011, 11:49 AM
Archie Meade
If the cereal box is guaranteed to have a CD in it
and each of the CDs have equal probability of being placed in a cereal box,
then you can find the probability of NOT finding all five CDS in the 12 boxes
by calculating the probabilities that
(1) one of the CDs is not in the 12 packets
(2) two of the CDs are not in the 12 packets
(3) three of the CDs are not in the 12 packets
(4) four of the CDs are not in the 12 packets
All 5 CDs cannot be missing of course.
The probability that one CD is missing is 5C1(4/5)^(12)
because one of the other 4 CDs are in each box for each missing one.
The probability that two CDs are missing is 5C2(3/5)^(12)
because one of the other 3 CDs is in each box for every missing pair.
The probability that three CDs are missing is 5C3(2/5)^(12)
Also, include four CDs missing.
Now the question is...
Does the probability of one being missing include the probability of 2, 3 or 4 also missing ?
Subtract the probability of at least one CD missing from 1. | {"url":"http://mathhelpforum.com/statistics/178245-word-problem-print.html","timestamp":"2014-04-21T03:10:02Z","content_type":null,"content_length":"7327","record_id":"<urn:uuid:bad08434-def6-4d0d-89cd-b60698131619>","cc-path":"CC-MAIN-2014-15/segments/1397609539447.23/warc/CC-MAIN-20140416005219-00120-ip-10-147-4-33.ec2.internal.warc.gz"} |
Square roots by hand: Divide and Average
Date: 8/31/96 at 8:10:55
From: Janet Rodgers
Subject: Calculating square roots
Dear Dr. Math,
My 7th grade son has been trying to find out how to calculate
square roots by hand. I remember learning to do this in the old days
before calculators, but I can't find the technique in any of my books.
Can you find out how to do this?
Date: 9/1/96 at 0:29:0
From: Doctor Jerry
Subject: Re: Calculating square roots
There are several methods, but the most ancient (dating back to the
Babylonians, ca. 1700 B.C.) and best is the divide-and-average method.
Here's the method.
Let A be the number whose square root is wanted. Determine in some
way a first guess g of the square root of A. The remaining part of
the method is to repeat the following step: Divide A by the current
guess g and then average the quotient with g. The average is the new
Repeat until you have calculated the square root to desired accuracy.
Roughly, the number of decimal places doubles with each repetition.
Suppose we want to calculate the square root of 2. Take the first
guess to be 1.5 (or 1 or ...). I'm going to copy the results directly
from my calculator screen. In a "production line" calculation the
intermediate results need not be written down - just keep them in the
1. 2/1.5 = 1.33333333333
(1.5+1.33333333333)/2 = 1.41666666666
2. 2/1.41666666666 = 1.41176470589
(1.41666666666+1.41176470589)/2 = 1.41421568628
Since the square root of 2 is (to calculator accuracy) 1.41421356237,
the algorithm is doing rather well. One more step gives the result to
11 significant figures!
-Doctor Jerry, The Math Forum
Check out our web site! http://mathforum.org/dr.math/
Date: 9/1/96 at 0:32:23
From: Doctor Ken
Subject: Re: Calculating square roots
There are a couple of other methods to calculate square roots that
you can find in our archives, or look under "Square roots..." in our
Dr. Math FAQ at
-Doctor Ken, The Math Forum
Check out our web site! http://mathforum.org/dr.math/ | {"url":"http://mathforum.org/library/drmath/view/52623.html","timestamp":"2014-04-19T00:30:36Z","content_type":null,"content_length":"7152","record_id":"<urn:uuid:6872edb6-b8e6-4a46-983a-aa90955209dd>","cc-path":"CC-MAIN-2014-15/segments/1397609538787.31/warc/CC-MAIN-20140416005218-00453-ip-10-147-4-33.ec2.internal.warc.gz"} |
Selected Publications
1. Sigman, K. (1988). Regeneration in Tandem Queues with Multi-Server Stations. Journal of Applied Probability, 25, 391-403.
2. Sigman, K. (1988). Queues as Harris Recurrent Markov Chains. Queueing Systems (QUESTA),3, 179-198.
3. Sigman, K. (1989). Notes on the Stability of Closed Queueing Networks. Journal of Applied Probability, 26, 678-682 [Correction: (1990), 27].
4. Chao, X., M. Pinedo and K. Sigman, (1989). On the Interchangeability and Stochastic Ordering of Exponential Queues in Tandem with Blocking, Probability in the Engineering and Informational
Sciences (PEIS), 3, 223-236.
5. Sigman, K., (1990). One-Dependent Regenerative Processes and Queues in Continuous Time, Mathematics of Operations Research, 15, 175-189.
6. Sigman, K., (1990). The Stability of Open Queueing Networks, Stochastic Processes and their Applications, 35,11-25.
7. Sigman, K.(1991). A Note On A Sample-Path Rate Conservation Law and its Relationship with H= l G. Advances in Applied Probability, 23, 662-665.
8. Gelenbe, E., P. Glynn and K. Sigman, (1991). Queues with Negative Arrivals, Journal of Applied Probability, 28, 245-250.
9. Sigman, K. (1992). Light Traffic For Work-Load In Queues. Queueing Systems (QUESTA), 4, 429-442.
10. Browne, S. and K. Sigman (1992) . Work-Modulated Queues with Applictions to Storage Processes. Journal of Applied Probability, 3, 699-712.
11. Glynn, P. and K. Sigman, (1992). Uniform Cesaro Limit theorems for Synchronous Processes with Applications to Queues. Stochastic Processes and their Applications, 40, 29-44.
12. Sigman, K. and G. Yamazaki (1992). Fluid Models with Burst Arrivals: A Sample-Path Analysis. Probability in the Engineering and Informational Sciences (PEIS), 6, 17-27.
13. Sigman, K., and D. Simchi-Levi (1992). A Light Traffic Heuristic for an M/G/1 Queue with Inventory. Annals of Operations Research, 40, 371-380.
14. Yamazaki, G., K. Sigman and M. Miyazawa (1992). Moments in Infinite Channel Queues. Computers and Mathematics with Applications, 24, 1/2, 1-6.
15. Kiang, S. and K. Sigman (1992). Burst Fluid Models with General Flow and Process Rates.(Technical Report, Columbia University, Department of IEOR)
16. Bardhan, I., and K. Sigman (1993). Rate Conservation Law for Stationary Semimartingales. Probability in the Engineering and Informational Sciences (PEIS), 7,1-17.
17. Sigman, K. and R. Wolff (1993). A Review of Regenerative Processes. SIAM Review, 2, 269-288.
18. Sigman, K. and D. Yao. (1993) Finite Moments for Inventory Processes. Annals of Applied Probability, 3,765-778.
19. Bardhan, I. and K. Sigman (1994). Stationary regimes for inventory processes. Stochastic Processes and their Applications, 56 , 77-86.
20. Sigman, K., Thorisson, H. and R.W. Wolff (1994). A note on the existence of regeneration times. J. of Applied Probability, 31,1116-1122.
21. Asmussen, S. and K. Sigman (1996). Monotone Stochastic Recursions and their Duals. Probability in the Engineering and Informational Sciences (PEIS),10, 1-20.
22. Jain, G. and K. Sigman (1996) . A Polleczek-Khintchine Formulation for M/G/1 Queues with Disasters. Journal of Applied Probability,33, 1191-1200.
23. Jain, G. and K. Sigman (1996) . Generalizing the Polleczek-Khintchine Formula to Account for Arbitrary Work Removal. Probability in the Engineering and Informational Sciences (PEIS) 10, 519-531.
24. Sigman, K. (1996) . Queues Under Preemptive LIFO and Ladder Height Distributions for Risk Processes: A Duality. Stochastic Models, 12,4, 725-736.
25. P. Glasserman, K. Sigman and D. Yao (Editors) (1996). Stochastic Networks: Stability and Rare Events. Springer Lecture Notes in Statistics , 117, Springer-Verlag, New York.
26. A. Scheller-Wolf and K. Sigman (1997) . Delay Moments for FIFO GI/GI/s Queues. Queueing Systems (QUESTA)(to appear)
27. A. Scheller-Wolf and K. Sigman (1997) . New Bounds for Expected Delay in FIFO GI/GI/c Queues Queueing Systems (QUESTA), 25, 77-96.
28. Boucherie, R.J., Boxma, O. and K. Sigman (1997). A note on negative customers, GI/G/ workload, and risk processes Probability in the Engineering and Informational Sciences (PEIS), 10, 305-311.
29. P. Glynn and K. Sigman (1999). Independent Sampling of a Stochastic Process Stoch. Proc. Appls., 74, 151-164.
30. A. Scheller-Wolf and K. Sigman (1998) . Moments in Tandem Queues. Operations Research ,46, 378-380.
31. Asmussen, S., Kluppelberg, C, and K. Sigman (1999). Sampling at subexponential times, with queueing applications Stoch. Proc. Appls., 79, 265-286.
32. T. Huang and K. Sigman (1999). Steady-state asymptotics for tandem, split-match and other feedforward queues with heavy-tailed service Queueing Systems (QUESTA), 33, 233-260.
33. K. Sigman (1999). A Primer on heavy-tailed distributions Queueing Systems (QUESTA), 33, 261-275.
34. B. Bl aszczyszyn and K. Sigman (1999). Risk and Duality in Multidimensions Stoch. Proc. Appls., 83, 331-356.
35. K. Sigman (1999)(Guest Editor). Special Volume on Queues with Heavy-tailed Distributions Queueing Systems (QUESTA), 33.
36. R. Ryan and K. Sigman (2000). Continuous-time monotone stochastic recursions and duality Adv. Appl. Prob, 32,426-445.
37. R. Erikson and K. Sigman (2000). A simple stochastic model for close US presidential elections.
38. R. Erikson and K. Sigman (2000). Gore favored in the Electoral College.
39. R. Erikson and K. Sigman (2000). A dead heat and the Electoral College (Bush verus Gore).
40. M. Miyazawa, G. Nieuwenhuis, and K. Sigman (2000). Palm theory for random time changes, JAMSA, 14, 55-74.
41. Mor Harchol-Balter, K. Sigman and Adam Wierman (2002). Asymptotic Convergence of Scheduling Policies with respect to Slowdown. Performance Evaluation, 49, 241-256. International Symposium on
Computer Modeling, Measurement and Evaluation.
42. L. Munasinghe and K. Sigman (2004). A hobo syndrome? Mobility, wages, and job turnover. Labour Economics, 11, 191-218.
43. M. Nakayama, P. Shahabuddin, and K. Sigman (2004). On finite exponential moments for branching processes and busy periods for queues. Journal of Applied Probability (Special Volume), 41A,
44. G. Iyengar and K. Sigman (2004). Exponential Penalty Function Control of Loss networks. Annals of Applied Probability, 14, 1698-1740.
45. K. Sigman (2004) Queueing theory. Encyclopedia of Actuarial Science (EoAS), 3, 1357-1361
46. J. Cosyn and K. Sigman (2004). Stochastic networks: admission and routing using penalty functions. Queueing Systems (QUESTA), 48, 237-262.
47. J. Cosyn and K. Sigman (2004). Admission control of the infinite server queue with applications to bandwidth control (submitted).
48. K. Sigman (2006). Stationary marked point processes. (Contributed invited chapter in Springer Handbook of Engineering Statistics (Part A), Springer-Verlag.
49. K. Sigman and U. Yechiali (2007). Stationary remaining service time conditional on queue length. Operations Research Letters, 35, 581-583.
50. Harchol Balter, Varun Gupta, K. Sigman and W. Whitt (2007). Analysis of join-the-shortest-queue routing for web server farms. Performance Eval., 64, 1062-1081.
51. K. Sigman and W. Whitt (2011). Heavy-traffic limits for nearly deterministic queues. Journal of Applied Probability, 48, 657-678.
52. K. Sigman and W. Whitt (2011) Heavy-traffic limits for nearly deterministic queues II: Stationary distributions. Queueing Systems, 69, 145-173.
53. K. Sigman (2011). Exact simulation of the stationary distribution of the FIFO M/G/c queue. Journal of Applied Probability, 48A,209-216.
54. J. Blanchet and K. Sigman (2011). On Exact Sampling of Stochastic Perpetuities. Journal of Applied Probability, 48A, 165-182.
55. V. Goyal and K. Sigman (2011) On simulating a class of Bernstein polynomials. ACM Transactions on Modeling and Computer Simulation (TOMACS), 22, Issue 2, (to appear). | {"url":"http://www.columbia.edu/~ks20/papers.html","timestamp":"2014-04-18T03:45:50Z","content_type":null,"content_length":"9623","record_id":"<urn:uuid:7f7af931-4ddd-4eaa-bf14-b1289dd57397>","cc-path":"CC-MAIN-2014-15/segments/1397609532480.36/warc/CC-MAIN-20140416005212-00474-ip-10-147-4-33.ec2.internal.warc.gz"} |
Finding similarities in a multidimensional array
up vote 0 down vote favorite
Consider a sales department that sets a sales goal for each day. The total goal isn't important, but the overage or underage is. For example, if Monday of week 1 has a goal of 50 and we sell 60, that
day gets a score of +10. On Tuesday, our goal is 48 and we sell 46 for a score of -2. At the end of the week, we score the week like this:
In this example, both Monday (0,0) and Thursday and Friday (0,3 and 0,4) are "hot"
If we look at the results from week 2, we see:
For week 2, the end of the week is hot, and Tuesday is warm.
Next, if we compare weeks one and two, we see that the end of the week tends to be better than the first part of the week. So, now let's add weeks 3 and 4:
From this, we see that the end of the week is better theory holds true. But we also see that end of the month is better than the start. Of course, we would want to next compare this month with next
month, or compare a group of months for quarterly or annual results.
I'm not a math or stats guy, but I'm pretty sure there are algorithms designed for this type of problem. Since I don't have a math background (and don't remember any algebra from my earlier days),
where would I look for help? Does this type of "hotspot" logic have a name? Are there formulas or algorithms that can slice and dice and compare multidimensional arrays?
Any help, pointers or advice is appreciated!
algorithm math statistics time-series
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7 Answers
active oldest votes
This data isn't really multidimensional, it's just a simple time series, and there are many ways to analyse it. I'd suggest you start with the Fourier Transform, it detects "rhythms" in
a series, so this data would show a spike at 7 days, and also around thirty, and if you extended the data set to a few years it would show a one-year spike for seasons and holidays.
up vote 2 That should keep you busy for a while, until you're ready to use real multidimensional data, say by adding in weather information, stock market data, results of recent sports events and
down vote so on.
Ouch, faster by a few seconds. – fortran Oct 1 '09 at 18:32
5 I had the whole thing hotkeyed. I knew someday someone would ask a question whose answer was "the Fourier Transform". That frees up [f10], only two more to go. – Beta Oct 1 '09 at
xDDD I'd vote up you twice for that comment if I could :p – fortran Oct 1 '09 at 21:29
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The following might be relevant to you: Stochastic oscillators in technical analysis, which are used to determine whether a stock has been overbought or oversold.
I'm oversimplifying here, but essentially you have two moving calculations:
• 14-day stochastic: 100 * (today's closing price - low of last 14 days) / (high of last 14 days - low of last 14 days)
• 3-day stochastic: same calculation, but relative to 3 days.
up vote 2 The 14-day and 3-day stochastics will have a tendency to follow the same curve. Your stochastics will fall somewhere between 1.0 and 0.0; stochastics above 0.8 are considered overbought
down vote or bearish, below 0.2 indicates oversold or bullish. More specifically, when your 3-day stochastic "crosses" the 14-day stochastic in one of those regions, you have predictor of momentum
of the prices.
Although some people consider technical analysis to be voodoo, empirical evidence indicates that it has some predictive power. For what its worth, a stochastic is a very easy and
efficient way to visualize the momentum of prices over time.
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It seems to me that an OLAP approach (like pivot tables in MS Excel) fit the problem perfectly.
up vote 1
down vote
2 I think this is the most useful suggestion for someone who's not trying to hack this data in R or something and get complex forecasting or statistical significance results. Pivot
tables in Excel are incredibly useful, not that hard to set up or use, and you're almost certain to already have the appropriate tool. – Harlan Oct 2 '09 at 19:52
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What you want to do is quite simple - you just have to calculate the autocorrelation of your data and look at the correlogram. From the correlogram you can see 'hidden' periods of your
data and then you can use this information to analyze the periods.
Here is the result - your numbers and their normalized autocorrelation.
10 1,000
-2 0,097
1 -0,121
7 0,084
6 0,098
-4 0,154
2 -0,082
-1 -0,550
4 -0,341
up vote 1 down 5 -0,027
vote -8 -0,165
-2 -0,212
-1 -0,555
2 -0,426
3 -0,279
2 0,195
3 0,000
4 -0,795
7 -1,000
I used Excel to get the values. But the sequence in column A and add the equation =CORREL($A$1:$A$20;$A1:$A20) to cell B1 and copy it then up to B19. If you the add a line diagram, you
can nicely see the structure of the data.
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You can already make reasonable guesses about the periods of patterns - you're looking at things like weekly and monthly. To look for weekly patterns, for example, just average all the
mondays together and so on. Same goes for days of the month, for months of the year.
up vote 0 Sure, you could use a complex algorithm to find out that there's a weekly pattern, but you already know to expect that. If you think there really may be patterns buried there that you'd
down vote never suspect (there's a strange community of people who use a 5-day week and frequent your business), by all means, use a strong tool -- but if you know what kinds of things to look for,
there's really no need.
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Daniel has the right idea when he suggested correlation but I don't think autocorrelation is what you want. Instead I would suggest correlating each week with each other week. Peaks in
your correlation--that is values close to 1--suggest that the values of the weeks resemble each other (I.e. are peiodic) for that particular shift.
For example when you cross correlate
the result would be
the highest value is 3, which corresponds to shifting (right) the second array by 4
up vote 0 0 0 0 1 1 0 --> 0 0 1 1 0 0
down vote
and thenn multiplying component wise
0 + 0 + 1 + 2 + 0 + 0 = 3
Note that when you correlate you can create your own "fake" week and cross-correlate all your real weeks, the idea being that you are looking for "shapes" of your weekly values that
correspond to the shape of your fake week by looking for peaks in the correlation result.
So if you are interested in finding weeks that are close near the end of the week you could use the "fake" week
-1 -1 -1 -1 1 1
and if you get a high response in the first value of the correlation this means that the real week that you correlated with has roughly this shape.
add comment
This is probably beyond the scope of what you're looking for, but one technical approach that would give you the ability to do forecasting, look at things like statistical
up vote 0 down significance, etc., would be ARIMA or similar Box-Jenkins models.
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Not the answer you're looking for? Browse other questions tagged algorithm math statistics time-series or ask your own question. | {"url":"http://stackoverflow.com/questions/1505607/finding-similarities-in-a-multidimensional-array","timestamp":"2014-04-17T22:48:06Z","content_type":null,"content_length":"92149","record_id":"<urn:uuid:c7e8ea18-fdd1-4f6b-be39-0423f6976565>","cc-path":"CC-MAIN-2014-15/segments/1397609532128.44/warc/CC-MAIN-20140416005212-00032-ip-10-147-4-33.ec2.internal.warc.gz"} |
Symmetry Extensions and Their Physical Reasons in the Kinetic and Hydrodynamic Plasma Models
SIGMA 4 (2008), 006, 7 pages arXiv:0801.2773 http://dx.doi.org/10.3842/SIGMA.2008.006
Contribution to the Proceedings of the Seventh International Conference Symmetry in Nonlinear Mathematical Physics
Symmetry Extensions and Their Physical Reasons in the Kinetic and Hydrodynamic Plasma Models
Volodymyr B. Taranov
Institute for Nuclear Research, 47 Nauky Ave., 03028 Kyiv, Ukraine
Received October 31, 2007, in final form January 14, 2008; Published online January 17, 2008
Characteristic examples of continuous symmetries in hydrodynamic plasma theory (partial differential equations) and in kinetic Vlasov-Maxwell models (integro-differential equations) are considered.
Possible symmetry extensions conditional and extended symmetries are discussed. Physical reasons for these symmetry extensions are clarified.
Key words: symmetry; plasma; hydrodynamic; kinetic.
pdf (166 kb) ps (124 kb) tex (9 kb)
1. Ibragimov N.H., Kovalev V.F., Pustovalov V.V., Symmetries of integro-differential equations: a survey of methods illustrated by the Benney equation, Nonlinear Dynam. 28 (2002), 135-165, math-ph/
2. Cicogna G., Ceccherini F., Pegoraro F., Applications of symmetry methods to the theory of plasma physics, SIGMA 2 (2006), 017, 17 pages, math-ph/0602008.
3. Taranov V.B., On the symmetry of one-dimensional high frequency motions of a collisionless plasma, Sov. J. Tech. Phys. 21 (1976), 720-726.
4. Gordeev A.V., Kingsep A.S., Rudakov L.I., Electron magnetohydrodynamics, Phys. Rep. 243 (1994), 215-465.
5. Meleshko S.V., Application of group analysis in gas kinetics, in Proc. Joint ISAMM/FRD Inter-Disciplinary Workshop "Symmetry Analysis and Mathematical Modelling", 1998, 45-60.
6. Horton W., Drift waves and transport, Rev. Modern Phys. 71 (1999), 735-778.
7. Taranov V.B., Drift and ion-acoustic waves in magnetized plasmas, symmetries and invariant solutions, Ukrainian J. Phys. 49 (2004), 870-874.
8. Taranov V.B., Symmetry extensions in kinetic and hydrodynamic plasma models, in Proceedinds of 13th International Congress on Plasma Physics (2006, Kyiv), 2006, A041p, 4 pages.
9. Cicogna G., Laino M., On the notion of conditional symmetry of differential equations, Rev. Math. Phys. 18 (2006), 1-18, math-ph/0603021. | {"url":"http://www.emis.de/journals/SIGMA/2008/006/","timestamp":"2014-04-17T04:16:17Z","content_type":null,"content_length":"6045","record_id":"<urn:uuid:7f1117ec-b1fd-4432-9636-8896c5f2da9f>","cc-path":"CC-MAIN-2014-15/segments/1397609526252.40/warc/CC-MAIN-20140416005206-00111-ip-10-147-4-33.ec2.internal.warc.gz"} |
use paradox in a sentence.
How is that paradoxical?
Google "paradox of thrift.
The Haskell Paradox, too.
So where's the paradox?
In what sense is the birthday paradox a paradox?
Your statement is a paradox!
It's called the birthday paradox :)
Your two statements are paradoxical.
This title is a paradox.
The paradox of choice - as a framework!
How is misapplied probability theory a paradox?
Recently Searched Words
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Prove x^n > x for all x > 1.
November 20th 2009, 05:54 PM
Prove x^n > x for all x > 1.
Suppose n > 1 is a positive integer. Prove $x^n > x$for all x > 1.
(hint: suppose we know f and g are differentiable on the interval (-c,c) and f(0) = g(0), if f'(x) > g'(x) for all $x \in(0,c)$, then f(x) > g(x) for all $x \in (0,c)$.
November 20th 2009, 06:38 PM
Do you know how to prove something by induction? If not I am unsure how to do this for general n.
November 20th 2009, 06:46 PM
November 20th 2009, 06:49 PM
What you want to do is use the hint on a base case (n=2). Then you will make the assumption that the problem hypothesis holds for n=k. Use this assumption to prove that it is true for n=k+1.
November 20th 2009, 06:52 PM
November 20th 2009, 06:54 PM
Because it is easy and the problem statement says n > 1.
November 20th 2009, 06:59 PM
November 20th 2009, 07:03 PM
Note that if $f(x)=x^n\quad n>1$ and $g(x)=x$. That $f(1)=1=g(1)$ and $f'(x)=nx^{n-1}\ge n>1=g'(1)$ and the concluso follows.
November 20th 2009, 07:05 PM
Okay, I will show you the base case.
Assume n=2.
$f(x) = x^2, \; g(x) = x$
Now let's show that these are equal at $x=0$
$f(1)=1^2 = 1 = g(1)$
Now we differentiate the two
$f'(x) = 2x$
$<br /> g'(x) = 1$
For $x>1$
$<br /> 2x > 2$
Therefore we see that $f'(x)=2x>1 =g'(x)$ for $x>1$.
November 20th 2009, 07:07 PM
November 20th 2009, 08:30 PM
November 20th 2009, 09:26 PM
November 20th 2009, 09:33 PM
We know $g(x)=x$ because you are trying to prove $x^n > x$ and the hint was suggesting you $f(x) > g(x)$
Wouldn't you agree $nx^{n-1} \ge n$ for $x>1$ and $n>1$? But we know that $f'(x) = nx^{n-1}$ and $g'(x) = 1$.
November 20th 2009, 09:35 PM
November 20th 2009, 09:37 PM
Yes, because the hint was to prove $f(x) > g(x)$ and we wanted to prove $x^n > x$ so it makes sense to assume $g(x)=x$. | {"url":"http://mathhelpforum.com/calculus/115829-prove-x-n-x-all-x-1-a-print.html","timestamp":"2014-04-20T01:43:14Z","content_type":null,"content_length":"24721","record_id":"<urn:uuid:d83e1704-144f-4572-ab0b-070bb0146d77>","cc-path":"CC-MAIN-2014-15/segments/1397609537804.4/warc/CC-MAIN-20140416005217-00058-ip-10-147-4-33.ec2.internal.warc.gz"} |
What Are the Assumptions of Classical Fault-Tolerance?
Oftentimes when we encounter the threshold theorem for quantum computation, it is pointed out that there are two assumptions which are necessary for fault-tolerant quantum computation. These two
assumptions are
1. A supply of refreshable pure or nearly pure ancillas into which the entropy generated by quantum errors can be dumped throughout the error correction procedure.
2. Parallel operations.
For point 1, an important reference is Aharonov, Ben-Or, Impagliazzo and Nisan, quant-ph/9611028, and for point 2, an important reference is Aharonov and Ben-Or, quant-ph/9611029. These two
assumptions are provably necessary. Usually when I see these two points presented, they are presented as if they are something unique to quantum computation. But is this really true? Well certainly
not for the first case. In fact quant-ph/9611028 is first about computation in a noisy model of classical reversible computation and second about a noisy model of quantum computation! What about
point 2? Well I can’t find a reference showing this, but it seems that the arguments presented in quant-ph/9611029 can be extended to reversible noisy classical computation.
Of course many of you will, by now, be grumbling that I haven’t included the other assumptions usually stated for the threshold theorem for fault-tolerant quantum computation. Okay, so I’ll put it
3. A noisy model in relation to control of your system which is not too ridiculous.
Well, okay, usually it is not stated this way, but that’s because I don’t have an easy way to encapsulate the noise models which lead to provable lack of ability to compute fault tolerantly. But
certainly there are similar noise constaints for classical computation which are provably necessary for fault-tolerant classical computation.
So what are the difference between the assumptions for fault-tolerant quantum and fault-tolerant classical computation? I strongly believe that the only differences here are difference arising simply
going from a theory of probabilities in classical computation to a theory of amplitudes in quantum computation. Of course this short changes the sweet and tears which is necessary to build the theory
of fault-tolerant quantum computation, and I don’t want to disparage this in any way. I just want to suggest that the more we learn about the theory of fault-tolerant quantum computation, the more we
recognize its similarity to probablistic classical computation. I call this general view of the world the “principal of the ubiquitous factor of two in quantum computing.” The idea being mostly that
quantum theory differs from probablistic theory only in the necessity to deal with phase as well as amplitude errors, or more generally, to deal with going from probabilities to amplitudes. This
point of view is certainly not even heuristic, but it seems at least that this is the direction we are heading towards.
Of course the above view is speculative, not rigorous, and open to grand debate over beers or the within the battleground of an academic seminar. But it certainly is one of the reasons why I’m
optimistic about quantum computing, and why, when I talk to a researcher in higher dimensional black holes (for example) and they express pessimism, I find it hard to put myself in their shoes. To
summarize that last sentence, I’m betting we have quantum computers before higher dimensional black holes are discovered
29 Responses to What Are the Assumptions of Classical Fault-Tolerance?
1. I think there is a long tradition in the cellular automaton community of noting the need for parallelism to achieve fault tolerance. It is certainly emphasized in the papers by Peter Gacs. Which
is why we referred to Gacs in noting that these requirements apply to classical as well as quantum fault tolerance in Sec. II of http://arxiv.org/abs/quant-ph/0110143
Like or Dislike: 0 0
2. I meant to add that the reason we should emphasize these conditions for fault tolerance when giving talks about quantum fault tolerance (even though the same conditions must be satisfied
classically) is that it is important, when considering the promise of any proposal for physically realizing quantum gates, to assess how well that realization will meet these conditions. That
parallel operation and a place to dump entropy are both necessary is sometimes overlooked.
Like or Dislike: 0 0
3. Either you’re secretly hosting a mirror of arxiv.org, or your URLs need fixing.
Like or Dislike: 0 0
4. Oops. Doh. Fixed.
Like or Dislike: 0 0
5. “I strongly believe that the only differences here are difference arising simply going from a theory of probabilities in classical computation to a theory of amplitudes in quantum computation.”
If run-of-the-mill classical computers were reversible, then I might agree.
What’s the largest scale classical reversible computer ever built?
Like or Dislike: 0 0
6. Ah but going to irreversible classical is like going to quantum theory with measurements, so I will restate:
“I strongly believe that the only differences here are differences arising simply going from a theory of probabilities in classical computation to a theory of amplitudes and probabilities in
quantum computation.”
Like or Dislike: 0 0
7. It appears to be correct that the requirements for fault tolerance in the quantum case are very similar (if not identical ) to those in the classical case. What is unclear is what is the correct
interpretation of this similarity. Does this means that since classical computation exists, quantum computers (and the “new phase of matter” they seem to represent, according to Dave) must be
feasible as well? Or is it that we should have a second look at “ridiculous” noise models of item 3. (I’d be happy to see many and detailed such examples even if not encapsulated.)
A technical point: as far as I know, the (indeed, important) result about noisy reversible computation still leaves open the
possibility that a noiseless classical computer
plus a noisy reversible quantum computer can factor in a polynomial time. Finding any model of low-rate noise (as ridiculous as one wishes) which (provably) does not enable even log-depth
computing seems difficult.
Like or Dislike: 0 0
8. Gil, I thought that the modular exponentiation part of Shor’s algorithm wasn’t log-depth. Am I misinterpreting what you are saying?
Also, of course, I’m a grumpy old man, so I will always maintain that the model of noiseless classical computers does not exist (nor does the model of noiseless quantum computers exist.) But
that’s just what I get for hanging out with too many experimental physicists.
Like or Dislike: 0 0
9. I meant that noiseless classical computers + log depth (noiseless) quantum computers are strong enough to enable factoring in polynomial time, (as far as I know, by a result of R. Cleve and J.
Watrous, quant-ph/0006004.)
Like or Dislike: 0 0
10. Ah right, thanks! Forgot about that. The idea is you compute the bitwise exponentiated terms classically in poly time and then do an interated multiplication a la Beame, Cook, and Hoover for the
quantum part of the circuit.
Like or Dislike: 0 0
11. Under the two assumptions that Dave mentioned the threshold theorem describes one important obstruction for fault tolerance:
(1) – The noise rate is large.
This obstruction deals with a single qubit/bit. We need low noise/signal ratio. Indeed there is essentially no difference between classical and quantum systems, and the feasibility of classical
computing indicates that intolerably high noise rate is not a universal property of noisy computing devices.
However, the following possible principal is also an obstruction
(2) – The correlation between the noise acting on pairs of correlated elements in a noisy device is high.
This obstruction deals with pairs of qubits. We need to require low signal/noise ratio also in terms of correlations. Again I see no difference between classical and quantum devices and am
curious if (2) generally applies to noisy physical devices (classical or quantum) whose behavior is stochastic. Such a principal appears to allow fault-tolerant classical computing, but be
harmful to quantum computing. (The assumptions of the threshold theorems exclude the possibility of (2))
Of course, one can ask how such a principal can come about. We do not want to seriously consider the fantastic scenarios of terrorist attacks Dave mentioned in his previous post, but rather
stochastic models for noise which are based on local operations, namely the noise being created by a sort of stochastic (rather primitive) quantum computer running aside our physical device. The
question is whether such models can create damaging patterns of noise. This looks interesting to me and starting with “ridiculous” models seems rather reasonable. Alternatively (or in parallel),
one can try to test (2) empirically.
Like or Dislike: 0 0
12. “I strongly believe that the only differences here are differences arising simply going from a theory of probabilities in classical computation to a theory of amplitudes and probabilities in
quantum computation.â€
Dave, I’m surprised you’re discounting the importance of the quantum tensor product structure (especially after you made such nice use of it in your “Operator Quantum Error Correcting Subsystems
for Self-Correcting Quantum Memories” paper). What about that old dictum “fight entanglement with entanglement”? It does not seem to apply to classical fault tolerance.
Like or Dislike: 0 0
13. Daniel,
The classical dictum is “fight correlation with correlation.” A classical noise process is simply getting correlated with a classical environment. If you don’t create correlated classical states
you won’t be able to protect your classical information (encoding into a classical error correcting code is nothing more than the limit of creating perfect correlation.)
Also I don’t think a tensor product structure alone makes the difference between the power of classical and quantum computers. The difference is when you combine complex amplitudes with the
tensor product.
Or at least in my myopic view of the world
Like or Dislike: 0 0
14. Gil,
I don’t understand what you mean that correlated errors seem to allow classical but not allow quantum computation. Certainly if I come up with ridiculous (to use my favorite word
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15. Dave,
As a Bell’s theorem entrepeneur I’m sure you’d agree there is a qualitative difference between classical correlation and quantum entanglement. (Well maybe your work on the communication cost of
simulating Bell correlations would actually indicate just a quantitative difference.)
And indeed, I meant to point out that there is more to quantum computing than complex amplitudes: the power comes from combining the latter with the tensor product structure.
Like or Dislike: 0 0
16. Right, so we agree that I that there is more to the the power of quantum computing than complex amplitudes. But I would argue (the original direction of your posting) that there is also more to
the difference between quantum and classical fault tolerance than amplitudes versus probabilities. And that more is (I suspect) at least in part the quantum tensor product structure (TPS).^*
I do agree that entanglement is not the single ingredient one must add to amplitudes to understand the quantum/classical difference. This is why I emphasized quantum TPS rather than entanglement.
Indeed, if you accept the “generalized entanglement” idea of Barnum et al. (quant-ph/0305023)^** then entanglement exists independently of a TPS. Namely, they have examples of generalized
entanglement where no subsystem partition exists (such as a single spin-1 particle). I suspect such entanglement plays no role in the quantum/classical speedup or fault tolerance distinction.
^* And that TPS is not an absolute property of the Hilbert space, but rather a property that is relative to the experimentally accessible set of observables: Quantum Tensor Product Structures are
Observable Induced
^** A pure state is generalized unentangled relative to a set of distinguished observables if its reduced state is pure, and generalized entangled otherwise.
Like or Dislike: 0 0
17. Daniel: see we agree!
I agree that there is more the the power of quantum computing than complex amplitudes. But there is more to the power of classical computing in classical computing, too!
It is an interesting question, why do quantum computers seem to outperform classical computers. But we can also ask, for example, why do classical computers outperform the Clifford group gate set
with computatinal basis preparation and measurement. Clearly Clifford group gates can generate entanglement, are quantum, but are also not universal for classical computation and can be
simulated. Why are classical computers more powerful than this type of quantum computer (which is the easiest model to implement fault-tolerantly, interestingly)?
I personally suspect that the reason we get into thinking about the power of quantum computation is that we are in the very early days of quantum algorithms. Certainly entanglement has something
to do with it, in the sense that it is “necessary
(in the Jozsa and Linden sense), but it is not sufficient. So trying to explain the power of quantum computers as coming just from a tensor product with a complex vector space, doesn’t seem to me
to be the answer.
And this is probably great! Because it means that there is much we need to explore. The answer, I suspect, won’t be a simple one, but more likely it will envolve some very sophisticated theory.
Can you tell that I’m sitting in a computer science department, now?
Like or Dislike: 0 0
18. We agree.
The difference may be that I think calling generalized entanglement “entanglement” is a bit heavy handed. I am very enthusiastic about the idea behind “generalized entanglement,” and especially
its usefulness in many settings (as, for example, in condensed matter systems where it leads to a definition of purity which corresponds at least in some systems to an order paramter
distinguishing different phases of the system), but I personally think that calling it entanglement is very confusing.
For example consider the example you site, a spin-1 particle. One takes a three dimensional irrep of su(2) and then looks at observables of the generators. These form a three dimensional sphere
and the extremal points on the boundary are the spin-coherent states. Then a state like |m=0> is not on the boundary and is therefore not a “pure” state with respect to these observables. They
then proceed to call this state a “generalized entangled” state, since entangled subsystem states are distinguished by being the states that are mixed when tracing over the full pure state. I do
agree that this is an interesting (and in many cases useful) definition, but what does this have to do with anything being entangled with anything else? I would instead talk about states which
are pure with respect to an observable algebra and states which are mixed with respect to an observable algebra. Then the interesting thing about “generalized entanglement” has nothing to do with
entanglement but everything to do about pure states with respect to one algebra which are mixed with respect to a different algebra.
But that’s just my opinion. And I just got my wisdom teeth out, so clearly I’m not thinking clearly.
Like or Dislike: 0 0
19. Dave, “Gil, I don’t understand what you mean that correlated errors seem to allow classical but not allow quantum computation. Certainly…they too will destroy classical computation”
Actually, there are some forms of highly correlated noise which allow classical computation and harm quantum computation but this was NOT the point I was making above. (Obstruction (2) from my
previous comment implies no correlations for errors in the classical case when the bits themselves are not correlated.)
“The only way I can see engineering worlds with classical but not quantum computation…”
Let us engineer a world, then, Dave. The requirement is this: (obstruction (2) to quantum fault tolerance from my comment above.)
(*) In any noisy system the noise “acting” on pairs of elements which are substantially correlated is also substantially correlated.
(Well, the word “noisy” refer to what happens when we approximate a small subsystem of a large system by neglecting the effect of the “irrelevant” elements (and by having an approximate “initial
conditions”). We suppose that we can have good approximations (this is what we mean by a small noise rate, and this assumption allows error correction and computation to start with,) but (*) is
an assumption about the nature of such approximations.)
1. (*) is just a specification, if we want to send our world plans for production we need more details how (*) comes about.
2. Maybe our present world already satisfies (*) BOTH for classical correlations and for quantum correlations. Can you think of a counter example? We may be able to examine (*) from the behavior
of pairs of qubits in small quantum computers.
3. (*) will allow classic (even randomized) computation (because in this case we do not need the bits to be correlated so (*) does not impose any condition on the noise being correlated.) But (*)
appears to be damaging
for quantum computation.
4. If (**) we have a noisy quantum computer with highly entangled qubits, and if (*) holds, then the nature of low-rate noise will be rather counter-intuitive. There will be a substantial
probability of a large proportion of all qubits being hit. (Hmmm, you called this t-error, nice!) I do not know if such a pattern for the noise should be considered ridiculous. But if you do,
note, that there are two “if”s here, and decide if you rather give up (**) or (*).
Returning to the main theme of this post: I agree that the issues of fault-tolerance are very similar for the classical and quantum case (in-spite of the recent Bacon-Lidar pact), I do not
understand why this should be a source of optimism (or pessimism).
Like or Dislike: 0 0
20. Without taking any position on the relative ease of classical-versus-quantum error correction, the “difficult-to-correct” postulate of Gil Kalai:
(*) In any noisy system the noise “acting†on pairs of elements which are substantially correlated is also substantially correlated.
… is a very reasonable postulate (to an engineer). For example, suppose two qubits are dipole-coupled with a coupling C~1 (in natural units set by the clock interval of the computation).
To an engineer, though, no qubit coupling C ever takes exactly its design value of unity. Rather, such couplings are known only within error intervals. And furthermore, such couplings are
generically subject to broad-spectrum time-dependent noise, originating e.g. within lattice vibrations of the device.
Doesn’t this generically introduce both systematic errors and stochastic errors into practical quantum computations, and aren’t these errors generically of precisely the form that Kalai’s post
regards as hard to correct?
The one area where I will venture an opinion is this: stochastic coupling noise is Choi-equivalent to a continuous weak measurement process, wherein the quantity being measured is the localized
qubit-dependent energy in the quantum computation.
In other words, for a quantum computer to operate successfully, it must necessarily be a device whose localized internal operations cannot be spied upon, not even by the physical noise mechanisms
of the device. And this absolute requirement of “hidden operation” is one feature that makes quantum computation essentially different from classical computation.
This is why we quantum system engineers are increasingly studying cryptography!
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21. Gil, I see, I misread your point (2). You have a noise model there were correlated noise only occurs between correlated (qu)bits. Oops.
In note 3 you say: 3. (*) will allow classic (even randomized) computation (because in this case we do not need the bits to be correlated so (*) does not impose any condition on the noise being
correlated.) This confuses me. Why don’t we require classical correlated bits for classical computation? When I perform classical error correction (as our hard drives and transistors are doing
for us) then aren’t the individual bits correlated? And what about during the classical computation? It seems to me that this will produce correlated bits along the computation.
As to optimism versus pessimism, I am an optimistic for many reasons. I am not an optimist when it comes to the philosophical question of creating computers which are perfectly fault-tolerant.
But for creating fault-tolerant quantum (and classical) computers which can solve interesting problems which will make our lives better: for this I am an optimist. Further, I am focused on the
main problems in the upcoming experiments. And they have nothing to do with crazy models of noise. Which is of course not an argument to say that such noise doesn’t exist: it’s just that right
now the discussions of these noise models are so far removed from the real problems of the lab, that I don’t see a reason to be pessimist. Further, perhaps my experience with decoherence-free
subspaces also shapes my optimism. Give me correlated noise and I am even happier. And I guess, on a fundamental level, I side with Einstein: the universe may be mysterious, but it is never
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22. John: the type of errors you are talking about are exactly the type of errors which fault-tolerant quantum computation is designed to deal with. Error which occur due to imprecision of couplings
for the gates you are trying to correct are fine for fault-tolerant quantum computation, as long as they are not strong.
The noise Gil seems to be arguing for (note: my understanding, not Gil’s menaing!) causes correlated errors when the (qu)bits are corelated, not coupled.
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23. Gee, I can’t even figure out how to insert paragraph breaks in these replies, so how can I figure out quantum error correction? Is there an XHTML code that is functionally similar ?
On hearing the words “solvable in principle”, a mathematician or physicist tends to focus mainly on the word “solvable”, while an engineer tends to hear “solvable in principle, but all-too-likely
to be NP-hard to solve in practice”.
The “Mike and Ike” book provides good examples. In the book, qubit gates are designed to be error-correcting via a beautiful and rigorous formalism. But in deriving the threshold theorem for the
overall computation, there seems to be an implicit assumption that whenever the deterministic part of a qubit coupling is turned off, the stochastic part of the coupling is turned off too.
The assumed noise turn-off is achievable in principle, but it is amazingly hard for engineers to achieve in practice. Even designs that physically shuttle qubits away from one another tend to
create enduring long-range couplings, e.g., via thermal fluctuations in the conduction bands of the confining walls of a shared ion trap, or via optical fluctuations in a shared Fabry-Perot
IMHO, mathematicians and physicists are right to ignore these couplings, for the sake of doing beautiful math and physics, and engineers are equally right to fear them, for the sake of building
hardware that works.
Note that this “noise turn off” problem is unique to quantum computers, in that classical computers have robust fan-out, such that any desired number of copies of any desired bit can be robustly
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24. Dave,
1. To the best of my understanding (or just instincts) the issue of correlated bits does not enter classical computation. So my (*) does allow classical computation. (But I will be happy to think
more about it off-line.)
(If there was a need to manipulate correlated bits (*) would have been as damaging, but there is no such need from a computational point of view.)
2. Well, I did not really present (unfortunately) a noise model, just an idea or a requirement for a noise model, where the noise (as you correctly put it) causes correlated errors when the
qubits are correlated. Another way to look at it (avoiding the optimism/pessimism issue which seems rather irrelevant) is this: The threshold theorem (which at present is the only game in town)
has spectacular consequences but we try to look at the simplest non-trivial consequence. Namely, that we can have two qubits in an entangled state (say, a cat) such that the noise for them is
(almost) independent – Inspite the arbitrary noise we allow on gates. (This is an ingredient of fault tolerance which is not similar to classical fault tolerance and it indeed looks a bit
This simple consequence of the threshold theorem seems to capture much of its power. So perhaps this is what we should try to attack (theoretically) or examine (empirically).
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25. Yeah, my line breaks are all fudged up. Need to fix that.
. But in deriving the threshold theorem for the overall computation, there seems to be an implicit assumption that whenever the deterministic part of a qubit coupling is turned off, the
stochastic part of the coupling is turned off too. Hm, I don’t think this is neglected. The stochastic part of the coupling, as you call it, is simply a quantum noise on a wire when the gate is
turned off.
The assumed noise turn-off is achievable in principle, but it is amazingly hard for engineers to achieve in practice. Even designs that physically shuttle qubits away from one another tend to
create enduring long-range couplings, e.g., via thermal fluctuations in the conduction bands of the confining walls of a shared ion trap, or via optical fluctuations in a shared Fabry-Perot
Like I said, the threshold theorem doesn’t depend on this, except in the strength of this problem, but like you said, it can be a major source of error. Certainly, it is a bigger problem in some
implementations than others. For example, while the patch potentials believed to cause heating in ion traps are a source of error, they are curerntly not the source of error which is keeping ion
traps above the fault-tolerant threshold. That error comes from the two-qubit (and to some extent one-qubit) gates.
Note that this “noise turn off†problem is unique to quantum computers, in that classical computers have robust fan-out, such that any desired number of copies of any desired bit can be
robustly cloned.
I don’t believe this. I believe that we think there is no such problem because we are dealing with already physically error corrected systems. Certainly if you try to engineer a SINGLE spin, for
example, to be a classical spin as a bit, it will have the same problems. We live in a world in which the ideas of error correction are sheilded from our eyes: our hard drives, our transistors,
are all fault-tolerant (or at least fault-tolerant with a lifetime long enough for our purposes: that data on your hard drive will eventually die…time to backup!) due to the physics of the
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We live in a world in which the ideas of error correction are shielded from our eyes: our hard drives, our transistors, are all fault-tolerant
That is very nicely put, IMHO! In fact, when we check the literature, our eyes themselves are very strongly error-corrected sensors, within which everal retinal molecule is strongly coupled to a
thermal reservoir of vibrating water molecules, with the functional consequence being that the signals from a single retinal molecule can be robustly amplified and fanned-out to the visual
From this point of view, we humans evolved at “hot” temperatures precisely so we could reliably observe and interact with a classical world, within which both our ideas and our genes can
reproduce Iand evolve). And yes, we humans do support embedded biological error correcting codes, which are found at every level of our metabolic function. In fact, from a combined physics/
engineering/biology perspective, it is a very striking fact that our own internal error-correcting processes are sufficient to create a classical world.
It is fun to imagine an alternative “cold” ecology, in which life has evolved (somehow) in the context of strongly entangled, robustly error-corrected quantum processes. It is hard to see how
concepts such as “individual”, “object”, “3D spatial localization”, and even “physical reality” could be well-defined in this quantum-entangled but still-Darwinian world. These cold quantum
creatures would perceive hot classical creatures like us as volcanoes of ecology-destroying raw entropy. Hey, we’re the “bad guys” in this scenario!
“Worst … plot … concept … ever”, as Comic Book Guy might say.
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27. Three remarks:
1. It is quite interesting that the sticky points that John sees from the engineering point of view are quite similar to the places I would seek the theoretic weakness of the quantum fault
tolerance hypothesis. The threshold theorem asserts that qubit coupling can be “purified” via fault-tolerance. John find it hard to implement in practice and I question if this purification is
not an artifact of too strong mathematical assumptions concerning the noise.
Another point of similarity is the advantage of the ability to clown and use majority that John mentioned. This allows error correction and classical computation to prevail in certain cases of
high-rate noise or low-rate noise with massive (“crazy”) correlations that would fail quantum computing.
2. If we agree that the matter of noise is essentially a matter of appropriate approximations we can see if in cases of dependence data we can expect approximation up to independent (or almost
independent) error. (I do not think we have in nature anything close to the massive (crazy?) forms of dependencies needed among qubits of a quantum computer, but we do have some examples we can
Take, for example, the weather. Suppose you wish to forecast the weather in 20 locations around Seattle. The weather in these places is clearly dependent. You can approximate the weather at time
T by the weather at time T-t, which is better and better as t gets small. You can also monitor and use sophisticated physics and mathematics to get some better forecasts.
Do you think we can have a forecast so the error terms will be independent among cities? or close to be?
3. So this brings me to the nice error-correction examples that Dave claims we take too much for granted. Consider the following hypothetical dialogue between Dave and me:
Dave: I can do something very nice. In the process there is an arbitrary matrix B and my method is based on my ability (using some physics you will probably not understand) to approximate it by a
matrix C so that E = B-C is small.
Gil: Sound good, cool.
Dave: And as it turns out, E would be(approximately) rank one matrix. (This is important and necessary.)
Gil: Sounds bad, forget it.
Dave: But why? We see such approximations all the time: in transistors and hard drives and many other natural and artificial cases that are simply shielded from your eyes.
Gil: Yes, but in ALL these cases you approximate a rank-one matrix to start with. I believe that you may be able to approximate a rank-one matrix up to a rank-one error. I do not believe that you
will be able to approximate an arbitrary matrix up to a rank one matrix.
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28. I will never look at rank one matrices the same
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29. Well, stretching perhaps the blog’s hospitality, here is a (finite) sequence of additional short comments on this interesting problem:
A. Dave’s analogy between classical and quantum fault tolerance.
1. The analogy between the classical and quantum case for error correction and fault tolerance is very useful. I also share the view that the underlying mathematics is very close.
2. Of course, the threshold theorem and various later approaches to fault tolerance give strong support to the hypothesis of fault-tolerant quantum computation (at present, this hypothesis is
“the only play in town”) and to the feasibility of computationally superior quantum computers.
3. A useful dividing line to draw is between deterministic information and stochastic correlated information (both classic and quantum).
4. The prominent role (sometimes hidden) of artificial and natural forms of error-correction is, of course, something to have in mind, but I do not share the view that it gives us substantial
additional reasons for optimism. (It does not give reasons for pessimism either.)
5. I am not aware of (and will be happy to learn about) cases (artificial or natural) where fault tolerance and error correction are used for stochastic correlated data, nor am I aware of natural
forms of error correction which uses a method essentially different than “clown and use majority”.
6. I also am not aware of cases of a natural system with stochastic highly correlated elements which admits an approximation up to an “almost independent” error term. This is the kind of
approximation required for fault tolerant quantum computation.
B. Correlated errors and spontaneous synchronization of errors.
7. A remark experts can skip: (Again it is better just to think about classical systems.) When you have a system with many correlated bits, an error in one bit will also effect the other bits. On
an intuitive level this looks like an obstacle to error correction but it is not. Independent errors effecting a substantial fraction of –even highly correlated bits, can be handled. (This is, I
believe, what Dave refers to as fighting correlations with correlations.) Correlated errors which allows, with substantial probabilities, errors that exceed the capacity of the error-corrector
are problematic. The crux of the matter (all the rank-one matrix stuff) is whether independent errors on highly correlated bits is a possible or even a physically meaningful notion. (So perhaps,
laymen concerns are correct after all.)
8. The postulate of correlated errors.
(*) In any noisy system, noise acting on pairs of elements which are substantially correlated is also substantially correlated.
Correlated errors can reflect the correlation of the actual bits/qubits which is echoed by the noise. (Another possible explanation for (*) for quantum computers is that the ability of the
physical device to create strong correlations needed for the computation may already cause it to be vulnerable to correlated noise regardless of the state of the computer.)
9. (We talk about pairs of bits/qubits, but let me remark, that while pairwise (almost ) independence appears necessary for fault tolerant quantum computatio(and is the first thing to challenge),
for the threshold theorem, pairwise (almost) independence would not suffice. Similar conditions on triples/quadruples etc. are also needed.)
10. The error correlation hypothesis and T-errors.
An appealing strengthening of (*) is
(*’) In a noisy system, the probabilities for noise to hit two strongly correlated elements have a substantial positive correlation.
Positive correlations for the event that the bits are hit for every pair of bits leads precisely to what Dave calls T-errors: There is a substantial probability for large portion of all bits to
be hit.
For example, if the probability for a bit flip is 1/1000, but for every pair the probability that they will both be flipped is > 1/3000, then there is a substantial probability that more than 25%
of the bits will be hit.
11. T-errors as spontaneous synchronization of errors.
Low rate T-errors amounts to the errors getting synchronized. Can we have a spontaneous synchronization of errors when the noise is expressed by a “local” coupled stochastic process? There is a
substantial literature (from physics, mainly) suggesting that the answer is yes.
For our purposes we want more: We want to show that for coupled processes, such as the process describing the noise of highly correlated physical system (especially, a quantum computer),
substantial amount of synchronization of errors will always take place.
12. Can coupled noise create interesting/harmful noise patterns?
Here is a related problem: Take a Zebra . Can the pattern of strips be the outcome of a locally defined stochastic process? This is a problem we used to discuss (me, Tali Tishby and others) in
our student years, 30 years ago, and we returned to related questions occasionally since then. To make the connection concrete think about the direction from tail to head as time, and the cells
in the horizontal lines as representing your qubits (white represent a defected qubit, say). Isn’t it a whole new way to think about a zebra?
13. Spontaneous synchronization and the concentration of measure phenomena.
A reason that I find T-errors/spontaneous synchronization an appealing possibility is that this is how a “random” random noise looks like. Actually talking about a random form of noise is easier
in the quantum context. If you prescribe the noise rate and consider the noise as a random (say unitary) operator (with this noise rate) you will see a perfect form of synchronization for the
errors, and this property will be violated with extremely low probability (when there are many qubits). Random unitary operators with a given noise rate appears to be unapproachable by any
process of a “local” nature, but their statistical properties may hold for such stochastic processes describing the noise. The fact that perfect error synchronization is the “generic” form of
noise suggests that stochastic processes describing the noise will approach this “generic” behavior unless they have a good reason (such as time-independence) not to.
14. The postulate of error synchronization.
(**) In any noisy system with many substantially correlated elements there will a strong effect of spontaneous error synchronization (T-errors).
In particular, for quantum computers:
(**) In a noisy quantum computer at a highly entangled state there will a strong effect of spontaneous error synchronization.
Both (*) and (**) can be tested, in principle, for quantum computers with a small number of qubits (10-20). I agree with Dave that even this is well down the road, (but much before the superior
complexity power of quantum computers will kick in). Maybe (*) and (**) can be tested on classical highly correlated systems (weather, the stock-market).
15. The idea that for the evolution of highly correlated systems, changes tends to be synchronized, and thus that we will witness rapid changes effecting large portion of the system (between long
periods of relative calm) is appealing and may be related to various other thing (sharp threshold phenomena, or, closer to themes of the Pontiff – evolution, the evolution of scientific thoughts,
etc. )
16. Clowning and taking majority:
suppose you have T-errors where a large proportion (99%) of bits are attacked in an unbiased way. (Namely, the probability to change a bit from 0 to 1 is the same as the probability from 1 to 0.)
Noiseless bits can still be extracted. This extends to the quantum case but only with the same conclusion: noiseless bits can still prevail. But there is no quantum error-correction against such
a noise.
17. But are (*) and (**) really true??
How would I know? These conjectures looks to me as expressing a rather concrete (although I would like to make them even more concrete) possible reason for why (superior) quantum computation may
be infeasible (in principle!). The idea of (forced) spontaneous synchronization of errors definitely sounds a bit crazy. Maybe it is true though.
C. Going below log-depth will not be easy.
18. Let me repeat the comment that even for noisy reversible computation, where the mathematical results support the physics intuition that computation is not possible, at present, we cannot
exclude log depth computation, (which allows together with noiseless classical computation to factor in polynomial time). So even assuming the pessimistic notions of noise, proving a reduction to
classical computation will not be an easy matter.
D. Optimism, skepticism
And there were the matters of skepticism and optimism and even the virtues of the universe; well, all these are extremely interesting issues but their direct relevance to the factual questions at
hand is not that strong. One comment is that we do not seem to have good ways to base interesting scientific research on skeptical approaches. (Maybe social scientists are somewhat better at
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Re: st: multiply imputed values for outcome
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Re: st: multiply imputed values for outcome
From Maarten buis <maartenbuis@yahoo.co.uk>
To statalist@hsphsun2.harvard.edu
Subject Re: st: multiply imputed values for outcome
Date Tue, 11 Jul 2006 08:33:33 +0100 (BST)
--- "Leslie R Hinkson asked:
> I have a data set that has multiply imputed values (5) for the
> outcome variable. I have previously used HLM software to conduct my
> analysis but I was told that with the new GLM features in Stata 9
> that it should be possible to do the same in Stata. Unfortunately, I
> haven't found that way yet. Any thoughts?
--- Austin Nichols answered:
> If you used HLM, you may want -xtmixed- or -gllamm- (-ssc install
> gllamm-) but I don't know about the "five plausible values".
The trick with multiple imputation is that you multiple plausible
values for each missing value, thus creating multiple "complete"
datasets. Next you estimate your model just as if you had a real
complete dataset on each "complete" dataset. In your case you would
than have five sets of estimates. The final point estimates are the
means over these five sets of estimates and the final standard error is
made with two components: the mean variance (squared standard error) of
each estimate and the variance between sets. The formulas can be found
on: http://www.stat.psu.edu/~jls/mifaq.html#howto . J. B. Carlin, N.
Li, P. Greenwood, & C. Coffey have writen tools for analyzing multiple
imputed datasets, that implement these formulas (-findit mifit-), but
I don't think they include -xtmixed- or -gllamm-. However, these
formulas are pretty simple and can be implemented by hand if needs be.
--- "Leslie R Hinkson also asked:
> Also, is it possible to conduct standard linear regression analysis
> with multiple plausible values for the dependent variable using
> Stata 9?
-mifit- can handle standard linear regression analysis.
Maarten L. Buis
Department of Social Research Methodology
Vrije Universiteit Amsterdam
Boelelaan 1081
1081 HV Amsterdam
The Netherlands
visiting adress:
Buitenveldertselaan 3 (Metropolitan), room Z214
+31 20 5986715
The all-new Yahoo! Mail goes wherever you go - free your email address from your Internet provider. http://uk.docs.yahoo.com/nowyoucan.html
* For searches and help try:
* http://www.stata.com/support/faqs/res/findit.html
* http://www.stata.com/support/statalist/faq
* http://www.ats.ucla.edu/stat/stata/ | {"url":"http://www.stata.com/statalist/archive/2006-07/msg00266.html","timestamp":"2014-04-16T16:04:43Z","content_type":null,"content_length":"8491","record_id":"<urn:uuid:8b091d00-3bf6-4501-8b8f-949322edfc73>","cc-path":"CC-MAIN-2014-15/segments/1397609539230.18/warc/CC-MAIN-20140416005219-00329-ip-10-147-4-33.ec2.internal.warc.gz"} |
Can't integrate this differential equation
May 4th 2010, 08:07 AM
[SOLVED] Integration of D.E.
Hi, having difficulty with this D.E. that relates to a tank of initially 400L of pure water, with 1L of salt water (concentration 4g/L) added per min, and 3L of the well-mixed tank solution
removed per min. This is what I have so far;
I'm trying to find an integrating factor though I keep ending up with a final differentiated x(t) function that doesn't make sense (based on t-200, resulting in the function only existing from t
> 200 when it should be from t > 0)
$P(t)=\frac{3}{2(200-t)}$ leading to
but then this doesn't seem to make the LHS equal to the derivative of $Q(t)I(t)$? Without the negative sign it would work fine but I'm sure that's the integral of P(t)... a little help? Maybe
I've done something stupid...
May 4th 2010, 09:47 AM
Hello, McChickenb!
$\frac{dQ}{dt} +\frac{3}{400-2t}\,Q\;=\;4$
We have: . $\frac{dQ}{dt} + \frac{3}{2}\cdot\frac{1}{200-t}\,Q \;=\;4$
Integrating factor: . $I \;=\;e^{\frac{3}{2}\!\int \!\frac{dt}{200-t}} \;=\; e^{-\frac{3}{2}\ln(200-t)} \;=\;e^{\ln(200-t)^{-\frac{3}{2}}} \;=\;(200-t)^{-\frac{3}{2}}$
Multiply by $I\!:\;\;(200-t)^{-\frac{3}{2}}\,\frac{dQ}{dt} + \frac{3}{2}\!\cdot\!\frac{1}{200-t}\!\cdot\!(200-t)^{-\frac{3}{2}} \,Q \;=\;4(200-t)^{-\frac{3}{2}}$
. . . . . . . . . . . . . . . . . . $(200-t)^{-\frac{3}{2}}+ \frac{3}{2}\,(200-t)^{-\frac{5}{2}}\,Q \;=\;4(200-t)^{-\frac{3}{2}}$
We have: . $\frac{d}{dt}\bigg[(200-t)^{-\frac{3}{2}}\,Q\bigg] \;=\;4(200-t)^{-\frac{3}{2}}$
Integrate: . $(200-t)^{-\frac{3}{2}}\,Q \;=\;8(200-t)^{-\frac{1}{2}} + C$
Multiply by $(200-t)^{\frac{3}{2}}\!:\;\;\;Q \;=\;8(200-t) + C(200-t)^{\frac{3}{2}}$
May 4th 2010, 10:13 AM
D'oh! That's perfect. I was confusing myself on the simple stuff. Thanks a lot! | {"url":"http://mathhelpforum.com/calculus/142988-cant-integrate-differential-equation-print.html","timestamp":"2014-04-21T02:23:41Z","content_type":null,"content_length":"8629","record_id":"<urn:uuid:ce1ab249-8e5d-4e62-8c07-1225c439be27>","cc-path":"CC-MAIN-2014-15/segments/1397609539447.23/warc/CC-MAIN-20140416005219-00197-ip-10-147-4-33.ec2.internal.warc.gz"} |
- -
Please install Math Player to see the Math Symbols properly
Click on a 'View Solution' below for other questions:
f The time taken by the first four swimmers in a 50 m free style swimming championship are 41.68, 41.65, 41.67, and 41.64 seconds. What is the correct order to represent the positions? View Solution
ccc The percentage change in the populations of the four cities Redfield, Rosedale, Jasper and Bisbee for the year 1990 - 2000 are 5.7, 5.5, 5.4 and 5.9. Which city has the highest View
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ccc What is the value of the digit 3 in the decimal number 6.36? View Solution
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ccc Find the number of decimal numbers between 4 and 5 that have only tenths. View Solution
ccc What are the values of each occurrence of the digit 6 in the number 61.16? View Solution
ccc Which of the following numbers has the same value of the digit 2 as in the number 53.123? View Solution
ccc The time taken by the first four swimmers in a 50 m free style swimming championship are 29.88, 29.85, 29.87, and 29.84 seconds. What is the correct order to represent the
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ccc The time taken by the first four swimmers in a 50 m free style swimming championship are 21.69, 21.66, 21.68, and 21.65 seconds. What is the correct order to represent the
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ccc The percentage change in the populations of the four cities Milan, Rosedale, Jasper and Rushville for the year 1990 - 2000 are 2.6, 2.4, 2.3 and 2.8. Which city has the highest View
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4.23 _____ 4.25 ccc
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Higher-Index Roots: Basic Operations
Higher-Index Roots: Basic Operations (page 6 of 7)
Sections: Square roots, More simplification / Multiplication, Adding (and subtracting) square roots, Conjugates / Dividing by square roots, Rationalizing denominators, Higher-Index Roots, A special
case of rationalizing / Radicals & exponents / Radicals & domains
Operations with cube roots, fourth roots, and other higher-index roots work similarly to square roots.
Simplifying Higher-Index Terms
• Simplify
Just as I can pull from a square (or second) root anything that I have two copies of, so also I can pull from a fourth root anything I've got four of:
If you have a cube root, you can take out any factor that occurs in threes; in a fourth root, take out any factor that occurs in fours; in a fifth root, take out any factor that occurs in fives; etc.
• Simplify the cube root:
• Simplify the cube root:
• Simplify: Copyright © Elizabeth Stapel 1999-2011 All Rights Reserved
• Simplify:
• Simplify:
Multiplying Higher-Index Roots
• Simplify:
• Simplify:
Adding Higher-Index Roots
• Simplify:
• Simplify:
Dividing Higher-Index Roots
• Simplify:
• Simplify:
I can't simplify this expression properly, because I can't simplify the radical in the denominator down to whole numbers:
To rationalize a denominator containing a square root, I needed two copies of whatever factors were inside the radical. For a cube root, I'll need three copies. So that's what I'll multiply onto
this fraction:
• Simplify:
Since 72 = 8 × 9 = 2 × 2 × 2 × 3 ×3, I won't have enough of any of the denominator's factors to get rid of the radical. To simplify a fourth root, I would need four copies of each factor. For
this denominator's radical, I'll need two more 3s and one more 2:
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Cite this article as: Stapel, Elizabeth. "Higher-Index Roots: Basic Operations." Purplemath. Available from
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Superintendent’s Memo #251-11
Department of Education
September 9, 2011
TO: Division Superintendents
FROM: Patricia I. Wright, Superintendent of Public Instruction
SUBJECT: Nominations for Standard Setting Committees for the Mathematics Standards of Learning Tests Based on the Revised 2009 Standards of Learning
The Office of Assessment Development is accepting nominations for standard setting committees for the Mathematics Standards of Learning (SOL) tests. Standard setting is necessary because of the
implementation of new mathematics tests based on the new 2009 Mathematics SOL in 2011-2012. The committee meetings will be held at the Wyndham Virginia Crossings Hotel and Conference Center, Glen
Allen, Virginia, as follows:
November 1-3, 2011
□ End-of-Course (EOC) Algebra I
□ EOC Algebra II
□ EOC Geometry
January 31-February 1-2, 2012
□ Grade 3 Mathematics
□ Grade 4 Mathematics
□ Grade 5 Mathematics
□ Grade 6 Mathematics
□ Grade 7 Mathematics
□ Grade 8 Mathematics
The standard setting committees are responsible for recommending cut scores that reflect students’ achievement status. These recommendations will be submitted to the Board of Education, which will
then decide the cut scores for fail basic (grades 3-8 only) pass/proficient and pass/advanced status. For Algebra II, a college ready cut score will be determined in lieu of a pass/advanced cut
score. Approximately 20 teachers will be selected to serve on each of the nine committees. School divisions are encouraged to nominate a representative for each committee.
Committee members will be chosen based on the following criteria:
• instructional training and experience in the mathematics content area;
• in-depth knowledge of the Mathematics Standards of Learning;
• instructional experience with students who have disabilities and/or students with limited English proficiency; and
• balanced regional representation.
Committee members will be provided the following:
• reimbursement for meals and travel expenses in accordance with state travel policy and guidelines;
• lodging; and
• attendance certificate for recertification points (pending local approval).
School divisions will be reimbursed for substitute teacher pay at a rate of $75 per day.
All nominees who wish to serve on the standard setting committees must complete the Web-based application using the Assessment Committee Application Processing System (ACAPS). The Web-based
application process requires a professional reference and division-level approval. Procedures for submitting the Web-based application are available at https://p1pe.doe.virginia.gov/acaps/.
Completed applications should be submitted to the Virginia Department of Education using ACAPS by September 30, 2011.
If you have questions, please contact the student assessment staff by e-mail at Student_Assessment@doe.virginia.gov or by phone at (804) 225-2102. | {"url":"http://www.doe.virginia.gov/administrators/superintendents_memos/2011/251-11.shtml","timestamp":"2014-04-21T10:10:29Z","content_type":null,"content_length":"4589","record_id":"<urn:uuid:0f2314fb-f235-4ec0-9942-242571d6ada8>","cc-path":"CC-MAIN-2014-15/segments/1397609539705.42/warc/CC-MAIN-20140416005219-00079-ip-10-147-4-33.ec2.internal.warc.gz"} |
Is the canonical morphism $\mathbb A^n \to\mathbb A^{n-1}$ a projective morphism?
up vote 1 down vote favorite
Let $\mathbb A^n$ be the n-dimensional affine space over a field K (algebraically closed if that makes it easier), so $\mathbb A^n= \text{Spec }K[x_1,...,x_n]$, and $\mathbb A^{n-1}$ the (n-1)
-dimensional analog. The inclusion $K[x_1,...,x_{n-1}]\to K[x_1,...,x_n]$ induces a map $\mathbb A^n \to\mathbb A^{n-1}$. Is this a projective map?
For it to be projective (in particular proper), the image of a closed set should be a closed set. Look at the case $n = 2$. Then the map $\mathbb{A}^2 \rightarrow \mathbb{A}^1$ with coordinates
2 $x,y$ and $x$ resp. The map on rings you wrote corresponds to the projection $(x,y) \mapsto x$. Now, $\mathbb{A}^2$ has the closed set $xy = 1$, the image of which is all of $\mathbb{A}^1$ except
the origin. Since this isn't a closed set, the map isn't proper. Similarly, you can rewrite the standard counterexample for larger $n$. (Btw, maybe you want to tag this "algebraic-geometry"). –
LMN Nov 17 '12 at 21:45
that was very useful, thanks a lot – kwkwkw Nov 17 '12 at 22:02
1 maybe you meant to ask if this were an affine map? (it makes less sense to ask about projective maps when the spaces are all affine.) – roy smith Nov 18 '12 at 6:48
It makes sense, it's just the same as talking about finite maps. – Will Sawin Nov 18 '12 at 7:24
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cylindrical equal-area projection
03-20-2009, 01:55 PM #2
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03-20-2009, 09:56 AM #1
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I have a bit of a problem with the cylindrical equal-area projection. I can't figure out how i can project it correctly from longitude , latitude and altitude data to x, y and altitude.
I need an equal-area projection, because i want to use this for a game map in which i want all the area's to have the same size (that would be fair for the players). But I also want to keep
realism, so i need a correct projection of the map.
Or can i better use an other projection method to achieve this (like for example the mollerweide projection)?
i hope my question is clear.
Last edited by haroim; 03-20-2009 at 11:31 AM.
Here are some formulas:
Each parallel has a length of 2*pi*cos(theta), this is the height of the map in y.
Each meridian has a length of 2*R/cos(theta), this is the width of the map in x.
The distance of each parallel to the equator is R*sin(phi)/cos(theta).
Where R is the radius of the generating globe of the chosen area scale, theta is the standard parallel, phi is the latitude of interest.
For a point on the globe (lat/lon), where lat is the latitude of the point and lon is the longitude of the point(in degrees) and the map is centered at latlon(0,0), west longitudes are negative
values as are south latitudes,
x = 2*pi*cos(theta) * (lon + 180)/360
y = R*sin(lat)/cos(phi)
The ratio of width/height of the map is determined from,
cos(theta) = acos((ratio/pi)^0.5)
Some standard projections and their standard parallels:
Lambert - pi/1 ratio, theta=0
Behrmann - ~2.356:1 ratio, theta=30
Gall-Peters - pi/2:1 ratio, theta=45
Hopefully this would be helpful.
edit: Info from Arthur Robinson, Randall Sale, Joel Morrison, "Elements of Cartography" 4th ed., 1978, John Wiley and Sons: New York.
The wikipedia describes this well under Gall-Peters projection. Even more generally, but in less detail under Lambert cylindrical equal-area projection.
Last edited by su_liam; 03-20-2009 at 02:05 PM. Reason: Added wikipedia references, 'cause the wikipedia adds an air of authority
Astrographer - My blog.
-How to Fit a Map to a Globe
-Regina, Jewel of the Spinward Main(uvmapping to apply icosahedral projection worldmaps to 3d globes)
-Building a Ridge Heightmap in PS
-Faking Morphological Dilate and Contract with PS
-Editing Noise Into Terrain the Burpwallow Way
-Wilbur is Waldronate's. I'm just a fan.
thanks for the reaction! i am now looking into it.
I have looked into it and i get out the following:
If i want to make a projection i will need for each (x,y) an altitude. I can find the altitude in a (lon,lat) form. So to get the altitude of (x,y) i will need to know the (lon,lat) of (x,y).
x = 2*pi*cos(theta) * (lon + 180)/360
lon = (x*360)/(2*pi*cos(theta))-180
y = R*sin(lat)/cos(phi)
so (x,y)=(lon,lat)=( (x*360)/(2*pi*cos(theta))-180, invsin(cos(phi)*y/R)) )
so i should be able to calculate the altitude of (x,y) by taking the altitude of ( (x*360)/(2*pi*cos(theta))-180, invsin(cos(phi)*y/R)) ).
is this correct? And what is phi? Is it the standard latitude, and what number should i enter?
Last edited by haroim; 03-21-2009 at 04:02 AM. Reason: invsin(cos(phi*y/R)) must be invsin(cos(phi)*y/R))
It looks correct to me... I'll have to check your algebra when I have more time and my brain is working.
Yes, phi is the standard latitude. It's a constant for a given map. For Lambert cylindrical equal-area it's 0, the equator. For Behrmann, it's 30degrees. For Gall-Peters it's 45 degrees.
Astrographer - My blog.
-How to Fit a Map to a Globe
-Regina, Jewel of the Spinward Main(uvmapping to apply icosahedral projection worldmaps to 3d globes)
-Building a Ridge Heightmap in PS
-Faking Morphological Dilate and Contract with PS
-Editing Noise Into Terrain the Burpwallow Way
-Wilbur is Waldronate's. I'm just a fan.
Thanks! So the phi is the same as the theta. Should i take the width of my map as R, or should i take the width, calculate it into a sphere, and calculate the radius of the sphere?
Last edited by haroim; 03-21-2009 at 04:37 AM.
03-20-2009, 04:00 PM #3
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03-20-2009, 06:53 PM #4
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03-21-2009, 12:08 AM #5
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03-21-2009, 03:00 AM #6
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Apr 2008 | {"url":"http://www.cartographersguild.com/regional-world-mapping/4825-cylindrical-equal-area-projection.html","timestamp":"2014-04-18T19:27:02Z","content_type":null,"content_length":"74747","record_id":"<urn:uuid:91c5c3a0-ea39-4e03-bac6-b078e5140f53>","cc-path":"CC-MAIN-2014-15/segments/1397609535095.7/warc/CC-MAIN-20140416005215-00062-ip-10-147-4-33.ec2.internal.warc.gz"} |
OpenMx - Advanced Structural Equation Modeling
Tue, 02/25/2014 - 02:02
unconformable arrays
what you say makes sense mike, but
mxAlgebra(-2 * sum(MZw.data.weight * log(MZ.objective) ), name = "mzWeigh")
generates the unconformable array error. Unfortunately the error doesn't tell one what the noncorforming array's sizes are...
figuring this might just be a column of weights from the data entering a row of likelihoods, i tried
mxAlgebra(-2 * sum(t(MZw.data.weight) * log(MZ.objective) ), name = "mzWeigh")
Also fail...
It shouldn't be necessary to place the weights into a matrix, should it?
OK.. tried making a matrix and placing the weight data in there. A benefit is that the errors are more informative. But same error. Would be great if the back end kicked back something like I was
doing "[300,1] %*% [300,1] when…"
Will look more in the morning.
PS: I assume the row objective is padded with NAs for all(is.na(rows)?
Tue, 02/25/2014 - 19:47
trying a simple example: vector alters estimates in model?
So, building a simple play model. After encountering some no-context errors :-(
This is just estimating the covariance of X and Y, which is around .5
#Simulate Data
rs = .5
nSubs = 1000
selVars <- c('X','Y')
nVar = length(selVars)
xy <- mvrnorm (nSubs, c(0,0), matrix(c(1,rs,rs,1),2,2))
testData <- data.frame(xy)
names(testData) <- selVars
m1 <- mxModel("vector_is_FALSE",
mxMatrix(name = "expCov", type = "Symm", nrow = nVar, ncol = nVar, free = T, values = var(testData)),
mxMatrix(name = "expMean", type = "Full", nrow = 1, ncol = nVar, free = T, values = 0),
mxExpectationNormal(covariance = "expCov", means = "expMean", dimnames = selVars),
mxFitFunctionML(vector = F),
mxData(observed = testData, type = "raw")
m1 <- mxRun(m1)
# summary shows all parameters recovered fine.
# Now: Switch to vector
m1 <- mxModel(m1, mxFitFunctionML(vector = T), name = "vector")
m1 <- mxRun(m1)
umxSummary(m1, showEstimates = "std" )
# what we get
# name Estimate Std.Error
# 1 vector.expCov[1,1] 0.4705 9.0e+07
# 2 vector.expCov[1,2] 0.6148 1.4e+08
# 3 vector.expCov[2,2] 1.0314 2.1e+08
# what it should be...
# XX 0.9945328
# XY 0.4818317
# YY 1.0102951
So when vector is on, something needs to change in the model to keep it driving toward good estimates?
Wed, 02/26/2014 - 00:53
Concerning ... but a solution
This simple example reliably causes my R to completely crash. Rolling back a couple of revisions does not fix the problem for me. Going back to r2655 still has the same crashing problem. For r2583, I
no longer crash, but I replicate your finding of rather bad estimates only when vector=TRUE. I should also note I haven't observed this problem with other models running vector=TRUE. However, those
other models do not optimize the fit function with vector=TRUE; they have that fit function as a component of another fit function. This got me thinking.
I think in this case OpenMx is trying to optimize the likelihood (not even the log likelihood) of the last row of data. Essentially, for the model you specified the row likelihoods do not collapse to
a single number for optimization. The developers should discuss if we should catch this, if it should case an error, a warning, or what.
The following code works like a charm.
#Simulate Data
rs = .5
nSubs = 1000
selVars <- c('X','Y')
nVar = length(selVars)
xy <- mvrnorm (nSubs, c(0,0), matrix(c(1,rs,rs,1),2,2))
testData <- data.frame(xy)
names(testData) <- selVars
m1 <- mxModel("vector_is_FALSE",
mxMatrix(name = "expCov", type = "Symm", nrow = nVar, ncol = nVar, free = T, values = var(testData)),
mxMatrix(name = "expMean", type = "Full", nrow = 1, ncol = nVar, free = T, values = 0, ubound=1),
mxExpectationNormal(covariance = "expCov", means = "expMean", dimnames = selVars),
mxFitFunctionML(vector = FALSE),
mxData(observed = testData, type = "raw")
m1 <- mxRun(m1)
# summary shows all parameters recovered fine.
# Now: Switch to vector
m1 <- mxModel(m1, mxFitFunctionML(vector = T), name = "vector")
mc <- mxModel("vector_is_TRUE", m1,
mxAlgebra(name="thefit", -2*sum(log(vector.fitfunction))),
mc <- mxRun(mc)
Wed, 02/26/2014 - 01:50
to fit or not to fit, that is the parameter
Very helpful mike H !
If a side effect of vector = TRUE is that the likelihoods are not optimised on, then perhaps mxFitFunctionML(vector = T) should have a fit = TRUE (default) parameter, which causes the ML values to be
fit, rather than just optimising one value in the vector?
mxFitFunctionML(vector = TRUE, optimiseMinus2log_all = TRUE) # note, i will find ML values based on -2 * sum(log(vector))
or perhaps a new function, followed by a fit algebra. Perhaps not... the complexity of this is already in the top 1 % of IQ :-)
mxLikelihoodsML(vector = T, name = "mylike") # note, you need to fit these...
Fri, 02/28/2014 - 18:17
weighted data tutorial
So, I made up a tutorial-type resumé of what I think I gathered here...
Does that seem correct?
Also, I noticed that these two give the same result: so the
mc <- mxModel("vector_is_TRUE",
mxMatrix(name = "expCov", type = "Symm", nrow = nVar, ncol = nVar, free = T, labels = c("XX","XY","YY"), values = var(testData)),
mxMatrix(name = "expMean", type = "Full", nrow = 1, ncol = nVar, free = T, labels = c("meanX","meanY"), values = 0, ubound=1),
mxExpectationNormal(covariance = "expCov", means = "expMean", dimnames = selVars),
mxFitFunctionML(vector = TRUE),
mxData(observed = testData, type = "raw")
mxAlgebra(name = "minus2LL", -2 * sum(log(vector.fitfunction)) ),
and this version with only one model containing two fit functions.
mxMatrix(name = "expCov", type = "Symm", nrow = nVar, ncol = nVar, free = T, labels = c("XX","XY","YY"), values = var(testData)),
mxMatrix(name = "expMean", type = "Full", nrow = 1, ncol = nVar, free = T, labels = c("meanX","meanY"), values = 0, ubound=1),
mxExpectationNormal(covariance = "expCov", means = "expMean", dimnames = selVars),
mxAlgebra(name = "minus2LL", -2 * sum(log(vector.fitfunction)) ),
mxFitFunctionML(vector = TRUE),
mxData(observed = testData, type = "raw")
Does that make any sense?
Fri, 02/28/2014 - 18:53
I like this. I'm surprised that the one with two fit functions in one model actually works, but pleased it does.
Your github link is incorrect (there's a trailing w).
It would be nice to have an example of selective sampling, and then recovery of population parameters by weighting.
Wed, 02/26/2014 - 10:30
Note example script in models/passing
Although it uses a mixture distribution, the principle for sample weights is very similar (just have a single component weighted mixture, essentially). It uses the technique of pulling variables out
of the data frame to use as weights.
I think this circumvents the issues with dimensionality, but I agree it is a bit clunky that the likelihoods are specified on an individual-wise basis (a la definition variable) but then you get the
whole vector of them back. It might be better to have a weight constructor that would compute an algebra (individual-wise) and weight the likelihoods on the way back.
Tue, 02/25/2014 - 09:41
Three things to try
With this example there aren't a lot of arrays available to be non-conformable. It's either -2, MZw.data.weight, or MZ.objective. First, I'd make sure the objective (fit function) for the MZ model
has vector=TRUE. Second, is MZw.data.weight a single column vector? The * operator is for elementwise multiplication, and it will not recycle either argument if they are different lengths like R. Use
%*% for matrix multiplication. Third, you could put MZw.data.weight into its own mxAlgebra and have a look at it that way.
Wed, 02/26/2014 - 08:04
True in this case not many sources of non-conformability
But, it would be SUPER HELPFUL to print out the dimensions of the matrices found not to be conformable along with the error. This is something classic Mx used to do, and it's sorely missed in OpenMx.
Pretty please?
Mon, 04/04/2011 - 10:25
Weighted objectives
This is possible in OpenMx; Mike was arguing for a shortcut.
To weight objective functions in OpenMx, you'll have to specify two models. The first model is your standard model as though there were no weights, with the "vector=TRUE" option added to the FIML
objective. This option makes the first model return not a single -2LL value for the model, but individual likelihoods for each row of the data.
That model is placed in a second MxModel object, which will also contain the data (if you put the data here, you don't have to put it in the first model, but you still can). That second model will
contain an algebra that multiplies the likelihoods from the first model by some weight. I'll assume that you want to weight the log likelihoods by a variable in your dataset. Your second model will
then look something like this, where "data.weight" is the weight from your data and "firstModel" is the name you assigned to your first model.
fullModel <- mxModel("ThisIsHowYouDoWeights",
mxFIMLObjective("myCov", "myMeans", dims, vector=TRUE)),
mxData(myData, "raw"),
mxAlgebra(-2 * sum(data.weight %x% log(firstModel.objective), name="obj"),
Edit: corrected the multiplication to a Kronecker product, as its a 1 x 1 definition variable matrix and a 1 x n likelihood vector.
Fri, 02/21/2014 - 00:17
Does this work to implement weighted analysis?
Following this thread and a previous one [1], I've been trying to implement differential weighting of subjects in a model by building another model along side which multiplies the vector of fits in
the first by the row weight in the data and optimizing that sum.
I thought it was working, but I am getting very different results from a colleague who is using MPlus's weight feature. Just deleting the low-weighted rows gives results more similar to his than to
wMZ = data.frame(weight = w[twinData$ZYG == "MZFF"]) # get weights for MZs
wDZ = data.frame(weight = w[twinData$ZYG == "DZFF"]) # and for DZs
# An MZ model with vector = T
mxModel(name = "MZ", mxData(mzData, type = "raw"), mxFIMLObjective("top.mzCov", "top.expMean", vector = bVector))
# model "MZw", which simply weights the MZ model
mxData(wMZ, "raw"),
mxAlgebra(-2 * sum(data.weight %x% log(MZ.objective) ), name = "mzWeightedCov"),
# ...
# In the container model, target an algebra which sums the weighted objectives
# of the MZw and DZw, which in turn target weighted functions of the individual row
# likelihoods of the underlying MZ and DZ ACE models
mxAlgebra(MZw.objective + DZw.objective, name = "twin"), mxAlgebraObjective("twin")
What (if anything) is wrong with this approach? (or my implementation)
[1] http://openmx.psyc.virginia.edu/thread/445
Mon, 02/24/2014 - 16:01
Unsure about product
mxAlgebra(-2 * sum(data.weight %x% log(MZ.objective) ), name = "mzWeightedCov"),
Is this really doing what we want, if MZ.objective is a vector of all the likelihoods? I'd be inclined to go for:
mxAlgebra(-2 * sum(mzDataWeight * log(MZ.objective) ), name = "mzWeightedCov"),
where mzDataWeight is the name of an mxMatrix explicitly loaded with the weights.
Mon, 04/04/2011 - 10:42
Umm, I'm not following the
Umm, I'm not following the previous explanation. The definition variable "data.weight" is a 1 x 1 matrix, and "firstModel.objective" is a n x 1 matrix when vector=TRUE. Another way to specify this is
to put the weights in a n x 1 MxMatrix, and then multiply the weights by the likelihood vector. I think there's a right-parenthesis missing the closing of the model "firstModel".
Mon, 04/04/2011 - 11:50
My fault for writing code
My fault for writing code without checking it. It should be a Kronecker product in this specification. You're right that a 1 x n matrix of weights may be included in lieu of the definition variable
approach. I fixed the parenthesis issue as well.
Mon, 04/04/2011 - 17:45
thank you
that was very helpful!
just a quick follow up question: I know Mplus uses a sandwich estimator to correct the standard errors in addition to using the weighted likelihood function. Is there a similar option in OpenMx? Can
I trust the standard errors? | {"url":"http://openmx.psyc.virginia.edu/thread/861","timestamp":"2014-04-21T04:39:33Z","content_type":null,"content_length":"96658","record_id":"<urn:uuid:775abad1-ccd8-403f-82dc-78dd6027c013>","cc-path":"CC-MAIN-2014-15/segments/1397609539493.17/warc/CC-MAIN-20140416005219-00543-ip-10-147-4-33.ec2.internal.warc.gz"} |
Singular Value Decomposition of Compact Operators: A Tool for Computing Frequency Responses of PDEs
Binh K. Lieu and Mihailo R. Jovanovic, 6th January 2012
(Chebfun example pde/SVDFrequencyResponse.m)
In many physical systems there is a need to examine the effects of exogenous disturbances on the variables of interest. The analysis of dynamical systems with inputs has a long history in physics,
circuit theory, controls, communications, and signal processing. Recently, input-output analysis has been effectively employed to uncover the mechanisms and associated spatio-temporal flow patterns
that trigger the early stages of transition to turbulence in wall-bounded shear flows of Newtonian and viscoelastic fluids.
Frequency response analysis represents an effective means for quantifying the system's performance in the presence of a stimulus, and it characterizes the steady-state response of a stable system to
persistent harmonic forcing. For infinite dimensional systems, the principal singular value of the frequency response operator quantifies the largest amplification from the input forcing to the
desired output at each frequency. Furthermore, the associated left and right principal singular functions identify the spatial distributions of the output (that exhibits this largest amplification)
and the input (that has the strongest influence on the system's dynamics), respectively.
We have employed Chebfun as a tool for computing frequency responses of linear time-invariant PDEs in which an independent spatial variable belongs to a compact interval. Our method recasts the
frequency response operator as a two-point boundary value problem (TPBVP) and determines the singular value decomposition of the resulting representation in Chebfun. This approach has two advantages
over currently available schemes: first, it avoids numerical instabilities encountered in systems with differential operators of high order and, second, it alleviates difficulty in implementing
boundary conditions. We refer the user to Lieu & Jovanovic 2011 [3] for a detailed explanation of the method.
We have developed the following easy-to-use Matlab function (m-file) [1]
which takes the system's coefficients and boundary condition matrices as inputs and returns the desired number of left (or right) singular pairs as the output. The coefficients and boundary
conditions of the adjoint systems are automatically implemented within the code. Thus, the burden of finding the adjoint operators and corresponding boundary conditions is removed from the user. The
algorithm is based on transforming the TPBVP in differential form into an equivalent integral representation. The procedure for achieving this is described in Lieu & Jovanovic 2011 [3]; also see T.
Driscoll 2010 [2]. Additional examples are provided at [1].
help svdfr
[Sfun,Sval] = svdfr(A0,B0,C0,Wa0,Wb0,LRfuns,Nsigs)
Given a two point boundary value representation of the frequency response
{ A0*phi = B0*d,
T: { u = C0*phi,
{ 0 = Wa0*phi(a) + Wb0*phi(b),
solve the eigenvalue problem
T*Ts*Sfun = Sval*Sfun, or Ts*T*Sfun = Sval*Sfun,
where Ts is the adjoint of the frequency response operator T
{ A0s*psi = B0s*f,
Ts: { g = C0s*psi,
{ 0 = Wa0s*psi(a) + Wb0s*psi(b).
LRfuns = 1 --> solve for left singular functions: T*Ts
--> determine spatial profile of the output
LRfuns = 0 --> solve for right singular functions: Ts*T
--> determine spatial profile of the input
Nsigs --> number of desired singular values (default: Nsigs = 1)
Sval --> singular values of T arranged in descending order
Sfun --> singular functions associated with Sval
written by: Binh Lieu, 2011
B.K. Lieu & M.R. Jovanovic, "Computation of frequency responses
for linear time-invariant PDEs on a compact interval", Journal of
Computational Physics, submitted (2011); also arXiv:1112.0579v1
Example: one-dimensional diffusion equation
We demonstrate our method on a simple one-dimensional diffusion equation subject to spatially and temporally distributed forcing d(y,t), homogenous Dirichlet boundary conditions, and zero initial
u_t(y,t) = u_{yy}(y,t) + d(y,t), y \in [-1,1]
u(-1,t) = u(+1,t) = 0, u(y,0) = 0.
Application of the temporal Fourier transform yields the following two point boundary value representation of the frequency response operator T,
( D^{(2)} - i*w )*u(y) = -d(y)
( [1; 0] E_{-1} + [0; 1] E_{1} )*u(y) = [0; 0]
w -- temporal frequency
D^{(2)} -- second-order differential operator in y
i -- imaginary unit
E_{j} -- point evaluation functional at the boundary y = j.
We note that svdfr.m only requires the system's coefficients and boundary conditions matrices to compute the singular value decomposition of the frequency response operator T. For completeness, we
will next show how to obtain the two point boundary value representations of the adjoint operator Ts and the composition operator T*Ts.
The two-point boundary value representation for the adjoint of the frequency response operator, Ts, is given by
( D^{(2)} + i*w )*v(y) = f(y)
g(y) = -v(y)
( [1; 0] E_{-1} + [0; 1] E_{1} )*v(y) = [0; 0].
The representation of the operator T*Ts is obtained by combining the two point boundary value representations of T and Ts. This can be achieved by setting d = g
( D^{(2)} - i*w )*u(y) - v(y) = 0
( D^{(2)} + i*w )*v(y) = f(y)
( [1; 0] E_{-1} + [0; 1] E_{1} )*u(y) = [0; 0]
( [1; 0] E_{-1} + [0; 1] E_{1} )*v(y) = [0; 0].
Note that svdfr.m utilizes the integral form of a two point boundary value representation of the operator T*Ts. This yields accurate results even for systems with high order differential operators
and poorly-scaled coefficients.
% The system parameters:
w = 0; % set temporal frequency to the value of interest
dom = domain(-1,1); % domain of your function
fone = chebfun(1,dom); % one function
fzero = chebfun(0,dom); % zero function
y = chebfun('x',dom); % linear function
The system operators can be constructed as follows:
A0{1} = [-1i*w*fone, fzero, fone];
B0{1} = -fone;
C0{1} = fone;
The boundary condition matrices are given by:
Wa0{1} = [1, 0]; % 1*u(-1) + 0*D^{(1)}*u(-1) = 0
Wb0{1} = [1, 0]; % 1*u(+1) + 0*D^{(1)}*u(+1) = 0
The singular values and the associated singular functions of the frequency response operator can be computed using the following code
Nsigs = 25; % find the largest 25 singular values
LRfuns = 1; % and associated left singular functions
[Sfun, Sval] = svdfr(A0,B0,C0,Wa0,Wb0,LRfuns,Nsigs);
Analytical expressions for the singular values and corresponding singular functions are given by:
Sa = (4./(([1:Nsigs].*pi).^2)).'; % analytical singular values
Sf1 = sin((1/sqrt(Sa(1))).*(y+1)); % analytical soln of 1st singular function
Sf2 = sin((1/sqrt(Sa(2))).*(y+1)); % analytical soln of 2nd singular function
The absolute error of the first 25 singular values
norm(Sval - Sa)
ans =
The 25 largest singular values of the frequency response operator: svdfr versus analytical results.
hold on
xlab = xlabel('n', 'interpreter', 'tex');
set(xlab, 'FontName', 'cmmi10', 'FontSize', 20);
h = get(gcf,'CurrentAxes');
set(h, 'FontName', 'cmr10', 'FontSize', 20, 'xscale', 'lin', 'yscale', 'lin');
The principal singular function of the frequency response operator: svdfr versus analytical results.
hold off; plot(y,-Sfun(:,1),'bx-','LineWidth',1.25,'MarkerSize',10)
hold on; plot(y,Sf1,'ro-','LineWidth',1.25,'MarkerSize',10);
xlab = xlabel('y', 'interpreter', 'tex');
set(xlab, 'FontName', 'cmmi10', 'FontSize', 20);
h = get(gcf,'CurrentAxes');
set(h, 'FontName', 'cmr10', 'FontSize', 20, 'xscale', 'lin', 'yscale', 'lin');
axis tight
The singular function of the frequency response operator corresponding to the second largest singular value: svdfr versus analytical results.
hold off; plot(y,Sfun(:,2),'bx-','LineWidth',1.25,'MarkerSize',10)
hold on; plot(y,Sf2,'ro-','LineWidth',1.25,'MarkerSize',10);
xlab = xlabel('y', 'interpreter', 'tex');
set(xlab, 'FontName', 'cmmi10', 'FontSize', 20);
h = get(gcf,'CurrentAxes');
set(h, 'FontName', 'cmr10', 'FontSize', 20, 'xscale', 'lin', 'yscale', 'lin');
axis tight
[1] http://www.ece.umn.edu/users/mihailo/software/chebfun-svd/
[2] T. A. Driscoll, Automatic spectral collocation for integral, integro-differential, and integrally reformulated differential equations, J. Comput. Phys. 229 (2010), 5980-5998.
If you are using this software please cite:
[3] B. K. Lieu and M. R. Jovanovic, "Computation of frequency responses for linear time-invariant PDEs on a compact interval", Journal of Computational Physics (2011), submitted; available at:
[4] L. N. Trefethen and others, Chebfun Version 4.0, The Chebfun Development Team, 2011, http://www.maths.ox.ac.uk/chebfun/ | {"url":"http://www.mathworks.com/matlabcentral/fileexchange/23972-chebfun/content/chebfun/examples/pde/html/SVDFrequencyResponse.html","timestamp":"2014-04-18T08:18:27Z","content_type":null,"content_length":"127879","record_id":"<urn:uuid:cfa619d3-6a41-4143-b915-a47acfda4b5b>","cc-path":"CC-MAIN-2014-15/segments/1397609533121.28/warc/CC-MAIN-20140416005213-00418-ip-10-147-4-33.ec2.internal.warc.gz"} |
ACCA F2 Cost classification part b
View all ACCA F2 / FIA FMA lectures >> This ACCA F2 / FIA FMA lecture is based on OpenTuition course notes, view or download here>>
1. Good afternoon!
I have a question regarding the variable and fixed costs per unit and I was hoping you could clear it out for me.
I looked over a table in my text book, where it says that the variable cost per unit is constant when the volume of production increases, and the fixed cost per unit decreases while production
increases. I am really confused about this. Shouldn’t the fixed costs be constant regardless of output and variable costs vary accordingly to the volume of production? Thank you!
□ I think maybe you are reading it too fast
The book is correct (and says the same as my lecture).
Variable costs are fixed per unit – it is the total variable cost that changes with the number of units.
Fixed costs are fixed in total, but the fixed cost per unit will fall with more units being produced.
(If the fixed cost is $10000 and you produce 10000 units, then each unit costs $1. If you produce 20000 units, then the total cost stays at $10000 but the cost per unit is then $0.50 )
☆ You are right-I didn’t think it through properly. I am clear now :D. Thank you again very much!
2. Hello everyone! Can someone please tell me what does: ” overtime is paid at a rate of time and a quarter” mean? For example: workers are paid $10 per hour and this week they have worked a total
of 20 hours overtime: 12 hours on specific orders and 8 hours on general overtime. How do I compute the specific overtime, keeping in mind that ” overtime is paid at a rate of time and a
quarter”? Thank you!
□ It means that for every hour of overtime they are paid for 1.25 hours.
So if they work 12 hours overtime they will be paid for 12 x 1.25 = 15 hours at $10 =
$ 150
(Alternatively you can get the same answer by paying the overtime hours at 1.25 x the normal rate. So 1.25 x $10 =
12 hours at $12.50 an hour = $150 )
☆ Thank you so much!!! It’s clear now:)
3. These lectures are so helpful! Thanks so much, they’re very clear and easy to understand!
4. most of the topics in f2 are missing in these lectures.plzzzzzzzzzzz included that topics
□ Rubbish! Most of the topics in F2 are included in the lectures – and all the more important topics are certainly covered.
Those that are not in the lectures are covered in the Course Notes for you to read yourself.
There will not be more lectures.
☆ For this examination we will assume that total variable costs vary linearly with the level of
production (or that the variable cost per unit remains constant). In practice this may not be the
case, but we will not consider the effect of this until later examinations.I NEED HELP :MEANING.:)
6. an organization has the following activity
total cost activity level
variable cost is constant within the activity range but there is a step up increase 4000 in total fixed cost when an activty level of 22500 is reached .. what are total fixed cost at an activty
level of 24000 units
A 40000
b 44000
c 60000
d 64000
answer please .
□ The Answer is D.
activity level total cost
High 24,000 136,000
low 20,000 120,000
difference 4,000 16,000
Extra fixed cost (4,000)
Total variable cost 12,000
Variable cost per unit 12,000/4000 = 3
Fixed cost at level 24,000 (which is the highest level)
= Total cost at that level – variable cost
= 136,000 – (24,000 x 3)
= 136,000 – 72,000
= 64,000
7. Question 5 says “Up to a given level o? activity in each period the purchase price per unit o? a raw material is constant”i cant see that constant part on that graph?
□ The graphs are showing the total cost. If the price per unit is constant, then the total cost will increase linearly with more units!
8. the lecture is great. am enjoying it.
9. i now have a better understanding of cost behavior. Thank you Opentuition
10. Guys what the best book publusher for ACCA, have anyone tried Get Through Guide?
11. Can anyone explain Q7 from chapter 4 please? Did not understand the first bit of question – guarantees 80% of time based pay….?
□ guaranteed 80% of time based pay means that although they are paid on the basis of what they produce, the minimum they can be paid is 80% of what it would be if they were simply paid per
So…the minimum they can be paid is 80% x 8 hours x $20 = $128
You have to calculate what they would be paid using piecework. If it comes to more than $128 then they get paid the piecework amount. If it comes to less than $128 they they get the
guaranteed $128.
The full answer is at the back of the course notes.
12. Two budgets are given below:
Output(units) 1,000 2,000
Budgeted cost 2,500 3.500
The total fixed costs estimated for the 2,000 units budget are 20% higher than the total fixed costs for the 1.000 units budget.
What is the budgeted variable cost/unit of output?
plzz someone help me with this question..!!!!!
□ The answer is $0.625.
Use simple algebra.
☆ I am getting $0.5…Can you please explain how it is $0.625?
This is how I worked:
Units Budgeted Cost
2000 3000 (as total fixed cost for 2000 units is 20% higher than that of 1000 units – hence 3500 – (20% of 2500) )
Using High low method then it comes $0.5
☆ Your workings are treating the whole of the $2500 as though it was a fixed cost.
If v is the variable cost per unit, and if f is the fixed cost per unit, then:
for 1000 units: 1000v + f = 2500
for 2000 units: 2000v + 1.2f = 3500
There are several ways of solving these equations (all obviously giving the same answer!), but here is one way:
Multiply the first equation throughout by 1.2
This gives: 1200v + 1.2F = 3000
The subtract this equation for the equation for 2000 units:
(2000v – 1200v) + 0 = 3500 – 3000
so 800v = 500
v = 500/800 = 0.625
13. Can some one help me with question 1 please I understand the high low method but I don’t understand the trick thank u
14. Can some one help me question one I know how to do high low method but I didn’t understand the trick AND I HOPE THIS QUESTION WONT COME TO THE EXAM. Thank u
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Round Lake, IL Geometry Tutor
Find a Round Lake, IL Geometry Tutor
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adding more laws and formula to their knowledge. I wish I can be helpful to the students to have a liking towards mathematics.
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Permittivity and Permeability of Free Space
Permittivity of a vacuum is a number arrived at beginning with a value for the speed of light in the vacuum and the permeability of the vacuum. NIST uses the term "electric constant" for what is
commonly known as the permittivity of free space:
Here's their official value:
http://physics.nist.gov/cgi-bin/cuu/Value?eqep0 http://physics.nist.gov/cuu/Constant.../gif/eqep0.gif
The permeability of a vacuum is called by NIST "the magnetic constant":
and has a value of 4pi x 10^-7 N A^-2, which is clearly
an experimentally established value. The speed of light in a vacuum, on the other hand,
experimentally determined (at least originally). This constant is called "the speed of light in vacuum by NIST":
As I understand the concept of permittivity, it represents something similar to a modulus of elasticity. It is a measure of how polarized a medium becomes when subjected to an electric field. Though
most discussions I've encountered on the topic tend to give permittivity and permeability of free space very little discussion, I have always been inclined to believe these concepts are of essential
relevance in correctly understanding Nature at a fundamental level.
Does anybody else agree or disagree with that suggestion? What do these concepts mean to you?
I am aware that the equations of EM can be written in units where these constants vanish. I guess if time were measured in meters, and permeability were set to 1, these constants could be dropped
from the expressions. Nonetheless, there seem to be concepts which underpin Maxwell's equations that are implicitly, if not explicitly assumed. Does free space become polarized in the presence of an
electric field? Put differently, one might ask if an electric field
the polarization of free space. | {"url":"http://www.physicsforums.com/showthread.php?t=122121","timestamp":"2014-04-19T04:33:03Z","content_type":null,"content_length":"54247","record_id":"<urn:uuid:9694bb39-0b45-4a50-9a8e-109500779fc8>","cc-path":"CC-MAIN-2014-15/segments/1398223203422.8/warc/CC-MAIN-20140423032003-00115-ip-10-147-4-33.ec2.internal.warc.gz"} |
MEP question: does MEP year 6 prepare for AoPS? - K-8 Curriculum Board
MEP question: does MEP year 6 prepare for AoPS?
Started by serendipitous journey , Mar 06 2012 11:52 PM
aops mep pre-algebra
15 replies to this topic
Posted 06 March 2012 - 11:52 PM
... just wondering if we'd need a transition btw. MEP Year6 and AoPS ...
Posted 06 March 2012 - 11:59 PM
... just wondering if we'd
need a transition btw. MEP Year6 and AoPS ...
Which AoPS? AoPS Introduction to Algebra would be a stretch for most 11 or 12yos, but pre-algebra, yes.
Posted 07 March 2012 - 01:26 AM
Which AoPS? AoPS Introduction to Algebra would be a stretch for most 11 or 12yos, but pre-algebra, yes.
Thanks ... did you go from MEP to AoPS? I'd be happy with pre-algebra; but I think we'll get there when Button is 8 or 9. Good grief, I realize I'm going 'round in circles about this; am just not
sure what to do with Button. He's in MEP Year 4 now, and is 6 1/2. Now that you mention the 11/12 yos, I remember that I was concerned about starting AoPS b/c of the reading/writing component, but I
think he'll be ready for algebra in the next couple of years.
At any rate: that's just what I needed to know! and anything you have to add is very much appreciated.
Posted 07 March 2012 - 01:59 AM
Thanks ... did you go from MEP to AoPS?
No, I didn't know about MEP when DD the Elder went through K6 math and AoPS Pre-Algebra came out too later for her. DD the Younger is almost through Y2, and I've looked ahead, comparing content to
Singapore. MEP Y6 will put the student in at least as good a position as Singapore (I've only seen the US edition, not Standards, so take that with a grain of salt).
I'd be happy with pre-algebra; but I think we'll get there when Button is 8 or 9. Good grief, I realize I'm going 'round in circles about this; am just not sure what to do with Button. He's in
MEP Year 4 now, and is 6 1/2. Now that you mention the 11/12 yos, I remember that I was concerned about starting AoPS b/c of the reading/writing component, but I think he'll be ready for algebra
in the next couple of years.
It's not just the readiness. AoPS
Introduction to Algebra
is *hard* and can't be done in 30 minutes a day.
The Art of Problem Solving Vol. 1
while working through
Life of Fred: Beginning Algebra
. I'll then have her do AoPS
Number Theory
Probability & Counting
before AoPs
Introduction to Algebra
, and proceeding from there.
(pasted from a previous post)
Things we've used in our great algebra put-off:
LoF: Fractions
Decimals & Percents
Pre-Algebra 1
(and soon 2)
Venn Perplexors A-D
(we use them without the pre-drawn charts)
Can You Count in Greek?
(highly enjoyable)
units, including
codes and ciphersIt's Alive,
It's Alive and Kicking
(found a couple errors in solutions, including one major error)
Logic Countdown, Logic Liftoff
Orbiting with LogicThe CryptoclubBecoming a Problem Solving Genius
Challenge MathBrain Maths
(puzzles, from SingaporeMath.com, we found a few errors in the solutions for Book 1, many for Book 2)
Mathematics 6
(Russian Math, selected sections and all starred problems; this text is a thing of beauty)
CWP 5 and 6
Alien Math
(working four operations in different number bases)
Piece of Pi
Patty Paper Geometry
(I highly recommend this)
Cryptoclub was a hoot. It takes awhile to get into the heavier math, but it was worth it
are some more short, free codes & ciphers units from CIMT (the MEP folks).
Posted 07 March 2012 - 02:36 AM
I'm having her continue with The Art of Problem Solving Vol. 1 while working through Life of Fred: Beginning Algebra. I'll then have her do AoPS Number Theory and Probability & Counting before
AoPs Introduction to Algebra, and proceeding from there.
We're doing AoPS Vol.1 now. I was going to get ItA, because that was what I had previously planned, but forget that she finished Dolciani in 7th grade, then LOF Algebra. I'm wondering if AoPS Vol. 1
and 2 (along with all the LOFs and a Geometry course by a guy with a vich at the end of his name
Posted 07 March 2012 - 02:43 AM
We're doing AoPS Vol.1 now. I was going to get ItA, because that was what I had previously planned, but forget that she finished Dolciani in 7th grade, then LOF Algebra. I'm wondering if AoPS
Vol. 1 and 2 (along with all the LOFs and a Geometry course by a guy with a vich at the end of his name
Posted 07 March 2012 - 03:05 AM
Posted 07 March 2012 - 07:08 AM
Things we've used in our great algebra put-off:
LoF: Fractions, Decimals & Percents and Pre-Algebra 1 (and soon 2)
Venn Perplexors A-D (we use them without the pre-drawn charts)
Can You Count in Greek? (highly enjoyable)
Selected MEP units, including codes and ciphers
It's Alive, and It's Alive and Kicking (found a couple errors in solutions, including one major error)
Logic Countdown, Logic Liftoff, Orbiting with Logic
The Cryptoclub
Becoming a Problem Solving Genius
Challenge Math
Brain Maths (puzzles, from SingaporeMath.com, we found a few errors in the solutions for Book 1, many for Book 2)
Mathematics 6 (Russian Math, selected sections and all starred problems; this text is a thing of beauty)
CWP 5 and 6
Alien Math (working four operations in different number bases)
Piece of Pi (meh)
Patty Paper Geometry (I highly recommend this)
Thanks for posting this!
Posted 07 March 2012 - 07:55 AM
THank you for posting that list! We've done some of them, but not all. I am thinking of spending another year on pre-pre-algebra stuff (MM6, CWP5-6) before heading into pre-algebra and these are
great resources.
Posted 07 March 2012 - 08:03 AM
THank you for posting that list! We've done some of them, but not all. I am thinking of spending another year on pre-pre-algebra stuff (MM6, CWP5-6) before heading into pre-algebra and these are
great resources.
Yeah, I think we're going to take a math sabbatical next year. My daughter wants to try Math on the Menu (it's in the math section of TWTM).
Posted 07 March 2012 - 10:13 AM
Posted 07 March 2012 - 12:54 PM
Thanks so much, Moira. I think the crypto stuff will be esp. engaging; the list is terrific to have.
Posted 07 March 2012 - 07:02 PM
Well, as I'm feeling optimistic, I'm not going to concern myself with it until later.
I am hoping that between the Instructor's Manual and the text, I will be prepared.
Posted 07 March 2012 - 07:25 PM
Well, as I'm feeling optimistic, I'm not going to concern myself with it until later.
I am hoping that between the Instructor's Manual and the text, I will be prepared.
... very dangerous. You go first.
(and share with me
Posted 07 March 2012 - 10:23 PM
Edited by Honoria Glossop, 20 November 2012 - 10:42 PM.
Posted 08 March 2012 - 07:06 PM
Dear Moira and Shawna,
(Apologies in advance for the derail, Ana!) I was liking the looks of the Solomonovich book, too, but am scared off by having no solutions....then I found this, by James Tanton:
There's a second volume of geometry, too--the two volumes amount to nearly a thousand pages, and the author has a PhD in math from Princeton, so I'm pretty hopeful! I wish there were a lengthier
sample, but it has definite possibilities, I think.
Anyway, thought I'd toss the link out there for other soon-to-be geometers....
ETA: Meant to mention that Tanton has a lot of geometry (among other topics) videos on his youtube channel, so it's possible to get a bit more of a feel for his approach there than is possible
from the brief samples at Lulu.
I sent the videos to my oldest so she could see if she liked him (she's quite loyal to Salman Khan, so she's glaring at me
I did google him looking for more samples of the geometry, but got sidetracked by his other work. I'm thinking about getting a few of the ebooks for my youngest because I really liked those. Thanks
for mentioning him.
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Explain a TTL NAND gate and its operation, Computer Engineering
Give the circuit of a TTL NAND gate and explain its operation in brief.
Operation of TTL NAND Gate: Fig.(d) Demonstrates a TTL NAND gate with a totem pole output. The totem pole output implies that transistor T[4] sits atop T[3] in order to give low output
impedance. The low output impedance means a short time constant RC therefore the output can change rapidly from one state to the other. T[1] is a multiple type emitter transistor. Such transistor
can be thought of like a combination of various transistors along with a common collector and base. Multiple emitter transistors along with about 60 emitters have been developed. In this figure, T[1]
has 3 emitters thus there can be three inputs A, B, C. The transistor T[2] functions as a phase splitter since the emitter voltage is out of phase along with the collector voltage. The transistors T
[3] and T[4] by the totem pole output, the capacitance CL shows the stray capacitance and so on. The diode D is added to make sure that T[4] is cut off while output is low. The voltage drop of diode
D remains the base-emitter junction of T[4] reverse biased therefore only T[3] conducts while output is low. The operation can be described briefly by three conditions as specified below:
Condition 1: At least one input is low (that is, 0). Transistor T[1] saturates. Thus, the base voltage of T[2] is almost zero. T[2] is cut off and forces T[3] to cut off. T[4] functions as an
emitter follower and couples a high voltage to load. Output is high (that is Y=1).
Condition 2: Each input is high. The emitter base junctions of T[1] are reverse biased. The collector base junction of T[1] is forward biased. Therefore, T[1] is in reverse active mode. The collector
current of T[1] flows in reverse direction. Because this current is flowing in the base of T[2], the transistors T[2] and T[3] saturate and then output Y is low.
Condition 3: The circuit is operating under II while one of the inputs becomes low. The consequent emitter base junction of T[1] starts conducting and T[1] base voltage drops to a low value. Thus, T
[1] is in forward active mode. The high collector current of T[1] shifts the stored charge in T[2] and T[3] and hence, T[2] and T[3] go to cut-off and T[1] saturates and then output Y returns to
Fig.(d) Logic Diagram of TTL NAND Gate with Totem Pole Output
Posted Date: 5/4/2013 12:46:03 AM | Location : United States
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Write shorts notes on Remote Procedure Call (RPC) The facility whihc was created to help the programmers write client server software is termed as Remote Procedure Call. Within
What is the protocol used by SAP Gateway process? The SAP Gateway method communicates with the clients based on the TCP/IP Protocol.
What is a customer-to-business transaction? C2B (customer-to-business): The most significant activity into e-commerce isn’t selling. That is buying. Rather often which do
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Network address prefixed by 1110 is a? Network address prefixed through 1110 is a multicast address.
Explain about the term false path? How it find out in circuit? What the effect of false path in circuit? By timing all the paths into the circuit the timing analyzer can find o
Q. Disk operating system? The operating system (OS) is the first program that should be loaded into the memory of your PC before you can use it for any application. You can st
Q. Illustrate working of T Flip-Flop? T (Toggle) flip-flop is attained from JK flip-flop by combining inputs J and K together. The implementation of T flip-flop is displayed in
Sample Program In this program we only display: Line Offset Numbers -----------------Source Code-----------------
The usability and user experience goalsprovidet heinteraction designer with high-level goals for the interactivep roduct. Thenextissueis how to design a product that satisfies thes | {"url":"http://www.expertsmind.com/questions/explain-a-ttl-nand-gate-and-its-operation-30161619.aspx","timestamp":"2014-04-20T21:23:25Z","content_type":null,"content_length":"32256","record_id":"<urn:uuid:65a78752-459e-4a0f-9d1e-2a2cbfb8f91d>","cc-path":"CC-MAIN-2014-15/segments/1398223205137.4/warc/CC-MAIN-20140423032005-00436-ip-10-147-4-33.ec2.internal.warc.gz"} |
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check for prime numbers using only IF-EL
Unless the user input is limited to a very small range of values, using only if-else is not really feasible. The amount of coding required could be huge (unless you used some sort of macro to
construct the code).
#include <iostream>
using namespace std;
int main ()
int n;
cout << "Enter an integer !"<<endl;
if (n%2==0)
cout << n<< " is not prime number" << endl;
else if (n%2==1)
cout<<n<<" is a prime number"<< endl;
return 0;
I think by using this will pretty much tell the user the number is prime or not
i already tried it , it giving me that '9' is prime number
its not going to work. you need a while loop and xsxs's will only test for evens and odds
Well, if it can be done using if-else + a recursive function, it could also be done using if-else + goto or if-else + while or if-else + something.
I actually think you can do it using template recursion too, the number of values you'd accept would be limited by how much recursion you did and compile time would increase as well, but it should be
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Class 9 - Maths - CH1 - Number Systems - Ex 1.2
NCERT Chapter Solutions
Is Pi rational or irrational?
Q1: State whether the following statements are true or false. Justify your answers.
(i) Every irrational number is a real number. ✓(TRUE)
Explanation: Real numbers are the collection of rational and irrational numbers.
(ii) Every point on the number line is of the form √m, where m is a natural number. ✗ (FALSE)
Explanation: Since -ve numbers cannot be expressed as square roots.
(iii) Every real number is an irrational number. ✗ (FALSE)
Explanation: Real numbers contains both rational and irrational numbers.
Q2. Are the square roots of all positive integers irrational? If not, give an example of the square root of a number that is a rational number.
Answer: No, the square roots of all positive integers can be rational or irrational. e.g. √9 = 3 which is a rational number.
Q3: Show how √5 can be represented on the number line.
We can write √5 in Pythagoras form:
= √(4 + 1) = √(22 + 1)
1. Take a line segment AB = 2 units (consider 1 unit = 2 cm) on x-axis.
2. Draw a perpendicular on B and construct a line BC = 1 unit length.
3. Join AC which will be √5 (Pythagoras theorem). Take A as center, and AC as radius draw an which cuts the x-axis at point E.
4. The line segment AC represents √5 units.
Q4. Construct the ‘square root spiral’ (Class room activity.)
Following video shows the square root spiral.
Q5: Who discovered √2 or disclosed its secret?
Answer: Hippacus of Croton. It is assumed Pythagoreans, followers of the famous Greek mathematician Pythagoras, were the first to discover the numbers which cannot be written in the form of a
fraction. These numbers are called irrational numbers.
Q6: Who showed that showed that "Corresponding to every real number, there is a point on the real number line, and corresponding to every point on the number line, there exists a unique real number"?
Answer: Two German mathematicians, Deddkind and Cantor.
Q6: Is π (pi) rational or irrational?
Answer: π is an irrational number. 22/7 is just an approximation.
Q7: How many rational numbers exist between any two rational numbers?
Answer: Infinite
Q8: Are irrational numbers finite?
Answer: No. There are infinite set of irrational numbers. Q9: The product of any two irrational numbers is :
(A) always an irrational number.
(B) always a rational number.
(C) always an integer.
(D) sometimes rational, sometimes irrational number.
Answer: (D) sometimes rational, sometimes irrational number e.g. √2 and √3 are two irrational numbers, √2×√3 = √6 an irrational number. √2 and √8 are two irrational numbers, √2×√8 = √(16) = 4 a
rational number.
⟵Go to CH1-Number Systems - Ex 1.1 Goto CH1 - Number Systems Ex 1.3→
2 comments:
1. great,,,
i want to share about irrational number
2. i want the solution for this question of rationalisation
sq root of (3+2*sq root of2) | {"url":"http://cbse-notes.blogspot.com/2012/06/class-9-maths-ch1-number-systems-ex-12.html","timestamp":"2014-04-19T12:00:54Z","content_type":null,"content_length":"123750","record_id":"<urn:uuid:fa008b9e-326b-493b-817c-d237aa283035>","cc-path":"CC-MAIN-2014-15/segments/1397609537186.46/warc/CC-MAIN-20140416005217-00310-ip-10-147-4-33.ec2.internal.warc.gz"} |
RE: st: using the first n observations in a dataset w/o evaluating the
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RE: st: using the first n observations in a dataset w/o evaluating the whole thing?
From "Rodini, Mark" <mrodini@compasslexecon.com>
To <statalist@hsphsun2.harvard.edu>
Subject RE: st: using the first n observations in a dataset w/o evaluating the whole thing?
Date Thu, 3 Apr 2008 17:28:04 -0700
Thanks for the replies, and the long explanation.
(Yes I did mean little _n: typo!)
Anyway, I tried the suggestion: use in 1/99 using mydata
and I did indeed find it took time. In fact, I tried to apply the idea
to a 20MB dataset after having only set the memory to 10MB, and it
completely froze up. It only worked on the 20MB dataset if I set memory
to >20MB, and it was slow --as though it were reading the whole thing
Oh well.
-----Original Message-----
From: owner-statalist@hsphsun2.harvard.edu
[mailto:owner-statalist@hsphsun2.harvard.edu] On Behalf Of David Kantor
Sent: Thursday, April 03, 2008 5:20 PM
To: statalist@hsphsun2.harvard.edu
Subject: Re: st: using the first n observations in a dataset w/o
evaluating the whole thing?
At 07:31 PM 4/3/2008, Mark Rodini wrote:
>Suppose I have a large Stata dataset (e.g. 3,000,000 observations) and
>only with to read in the first, say, 100 observations.
>I have tried the code, which works:
>use mydata if ( _N<100 )
>However, evidently, this code goes through ALL 3 million observations
>evaluate the expression in parentheses, which can be very time
>(and sort of defeats the purpose). Is there a way to only read the
>first 100 observations without having to evaluate the entire dataset?
>Perhaps some application of the "set obs 100"? But I have not been
>Thank you.
First, that is not officially valid syntax, though it is accepted. I
find that it gets you 0 observations, though it does read through the
whole file.
You probably mean _n (little n), rather than _N. (I suppose _N is .
during the loading process, so _N <100 is false).
Officially correct syntax is,
use if _n <100 using mydata
or, better yet,
use in 1/99 using mydata
(This latter syntax is much more efficient.)
But in any case, my experience has been that it always reads through
the whole file. And you can tell it's dong that if you have 3000000
observations. The reason is that, in the file structure, there are
some important elements that come after the data (values labels, I
believe, for example), so there is a reason to have to read the whole
file. At least that's how it's been as far as I know; I don't know
if they've changed the file structure in that regard in version 10.
I may have written to Stata Corp. about this some time in the past;
if I had my way, there would either...
be nothing after the end of the data segment, or
be some way to jump directly to the part of the file that lies
after the data.
(The latter idea may or may not work, depending on file-system issues.)
In either case, I would want it to not read the whole file if you
asked for an initial subset.
But as things stand now, we are stuck with this behavior.
The only thing you can do is, if you plan to experiment on a small
segment of the file (and want to load it many times), load a small
segment and save it under a different name. Thus, you go through the
lengthy process just once.
use in 1/99 using mydata
save mydata_shortversion
use mydata_shortversion
-- should load quickly.
Hope this helps.
* For searches and help try:
* http://www.stata.com/support/faqs/res/findit.html
* http://www.stata.com/support/statalist/faq
* http://www.ats.ucla.edu/stat/stata/
* For searches and help try:
* http://www.stata.com/support/faqs/res/findit.html
* http://www.stata.com/support/statalist/faq
* http://www.ats.ucla.edu/stat/stata/ | {"url":"http://www.stata.com/statalist/archive/2008-04/msg00191.html","timestamp":"2014-04-18T08:21:35Z","content_type":null,"content_length":"9258","record_id":"<urn:uuid:b57f8639-7ba4-40d8-846a-4c0b95fea939>","cc-path":"CC-MAIN-2014-15/segments/1397609533121.28/warc/CC-MAIN-20140416005213-00197-ip-10-147-4-33.ec2.internal.warc.gz"} |
Re: st: adding observation of means of variables
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: adding observation of means of variables
From Abhimanyu Arora <abhimanyu.arora1987@gmail.com>
To statalist@hsphsun2.harvard.edu
Subject Re: st: adding observation of means of variables
Date Thu, 16 Feb 2012 14:09:40 +0100
Thanks a lot, Nick, very reassuring indeed. All points accepted.
On Thu, Feb 16, 2012 at 1:56 PM, Nick Cox <n.j.cox@durham.ac.uk> wrote:
> I don't see what is puzzling you here.
> The referees can be shown a Stata log, or given a do-file that contains the Stata instructions to replicate the results in your paper. The means can be obtained directly by (e.g.) -summarize- and just displayed in the log or by the do-file.
> Adding them to the dataset is only needed if the referees insist that results be sent to them as a dataset, and even then, means can be added as extra variables. Adding them as observations is a choice, and only that. If you are doing nothing further with the dataset, the main disadvantage I emphasised in my first reply will indeed not bite anyone.
> Nick
> n.j.cox@durham.ac.uk
> Abhimanyu Arora
> I see your point if it were the original dataset, true, but while I do
> start with it in the -do- file, it had to be modified on the way.
> But sure, if you have better suggestions, would love to hear them.
> On Thu, Feb 16, 2012 at 1:34 PM, Nick Cox <n.j.cox@durham.ac.uk> wrote:
>> OK, but there is no need to add the means to the dataset to do that.
> Abhimanyu Arora
>> Thanks very much for your conscientious advice.
>> Basically, I had created tables for in my paper (which had averages in
>> the last row). Now that the analysis is complete, I just would like to
>> make sure the numbers are replicable from A-Z (in stata itself---I
>> thought bringing them out as datasets would be ok for my purpose), in
>> case the referees would like to see where they come from.
>> On Thu, Feb 16, 2012 at 1:04 PM, Nick Cox <n.j.cox@durham.ac.uk> wrote:
>>> Phil gives accurate advice, and as he said there are other ways to do it.
>>> Here's another:
>>> set obs `=_N + 1'
>>> ds, has(type numeric)
>>> qui foreach v in `r(varlist)' {
>>> su `v', meanonly
>>> replace `v' = r(mean) in L
>>> }
>>> That said, I think this is a bad idea for working with Stata. No, let me rephrase that: it's a very bad idea. A rule of thumb, blunt though it will seem, is that if you have to ask how to do this you don't yet understand Stata well enough to use it safely.
>>> My advice is not to do this.
>>> It's a spreadsheet practice that matches the way spreadsheets are set-up. It's not a good idea for working with statistical software like Stata, The problem is that once those extra observation(s) are added, you _must_ always exclude them from further analyses with the same dataset. Otherwise you just get nonsense results. Add to that the fact that if you -sort- your dataset, or some program or command -sort-s your data as a side-effect (now rare but not impossible), those observations with summaries will typically no longer be at the end of your dataset, so you need to invent extra machinery to keep track of where they are.
>>> Better advice would depend on knowing quite why you want to this. Keeping means in variables, although there can be redundancy, can be a reasonable idea for some purposes.
>>> Nick
>>> n.j.cox@durham.ac.uk
>>> Phil Clayton
>>> Not as far as I know, but it's easy to program. Here's one solution:
>>> preserve
>>> collapse (mean) *
>>> tempfile means
>>> save `means'
>>> restore
>>> append using `means'
>>> The above assumes that all variables are numeric. If they're not, you could replace:
>>> collapse (mean) *
>>> with:
>>> ds, has(type numeric)
>>> collapse (mean) `r(varlist)'
>>> On 16/02/2012, at 10:33 PM, Abhimanyu Arora wrote:
>>>> Is there a direct command that appends an observation to the dataset,
>>>> giving the means of all the numeric variables?
>>>> Perhaps I am using -findit- not that efficiently, but if I am not
>>>> mistaken there was one...
> *
> * 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/
* 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/ | {"url":"http://www.stata.com/statalist/archive/2012-02/msg00783.html","timestamp":"2014-04-20T11:24:53Z","content_type":null,"content_length":"13884","record_id":"<urn:uuid:51708472-5379-4613-8b59-df420d5debd4>","cc-path":"CC-MAIN-2014-15/segments/1398223201753.19/warc/CC-MAIN-20140423032001-00289-ip-10-147-4-33.ec2.internal.warc.gz"} |
On Tue, Mar 6, 2012 at 16:23, Tim Prince
On 03/06/2012 03:59 PM, Kharche,
I am working on a 3D ADI solver for the heat equation. I have implemented it as serial. Would anybody be able to indicate the best and more straightforward way to parallelise it. Apologies if
this is going to the wrong forum.
If it's to be implemented in parallelizable fashion (not SSOR style where each line uses updates from the previous line), it should be feasible to divide the outer loop into an appropriate number
of blocks, or decompose the physical domain and perform ADI on individual blocks, then update and repeat.
True ADI has inherently high communication cost because a lot of data movement is needed to make the _fundamentally sequential_ tridiagonal solves local enough that latency doesn't kill you trying to
keep those solves distributed. This also applies (albeit to a lesser degree) in serial due to way memory works.
If you only do non-overlapping subdomain solves, you must use a Krylov method just to ensure convergence. You can add overlap, but the Krylov method is still necessary for any practical convergence
rate. The method will also require an iteration count proportional to the number of subdomains across the global domain times the square root of the number of elements across a subdomain. The
constants may not be small and this asymptotic result is independent of what the subdomain solver is. You need a coarse level to improve this scaling.
Sanjay, as Matt and I recommended when you asked the same question on the PETSc list this morning, unless this is a homework assignment, you should solve your problem with multigrid instead of ADI.
We pointed you to simple example code that scales well to many thousands of processes. | {"url":"http://www.open-mpi.org/community/lists/users/att-18712/attachment","timestamp":"2014-04-20T01:51:44Z","content_type":null,"content_length":"3616","record_id":"<urn:uuid:b643b396-4d4c-4ef3-8aef-6443404de523>","cc-path":"CC-MAIN-2014-15/segments/1398223211700.16/warc/CC-MAIN-20140423032011-00194-ip-10-147-4-33.ec2.internal.warc.gz"} |
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bayes' theorem
March 28th 2010, 09:02 AM #1
Junior Member
Oct 2009
bayes' theorem
The probability that a person has a deadly virus is 5 in 1000. A test will correctly diagnose this disease 95% of the time and incorrectly on 20% of occasions.
Find the probability of this test giving correct diagnosis. <-- not sure how to set up this expression.
Pr(correct diagnosis) = Pr(Correct diagnosis | Person has disease) x Pr(Person has disease) + Pr(Correct diagnosis | Person does not have disease) x Pr(Person does not have disease).
March 28th 2010, 01:02 PM #2 | {"url":"http://mathhelpforum.com/statistics/136081-bayes-theorem.html","timestamp":"2014-04-17T23:10:21Z","content_type":null,"content_length":"32999","record_id":"<urn:uuid:d91471c2-2dc4-40f9-8f4d-a4ac46c088cb>","cc-path":"CC-MAIN-2014-15/segments/1397609532128.44/warc/CC-MAIN-20140416005212-00443-ip-10-147-4-33.ec2.internal.warc.gz"} |
Space and Matrix Transformations - Building a 3D Engine
This article is my second of many articles describing the mechanics and elements of a 3D Engine. This article describes how to navigate 3D space programmatically through 4x4 matrix transforms. I
discuss basic transforms, model transforms, view transformer and projection transforms. The purpose of the series is intended to consolidate a number of topics written in C# allowing non-game
programmers to incorporate the power of 3D drawings into their application. While the vast majority of 3D Engines are used for game development, this type of tool is very useful for visual feedback
and graphic data input. This tool began with a focus on simulation modeling but the intent is to maintain a level of performance that will allow real-time graphics.
Fortunately work has been really good; consequenty has work has pushed writing articles down the list priorities. This has took significantly longer than I had planned. I am not a computer programmer
by profession and as such I would expect that ideas and algorithms expressed in this series to be rewritten and adapted to your application by “real” programmers.
Topics To Be Discussed
Basics: Drawing:
• Drawing Points, Lines and Triangles
• Textures and Texture Coordinates
• Loading a Static Mesh Files (.obj)
• SceneGraphs
Standard 3D Objects and Containment: Sorting Tree and Their Application:
• Spheres and Oriented Boxes • Octree Tree Sort
• Capsules and Cylinders • BSP Tree Sort
• Lozenges and Ellipsoids • Object Picking and Culling
• Lights and Special Effects
• Distance Methods
• Intersection Methods
I am currently rewriting one of my company’s software models and have proposed revamping our GUI interface providing users a drawing interface. This series is a product of a rewrite of my original
proof of concept application.
I am an Electrical Engineer with Lea+Elliott and as part of my responsibilies I develop and maintain our company's internal models. That being said, I am not a professional programmer so please keep
this in mind as you look through my code.
Prior to Using the Code
Before you can use this code you need to download the Tao Framework to access the Tao namespaces. You also need to reference the AGE_Engine3D.dll, add the AGE_Engine3D to your project or copy the
applicable .cs file and add them to your project.
The Basics
Both 3D APIs (DirectX and OpenGL) work with 4D vectors and 4x4 matrixes. I have seen different explanations but this is how I compose my matrix transforms. The basic 4x4 Matrix is a composite of a
3x3 matrixes and 3D vector.
These matrix transformations are combined to orient a model into the correct position to be displayed on screen. Unlike normal multiplication, matrix multiplication is not commutative. With matrixes,
A*B does not necessary equal B*A. That being said, the order that these transforms are applied is extremely important. This is discussed more in the subsequent sections. [Note: These samples are
written implementing the Tao OpenGL interface, but could very easily be modified to service DirectX or XNA interface.]
All transforms listed in this article are implemented in the AGE_Matrix44 class through static methods. The method's name is a good indication of its propose such as:
public static AGE_Matrix_44 HProjGL(float left,
float right,
float top,
float bottom,
float near,
float far)
Creates a 4 x 4 Projection Matrix
Simple Transforms
Scale Matrix (Hs)
The Scale Matrix is used to scale a model in one, two or three dimensions. It is composed of a 4x4 matrix with a 3D scaling vector on the diagonal. The scaling vector components represent a scaling
in their respective dimension. It is in the form of:
It is implemented in static AGE_Matrix44.HWorld method.
public static AGE_Matrix_44 HWorld(ref AGE_Vector3Df Location,
ref AGE_Vector3Df Scale)
AGE_Matrix_44 result = AGE_Matrix_44.Identity();
result.col[0][0] = Scale.X;
result.col[1][1] = Scale.Y;
result.col[2][2] = Scale.Z;
return result;
Rotation Matrix (Hr)
There are three rotation matrixes that can be used to rotate a model around the X-Axis, Y-Axis, and Z-Axis. There are three variations of a 4x4 matrix with various arrangements on the M matrix
mentioned above. It is in the form of:
It is implemented in static AGE_Matrix44.HRotation method.
public static AGE_Matrix_44 HRotation(float theta_X, float theta_Y, float theta_Z)
AGE_Matrix_44 result = AGE_Matrix_44.Identity();
//Rotate about the X-Axis
if (theta_X != 0)
AGE_Matrix_44 H_Rot_X = AGE_Matrix_44.Identity();
H_Rot_X.col[1][1] = H_Rot_X.col[2][2] = (float)System.Math.Cos(theta_X);
H_Rot_X.col[1][2] = (float)System.Math.Sin(theta_X);
H_Rot_X.col[2][1] = -H_Rot_X.col[1][2];
result = result * H_Rot_X;
//Rotate about the Y-Axis
if (theta_Y != 0)
AGE_Matrix_44 H_Rot_Y = AGE_Matrix_44.Identity();
H_Rot_Y.col[0][0] = H_Rot_Y.col[2][2] = (float)System.Math.Cos(theta_Y);
H_Rot_Y.col[2][0] = (float)System.Math.Sin(theta_Y);
H_Rot_Y.col[0][2] = -H_Rot_Y.col[2][0];
result = result * H_Rot_Y;
//Rotate about the Z-Axis
if (theta_Z != 0)
AGE_Matrix_44 H_Rot_Z = AGE_Matrix_44.Identity();
H_Rot_Z.col[0][0] = H_Rot_Z.col[1][1] = (float)System.Math.Cos(theta_Z);
H_Rot_Z.col[0][1] = (float)System.Math.Sin(theta_Z);
H_Rot_Z.col[1][0] = -H_Rot_Z.col[0][1];
result = result * H_Rot_Z;
return result;
Rotation Matrix via Quaternion (Hq)
I often use quaternion for creating my rotation matrixes. It is simple and intuitive. Creating a quaternion for rotation requires a vector identifying the axis of rotation and the angle of rotation.
I believe it is commonly used in ArcBall(add hyperlink to) and other orbiting camera schemes. I will discuss it in another article but if you want to look at the code it is included.
public static AGE_Matrix_44 HRotation(ref AGE_Quaternion Rotation)
Transpose Matrix (Ht)
The Transpose Matrix is used to move a model from one position to another. It is composed of a 4x4 identity matrix with a 3D translation vector in the 4th column. The translation vector represents a
change in location. It is in the form of:
It is implemented in static AGE_Matrix44.HWorld method.
public static AGE_Matrix_44 HWorld(ref AGE_Vector3Df Location,
ref AGE_Vector3Df Scale)
AGE_Matrix_44 result = AGE_Matrix_44.Identity();
result.col[3][0] = Location.X;
result.col[3][1] = Location.Y;
result.col[3][2] = Location.Z;
return result;
World Space (Hw)
The World Space Transform is the first transform usually applied to a model. This transform is normally used to scale and orient the model relative to its world. The Model is defined in a model space
coordinate system and needs to be translated to the world coordinate system. Let's assume you have a model of a person and it normalized such that the model dimensions are within the range [-1, 1]
with an origin of <0,0,0>. You could populate a world with actors referencing this single resource by applying different World Transforms. The World Transform is composed of basic transforms typical
applied in this order.
The table (a.k.a. TheTable in the code below) shown in the animation above is a SpatialBaseContainer object. The follow code shows its creation and positioning in 3D space. Note that TheTable only
references the table geometry, so if we could have many tables referencing the same geometry using different world transform populating.
override public void InitializeSimulation()
if (ResourceManager.LoadMeshResouse("Table.obj"))
TheTable = new SpatialBaseContainer();
TheTable.Geometry = ResourceManager.GetMeshObject("Table.obj");
TheTable.NowRotate(0, 0, AGE_Engine3D.Math.AGE_Functions.PI / 2);
TheTable.NowMove(new AGE_Vector3Df(48.1973f, 222.4320f, 0f));
The order transforms that are applied is important. Had I placed the table in the center of the room then rotated the table, it would be in a completely different spatial place. It would equate to a
322 feet error.
Child Models and Local Transforms
There are many cases where a model may have sub, child or leaf models. These are kin to a glass relative to its parent table. If the table is not already scaled and aligned with the parent node (a
room) the table will have to be scaled, rotated and translated to its proper place via a local transform. This is accomplished by combining the basic transform in a specific order in the same matter
as the world space transform. If a child model is allowed to change relative to its parent then each child model will have a separate the overall World Transform and its local transform. The Total
World Transform would look something like this for various objects.
By specifying that the TheGlass is a child to TheTable, I only have to locate TheGlass relative to the table. The transforms applied to TheTable will be carried forward to TheGlass. I can also create
children to TheGlass and locate them relative to TheGlass. The Following code and image demonstrate this concept.
override public void InitializeSimulation()
if (ResourceManager.LoadMeshResouse("Glass.obj"))
TheCups = new SpatialBaseContainer();
TheCups.Geometry = ResourceManager.GetMeshObject("Glass.obj");
TheCups.NowMove(new AGE_Vector3Df(0.0f, 0.0f, 3.083f));
//Child Cup #1
SpatialBaseContainer NewKid = new SpatialBaseContainer();
NewKid.Geometry = TheCups.Geometry;
NewKid.NowScale(new AGE_Vector3Df(1.0f, 0.75f, 2));
NewKid.NowMove(new AGE_Vector3Df( 3f,0, 0));
//Child Cup #2
NewKid = new SpatialBaseContainer();
NewKid.Geometry = TheCups.Geometry;
NewKid.NowScale(new AGE_Vector3Df(1.0f, 3.0f, .5f));
NewKid.NowMove(new AGE_Vector3Df(-3f,0, 0));
View Space (Hv)
After a model is transformed to its position into World Space it will then be transformed in to View Space or Camera Space. The transform is characterized by Camera Location and the View direction.
The Transform is in the form of:
public static AGE_Matrix_44 HView(ref AGE_Vector3Df Eye, ref AGE_Vector3Df Target,
ref AGE_Vector3Df UpVector)
AGE_Matrix_44 result = AGE_Matrix_44.Zero();
AGE_Vector3Df VDirection = (Target - Eye);
VDirection = VDirection.UnitVector();
AGE_Vector3Df RightDirection = AGE_Vector3Df.CrossProduct(ref VDirection,
ref UpVector);
RightDirection = RightDirection.UnitVector();
AGE_Vector3Df UpDirection = AGE_Vector3Df.CrossProduct(ref RightDirection,
ref VDirection);
UpDirection = UpDirection.UnitVector();
result.col[0][0] = RightDirection.X;
result.col[1][0] = RightDirection.Y;
result.col[2][0] = RightDirection.Z;
result.col[0][1] = UpDirection.X;
result.col[1][1] = UpDirection.Y;
result.col[2][1] = UpDirection.Z;
result.col[0][2] = -1 * VDirection.X;
result.col[1][2] = -1 * VDirection.Y;
result.col[2][2] = -1 * VDirection.Z;
result.col[3][0] = -1* AGE_Vector3Df.DotProduct(ref RightDirection, ref Eye);
result.col[3][1] = -1 * AGE_Vector3Df.DotProduct(ref UpDirection, ref Eye);
result.col[3][2] = AGE_Vector3Df.DotProduct(ref VDirection, ref Eye);
result.col[3][3] = 1;
return result;
Because the OpenGL uses a right-handed coordinate system the D vector is multiplied by -1 in the matrix Q construction, but for left-handed coordinate system this would not be necessary.[1]
Provided in the zip file are simple camera classes. In the sample application I used the AGE_WalkThroughCamera class allowing me to create a camera that could move through the building and actively
update the camera's focus. This is accomplished by updating the Target and Eye locations, then recalculating the View Matrix. This creates the animation seen in the video above.
Projection Matrixes (Hproj and Hortho)
Perspective Transform
After a model is transformed to its position into View Space, it will then be transformed in to its final viewed position via projection transformation. There are two basic types of projection
transforms that I am aware of: Orthographic, and Perspective. The Perspective projection mimics the way we perceive the real world. Objects that are closer appear larger and parallel lines converge
at the horizon. Here is how the Perspective transform is constructed.
It is implemented in static HProjGL method.
public static AGE_Matrix_44 HProjGL(float left,
float right,
float top,
float bottom,
float near,
float far)
float invRDiff = 1 / (right - left + float.Epsilon);
float invUDiff = 1 / (top - bottom + float.Epsilon);
float invDDiff = 1 / (far - near + float.Epsilon);
AGE_Matrix_44 result = AGE_Matrix_44 .Zero();
result.col[0][0] = 2 * near * invRDiff;
result.col[1][1] = 2 *From Subject Received Size Categories
Newton, Curtis RE: Accident on the DCC System at the Mexico City Airport
1:35 PM 8 KB near * invUDiff;
result.col[2][0] = (right + left) * invRDiff;
result.col[2][1] = (top + bottom) * invUDiff;
result.col[2][2] = -1 * (far + near) * invDDiff;
result.col[2][3] = -1;
result.col[3][2] = -2 * (far * near) * invDDiff;
return result;
Orthographic Transform
The Orthographic projection is the somewhat the opposite of the Perspective projection. An object’s depth in the view plane has no bearing of size of the object and parallel lines remain parallel.
Here is how the Orthographic transform is constructed.
It is implemented in static HOrtho method.
public static AGE_Matrix_44 HOrtho(float left,
float right,
float top,
float bottom,
float near,
float far)
float invRDiff = 1 / (right - left);
float invUDiff = 1 / (top - bottom);
float invDDiff = 1 / (far - near);
AGE_Matrix_44 result = AGE_Matrix_44.Zero();
result.col[0][0] = 2 * invRDiff;
result.col[3][0] = -1 * (right + left) * invRDiff;
result.col[1][1] = 2 * invUDiff;
result.col[3][1] = -1 * (top + bottom) * invUDiff;
result.col[2][2] = -2 * invDDiff;
result.col[3][2] = -1 * (far + near) * invDDiff;
result.col[3][3] = 1;
return result;
How to use the Code
Once we have all of these different types of transforms how are they used? In each call to the rendering function the sample application [Implementing the OpenGL interface], the application resets
the Projection Matrix (GL_PROJECTION) and the Model/View Matrix (GL_MODELVIEW). Since the Camera and Projection rarely change during each render call, I load both the Projection Matrix and View
Matrix into the OpenGL’s Projection Matrix. I also load the identity matrix in the Model/View matrix which will be overloaded as each entity is rendered.
static public void RenderScence()
Object LockThis = new Object();
lock (LockThis)
Gl.glClear(Gl.GL_COLOR_BUFFER_BIT | Gl.GL_DEPTH_BUFFER_BIT);
//Verify the Current Camrea is Initalized
if (!CurrentCamera.IsInitalized)
CurrentCamera.ResizeWindow(ScreenWidth, ScreenHeight);
//Get the Combined Projection and View Transform
AGE_Matrix_44 HTotal = CurrentCamera.HTotal;
//Load it in to OpenGl
//If there are lights turn lighting on
//Render each visable Spatial Object
foreach (ISpatialNode obj in WorldObjectList)
if (obj.IsVisable) obj.RenderOpenGL();
//Render HUD
//Display the new screen
Now that the setup is complete the application cycles through all of my render-able objects and calls their rendering function. Each object is responsible for setting the Model/View Matrix
appropriate to the objects requirements.
//In the SpatialBaseContainer Class
public void RenderOpenGL()
if (this.IsVisable)
//Verify the Model Transform is Updated
if (!this.IsHModelUpdated)
//Verfiy any Parent Transfor is Included
if (!this.IsHTotalUpdated)
//Load the Model Transform in to OpenGL
//Render Parent Geometry
if (Geometry != null)
//Render any children
foreach (IRenderOpenGL ChildGeometry in this.Children)
There is a lot more included in the .zip file not discussed here, but I will go into the other classes and concepts in other articles.
Further Reading
• 2009-09-02 - Second Article Released. | {"url":"http://www.codeproject.com/Articles/42086/Space-and-Matrix-Transformations-Building-a-3D-Eng?fid=1548853&df=90&mpp=10&noise=1&prof=True&sort=Position&view=Quick&spc=Relaxed","timestamp":"2014-04-20T05:56:21Z","content_type":null,"content_length":"101035","record_id":"<urn:uuid:9bc6689f-f22a-4ea4-b5d9-e41861bdc08e>","cc-path":"CC-MAIN-2014-15/segments/1397609538022.19/warc/CC-MAIN-20140416005218-00423-ip-10-147-4-33.ec2.internal.warc.gz"} |
Physics Forums - View Single Post - Activation Energy
Also, since k is a function of temperature, if [tex]k_{1}[/tex] and [tex]k_{2}[/tex] are rate constants measured at [tex]T_{1}[/tex] and [tex]T_{2}[/tex] we have from the Arrhenius Equation,
\frac{k_{2}}{k_{1}} = e^{\frac{E_{a}}{R}(\frac{1}{T_{1}}-\frac{1}{T_{2}})}
So actually this eliminates the need to know the frequency factor A. All you need to know is the rate constants measured at two different temperatures or even their ratio [tex]\frac{k_{2}}{k_{1}}[/
tex] and that is enough to get [tex]E_{a}[/tex], which turns out to be,
E_{a} = R\frac{T_{1}T_{2}}{T_{2}-T_{1}}log_{e}\frac{k_{2}}{k_{1}}[/tex]
Please make note of this correction in my previous post. | {"url":"http://www.physicsforums.com/showpost.php?p=246185&postcount=3","timestamp":"2014-04-16T22:14:35Z","content_type":null,"content_length":"7852","record_id":"<urn:uuid:0da6979f-e4d6-4bd5-90b5-a2ee4e150954>","cc-path":"CC-MAIN-2014-15/segments/1397609525991.2/warc/CC-MAIN-20140416005205-00395-ip-10-147-4-33.ec2.internal.warc.gz"} |
This Article
Bibliographic References
Add to:
A Short Proof that Phylogenetic Tree Reconstruction by Maximum Likelihood Is Hard
January-March 2006 (vol. 3 no. 1)
pp. 92-94
ASCII Text x
Sebastien Roch, "A Short Proof that Phylogenetic Tree Reconstruction by Maximum Likelihood Is Hard," IEEE/ACM Transactions on Computational Biology and Bioinformatics, vol. 3, no. 1, pp. 92-94,
January-March, 2006.
BibTex x
@article{ 10.1109/TCBB.2006.4,
author = {Sebastien Roch},
title = {A Short Proof that Phylogenetic Tree Reconstruction by Maximum Likelihood Is Hard},
journal ={IEEE/ACM Transactions on Computational Biology and Bioinformatics},
volume = {3},
number = {1},
issn = {1545-5963},
year = {2006},
pages = {92-94},
doi = {http://doi.ieeecomputersociety.org/10.1109/TCBB.2006.4},
publisher = {IEEE Computer Society},
address = {Los Alamitos, CA, USA},
RefWorks Procite/RefMan/Endnote x
TY - JOUR
JO - IEEE/ACM Transactions on Computational Biology and Bioinformatics
TI - A Short Proof that Phylogenetic Tree Reconstruction by Maximum Likelihood Is Hard
IS - 1
SN - 1545-5963
EPD - 92-94
A1 - Sebastien Roch,
PY - 2006
KW - Analysis of algorithms and problem complexity
KW - probability and statistics
KW - biology and genetics.
VL - 3
JA - IEEE/ACM Transactions on Computational Biology and Bioinformatics
ER -
Maximum likelihood is one of the most widely used techniques to infer evolutionary histories. Although it is thought to be intractable, a proof of its hardness has been lacking. Here, we give a short
proof that computing the maximum likelihood tree is NP-hard by exploiting a connection between likelihood and parsimony observed by Tuffley and Steel.
[1] L. Addario-Berry, B. Chor, M.T. Hallett, J. Lagergren, A. Panconesi, and T. Wareham, “Ancestral Maximum Likelihood of Evolutionary Trees is Hard,” J. Bioinformatics and Computational Biology,
vol. 2, no. 2, pp. 257-271, 2004.
[2] R. Agarwala, V. Bafna, M. Farach, B. Narayanan, M. Paterson, and M. Thorup, “On the Approximability of Numerical Taxonomy (Fitting Distances by Tree Metrics),” SIAM J. Computing, vol. 28, pp.
1073-1085, 1999.
[3] G. Ausiello, P. Crescenzi, G. Gambosi, V. Kann, A. Marchetti-Spaccamela, and M. Protasi, Complexity and Approximation. Berlin: Springer, 1999.
[4] J. Cavender, “Taxonomy with Confidence,” Math. Biosciences, vol. 40, pp. 271-280, 1978.
[5] B. Chor and T. Tuller, “Maximum Likelihood of Evolutionary Trees is Hard,” Proc. Ninth Int'l Conf. Computational Molecular Biology (RECOMB 2005), 2005.
[6] A. Clementi and L. Trevisan, “Improved Non-Approximability Results for Minimum Vertex Cover with Density Constraints,” Theoretical Computer Science, vol. 225, nos. 1-2, pp. 113-128, 1999.
[7] W. Day, D. Jonhson, and D. Sankoff, “The Computational Complexity of Inferring Rooted Phylogenies by Parsimony,” Math. Biosciences, vol. 81, pp. 33-42, 1986.
[8] W. Day and D. Sankoff, “The Computational Complexity of Inferring Phylogenies by Compatibility,” Systematic Zoology, vol. 35, pp. 224-229, 1986.
[9] A.W.F. Edwards and L.L. Cavalli-Sforza, “Reconstruction of Evolutionary Trees,” Phenetic and Phylogenetic Classification, V.H. Heywood and J. McNeill, eds. Systematics Assoc., London, vol. 6, pp.
67-76, 1964.
[10] J.S. Farris, “A Probability Model for Inferring Evolutionary Trees,” Systematic Zoology, vol. 22, pp. 250-256, 1973.
[11] J. Felsenstein, “Evolutionary Trees from DNA Sequences: A Maximum Likelihood Approach,” J. Molecular Evolution, vol. 17, pp. 368-376, 1981.
[12] J. Felsenstein, Inferring Phylogenies. Sunderland: Sinauer Assoc., 2004.
[13] L. Foulds and R. Graham, “The Steiner Problem in Phylogeny is NP-Complete,” Advances in Applied Math., vol. 3, pp. 43-49, 1982.
[14] M.R. Garey and D.S. Johnson, Computers and Intractability. A Guide to the Theory of NP-Completeness, San Francisco: W.H. Freeman, 1976.
[15] J. Neyman, “Molecular Studies of Evolution: A Source of Novel Statistical Problems,” Statistical Decision Theory and Related Topics, S.S. Gupta and J. Yackel, eds. New York: Academic Press, pp.
1-27, 1971.
[16] C. Semple and M. Steel, Phylogenetics. Oxford Univ. Press, 2003.
[17] C. Tuffley and M. Steel, “Links between Maximum Likelihood and Maximum Parsimony under a Simple Model of Site Substitution,” Bull. Math. Biology, vol. 59, no. 3, pp. 581-607, 1997.
[18] H.T. Wareham, On the Computational Complexity of Inferring Evolutionary Trees, MSc thesis, Technical Report no. 9301, Dept. of Computer Science, Memorial Univ. of Newfoundland 1993.
Index Terms:
Analysis of algorithms and problem complexity, probability and statistics, biology and genetics.
Sebastien Roch, "A Short Proof that Phylogenetic Tree Reconstruction by Maximum Likelihood Is Hard," IEEE/ACM Transactions on Computational Biology and Bioinformatics, vol. 3, no. 1, pp. 92-94,
Jan.-March 2006, doi:10.1109/TCBB.2006.4
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Post a reply
Hello everybody, I have a probability question.
Could you please help me prove that "Conditional Independence does not imply Independence", and vice-versa "(Absolute) Independence does not imply Conditional Independence"?
Every book and online resource seems to take it for granted. They only offer simple counter examples, but I haven't found a formal proof, e.g., by contradiction.
If you can reference a book chapter or post the basics of the proof(s) it would help a lot :-)
Thank you for your time! | {"url":"http://www.mathisfunforum.com/post.php?tid=18932&qid=251759","timestamp":"2014-04-21T12:33:43Z","content_type":null,"content_length":"16291","record_id":"<urn:uuid:045c132d-cff0-4a0a-8d15-ed0f60421447>","cc-path":"CC-MAIN-2014-15/segments/1397609539776.45/warc/CC-MAIN-20140416005219-00646-ip-10-147-4-33.ec2.internal.warc.gz"} |
Analyzing 4:1 TLTs For Optical Receivers
Transmission-line transformers can provide reliable interfaces between different impedances for broadband applications in RF, wireless, and optical systems.
Transmission-line transformers (TLTs) provide broadband impedance transformation in a wide variety of RF circuits. When using TLTs, analysis of the frequency response is an important consideration.^
1,2 Such analyses usually assume that the load and source are impedance matched through the transformer, although this is not always the case.
At times, matching between the primary and secondary impedances of a TLT is either impractical or undesirable. An example of this is the interface circuit between a photodiode and an amplifier used
in a particular optical receiver architecture. Based on this example, Guanella and Ruthroff TLTs were analyzed, although TLTs are useful in many applications.
A number of architectures are used for optical receivers.^3 This article concerns a special case (Fig. 1) with a broadband 75-Ω power amplifier. The design challenge is to find an interface circuit
for optimal system sensitivity without degrading the system bandwidth and amplitude flatness.
From the viewpoint of the RF circuitry, the photodetector can be modeled as a current source with a high source impedance. When the resistors used for the DC bias are taken into account, the
photodetector and bias circuit can be modeled as a current source with a source impedance on the order of kiloohms. On the amplifier side, it can be replaced by a load impedance, R[L], which is equal
to its input impedance, R[A]. This reduces Fig. 1 to the equivalent circuit shown in Fig. 2. The load and source impedances are chosen as 75 Ω and 2 kΩ, respectively, for the purposes of numerical
analysis. TLTs are often selected as the interface in such cases when wide bandwidths are required.
From a power-transfer consideration alone, a stepdown transformer that matches R[L] (75 Ω) to R[s] (2 kΩ) would deliver maximum system responsivity. However, such a high-ratio impedance
transformation (26:1) is not only difficult to implement but results in an unacceptable reduction in bandwidth due to the photodiode's parasitic capacitance. As a compromise, a 4:1 TLT can provide
enough gain and bandwidth, even though it will result in a condition of mismatched impedances.
A TLT generally has a reactive component and exhibits frequency-dependent operation. Thus, the transformed impedance Z[in], seen from the diode side as defined in Fig. 2, can be expressed in general
Assuming a lossless transformer, power dissipated in the real part of Z[in], denoted as P[in], is equal to that in R[L], which in turn is proportional to the system responsivity. A simple circuit
calculation shows that:
Thus, through Eq. 1, the issue of frequency dependence of the system responsivity is reduced to finding Z[in] as a function of frequency.
In a TLT, parasitic effects are greatly suppressed because most stray capacitance is absorbed by the line inductance to form the characteristic impedance of the transmission line. This suppression of
parasitic effects is the main reason that TLTs have higher frequency limits than conventional transformers. Although in some applications the ultimate high-frequency limit is set by the residual
parasitic effects, this article will consider only the frequency response due to the intrinsic circuit parameters such as the length and characteristic impedance of the transmission line and load and
source impedances. As part of this idealized analysis, the TLT is considered lossless and free of parasitic effects. Furthermore, to get an insight on how the frequency dependence arises, two
conditions are listed below.^1 When these conditions are met, a TLT is considered ideal in that it has a frequency independent response regardless of the circuit parameters.
1. Only odd mode excitation is allowed, i.e. the currents in the conductors making up the transmission line are in opposite directions. High-permeability toroids are employed in many TLTs for this
2. The transmission line is short compared with the wavelength (say, less than l/8), which leads to a situation in which the voltages at two ends of a transmission line are the same and the currents
at each end of an individual conductor are equal.
Validity of the first condition usually sets a low-frequency limit on the ideal TLT, while validity of the second condition sets the high-frequency limit on the ideal TLT.
For two types of TLTs, Guanella and Ruthroff, the analysis is first performed under these ideal conditions and then focused on the non-ideal situation at high frequencies.
Figure 3 shows a Guanella transformer. Essentially, it consists of two pairs of transmission lines that are connected in series at the high-impedance side (the input in this case) and in parallel at
the low-impedance side (the output). The assignments of the voltage and current on the transmission lines, V[i] and I[i], are only correct for the two ideal conditions mentioned earlier, and under
these conditions the following relationships can be easily established:
The series connection at the input leads to V[in] = 2V[i] and I[in] = I[i]. The parallel connection at the output leads to V[out] = V[i] and I[out] = 2I[i]. Therefore,
Then R[in] = 4 R[out], confirming the 4:1 impedance transformation ratio and the frequency independency. When the length of the transmission line becomes relatively long, the second condition is no
longer valid and modifications are needed.
In Fig. 3, the input impedance Z[in] is simply the sum of two serial impedances transformed from the same load through two identical pairs of transmission lines respectively (true only when the first
ideal condition holds). These two impedances are obviously identical and designated as Z[L]'. Because they are identical, an analysis on the impedance transformation through one pair of transmission
lines is sufficient.
In this configuration, the current through load Z[L] is the sum of currents from each pair and therefore is twice that from a single pair. Hence, if a single TLT is used for analysis, the equivalent
load should be doubled from the real value of Z[L] to give the same current and voltage at the load. The pertinent equation for Z[L]' can be found in any textbook on microwave engineering:
Z[L] = the load impedance (in Fig. 2),
Z[c] = the characteristic impedance of the transmission line, and
bl = the phase angle.
From Eq. 2, the frequency dependence of Z[in] (=2Z[L]') is solely through the phase angle bl when Z[L] is purely resistive. Numerical results using Eq. 1 and 2 for several values of Z[c] and R[s]=2
kΩ, R[L]=75 Ω are shown in Fig. 4. The normalized power P(bl)/P(0) in logarithmic scale is used in the plot, where P(0) is the low frequency limit of P(bl) and given by:
The plot shows that the curve for Z[c]=2R[L]=150 Ω is flat, indicating that it is not dependent on frequency. This can be easily concluded by examining Eq. 2. Also, the condition for perfect flatness
is independent of the source impedance, R[s].
Page Title
Figure 5 shows the circuit schematic of another popular 4:1 transmission line transformer, the Ruthroff transformer. Unlike the Guanella TLT, which requires two pairs of transmission lines, a 4:1
impedance transformation can be achieved with a single transmission line in the Ruthroff topology. For the case when a balanced-to-balanced (with a center ground tab) impedance transformation is
required, however, two parallel pairs of transmission lines are still needed. The single and two-TLT Ruthroff configurations are shown in the inserts in Fig. 6 and 7, respectively. It is evident that
an analysis of the single-TLT configuration is sufficient.
Under ideal conditions, voltages and currents at input and output ends are:
and therefore V[in]/I[in] = 4(V[out]/I[out]), indicating a 4:1 impedance transformation.
When the transmission line is long, voltages V[1] and V[2] and currents I[1] and I[2] are related through:
where V^+ and V^− are forward and backward traveling waves at port 2, respectively. Solving Eqs. 3 through 6 gives V[1] and I[1] expressed in terms of V[2], I[2], and the transmission-line
parameters, Z[c] and bl:
In addition,
Using Eqs. 7 through 11, after some algebra, the input impedance Z[in] is:
Figures 6 and 7 show plots of numerical results for single and two-TLT configurations, respectively, for R[s] = 2 kΩ, R[L] = 75 Ω, and Z[c] as a variable. As shown in the insert of Fig. 7, for the
two-TLT configuration the actual values for R[s] and R[L] used in Eq. 12 are 1 kΩ and 37.5 Ω, respectively, to account for the fact that Eq. 12 is for the single-TLT configuration. A close comparison
of two groups of curves in Figs. 6 and 7 reveals that the curve with a Z[c] in the single-TLT configuration is identical to that with 2Z[c] in the two-TLT configuration. In fact, this can be readily
seen from Eq. 12, since R[L] used in the two-TLT configuration is simply one-half that in the single-TLT configuration. This feature is of significance in practical design because a TLT with high Z
[c] is often difficult to realize in practice.
The following can be concluded:
1. Even with a limited number of curves plotted in Figs. 4, 6, and 7, one can see that the frequency response curves are greatly affected by the choice of Z[c], Z[L], and Z[s]. For the 4:1 Guanella
transformer there is a condition (Z[c] =2Z[L]) for which the response is completely frequency independent whereas the frequency response of the 4:1 Ruthroff transformer always starts to roll off
at certain frequency point.
2. It has been assumed that transmission lines are lossless, aslthough in reality they exhibit loss. As shown in Figs. 4, 6, and 7, for both types of TLT when Z[c] is chosen properly the response
curve increases with frequency up to a certain point (around βl=1.5). This feature can be utilized in practice to compensate the loss in the transmission line and can it also provide system gain
with an upward slope.
3. The frequency response of the Ruthroff transformer monotonically decreases at high frequencies, while the Guanella's response is periodic with frequency.
4. For the load and source impedances considered in this article, the Ruthroff transformer with the two-TLT configuration requires the lowest Z[c] to have an upward response curve. The approach also
supports differential operation, but usually requires two parallel toroidal transformers.
The authors would like to thank Dr. Derald O. Cummings for introducing some basic concepts and techniques on the subject at the early stage of this project.
1. W. Alan Davis and Krishna K. Agarwal, Radio Frequency Circuit Design, Wiley, New York, 2001.
2. Jerry Sevick, Transmission Line Transformers, 4th Ed., Noble Publishing, Norcross, GA, 2001.
3. Stephen B. Alexander, Optical Communication Receiver Design, Bellingham, SPIE Optical Engineering Press, San Jose, CA, 1997. | {"url":"http://mwrf.com/components/analyzing-41-tlts-optical-receivers","timestamp":"2014-04-18T21:08:16Z","content_type":null,"content_length":"89527","record_id":"<urn:uuid:843e644f-6676-47be-8ee2-8857751fe793>","cc-path":"CC-MAIN-2014-15/segments/1397609535095.9/warc/CC-MAIN-20140416005215-00193-ip-10-147-4-33.ec2.internal.warc.gz"} |
Monochromatic cows
September 1998
The following is a fallacious proof that "all cows in a field are the same colour". A quick trip out into the countryside will soon provide a counter-example to the Theorem - but where's the flaw in
the argument?
Theorem: All cows in a field are the same colour
Proof by induction on the number of cows
Induction hypothesis: n cows in a field are the same colour, for all n=1,2,3,4,...
Initial Step Clearly one cow in a field is the same colour as itself, so the induction hypothesis is true for n=1.
Induction Step Now suppose n is at least 1, we have n cows in a field F, and that the induction hypothesis has been proved for all fields containing at most n cows. By the induction hypothesis all
the cows in F are the same colour.
Firstly, remove any cow from F and put it to one side. Now take a cow from some other field and put it in field F. We again have n cows in field F, so by the induction hypothesis they are all the
same colour. Finally, put back the first cow you thought of. We already know that it's the same colour as all the other cows in F, and so now we have n+1 cows in field F, and they're all the same
colour. This completes the induction step!
Submitted by Anonymous on July 26, 2013.
Induction hypothesis is governed by principle ordering which has been neglected when we remove 1 cow from the set and replace with another random one outside the set. | {"url":"http://plus.maths.org/content/monochromatic-cows","timestamp":"2014-04-16T16:07:18Z","content_type":null,"content_length":"23913","record_id":"<urn:uuid:572e3246-70d2-4cd0-8023-3799e5e58c65>","cc-path":"CC-MAIN-2014-15/segments/1398223206120.9/warc/CC-MAIN-20140423032006-00375-ip-10-147-4-33.ec2.internal.warc.gz"} |
Day 1
I wrote my First Day plan about a month before school actually started, and surprisingly, it didn’t change much. (I was right on target when I said we would probably run out to time and not be able
to even start the “My Math Stuff.”)
My Algebra students tried to count “Squares on a Chessboard” and became slightly overwhelmed by the enormity of the task after initially being sure it was just 64!
My Math 8 students investigated “Rice on a Chessboard” that will lead nicely into our first unit on Exponents and Scientific Notation.
Day 2
We spent a good chunk of this day making magnets, math portfolios to store their assessments and tracking forms, and Stars for Standards, but also did a “Get to Know You” activity (modified from an
idea by MathMamaWrites) requiring students to think about positive and negative in the coordinate plane. Topics were randomly chosen from their “Know Me Notecards” and groups placed their magnets
Day 3
In Algebra, we went back to the chessboard squares and made some more progress. Some groups studied the relationship between the size of the squares and the number of squares. Other groups looked at
the size of the board and the number of squares. Patterns were emerging!
For the daily Math Greeting, students placed their magnets on the Venn Diagram regarding their thoughts on their relationship to math. This is an Algebra class. Some for the other periods were more
interesting, but I neglected to take pictures :(
The glare (and an old pen) made it hard to read. Top circle says “enjoy,” left one says “understand,” and right one is “work hard.”
msSunFun: Musical Math Partners
One of the “math games” we play in my classes that I would like to share for
It is quite flexible, gets kids out of their seats, and gives students an opportunity to use mental math skills.
Each student receives some sort of “card” depending on the topic of the day.
Some sort of “path” is designed for students to travel around the room without too many “log jams.”
Here is a diagram of how I make it work in my room:
Students follow a path around a “row” of four pairs of desks, but at the end of the row they may choose to turn either left or right and continue around that “row.” (Even though there are two arrows
in the diagram, students are single file when walking between the desks.)
As the teacher plays their choice of music, students “travel” around the path – usually only 10-15 seconds. Once the music stops, students take a seat in the nearest desk and become “partners” with
the person in the adjacent seat. (If you have more desks than students, you might want to identify some of the desks as “out of the game.” As it is, you will often have students that must continue to
“travel” after the music stops in order to find a partner. If you have an odd number of students, the “leftover” person becomes partners with the teacher.)
Students “do the math” (more details below) – usually trying to finish more quickly than the other person – followed by a short debriefing regarding how they solved the problem, and then they trade
cards so that they have different experiences with each new partner.
Possibilities. . .
The possibilities are only limited by your imagination / ingenuity. :)
Integer Operations
Each person holds a card with an integer. (A deck of playing cards work well for this: black = positive and red = negative.) When the music stops, the teacher randomly flips an operation card and
students perform the operation from “left card” to “right card.” (I would only recommend division if you are interested in also focusing on fractions and mixed numbers.)
Fraction/Decimal Operations
Similar to the Integer Operations, only cards have fractions (or decimals.) I am fairly certain I would not include multiplication if decimals were involved, unless they were single digits. :) For
subtraction, if students are not yet familiar with negative numbers, students can subtract the lesser value from the greater one.
Fraction/Decimal/Percent Comparisons
Each student has a card containing a number. (You could do all fractions, all decimals, or a mix.) When the music stops, students race to decide which value is greater.
Evaluating Algebraic Expressions
Each student has a card containing two pieces of information: an algebraic expression (as simple or complex as you wish) and a numerical value (also as simple or complex as you wish – including
negatives, decimals, or fractions – but remember the goal is for students to evaluate fairly quickly.) When the music stops, students must evaluate the expression of the left card by the number on
the right card and then vice versa.
Operations on Algebraic Expressions
Each student has a card with an algebraic expressions. When the music stops, teacher randomly selects add or subtract and students perform the operation from left to right. If the expressions are
binomials, multiplication could also be included. (I am not sure I want students multiplying trinomials using mental math.)
Solving Linear Equations
Two options using cards throw the previous two “games” if the expressions are fairly simple.
Option 1: For the cards that have the expression and the number, students set the left expression equal to the number on the right card and solve, then switch to number on the left equal to
expression on the right.
Option 2: For the cards that all have algebraic expressions, students set the two expressions equal and solve. (This would work best if the expressions were fairly simple, but again, it is up to the
teacher depending on the level of students and the current unit of study.)
Measures of Center and Spread
Each student has a card containing a fairly small data set. When the music stops, the two data sets are combined, and students are to find median, mode, and range. Depending on the size of the sets
and the values, you could also have students find mean or the quartiles and interquartile range.
Plotting Points
Each student has a card with an integer and each desk pair contains a coordinate grid. When the music stops, students point to the location on the coordinate grid (of course, the left card is x
coordinate, and the right card is y coordinate.)
Pythagorean Theorem
Each student has a card with a value representing a side of a right triangle. When the music stops, the teacher either chooses “two legs” or “hypotenuse and leg.” Students find the remaining side.
(Since relatively small values should be chosen in order for students to compute mentally, there would be some duplication. If students with equal values became partners, the second option could
provide some interesting discussion!)
Distance on a Coordinate Plane
Each student has a card containing the coordinates of a point. When the music stops, student locate the two points and find the distance between them.
That’s about all for now. Feel free to add more ideas in the Comments. This is certainly an activity to use only after students have a fairly strong understanding of the concepts used in the tasks.
Teachers need to be careful when choosing the values / expressions on the cards so that mental calculation is fairly accessible.
Made4Math Monday: Formative Assessment Forms
I definitely left some things “hanging” on my last SBG post, so I thought I would “kill two birds with one stone,” so to speak, and write about some of the ways I incorporate formative assessment
into my daily routine.
A lot of people have written about “Exit Tickets,” and I guess, in a way, that’s what this is. I LOVE Sarah’s recent post at Everybody is a Genius (I’m not sure about everybody, but she IS a genius!)
about the laminated dry erase ones :) I think I might try that for my more “reflective” tasks to end the period, but I like the “permanence” of the system that I have set up.
Here is my:
Question of the Day
After we have spent a bit of time on a concept (but before it is time for a Mini-Assessment (Quiz)) I use Question of the Day to assess where students are at on a concept and us it to guide my
instruction for the next few days. I will put up a slide such as this:
at the end of class and given them about five minutes to work on it. (Depends on the task.) Rather than just using quarter sheets of paper (or anything dry erase) I wanted to have a tool that I could
use to provide feedback and also to have students keep as a record of their growth. I came up with the following simple “form”:
I print out four copies and then use the “booklet” app on our copy machine so that it shrinks it down and prints all four onto an 8.5 x 11 sheet. The original copy has more “space” on the top section
because when it shrinks down there is an extra “gap” at the bottom of the page. This provides opportunities for 8 QOD’s before the students need a new sheet. They will store them in the “pocket” of
their Math Notebook once they receive them back the following day. I use colored markers to write their number at the top so it’s easier to sort and hand back to color or number groups. The specific
standard (CCSS) is written on the form, and we update scores on their Tracking Sheets (see SBG post) fairly regularly. When the forms are “full” students store them in their Math Portfolio – a
hanging file folder in a crate that contains their tracking sheet as well as other “larger” formative assessments. Since there is only one question (ok, there’s usually two to three parts though)
they don’t take long to grade and record. I can also “sort” the forms to create “Just for Today” seats (also on Sticks and Seats post) to either differentiate instruction or provide support for
students who have not yet mastered the concept.
Writing About Math
I have students write in math class quite often,
but I haven’t always spent the time reading and commenting on their responses as I should. This year we are implementing the Common Core State Standards, and there are more than a few that employ
the use of the verb “explain” or “describe.” I am looking forward to challenging students to meet standard on those in addition to the more skill-based ones. I decided to modify my Question of the
Day form to use it for Writing About Math prompts as well.
Sometimes the prompt will be tied to a particular standard and graded/recorded as such, but other times I am just looking into their thinking about a concept and wanting a way to provide feedback.
Recording System
Since I record multiple scores for each standard (and multiple standards on each assessment) I needed a way to organize that information so that I can see the progressions of scores in each area. I
created a “grade book” form that allows for up to four scores (more if I sneak in a re-assessment score next to the original) for each standard with room for three standards on each sheet:
Once I have my class lists I will enter them on a blank copy, make four copies of that, and again use the booklet feature on the copy machine. This time I transfer them to the 8.5 x 14 size,
otherwise it shrinks down more than I want. When folded they create a nice size for up to twelve standards. I usually don’t have more than that in any one unit, so I create a new “grade sheet” for
each unit. When I am entering grades into the online Gradebook, I just have to look at the most recent column and update scores that have changed from previous assessments.
Final Note: QOD and WAM are not always relegated to “end of period tasks.” I also use them at the beginning of the period. After I collect them we discuss possible responses. I am excited to use “My
Favorite No” either when we do them at the beginning of the period, or the following day as a re-cap / remediation activity!
New Blogger Initiative: Mystery Number Puzzles
For the third prompt of the New Blogger Initiative, I have decided to choose Option 1, if not purely for the sake of wanting to learn more about how to use LaTeX, especially within a blog post.
My Algebra students will be diving into solving linear equations within the first week or so of the start of school. Since they have already had experience with 1-step and 2-step equations in sixth
and seventh grade, I wish to quickly expand their repertoire into solving multi-step equations by first taking on those involving just a single “x” term. (Ok, seriously, I’m not going to use LaTeX
for that! Would it just be $x$? Soooo frustrating not to know until it’s published!) (Aha! If I publish it as “Private” I can look at the preview of the actual online version – success!)
Initially, the tasks will not involve equations at all, but those goofy “Mystery Number Puzzles” such as:
I am thinking of a number. . . .
When I multiply it by 2 and then add three, multiply the result by 4 and divide by 6, then subtract 5, my answer is 1. WHAT is my number?!
Of course, if this task is just given verbally, it would be EXTREMELY challenging to solve – who can remember back that many steps?!
My lesson plan involves posting pages all over the classroom (in those lovely page protectors of course.) Students working in pairs would all start at a different spot in the room and complete a
Scavenger Hunt. (I know I read something about this idea somewhere, but I don’t have the link. Edit: Found the link to Math-In-Spire thanks to @msrubinteach.) The bottom of the page will have a
“Mystery Number Puzzle.” Once the pair has “solved the puzzle” they search around the classroom for that “solution” on the top half of another page. There will also be a “symbol” on the page for them
to record. On the bottom will be a new puzzle to solve, and so on, and so on. There will be three different “sets” of six puzzles, so after six problems they SHOULD be back where they started.
(Unless, of course, they solved a problem incorrectly and moved onto another “track” – uh oh!) The “symbols” they record will also be in the proper order if the problems were solved correctly. A
class discussion involving strategies used to solve the puzzles would ensue.
After this opening activity, we will take a look at how to record this information mathematically. The first step would be to determine how to actually write down the original puzzle in mathematical
form. After a brief partner brainstorming session the following “should” be agreed upon. (It will be interesting to see whether the use of parentheses will be remembered and/or emphasized.)
I am thinking of a number. . .
When I multiply it by 2: $2x$
and then add three: $2x+3$
multiply the result by 4: $4(2x+3)$
and divide by 6: (here’s the tricky LaTeX part) $\dfrac{4(2x+3)}{6}$
then subtract 5: $\dfrac{4(2x+3)}{6}-5$
my answer is 1: $\dfrac{4(2x+3)}{6}-5=1$
WHAT is my number?!
It is important to note here that some students may alternatively use: $(4(2x+3))\div6-5=1$ instead of the fraction bar, and that is totally acceptable.
Next phase: pairs return to their “starting page,” flip up the page, and write out the mathematical representation on the top half of the BACK of the page using a dry erase marker (or dry erase
crayon.) After returning their page to its original position, students do a Gallery Walk around the room and mentally conjure up the proper representation before peeking on the back side to see what
other students wrote.
The last step involves representing the solution process mathematically. I am not “picky” when it comes to solving these types of linear equations. As long as there is only one term involving $x$
(wow, it just starts to roll right out of your fingers) the entire process can be done using inverse operations. In fact, my goal would be for students to be able to write the solution for THIS
equation in the following way: $x=\dfrac{6(1+5)/4-3}{2}=3$. I certainly do not expect that right off the bat, but I am very comfortable with something along the lines of:
I would hope that students would begin to complete at least a few steps at a time while still showing the computation appropriately, such as:
As students begin to describe their steps/explain their thinking, I would introduce the term “inverse” to move along their mathematical vocabulary.
Finally, students return to their “page” and write their mathematical solution process using as few “lines” as they feel comfortable using, as long as ALL of the steps in the process are shown.
Gallery Walk THIS time involves making sure each pair DID show each operation in the process!
(Ok, it’s not really over yet, but probably for that day. Given a multi-step equation, students will write – and solve – the Mystery Number Puzzle associated with it. Students will create their own
puzzles and challenge others to solve them. THEN, what to do when the puzzle involves a step like: subtract your original number. . .? JUST when they’re feeling all proud and confident you set the
bar a bit higher and off they go again!)
Final note: found a link on (someone’s blog – I think it was a new blogger – yeah, that really narrows it down- I was just so awestruck when I clicked the link that I never went back) post to an
AWESOME WebApp called WebEquation, at least for iOs. (Maybe Android?) You can literally just WRITE a mathematical expression/equation/whatever and it will turn it into LaTeX and “Math Script”
While I totally appreciate Sam’s link to more info on LaTeX syntax, I seriously would just go to WebEquation and handwrite the expression to find the proper way to “type” it. Unfortunately, you can’t
just copy and paste the LaTeX script – it comes out in HTML or something. However, you CAN press on the LaTeX until it opens a new window showing the “Math Type” version and then click to save that
as an image:
(Ok, technically I cheated. It would not actually paste the image of the equation, but I pasted it onto a Pages document and then took a picture to insert here. Not sure it’s worth the effort here.
It does make a nice work around for Pages and Keynote though!)
So. . . depending on the complexity of the expression, I might just skip the LaTeX altogether!
msSunFun: Homework Hassles
Somewhere along the clogged up Google Reader, I missed the topic for this week’s:
It didn’t take more than opening up Flipboard to find out that the dreaded “Homework” is what’s on the menu today.
I have taught Middle School Math for the past four years. Until this year I have always taught sixth along with “something else.” Beginning Wednesday I will have three classes of Algebra (8th
graders) and three classes of Math 8. I must admit that I am not exactly looking forward to the “Homework Hassles.”
I use Standards Based Grading in all of my classes, and one of the tenets that I feel strongly about is that a student’s grade is based purely on their understanding of math concepts and not on
“participation,” “effort,” “behavior,” or “homework.” Therefore, even though I assign homework fairly regularly in my classes, it does not factor into their overall grade. Let me rephrase that: There
are no “points” from homework involved in their grade, but I do feel there is a fairly strong correlation between a student’s homework completion rate and their overall grade. Unfortunately,
(surprisingly enough,) not all middle school students necessarily have that same philosophy.
From my (albeit brief) observation of student behavior, sixth graders are much more “regular” in regards to homework completion. I think that whatever habit they developed in Elementary School tends
to stay with them as the begin their life as a middle schooler. At some point, for some of the students, “grades” start to get in the way. If an assignment for another class counts as a part of their
“grade” then it surely takes precedence over their math homework that “doesn’t really count” for their grade. By the time they are eighth graders, the percentage of the student population who find
“more important things to do with their time” has definitely increased. Instead of completing a set of practice problems that your teacher has assigned in order to help you learn and retain the math
concepts, school has become a series of earning “points” (or not earning them) to reach a desired “grade.”
Enough ranting, here is my “multi-pronged” approach for this year.
Math@Home Logs
For at least the first month of school, all students will be required to fill out a log, tracking the date, the assignment, time spent, whether it was completed, student initial, parent initial, and
teacher initial.
This will be kept in the front “pocket” of their math notebook. I will stamp (or not stamp) daily and collect the log on Fridays. A “bulk” email will be sent home to any students who were negligent
in completing the assignments and/or having their parent/guardian sign off on their log.
If a student has been successful at completing their homework for the first month of school they will be excused from completing the log unless their habits begin to change.
No Homework Notice
As I circulate and check off (stamp) homework, student who did not complete the assignment (didn’t start, didn’t finish, or didn’t show process) will be given the following reminder:
and will then be required to fill out a GoogleDocs form detailing the reason why they didn’t complete the homework along with a plan for completion.
I can print off the spreadsheet whenever necessary. As students do complete the assignment I can delete that row from the sheet.
Morning Math / Learn @ Lunch / Afternoon Academy
These are just my fancy ways of saying before school / during lunch / after school. Each Monday, students with assignments still missing will need to choose 15-60 minutes (depending on how much is
missing) of time in which they will come to class and work on their missing work. (I have a cool grid on my board all ready to go for this. Students will each have a magnet with their name that can
be placed in the location of their “first appointment.” No picture today – maybe later this week when I am at school.)
Making Homework Accessible and Appropriate.
Why have homework? What is my goal for my students when I assign it? These are important questions to consider when making decisions about a lesson each day. One goal I have for this year is to make
sure I don’t assign problems “too early” in the concept development cycle. Just because a topic was introduced in class that day, doesn’t mean that students are really ready to “practice” on their
own. Instead an assignment might focus on previous skills that will be helpful in solidifying the current concepts. This will take careful consideration, since I am not always sure how much progress
will be made in class each day.
For some units I design quite a bit of the homework assignments myself, trying to incorporate some “puzzle/problem” solving activities that require students to come up with strategies that will, in
the long run, help develop their understanding of math concepts. (See the example in this “Tricky Tables” post.)
In addition, I hope to make textbook homework less “rote” and more “reflective” at least SOME of the time. A few posts by David Coffey that I read recently share how to “flip homework” so that
students are really analyzing a set of problems instead of just “cranking out answers.”
I certainly don’t have the “magic bullet.” I forced myself to refrain from reading today’s posts before composing this one, but I am looking forward to finding some ideas that will be useful. :)
Tech Talk: Keynote + Absolute Board = Awesome!
Continuing my series on iPad Apps. . .
When I first received my iPad a year ago last spring, I was DESPERATE to use it in the classroom. My school laptop was getting old and cranky, freezing up at the most inopportune moments. I have used
Power Point in the classroom for quite awhile – not to “tell” students information, but to “ask” them questions. I searched through the App store quite extensively, looking for something cheap to do
the job, but nothing looked too promising, so I bit the bullet, forked out $9.99 + tax and bought. . .
My kids use an Apple laptop at home, so I was familiar with the program. (We actually bought a “five-user pack” of Keynote/Pages/Numbers way back when. . . I cringe when I think of how much we spent.
Who knew what would happen to the App industry?!) The iPad version doesn’t quite have as many features as the Mac version, but an upgrade sometime during the past year brought the two closer
together. I have been able to email my old PowerPoints to myself and open them in Keynote. Occasionally there are a few glitches – borders on a table don’t show up, or a cool font isn’t available –
but for the most part it has worked out well. I have different slide backgrounds that I use for different activities in the class, and have added to my collection with the ones in Keynote. Working on
the iPad is more enjoyable than on the laptop, and moving or modifying slides is a snap. You can “nest” slides under one another, so this summer I combined all of my slides for each day of a unit
together, with a general lesson plan as the “top” slide.
Clicking on the triangle by the slide will “collapse” all of the slides underneath.
Drawback #1
One glaring absence in Keynote is the ability to use superscript (and subscript, but I don’t need that NEARLY as often.) It is really unfathomable to me that this is not available in the font
modifications. HOWEVER, I have found a work around. Remember when I said I emailed PowerPoints and opened them within Keynote. The superscript STAYS so I just have a “fall back” slide that I go to
when I need to use an exponent. I can change the font/color/size and the superscript will change along with it! (Strange, but true!) I use a similar “shortcut” with square roots, as the keyboard
shortcut for the radical symbol is non-existant, but I’ve copied it over from PowerPoint. Since the highest level of math I teach is Algebra, it’s not as if I need a full-fledged Math Editor
(although it would be nice!) I am playing around with Mathbot and TeXit and learning a bit of LaTex so that I can paste in some more complex equations later on this year.
Drawback #2
I always though it would be nice to be able to “draw” on the slides! It is, after all, an iPad App! There is really no “freehand tool.” You can created shapes and lines and curves, but you can’t just
“scribble something out” on a slide. At first I was really bummed about this, but I have come to realize that maybe I wouldn’t really want to, since I would need to use the slide again the following
period. It would be kind of a pain to make multiple copies of each of the slides I wish to write on, so maybe not such a deal-breaker. (Although, it still would be an awesome feature just for
CREATING the slides, but I doubt that Apple is listening.) So. . .from the multitude of options that I have downloaded, played with, and even used for awhile, the winner in the end is the FREE App. .
It really IS free!
The way I use this in conjunction with Keynote on my iPad is that I will take a screen shot of the PowerPoint slides I plan on using so that they are stored in the Camera Roll. When I pull up
Absolute Board I can quickly pick the slide and it will fill the screen. (For awhile I did it ahead of time, but it doesn’t take any longer to grab the slide than it does to select it from the pages
stored in Absolute Board.) I can zoom in and out, change pen size and color, and write down a solution process as a student shares it aloud. (I have found that it is a BIG time saver to have me
record rather than a student “write” on the iPad. There are other opportunities in class for them to write.)
The “marked up” slides are then saved in Absolute Board, and I can pull them up later. I am not sure what the limit is on the number of “pages” you can store. Every once in awhile I will “purge” the
old drawings, but I can also save them to my photo album if I wish!
These two together make a great combination for me!
Made4Math Monday: Sticks and Seats
One of the benefits of working part time for a NUMBER of years was that I could find the time to be a “parent volunteer” while my own kids were in Elementary School. I must say that it was a valuable
experience as a teacher as well. Most of my teaching experience up until that point was at the high school level, so many of the classroom routines were new to me:) One of those ideas was “picking
with Popsicle sticks.”
The question is. . .how do you pick sticks when you have six different classes? Do you have six different cups of sticks? After a few years of “refinement” I have a method that works well for me. I
picked up a package of big, brightly colored “craft sticks” at the dollar store. There are actually six different colors, but I generally only use four. I wrote the numbers 1-8 on the bottom of the
sticks and I am good to go for a class of up to 32 students. Each student is assigned a color and a number (which they remember quite readily after the first few days.) I, on the other hand, have a
“cheat sheet” that I post on the board and “borrow” during class so I can actually call on the student’s name instead of just the “color-number combination.”
Here is my “Wizarding World of Harry Potter” butterbeer cup that I will being using this year. (I am finally retiring my University of Minnesota cup with the lettering all worn off, but I am still
using it to hold my little six inch rulers.)
Here is one of the (currently empty) “cheat sheets” that will be filled with names once our class lists are finalized.
(Ha! I just noticed the Yellow column twice. I must have used my class that only had three colors last year and copied and pasted a new column. It’s supposed to say Green!)
Soooo. . . what if you have more than 32 students in a class? I’m glad that you asked. There are a number of different ways that you can deal with this issue. Since there are more than four colors
available, you can certainly use more colors and fewer sticks of each color if you wish. Last year I had a class of 34 students and I included two blue sticks (in addition to the red, orange, yellow
and green ones) numbered 1-2. If you have far fewer than 32 students (less than 24, for instance) you can eliminate one color all together, and never “pick” that color during that class. I have an
additional “rule” that I follow as well. If I pick a stick that does not “belong” to anyone in the class (not just because they are absent) then I am the one who must answer the question!
(Occasionally I have a class with lots of “unclaimed sticks” and I end up “calling on myself” more often than I wish, so I will put a limit on how many times I can answer in a period.)
I rarely use the sticks for “total cold calls” in class. Quite often students work on a problem, discuss it with a partner, and then I pick a stick to share their response. Other times I will have
students work on five or six problems and as I begin to pick sticks, the first person chosen is allowed to decide WHICH of the five or six problems he or she wishes to share. (If no choice is made, I
will choose!) By the time the last person is chosen, at least we have already discussed the other problems – even if the most challenging one was left for last. However, this is certainly not always
the case. Some students definitely WANT to share their response to the toughest problem! I do not allow students to “pass,” even if they have not completed the problem. I will instead ask questions,
questions, and more questions, to help them reach a solution. I will also call on raised hands after a problem has been shared if students wish to add more information, an alternative process, or an
alternative solution. I generally leave the stick out of the cup for the rest of the period. I am not sure this is a “good thing.” Some students then “relax” knowing they won’t be called again.
Others are disappointed that they won’t get “picked” another time.
I have to chuckle sometimes at the responses I get when I start to pick a stick. (Some times I will have already grabbed it, just waiting for the time to call.) Since they know their color, some are
happily anticipating that it might be them, while others are nervously hoping it WON’T be them! Often the people up front will see the number as the stick is drawn (apparently I’m not very adept at
hiding it) and actually KNOW the particular student before I can even look it up! Occasionally a student professes “disbelief” because they (let’s be honest, it’s usually a “he!”) “called it” that he
would be picked. Oh, sixth graders – I will miss them this year :(
Why all this hassle just to assign a stick to a student? I have ulterior motives. :) The color-number combination is also often used to assign seats! On a daily basis, students enter the room and
need to figure out where they are sitting and who they are sitting with for that particular day. (I alluded to this in my First Day post, but here is the full text version.) Some days the desks will
be in groups of four and they might be sitting with their number groups or “half” of their color groups (evens and odds or highs and lows.)
Other days they will be sitting in two’s where they are generally paired up with someone else in their color group.
Since there are usually six or seven other people in their color group, this partner will also change from day to day. Part of my reason for posting the lists is that they act as a “cheat sheet” for
the students as well.
You may have noticed signs above the class listings that notify students the groups for the day. I have laminated (double-sided so I can just flip for a new option) all of the different seating
choices available. For the “color pairs” there are seven different signs that all have each number paired with a different “partner.” (Now there’s a math problem for you!) I also have laminated signs
for each number group that I set out on the groups of four, and for each color group that I place out to determine the “row” or “section” for that color. These are all stored in a basket right under
the section of the board I use for group assignments.
Just to make things even MORE confusing, I also have a few additional ways of picking the groups for the day. One is “Find Your Match” (pairs or trios) that I described in my post on Math Cards. The
other is “Just for Today” groups (pairs, trios, or quads.) I often use this option after a formative assessment or during review activities where I purposely group students so that at least one
person in the group has a strong understanding of the concepts. I sort their formative assessments and put them in groups, then jot down the names on a blank seating chart in a page protector using
an overhead pen (I guess they still have their uses!)
I assign new “color groups” every quarter. (I used to do it more frequently, but it can be time consuming, and with eight people in group, plus the other options, they get quite a variety throughout
the quarter.) Initially the groups are usually alphabetical. By the second quarter I put some work into assigning groups. I often have the highest performing students with the same number (or two to
three numbers) so that I can choose to use “Number Groups” when I want to differentiate a bit. (Those groups would “receive” a more challenging problem than some of the other groups.) I am also aware
of when I end up with “high-low” pairings so that I either take advantage of built in “tutors” or at least I do not plan an activity in which partners might be “competing” against each other. A
definite part of my lesson planning involves deciding how students will be grouped for the day. Finally, for the last quarter I usually allow some student input regarding who they would like to have
in their group. I take requests of 2-3 people for each student and I can usually place them all in a group with at least ONE of the people they requested, often with two (or another in their number
group.) Again, THIS is a challenge as well!
Back to Sticks
There were blue sticks I used in my class of 34 last year. When in “Number Groups,” they just created a group of five. When in color groups pairs they were my “Wild Cards.” They took the place of
students who were absent for the day or sat together as a pair. (I also randomly drew sticks that they would switch with so each day there were different student’s sitting in the “Wild Card” seats.)
For odds/evens or high/lows, if everyone was present, I would have them join a “convenient group” with the most extra space to form groups of five. I would also probably do this for a class of 25 or
26 instead of having four color groups with “lots of empty seats.”
Last year I came up with a new way to use the sticks during group work. If the students were in Number Groups, I would walk around with one stick of each color in my hand. When stopping to check on a
group or to answer a question, I would place the sticks behind my back, mix them up and pick one to determine who to call on to either ask or answer a question. If students were in Color Groups, a
collection of sticks with different numbers could be used. (Although I also used an octahedron, but this sometimes resulted in MANY rolls if I was at an “even” group and the numbers kept coming up
Whatever For!?
Sticks: I am TERRIBLE at calling on “purely random” students, so the sticks help me to do that on a more regular basis. (We also have class discussions that don’t involve sticks – it depends on the
particular activity.)
Seats: I want every student in the class to be comfortable and familiar working with every other student in the class. We all have our strengths and weaknesses that we bring to a group and I want
student to be aware of that fact, and looking for opportunities to share their strengths while acknowledging and working on their weaknesses. Not every student “enjoys” working with every other
student in the class, but they know it will change the next day. Sometimes we will stay in the same groups for two consecutive days, but really no more than that. Some students swear to me that they
have been with the same partner waaaay too often because they “happen” to end up partners in find your match (over and over. . .), they are IN the same color group so they are in pairs and quads with
that person, AND I even put them together in a “Just for Today” group!! Oh, the injustices of being a middle school student!
Minor Pitfalls
Last year my sixth graders were my first class of the day. The buses arrive by 7:30 and class starts at 8:00, so more than a few of them would “hang out” in class for awhile. Then, I started noticing
some patterns. Within a “column” of eight seats, the pairs can choose their spots on a first come – serve basis. Some students would quickly “claim” spots so that they could be just across the aisle
from their “BFF” – who they were not actually in a group with (on purpose, from my point of view) but wished to sit near them anyways. I began to be quite careful about where each color groups was
assigned, or where each number group was placed to try to avoid the “cliques.” I was not always successful. This year I have 8th grade Algebra students first thing in the morning. I am not sure they
will “hang out” in class before school, but if so, I think I will wait until closer to 7:55 to make the group placements for the day!
Whew! “Leadership Team” meeting this morning, working in my room all afternoon, and I still finished this post at a decent hour – West Coast Time! Still nine more days ’till students arrive – unless
you count out Open House on Wednesday, but I think I’m ready for that :)
Reflections: Lurking to Learning
Today is my one month “blogversary!” I don’t have much to add to the conversation on Advisory for msSunFun, but I do have an item on my To Do page that I would like to tackle: My journey to becoming
a blogger.
I first “lurked” onto the “mathedublogger” scene just over SEVEN years ago! At the time I was teaching at an Alternative High School, and we had just been “chosen” to participate in a 1-1 program
beginning during the 2005-6 school year. We (the teachers) had some training at the end of the school year and then took our laptops home over the summer. Wow! The Internet in your lap! During the
hours/days/weeks spent searching for online tools and resources I ran across some “bloggers” talking about what they do in the classroom – especially in regards to technology. It was still a fairly
new idea from my perspective. Every once in awhile I would wander back and take a look at what was happening. Often one site would link me to another, and so on, and so on. . .
Meanwhile, I moved to one of the middle schools in the district and was no longer involved in the 1-1 program. However, our Computer Lab teacher shared GoogleReader with us, and my lurking seriously
went up a notch or two. My tiny iPod touch became my window into the “mathedubloggosphere.” I don’t have the data to back it up, but I really do believe that the MathEd blog scene has grown
exponentially – meaning that the growth was actually quite gradual to begin with, but then started to take off! I remember running across dy/dan (what a cool blog name, I thought) before he WAS dy/
dan! I “lurked” as some of my favorites, Continuous Everywhere but Differentiable Nowhere, f(t), and MathMamaWrites built their followings. ThinkThankThunk next slammed onto the scene along with his
SBG cohort Point of Inflection. Even though most of the bloggers were teaching at the high school level (or higher) and I was now working with sixth graders, I could relate to a lot of what they were
saying – especially in regards to SBG. I thought, hmmmmm, maybe I should do this. However, I am the LAST one in a large group of people that I don’t really know to actually speak up. . .And I wonder
why my own kids are so shy. . .
The next “jump” in my lurking occurred when I got my iPad and found Flipboard, that I shared in this post. It makes scrolling through blogposts pure pleasure! I would look at the blog rolls of the
bloggers I was reading, check out new writers, subscribe to their posts, and so on, and so on. . .
After returning from our vacation this summer, I read some posts about Twitter Math Camp. Seriously? These people just planned their own “retreat/workshop/conference!?” Wow! More new bloggers were
added to my feed. . .and then one afternoon I was weeding out in the yard, listening to my iPod when I heard the lyrics, “This is your life, are you who you want to be. . .” and something just
clicked. Yeah, I can do this! You know, it makes so much sense to do it NOW, when I am just going back to teaching full time after being part time for 16 years. Sure, I have PLENTY of time n my
hands! Oh well, I downloaded the WordPress app (recommended by Sam) and signed up for Twitter to boot! After all, I had already picked out a name and everything. The rest, as they say, is history. .
.but not really what I expected.
It has been a roller coaster ride over the past month. I started out with ideas about what I wanted to “say,” but I find myself “saying” a lot of other things too. msSunFun came onto the scene
shortly after I first started. “Ok, I’ll try that.” Then I noticed the Made4Math Monday. “Suuuure, that too.” Sam posted his New Blogger Initiative. “Hey, that’s me! I’ll sign up.” All of a sudden I
have more ideas to share than there are hours in the day. How to keep up? Oh yeah, and school is starting up soon, too! My Start, Stop, Continue post includes blogging goals that I hope I can keep. I
even made a To Do’s page to keep myself honest.
Posting comments on other blogs was one of my first “baby steps” after I started my blog. I really appreciated it when the blogger would then reply back. Especially now that I added all of the NBI
bloggers to my feed, I find myself overwhelmed with how much there is to read. I want to make “meaningful comments,” as opposed to “I really love that idea.” Maybe that’s more of a “Twitter” response
to a post.
Twitter has been hard to “jump into.” I once tweeted that I felt like the “new kid at school,” just listening in on other conversations – except they may have happened hours ago! It is very strange
sometimes. I will “reply” and then realize that the person may have already moved waaaay beyond that part of the conversation. I don’t know how people can even BEGIN to follow as many people as they
When I was lurking it was so much more of a passive experience. Now that I am “in it,” I have been learning sooooo much more from others. I am reading posts and tweets from Middle School Math
teachers who are out there in force, (@jruelbach, @4mulafun, @fawnpnguyen, @mr_stadel, @Borschtwithanna, @mathbratt, the list goes on, and on, and on) and I didn’t really know about them before. I
LOVE the posts and twitter conversations with/between those involved in Math Education (@delta_dc, @mathhombre, @ChrisHunter36, @trianglemancsd) that make me think more deeply about learning
mathematics! I feel a kinship with other “newbies” like me (@danbowdoin – although he is on the fast track to blogger stardom, @G8rAli – who should really start a blog, @aekland – who has such
thoughtful posts, @ray_emily – who has an abundance of enthusiasm, and Pai Intersect – who I haven’t seen on Twitter, but has great insights on his blog.)
I vacillate between thinking that “nothing I have to say has any value when compared to all of the ideas that others have shared” and “oh, I really want to chime in,” or “I think I should share that.
. .”
I am surprised at how much I have learned about myself and my teaching from writing posts for the blog. @ray_emily tweeted earlier today: “I’m finding I have a new clarity / fresh eyes on a topic
after blogging about it.” I entirely agree! I am especially looking forward to learning even more as I blog about my experiences in the classroom :)
New Blogger Initiative: Integer Context Cards
For Week 2 of the “New Math Blogger Initiative” (or is it an “initiation?” hmmmmm) I decided to learn how to embed a document using Scribd and show something that I am proud to share!
If you read my Made4Math post from last week (Math Cards) you know I like “multi-purpose” tools. I haven’t been teaching Middle School too long, in the grand scheme of things, and when I first had
the opportunity to introduce Integers to a class of very low sixth graders (all Level 1 on the state test) I knew that putting them into context would make all the difference. So. . . what are some
contexts for integers? Well, there’s temperature, and altitude, and money. . .? I brainstormed long and hard and came up with quite a list.
Without further ado, my very first Scribd document!
Integer Context Cards
(Hmmmm. Rats! The font changed when I uploaded the document. The original was in Herculanum, a very cool looking ALL CAPS font, so now it appears that I don’t know my capitalization rules – oh well.)
There are a total of eighteen different contexts with six cards for each one. Some are “stretching it” a bit, but still reasonable. Note: for altitude, one of the cards is for Arlington, Washington
where my school is located, so you might want to “personalize” that one. :)
I made one copy of each page on a different color of copy paper to make them easier to sort and then had them laminated, cut, and paperclipped in sets of six. (I only made one set for the entire
class, but you can certainly create duplicates, especially if you have a very large class.)
Order, Order, Order
Phase 1: After a brief exposure to a few of the cards, we started our first activity with the cards. My students were already assigned to groups of six. We actually went out into a space in the hall
for this.) Each group received a different set of cards, passed them out amongst the group members, and silently “raced” to put themselves in order from least to greatest. Once a group was done, they
ALL had to raise their hands. After I checked for accuracy, they turned in their set and grabbed a new one. (The first year I did this I checked the groups off on a master sheet, but the following
years I just trusted their memory – “We already did that set.”) The goal was to accurately get through as many sets as they could in the time allotted. Often enough, groups were in too much of a
hurry to read carefully enough to identify the “key words” that signified whether their value was positive or negative. Getting a “no” when their hands were raised definitely encouraged them to take
their time a bit more.
Phase 2: The next activity took it just a bit further. Groups (of three this time) had a sheet of number lines. Each time they received a new set of cards they had to “fairly accurately” plot and
label all six values (along with zero if it wasn’t in the set) on a number line. Choosing a scale was challenging for some of the sets (especially with the first group of students.)
(See the cool font? Oh, well.)
Integer Operations in Context
A few weeks later in the year, my seventh grade class was working on operations with integers. I printed the Integer Card sheets four to a page and created little packets for students to share. (See
the image above.) Using a “Think Pair Share” type of model, I posed questions using the people, places, or things on the cards and students had to write a math sentence, model the problem on a number
line, and find the value (first on their own paper, then with a partner on a mini-whiteboard.) A big key was writing the math sentence as opposed to just finding the value. I wanted them to make the
connection so that when they saw a “naked numbers” problem they could try to connect it to the contexts we worked with in class.
We initially focused on addition and subtraction situations:
(**The white text is shown first. After the Think-Pair-Share on whiteboards I reveal the number sentence and diagram and move onto the next problem.**)
Next we moved onto subtraction in “finding differences” contexts, as well as multiplication and division. I dropped the “number line” requirement, although we did end up sharing it on some of the
more challenging multiplication/division situations. (Again, the white text for each problem is given first, TPS, then the yellow answer is revealed, discussed, and we move on to the next problem.)
In subsequent years I used the Integer Operation activities with my sixth grade classes as we introduced this seventh grade concept in the Spring after the state test. During “Review Activities” time
in class I had small groups create original “stories” along with their associated number sentences, from the cards, but I think I would like to make that a more integral part of the initial learning
as well.
Ahhhhh, context! Even if it is really “pseudo context,” in this situation students have something to grab onto that helps them make sense of the integer operations as opposed to a mercurial “set of
rules to follow.” On the other hand, it is THROUGH these experiences that students begin to create their OWN rules about the patterns involved in integer operations, but only when the concepts “make
sense” to THEM! I think we are finding a bit of EMU right there. :)
Note: There are multiple slides for each operation, and if I got my act together I might copy and paste them all into one slide show and try to attach them, but that would take a higher level of
understanding on Scribd than I currently possess, as the slides are all now on my iPad, and I am just sharing screenshots.
Made4Math Monday: Monster Whiteboards
Yesterday I finished getting my “monster” whiteboards ready and I brought them into my classroom today. I am far from the first to have created these learning tools. Frank Nochese sang their praises
here, and Anna followed up with another post as well. I use the little mini-whiteboards quite regularly with pairs, and they will still have a place in my class, but the opportunity for groups of
three or four is really exciting! I especially envision using them for problem solving, such as the chessboard problems for my First Day Activities. I am concerned that we will not always have enough
time to complete the “solving” as well as the “sharing” in one class period, but my solution will be to at least take photos of all of the boards and project them on the screen the following day as
groups present. Another intriguing use will be the Mistake Game, as described by Kelly O’Shea. (Go there! Read it!) Sooooooo looking forward to trying this out – especially with my Algebra students
because I think they will thrive on it. However, in the long run I predict it will be incredibly valuable to my Math 8 students in freeing them from fear of failure. The classes on the whole are not
full of students who have been successful in the past, and developing a classroom culture where mistakes are acceptable and even celebrated as ways to learn concepts more deeply (EMU!) will be a huge
step for their learning.
I bought mine at the local Home Depot for about $13.50 per sheet. I picked up two that they cut (for free!) into the six 24″ x 32″ pieces recommended by Frank. (I considered going 2′ x 2′, but I am
very glad I went with the extra inches on the length.)
My next task was to put duct tape (or duck tape) on the edges to keep them from degrading. This is where Anna gets all the credit! I had planned on “buying” some of my daughter’s stash
until I “did the math!” Each board requires almost ten feet of tape (112 inches of perimeter, plus a bit extra on either end that I cut off while taping.) Since I had twelve boards, the 120 feet of
tape would have decimated her supply. (As it was she wasn’t going to give me the “fancy” stuff anyways.) I decided to go with my school colors and picked up a blue roll and a yellow roll of 20 yds
each. Needless to say, there’s not much left. (Maybe just enough to use for periodic “repairs.”) The tape cost $7 for the two rolls, for a grand total of $34 + tax, or about $3 a board. (The cuts
were free, and didn’t take long at all, but the taping process took me over an hour!)
Here they are!
Notice as well, the PERFECT storage space that is holding the other ten!
Now, for the writing instruments. I am always frustrated by how quickly the dry erase markers die. Last year I picked up some of the Crayola Dry Erase Crayons. Other than the periodic breakage, they
seem to “last” much longer. (Hey, when they break, one crayon turns into two!) In addition, I have to watch out for pieces that accidentally “chip” off, are left on the floor, and then get ground
into the carpet.😞 The crayons look pretty sharp (sharp = awesome, not sharp = pointed) on these boards, and the color variety is great, with the exception of the yellow crayon – whoever thought it
would work well on a whiteboard was sadly mistaken. I guess they sell yellow markers as well – maybe it’s for those black dry erase surfaces. I “inherited” a number of boxes of the crayons (eight
colors in a box plus an eraser “mitt” and sharpener) when I moved into this classroom – enough that each group could have their own box! (I will generally only use eight boards at one time.) The kids
liked markers better, but maybe the different colors will “sell it.”
One more note about the duct tape. In hindsight I don’t think yellow was a great choice. As I was erasing one of the boards today I realized the dry erase particles will accumulate along the edges of
the tape, so lighter colors will start to look “grungy” after awhile. Oh, well.
One “short” story about my whiteboard history. In the early nineties (yes. . .I am dating myself, but I have already done that on a previous post) I was teaching Calculus. I had one particular
student who repeatedly asked if he could go up and work out problems on the corner of the board while the class was working on a set of problems. It was certainly fine with me. He commented on how
his thoughts seemed to “flow” from his brain when he used a whiteboard. It was not long before he bought himself a 1′ x 2′ framed board to use at home, as well as another one that he brought to
school (and stored in the classroom) to use at his desk. I have no idea where he found these boards, as I had never seen them in stores. At the end of the year he gave me the one he had brought to
school and I still have it to this day. (It was one of the last years that I taught Calculus. Not many years later, I went on maternity leave and then came back to teaching just part time. My own
kids scribbled and doodled on it while they were toddlers.) Jump forward about ten years, when I was volunteering in my son’s first grade class, I observed his teacher using a class set of
mini-whiteboards with her students. Within about a year, I had a set for my classes. Now I am on to the next phase – MONSTER whiteboards!
(Ok, it wasn’t so short. Have I mentioned that I ramble . . .?) | {"url":"http://findingemu.wordpress.com/","timestamp":"2014-04-20T05:43:39Z","content_type":null,"content_length":"110552","record_id":"<urn:uuid:ea3577d2-c5f5-41d7-90a8-4ec8aa3b57fe>","cc-path":"CC-MAIN-2014-15/segments/1397609538022.19/warc/CC-MAIN-20140416005218-00260-ip-10-147-4-33.ec2.internal.warc.gz"} |
Quantum Groups, Quantum Categories and Quantum Field Theory
Results 1 - 10 of 50
- Jour. Math. Phys , 1995
"... For a copy with the hand-drawn figures please email ..."
- Annals of Mathematics
"... Abstract. In this paper we extend categorically the notion of a finite nilpotent group to fusion categories. To this end, we first analyze the trivial component of the universal grading of a
fusion category C, and then introduce the upper central series ofC. For fusion categories with commutative Gr ..."
Cited by 76 (17 self)
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Abstract. In this paper we extend categorically the notion of a finite nilpotent group to fusion categories. To this end, we first analyze the trivial component of the universal grading of a fusion
category C, and then introduce the upper central series ofC. For fusion categories with commutative Grothendieck rings (e.g., braided fusion categories) we also introduce the lower central series. We
study arithmetic and structural properties of nilpotent fusion categories, and apply our theory to modular categories and to semisimple Hopf algebras. In particular, we show that in the modular case
the two central series are centralizers of each other in the sense of M. Müger. Dedicated to Leonid Vainerman on the occasion of his 60-th birthday 1. introduction The theory of fusion categories
arises in many areas of mathematics such as representation theory, quantum groups, operator algebras and topology. The representation categories of semisimple (quasi-) Hopf algebras are important
examples of fusion categories. Fusion categories have been studied extensively in the literature,
, 1996
"... It has been discussed earlier that ( weak quasi-) quantum groups allow for conventional interpretation as internal symmetries in local quantum theory. From general arguments and explicit
examples their consistency with (braid-) statistics and locality was established. This work addresses to the reco ..."
Cited by 49 (8 self)
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It has been discussed earlier that ( weak quasi-) quantum groups allow for conventional interpretation as internal symmetries in local quantum theory. From general arguments and explicit examples
their consistency with (braid-) statistics and locality was established. This work addresses to the reconstruction of quantum symmetries and algebras of field operators. For every algebra A of
observables satisfying certain standard assumptions, an appropriate quantum symmetry is found. Field operators are obtained which act on a positive definite Hilbert space of states and transform
covariantly under the quantum symmetry. As a substitute for Bose/Fermi (anti-) commutation relations, these fields are demonstrated to obey local braid relation. Contents 1 Introduction 1 2 The
Notion of Quantum Symmetry 5 3 Algebraic Methods for Field Construction 9 3.1 Observables and superselection sectors in local quantum field theory . . . . 10 3.2 Localized endomorphisms and fusion
structure . . . . . ....
, 1999
"... The geometry of D-branes can be probed by open string scattering. If the background carries a non-vanishing B-field, the world-volume becomes noncommutative. Here we explore the quantization of
world-volume geometries in a curved background with non-zero Neveu-Schwarz 3-form field strength H = dB. U ..."
Cited by 44 (6 self)
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The geometry of D-branes can be probed by open string scattering. If the background carries a non-vanishing B-field, the world-volume becomes noncommutative. Here we explore the quantization of
world-volume geometries in a curved background with non-zero Neveu-Schwarz 3-form field strength H = dB. Using exact and generally applicable methods from boundary conformal field theory, we study
the example of open strings in the SU(2) Wess-Zumino-Witten model, and establish a relation with fuzzy spheres or certain (non-associative) deformations thereof. These findings could be of direct
relevance for D-branes in the presence of Neveu-Schwarz 5-branes; more importantly, they provide insight into a completely new class of world-volume geometries.
"... A 2-Hilbert space is a category with structures and properties analogous to those of a Hilbert space. More precisely, we define a 2-Hilbert space to be an abelian category enriched over Hilb
with a ∗-structure, conjugate-linear on the hom-sets, satisfying 〈fg,h 〉 = 〈g,f ∗ h 〉 = 〈f,hg ∗ 〉. We also ..."
Cited by 43 (13 self)
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A 2-Hilbert space is a category with structures and properties analogous to those of a Hilbert space. More precisely, we define a 2-Hilbert space to be an abelian category enriched over Hilb with a
∗-structure, conjugate-linear on the hom-sets, satisfying 〈fg,h 〉 = 〈g,f ∗ h 〉 = 〈f,hg ∗ 〉. We also define monoidal, braided monoidal, and symmetric monoidal versions of 2-Hilbert spaces, which
we call 2-H*-algebras, braided 2-H*-algebras, and symmetric 2-H*-algebras, and we describe the relation between these and tangles in 2, 3, and 4 dimensions, respectively. We prove a generalized
Doplicher-Roberts theorem stating that every symmetric 2-H*-algebra is equivalent to the category Rep(G) of continuous unitary finite-dimensional representations of some compact supergroupoid G. The
equivalence is given by a categorified version of the Gelfand transform; we also construct a categorified version of the Fourier transform when G is a compact abelian group. Finally, we characterize
Rep(G) by its universal properties when G is a compact classical group. For example, Rep(U(n)) is the free connected symmetric 2-H*-algebra on one even object of dimension n. 1
- Rev. Math. Phys , 1999
"... A two-sided coaction δ: M → G ⊗M⊗G of a Hopf algebra (G, ∆, ǫ, S) on an associative algebra M is an algebra map of the form δ = (λ ⊗ idM) ◦ ρ = (idM ⊗ ρ) ◦ λ, where (λ, ρ) is a commuting pair of
left and right G-coactions on M, respectively. Denoting the associated commuting right and left actions ..."
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A two-sided coaction δ: M → G ⊗M⊗G of a Hopf algebra (G, ∆, ǫ, S) on an associative algebra M is an algebra map of the form δ = (λ ⊗ idM) ◦ ρ = (idM ⊗ ρ) ◦ λ, where (λ, ρ) is a commuting pair of left
and right G-coactions on M, respectively. Denoting the associated commuting right and left actions of the dual Hopf algebra ˆ G on M by ⊳ and ⊲, respectively, we define the diagonal crossed product M
⊲ ⊳ ˆ G to be the algebra generated by M and ˆ G with relations given by ϕm = (ϕ (1) ⊲m ⊳ ˆ S −1 (ϕ (3)))ϕ (2), m ∈ M, ϕ ∈ ˆ G. We give a natural generalization of this construction to the case where
G is a quasi–Hopf algebra in the sense of Drinfeld and, more generally, also in the sense of Mack and Schomerus (i.e., where the coproduct ∆ is non-unital). In these cases our diagonal crossed
product will still be an associative algebra structure on M ⊗ ˆ G extending M ≡ M ⊗ ˆ1, even though the analogue of an ordinary crossed product M ⋊ ˆ G in general is not well defined as an
associative algebra. Applications of our formalism include the field algebra constructions with quasi-quantum
- Commun. Contemp. Math
"... Abstract. We classify all unitary modular tensor categories (UMTCs) of rank ≤ 4. There are a total of 35 UMTCs of rank ≤ 4 up to ribbon tensor equivalence. Since the distinction between the
modular S-matrix S and −S has both topological and physical significance, so in our convention there are a tot ..."
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Abstract. We classify all unitary modular tensor categories (UMTCs) of rank ≤ 4. There are a total of 35 UMTCs of rank ≤ 4 up to ribbon tensor equivalence. Since the distinction between the modular
S-matrix S and −S has both topological and physical significance, so in our convention there are a total of 70 UMTCs of rank ≤ 4. In particular, there are two trivial UMTCs with S = (±1). Each such
UMTC can be obtained from 10 non-trivial prime UMTCs by direct product, and some symmetry operations. Explicit data of the 10 non-trivial prime UMTCs are given in Section 5. Relevance of UMTCs to
topological quantum computation and various conjectures are given in Section 6. 1.
- In
"... Abstract. After a brief review of recent rigorous results concerning the representation theory of rational chiral conformal field theories (RC-QFTs) we focus on pairs (A, F) of conformal field
theories, where F has a finite group G of global symmetries and A is the fixpoint theory. The comparison of ..."
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Abstract. After a brief review of recent rigorous results concerning the representation theory of rational chiral conformal field theories (RC-QFTs) we focus on pairs (A, F) of conformal field
theories, where F has a finite group G of global symmetries and A is the fixpoint theory. The comparison of the representation categories of A and F is strongly intertwined with various issues
related to braided tensor categories. We
- In preparation
"... We show that the author’s notion of Galois extensions of braided tensor categories [22], see also [3], gives rise to braided crossed G-categories, recently introduced for the purposes of
3-manifold topology [31]. The Galois extensions C ⋊ S are studied in detail, and we determine for which g ∈ G non ..."
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We show that the author’s notion of Galois extensions of braided tensor categories [22], see also [3], gives rise to braided crossed G-categories, recently introduced for the purposes of 3-manifold
topology [31]. The Galois extensions C ⋊ S are studied in detail, and we determine for which g ∈ G non-trivial objects of grade g exist in C ⋊ S. 1
- J. reine angew. Math
"... Abstract. We give a full classification of all braided semisimple tensor categories whose Grothendieck semiring is the one of Rep ` O(∞) ´ (formally), Rep ` O(N) ´ , Rep ` Sp(N) ´ or of one of
its associated fusion categories. If the braiding is not symmetric, they are completely determined by th ..."
Cited by 9 (0 self)
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Abstract. We give a full classification of all braided semisimple tensor categories whose Grothendieck semiring is the one of Rep ` O(∞) ´ (formally), Rep ` O(N) ´ , Rep ` Sp(N) ´ or of one of its
associated fusion categories. If the braiding is not symmetric, they are completely determined by the eigenvalues of a certain braiding morphism, and we determine precisely which values can occur in
the various cases. If the category allows a symmetric braiding, it is essentially determined by the dimension of the object corresponding to the vector representation. 1. | {"url":"http://citeseerx.ist.psu.edu/showciting?cid=1654348","timestamp":"2014-04-18T12:26:43Z","content_type":null,"content_length":"36708","record_id":"<urn:uuid:981cc65b-cc16-4019-b702-a8a4348d35e8>","cc-path":"CC-MAIN-2014-15/segments/1397609533308.11/warc/CC-MAIN-20140416005213-00012-ip-10-147-4-33.ec2.internal.warc.gz"} |
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If you Log in you could create a "neck problems" node. If you don't already have an account, you can register here. | {"url":"http://everything2.com/title/neck+problems","timestamp":"2014-04-19T13:23:06Z","content_type":null,"content_length":"29640","record_id":"<urn:uuid:6c43f3c6-b8a3-40e1-bdc7-c3c3874e50ca>","cc-path":"CC-MAIN-2014-15/segments/1397609537186.46/warc/CC-MAIN-20140416005217-00163-ip-10-147-4-33.ec2.internal.warc.gz"} |
What is the coalescence factor?
I am reading an article on experimental nuclear physics. The article is about deuteron and triton production in Pb + Pb collisions. In the article they mention the coalescence factor which is given
[itex]B_{A}=A\frac{2s_{A}+1}{2^{A}}R^{N}_{np}\left(\frac{h^{3}}{m_{p}\gamma V}\right)^{A-1}[/itex]
The coalescence factor has something to do with the formation of light clusters A(Z,N). So A is the mass number of the cluster, the R_np thing is the ratio of neutrons to protons participating in the
collision, gamma is just the lorentz factor related to the velocity of the cluster and V is the volume of of particles at freeze-out (after hadronization, when there are no more strong interactions
between the nucleons). I don't know what s_A is though. I have worked out the units for A = 2 to be:
[itex]s_{A}\cdot \frac{ev^{2}s}{m}+\frac{ev^{2}s}{m}[/itex]
But i don't know the units of s_A. I was hoping for this to turn out unit-less, since it could then be interpreted as a sort of probability of a cluster A(Z,N) to form. Now i'm not sure what it is.
I apologize if this post was supposed to go in homework. Technically it is a sort of homework question, since i am supposed to present the article as an exam tomorrow. But i figured that there was a
bigger chance that someone could help me with this on the HEP board. | {"url":"http://www.physicsforums.com/showthread.php?p=4150536","timestamp":"2014-04-17T03:50:12Z","content_type":null,"content_length":"31111","record_id":"<urn:uuid:d8aa2fd1-b39d-4a70-b205-5095bc83b2d4>","cc-path":"CC-MAIN-2014-15/segments/1398223202774.3/warc/CC-MAIN-20140423032002-00504-ip-10-147-4-33.ec2.internal.warc.gz"} |
from The American Heritage® Dictionary of the English Language, 4th Edition
• n. The general form or a quantity indicative of the general form of a statistical frequency curve near the mean of the distribution.
from Wiktionary, Creative Commons Attribution/Share-Alike License
• n. A measure of "peakedness" of a probability distribution, defined as the fourth cumulant divided by the square of the variance of the probability distribution.
Greek kurtōsis, bulging, curvature, from kurtos, convex; see sker-^2 in Indo-European roots.
(American Heritage® Dictionary of the English Language, Fourth Edition)
From Ancient Greek κύρτωσις ("bulging, convexity"), from κυρτός (kurtos, "bulging"). (Wiktionary)
Log in or sign up to get involved in the conversation. It's quick and easy. | {"url":"https://wordnik.com/words/kurtosis","timestamp":"2014-04-21T08:21:22Z","content_type":null,"content_length":"34435","record_id":"<urn:uuid:fa562fd3-1204-464f-94c3-8319ad579fc2>","cc-path":"CC-MAIN-2014-15/segments/1397609539665.16/warc/CC-MAIN-20140416005219-00142-ip-10-147-4-33.ec2.internal.warc.gz"} |
Circular Logic not worth a Millikelvin
Guest post by Mike Jonas
A few days ago, on Judith Curry’s excellent ClimateEtc blog, Vaughan Pratt wrote a post “Multidecadal climate to within a millikelvin” which provided the content and underlying spreadsheet
calculations for a poster presentation at the AGU Fall Conference. I will refer to the work as “VPmK”.
VPmK was a stunningly unconvincing exercise in circular logic – a remarkably unscientific attempt to (presumably) provide support for the IPCC model[s] of climate – and should be retracted.
The background to VPmK was outlined as “Global warming of some kind is clearly visible in HadCRUT3 [] for the three decades 1970-2000. However the three decades 1910-1940 show a similar rate of
global warming. This can’t all be due to CO2 []“.
The aim of VPmK was to support the hypothesis that “multidecadal climate has only two significant components: the sawtooth, whatever its origins, and warming that can be accounted for 99.98% by the
AHH law []“,
· the sawtooth is a collection of “all the so-called multidecadal ocean oscillations into one phenomenon“, and
· AHH law [Arrhenius-Hofmann-Hansen] is the logarithmic formula for CO2 radiative forcing with an oceanic heat sink delay.
The end result of VPmK was shown in the following graph
Fig.1 – VPmK end result.
· MUL is multidecadal climate (ie, global temperature),
· SAW is the sawtooth,
· AGW is the AHH law, and
· MRES is the residue MUL-SAW-AGW.
As you can see, and as stated in VPmK’s title, the residue was just a few millikelvins over the whole of the period. The smoothness of the residue, but not its absolute value, was entirely due to
three box filters being used to remove all of the “22-year and 11-year solar cycles and all faster phenomena“.
If the aim of VPmK is to provide support for the IPCC model of climate, naturally it would remove all of those things that the IPCC model cannot handle. Regardless, the astonishing level of claimed
accuracy shows that the result is almost certainly worthless – it is, after all, about climate.
The process
What VPmK does is to take AGW as a given from the IPCC model – complete with the so-called “positive feedbacks” which for the purpose of VPmK are assumed to bear a simple linear relationship to the
underlying formula for CO2 itself.
VPmK then takes the difference (the “sawtooth”) between MUL and AGW, and fits four sinewaves to it (there is provision in the spreadsheet for five, but only four were needed). Thanks to the box
filters, a good fit was obtained.
Given that four parameters can fit an elephant (great link!), absolutely nothing has been achieved and it would be entirely reasonable to dismiss VPmK as completely worthless at this point. But, to
be fair, we’ll look at the sawtooth (“The sinewaves”, below) and see if it could have a genuine climate meaning.
Note that in VPmK there is no attempt to find a climate meaning. The sawtooth which began life as “so-called multidecadal ocean oscillations” later becomes “whatever its origins“.
The sinewaves
The two main “sawtooth” sinewaves, SAW2 and SAW3, are:
Fig.2 – VPmK principal sawtooths.
(The y-axis is temperature). The other two sinewaves, SAW4 and SAW5 are much smaller, and just “mopping up” what divergence remains.
It is surely completely impossible to support the notion that the “multidecadal ocean oscillations” are reasonably represented to within a few millikelvins by these perfect sinewaves (even after the
filtering). This is what the PDO and AMO really look like:
Fig.3 – PDO.
(link) There is apparently no PDO data before 1950, but some information here.
Fig.4 – AMO.
Both the PDO and AMO trended upwards from the 1970s until well into the 1990s. Neither sawtooth is even close. The sum of the sawtooths (SAW in Fig.1) flattens out over this period when it should
mostly rise quite strongly. This shows that the sawtooths have been carefully manipulated to “reserve” the 1970-2000 temperature increase for AGW.
Fig.5 – How the sawtooth “reserved” the1980s and 90s warming for AGW.
VPmK aimed to show that “multidecadal climate has only two significant components”, AGW and something shaped like a sawtooth. But VPmK then simply assumed that AGW was a component, called the
remainder the sawtooth, and had no clue as to what the sawtooth was but used some arbitrary sinewaves to represent it. VPmK then claimed to have shown that the climate was indeed made up of just
these two components.
That is circular logic and appallingly unscientific. The poster presentation should be formally retracted.
[Blog commenter JCH claims that VPmK is described by AGU as "peer-reviewed". If that is the case then retraction is important. VPmK should not be permitted to remain in any "peer-reviewed"
1. Although VPmK was of so little value, nevertheless I would like to congratulate Vaughan Pratt for having the courage to provide all of the data and all of the calculations in a way that made it
relatively easy to check them. If only this approach had been taken by other climate scientists from the start, virtually all of the heated and divisive climate debate could have been avoided.
2. I first approached Judith Curry, and asked her to give my analysis of Vaughan Pratt’s (“VP”) circular logic equal prominence to the original by accepting it as a ‘guest post’. She replied that it
was sufficient for me to present it as a comment.
My feeling is that posts have much greater weight than comments, and that using only a comment would effectively let VP get away with a piece of absolute rubbish. Bear in mind that VPmK has been
presented at the AGU Fall Conference, so it is already way ahead in public exposure anyway.
That is why this post now appears on WUWT instead of on ClimateEtc. (I have upgraded it a bit from the version sent to Judith Curry, but the essential argument is the same). There are many commenters
on ClimateEtc who have been appalled by VPmK’s obvious errors. I do not claim that my effort here is in any way better than theirs, but my feeling is that someone has to get greater visibility for
the errors and request retraction, and no-one else has yet done so.
118 Responses to Circular Logic not worth a Millikelvin
1. Assume AGW is a flat line and repeat the analysis. When the fit remains near perfect, trumpet the good news that AGW is no more!
2. Another failed attempt to force dynamic, poorly understood, under-sampled, natural process into some kind of linear deterministic logic system. It simply will not work. This foolishness is not
worth any further time or effort on the part of serious scientists.
3. What VP has said repeatedly on JC’s site is basically that if you can provide a better fit, please do.
Nobody that I’ve seen has yet done so. Now I’m not in any way suggesting that what VP has done is in any way useful science. But still, can you not simply alter the 4 or 5 sines waves to show
that you can provide just as good a fit without the AHH curve?
If you can, please present it here.
And if you cannot, then VP remains uncontested.
4. Why is it that the rationalizations of the Warmistas are beginning to remind me of Ptolemy and The Almagest?
5. I worked in the Banking Industry for most of my adult life. During that time, many people would be applying for finance for this business or that business – maybe a mortgage, maybe a loan.
All would arrive with their shiny spreadsheet proving their business model was viable and would soon show profitability
I never saw any proposed business plan to the Bank that didn’t show remarkable profit – certainly none ever predicted a loss.
Nonetheless, the vast majority of those business plans would fail abysmally.
Just goes to show, any spreadsheet can be made to produce whatever results the author wants – just tweak here or tweak there.
Now about milli-kelvins ?
6. ‘VPmK was a stunningly unconvincing exercise in circular logic – a remarkably unscientific attempt to (presumably) provide support for the IPCC model[s] of climate – and should be retracted.”
1. its not circular.
2. its not a proof or support for models.
3. you cant retract a poster.
4. This is basically the same approach that many here praise when scafetta does it.
basically he is showing that GIVEN the truth of AGW, the temperature series can be explained by a few parameters. GIVEN, is the key you misunderstand the logic of his approach.
7. “[Blog commenter JCH claims that VPmK is described by AGU as "peer-reviewed". If that is the case then retraction is important. VPmK should not be permitted to remain in any "peer-reviewed"
Describing VPmk as peer-reviewed is incorrect. Abstracts published by AGU for either poster sessions or presentations made at the meeting are not peer-reviewed. There are quite a few comments at
the blog after JCH on this topic.
8. VPmK was a stunningly unconvincing exercise in circular logic – a remarkably unscientific attempt to (presumably) provide support for the IPCC model[s] of climate – and should be retracted.
That is over-wrought. Vaughan Pratt described exactly what he did and found, and published the data that he used and his result. If the temperature evolution of the Earth over the next 20 years
matches his model, then people will be motivated to find whatever physical process generates the sawtooth. If not, his model will be disconfirmed along with plenty of other models. Lots of
model-building in scientific history has been circular over the short-term: in “The Structure of Scientific Revolutions” Thomas Kuhn mentions Ohm’s law as an example, and Einstein’s special
relativity; lots of people have noted the tautology of F = dm/dt where m here stands for momentum.
Pratt merely showed that, with the data in hand, it is possible to recover the signal of the CO2 effect with a relatively low-dimensional filter. No doubt, the procedure is post hoc. The validity
of the approach will be tested by data not used in fitting the functions that he found.
9. Steven Mosher wrote: 4. This is basically the same approach that many here praise when scafetta does it.
I agree with that.
10. Steven Mosher:
You innumerate four points in your post at December 13, 2012 at 9:04 am. I address each of them in turn.
1. its not circular.
(Clearly, it is “circular” in that it removes everything from the climate data except what the climate models emulate then says the result of the removal agrees with what the climate emulate when
tuned to emulate it.)
2. its not a proof or support for models.
(Agreed, it is nonsense.)
3. you cant retract a poster.
(Of course you can! All you do is publish a statement saying it should not have been published, and you publish that statement in one or more of the places where the “poster” was published; e.g.
in this case, on Judith Curry’s blog.)
4. This is basically the same approach that many here praise when scafetta does it.
(So what! Many others – including me – object when Scafetta does it. Of itself that indicates nothing.)
The poster by Vaughan Pratt only indicates that Pratt is a prat: live with it.
11. OOOps! I wrote
(Clearly, it is “circular” in that it removes everything from the climate data except what the climate models emulate then says the result of the removal agrees with what the climate emulate when
tuned to emulate it.)
Obviously I intended to write
(Clearly, it is “circular” in that it removes everything from the climate data except what the climate models emulate then says the result of the removal agrees with what the models emulate when
tuned to emulate it.)
12. I agree. I commented about it on Curry’s blog and called it worthless. I was particularly annoyed that he used HadCRUT3 which is error-ridden and anthropogenically distorted. I could see that he
was using his computer skills to create something out of nothing and did not understand why that sawtooth did not go away. That millikelvin claim is of course nonsense and was simply part of his
applying his computer skills without comprehending the data he was working with. Suggested that he write a program to find and correct those anthropogenic spikes in HadCRUT and others.
13. On the sidelines of the V.Pratt’s blog presentation there was a secondary discussion between myself and Dr. Svalgaard about the far more realistic causes of the climate change. Since Dr.S. often
does peer review on the articles relating to solar matters, leaving the trivia out, I consider our exchanges as an ‘unofficial peer review of my calculations’, no mechanism is considered in the
article, just the calculations. This certainly was not ‘friendly’ review, although result may not be conclusive, I consider it a great encouragement.
If there are any scientists who are occasionally involved in the ‘peer review’ type processes, I would welcome the opportunityto submit my calculations. My email as in my blog id followed by
14. Vaughan presented an interesting idea which has been roundly tested by many commenters in a spirit of science and hotly contested views within a framework of courtesy by Vaughan defending his
ideas. Personally I’m not convinced that CET demonstrates his theory , in fact I think it shows he is wrong, but if every post, whether here or at climate etc, was discussed in such a thorough
manner everyone would gain, whatever side of the fence they are on.
15. vukcevic says:
December 13, 2012 at 9:33 am
This certainly was not ‘friendly’ review, although result may not be conclusive, I consider it a great encouragement
It seems that In the twisted world of pseudo-science even a flat-out rejection is considered a great encouragement.
16. Steveta_uk
Re Pratt’s “if you can provide a better fit, please do.”
Science progresses by “kicking the tires”. Models are only as robust and the challenges put to them and their ability to provide better predictions when compared against hard data – not politics.
The “proof the pudding is in the eating”. Currently the following two models show better predictive performance than IPCC’s models that average 0.2C/decade warming:
Relationship of Multidecadal Global Temperatures to Multidecadal Oceanic Oscillations Joseph D’Aleo and Don Easterbrook, Evidence-Based Climate Science. Elsevier 2011, DOI: 10.1016/
Nicola Scafetta, Testing an astronomically based decadal-scale empirical harmonic climate model versus the IPCC (2007) general circulation climate models Journal of Atmospheric and
Solar-Terrestrial Physics 80 (2012) 124–137
For others views on CO2, see Fred H. Haynie The Future of Global Climate Change
No amount of experimentation can ever prove me right; a single experiment can prove me wrong. Albert Einstein
17. “22-year and 11-year solar cycles and all faster phenomena“.
What happened to longer term cycles?
Oh, they must be AGW./sarc
18. Curve-fitting is not proof of anything, especially when the input data is filtered. That heat-sink delay also needs some scrutiny. Worse, the data time range is fairly short, in geological terms.
On top of that, a four component wave function? Get real.
It’s wiggle-matching, with some post hoc logic thrown in. I’m underwhelmed.
19. Mosh: 4. This is basically the same approach that many here praise when scafetta does it.
Sorry what N. Scaffeta does is fit all parameters freely and see what results. What Pratt did was fit his exaggerated 3K per doubling model ; see what’s left , then make up a totally unfounded
waveform to eliminate it. Having thus eliminated it, he screwed up his maths and managed to also eliminate the huge discrepancy that all 3K sensitivity model have after 1998.
Had he got the maths correct it would have been circular logic. As presented to AGU it was AND STILL IS a shambles.
Attribution to AMO PDO is fanciful. The whole work is total fiction intended to remove the early 20th c. warming that has always been a show-stopper for CO2 driven AGW.
At the current state of the discussion on Curry’s site, he has been asked to state whether he recognises there is an error or stands by the presentation as given to AGU and published on Climate
At the time of this posting , no news from Emeritus Prof Vaughan Pratt.
20. The IPCC says that the current total forcing (all sources – RCP 6.0 scenario) is supposed to be about 2.27 W/m2 in 2012.
On top of that, we should be getting some water vapour and reduced low cloud cover feedbacks from this direct forcing so that there should be a total of about 5.0 W/m2 right now.
The amount of warming, however, (the amount that is accumulating in the Land, Atmosphere, Oceans and Ice) is only about 0.5 W/m2.
Simple enough to me.
Climate Science is much like the study of Unicorns and their invisibility cloaks.
21. I want to draw people’s attention to the frequency content of VPmK SAW2 and SAW3 wave forms. Just by eye-ball, these appear to be 75 year and 50 year frequencies. As Mike Jonas points out that
early in the paper VP posits they come from major natural ocean oscillations but later a more flexible “whatever its origins.”
I am not going to debate the origins of the low frequency. Take only from VPmK the temperature record contains significant very low frequency wave forms, wavelengths greater than 25 years, needed
to match even heavily filtered temperatures records where
three box filters being used to remove all of the “22-year and 11-year solar cycles and all faster phenomena“.
All that is left in VPmK data is very low frequency content and there appears to be a lot of it.
My comment below takes the importance of low frequency in VPmK and focuses on BEST: Berkley Earth and what to me appears to be minimally discussed wholesale decimation and counterfeiting of low
frequency information happening within the BEST process. If you look at what is going on in the BEST process from the Fourier domain, there seems to me to be major losses of critical information
content. I first wrote my theoretical objection to the BEST scalpel back in April 2, 2011 in “Expect the BEST, plan for the worst.” I expounded at Climate Audit, Nov. 1, 2011 and some other
My summary argument remains unchanged after 20 months:
1. The Natural climate and Global Warming (GW) signals are extremely low frequency, less than a cycle per decade.
2. A fundamental theorem of Fourier analysis is frequency resolution dw/2π Hz = 1/(N*dt) .where dt is the sample time and N*dt is the total length of the digitized signal.
3. The GW climate signal, therefore, is found in the very lowest frequencies, low multiples of dw, which can only come from the longest time series.
4. Any scalpel technique destroys the lowest frequencies in the original data.
5. Suture techniques recreate long term digital signals from the short splices.
6. Sutured signals have in them very low frequency data, low frequencies which could NOT exist in the splices. Therefore the low frequencies, the most important stuff for the climate analysis,
must be derived totally from the suture and the surgeon wielding it. From where comes the low-frequency original data to control the results of the analysis ?
Have I misunderstood the BEST process? Consider this from Muller (WSJ Eur 10/20/2011)
Many of the records were short in duration, … statisticians developed a new analytical approach that let us incorporate fragments of records. By using data from virtually all the available
stations, we avoided data-selection bias. Rather than try to correct for the discontinuities in the records, we simply sliced the records where the data cut off, thereby creating two records
from one.
“Simply sliced the data.” “Avoided data-selection bias” – and by the theorems of Fourier embraced high frequency selection bias and created a bias against low frequencies. There is no free lunch
here. Look at what is happening in the Fourier Domain. You are throwing a way signal and keeping the noise. How can you possibly be improving signal/noise ratio?
Somehow BEST takes all these fragments lacking low frequency, and “glues” them back together to present a graph of temperatures from 1750 to 2010. That graph has low frequency data – but from
where did it come? The low frequencies must be counterfeit – contamination from the gluing process, manufacturing what appears to be a low frequency signal from fitting high frequency from
slices. This seems so fundamentally wrong I’d sooner believe a violation of the 1st Law of Thermodynamics.
A beautiful example of frequency content that I expect to be found in century scale un-sliced temperature records is found in Lui-2011 Fig. 2. reprinted in WUWT In China there are no hocky sticks
Dec. 7, 2011 The grey area on the left of the Fig. 2 chart is the area of low frequency, the climate signal. In the Lui study, a lot of the power is in that grey area. It is this portion of the
spectrum that BEST’s scalpel removes! Fig. 4 of Lui-2011 is a great illustration of what happens to a signal as you add first the lowest frequency and successively add higher frequencies.
Power vs Phase & Frequency is the dual formulation of Amplitude vs Time. There is a one to one correspondence. If you apply a filter to eliminate low frequencies in the Fourier Domain, and a
scalpel does that, where does it ever come back? If there is a process in the Berkley glue that preserves low frequency from the original data, what is it? And were is the peer-review discussion
of its validity?
If there is no preservation of the low frequencies the scalpel removes, results from BEST might predict the weather, but not explain climate.
22. The real shame here lies in The Academy. As the author did supply all details behind the work when asked, I must assume it was done in good faith. The problem is that PhD’s are being awarded
without the proper training/education in statistical wisdom. Anybody can build a model and run the numbers with a knowledge of mathematical nuts and bolts and get statistical “validation.” But a
key element that supports the foundation upon which any statistical work stands seems to be increasingly ignored. That element has a large qualitative side to it which makes it more subtle thus
less visible. Of course I am speaking of the knowing, understanding, and verifying of all the ASSUMPTIONS (an exercise with a large component of verbal logic) demanded by any particular
statistical work to be trustworthy. I had this drilled into me during my many statistical classes at Georgia Tech 30 years ago. Why this aspect seems to be increasingly ignored I can’t say, but I
can say, taking assumptions into account can be a large hurdle to any legitimate study, thus very inconvenient. I imagine publish or perish environments and increasing politicization may have
much to do here. The resultant fallout and real crime is the population of scientists we are cultivating are becoming less and less able to discriminate between the different types of variation
that need to be identified so that GOOD and not BAD decisions are more likely. Until the science community begins to take the rigors of statistics seriously, its output must be considered
seriously flawed. To do otherwise risks the great scientific enterprise that has achieved so much.
23. lsvalgaard says:
December 13, 2012 at 10:07 am
Currently I am only concerned with the calculations, no particular mechanism is considered in my article, just volumes of data, AMO, CET, N.Hemisphere, Arctic, atmospheric pressure, solar
activity, the Earth’s magnetic variability, comparisons against other known proxies and reconstructions.
Since you couldn’t fail my calculations, you insisted on steering discussion away from the subject (as shown in this condensed version) with all trivia from both sides excluded:
Let’s remember:
Dr. L. Svalgaard :If the correlation is really good, one can live with an as yet undiscovered mechanism.
I do indeed do consider it a great encouragement that you didn’t fail calculations for
One step at the time. Thanks for the effort., its appreciated. Soon I’ll email Excel data on the
using 350 years of geological records instead of geomagnetic changes .Two reinforce each other.
We still don’t exactly understand how gravity works, but maths is 350 years old.
I missed your usually ‘razor sharp dissection’ of Dr. Pratt’s hypothesis
24. I don’t understand the objections to simplifying models until the correct outcome is achieved. After all if the sun really had anything to do with temperature it would get colder at night and
warmer during the day.
25. vukcevic says:
December 13, 2012 at 11:11 am
Since you couldn’t fail my calculations
Of course, one cannot fail made-up ‘data’. What is wrong with your approach is to compute a new time series from two unrelated time series, and to call that ‘observed data’.
26. Stephen Rasey:
re your post at December 13, 2012 at 11:00 am.
Every now and then one comes across a pearl shining on the sand of WUWT comments. The pearls come in many forms.
Your post is a pearl. Its argument is clear, elegant and cogent. Thankyou.
27. Should it not be milliKelvin?
28. Steveta_uk says:
What VP has said repeatedly on JC’s site is basically that if you can provide a better fit, please do..
A “better fit” is not useful. In cases like this the correct model will not provide a better fit, because the correct model has deviations between theory and reality due to noise..
If I have a 100% normally distributed population and take 100 samples then the result will never be an exact normal distribution. I could model a “distribution” that better matched my samples,
but it would most certainly not tell me anything useful. In fact it would lead me to believe my actual population was not normal.
This is why I am highly suspicious of any model that is trained on old data. It is basically an exercise in wiggle matching, not an exercise in getting the underlying physics correct. The best
climate models will have pretty poor fit to old temperatures.
29. Substitute a 1300 year wave length sine with a max at MWP and min at LIA for the AGW function and you will get similar results.
30. lsvalgaard says:
December 13, 2012 at 11:21 am
vukcevic says:
December 13, 2012 at 11:11 am
Since you couldn’t fail my calculations
Of course, one cannot fail made-up ‘data’. What is wrong with your approach is to compute a new time series from two unrelated time series, and to call that ‘observed data’.
Wrong Doc.
Magnometer at the Tromso does it every single minute of the day and night.
In red are Incoming variable solar magnetic field sitting on the top of the variable Earth’s magnetic field.
Rudolf Wolf started it with a compass needle, Gauss did it with a bit more sophisticated apparatus, and today numerous geomagnetic stations do it as you listed dozens in your paper on IDV.
So it is OK for Svalgaard of Stanford to derive IDV from changes of two combined magnetic fields, but is not for Vukcevic.
Reason plain and obvious, it would show that the SUN DOES IT !
Here is how geomagnetic field (Eart + solar) is measured and illustrated by our own Dr. Svalgaard
and he maintains they are not added together in his apparatus.
Can anyone spot 3 magnets?
Dr. S are you really serious to suggest that no changes in the Earth field are registered by your apparatus?
Case closed!
31. Steveta_uk says:
December 13, 2012 at 8:55 am
What VP has said repeatedly on JC’s site is basically that if you can provide a better fit, please do.
Nobody that I’ve seen has yet done so. Now I’m not in any way suggesting that what VP has done is in any way useful science. But still, can you not simply alter the 4 or 5 sines waves to show
that you can provide just as good a fit without the AHH curve?
If you can, please present it here.
And if you cannot, then VP remains uncontested.
Let me get this correct.
I publish a paper showing that the rise and fall of women’s skirts plus a saw tooth pattern provides a good fit to the curve. Since no one can provide a better ‘fit’ than that the paper has to
32. vukcevic says:
December 13, 2012 at 12:22 pm
So it is OK for Svalgaard of Stanford to derive IDV from changes of two combined magnetic fields, but is not for Vukcevic.
It is OK to derive two time series of the external field driven by the same source, but not to confuse and mix the external and internal fields that have different sources and don’t interact.
Case is indeed closed, as you are incapable of learning.
33. As mentioned by Steveta_uk and others ….
Rather than engage in histrionics, the way to refute the Vaughan Pratt poster is to create a similar spreadsheet (or modify his spreadsheet) to show a nominal AGW signal.
As nearly as I can tell, Dr. Pratt has done everything responsible skeptics ask:
- Formulated a hypothesis
- Presented all of the supporting data
- Published in an accessible forum
- Asked for feedback.
I have not dropped in on the thread for a couple of days. However, I suspect that if someone has published a spreadsheet model that refutes Dr. Pratt’s, then it would have been mentioned here.
I do not care whether four parameters can fit an elephant. I would like to see someone mathematically refute Dr. Pratt’s model. I took a look and realized I do not have the time to reacquire the
expertise to do it. (I had the expertise years ago and even have the optimization code I wrote for my AI class that could be adapted to this problem).
In my case, I strongly believe there are contradictory cases (it is just math and there are a lot of variables), but until someone devotes the mental sweat to create one (maybe Nick Scafetta has
per Steven Mosher @ 9:04 am), Dr. Pratt’s result stands as he has described it. He asks people to show the contradictions.
Finally re circularity. I agree that that post hoc curve fitting can be described as circular. All of the GCMs do it to reproduce historical temperature. What Dr. Pratt has done is simplify the
curve fitting to a spreadsheet we can all use.
34. I am with Steveta, on this one. Unless someone comes up with better numerology, Professor Pratt’s numerology stands.
35. Steveta_uk says: “What VP has said repeatedly on JC’s site is basically that if you can provide a better fit, please do.”
Why would anyone want spend time searching for a “better fit” of an exaggerated exponential, bend down my a broken filter, plus a non physically attributable wiggle to ANYTHING?
Please explain the motivation and rewards of such an exercise.
36. RobertInAz: I have not dropped in on the thread for a couple of days. …. Dr. Pratt’s result stands as he has described it. He asks people to show the contradictions.
Then you ought to do so before commenting , no?
He asks for criticisms but it’s fake openness. He clearly has no intent of admitting even the most blatant errors in his pseudo-paper-poster.
Oops is not in the vocabulary of this great scientific authority.
37. We have a few people with logic problems here.
Steven Mosher:
1. “its not circular“. Of course it’s circular. Circular Logic is when you claim to have proved something that you assumed in the first place.
2. “its not a proof or support for models“. It certainly looks like an attempt to support the models. But VP’s motives are private, which is why I used “(presumably)“.
3. “you cant retract a poster“. As Richard Courtney has pointed out, of course you can retract a poster.
4 “This is basically the same approach that many here praise when scafetta does it“.. What people think about Scafetta or anyone else is irrelevant to the content of VP’s poster. However, if it
illogically helps you to accept what I say now, here is what I once commented on a Loehle and Scafetta paper: “I’m calling BS on this paper.“. http://tinyurl.com/cxxo4lw
You say “if you cannot [provide a better fit], then VP remains uncontested“. Utter tosh. I have just contested VP’s poster. Therefore VP’s poster has indeed been contested. You do not have to put
up alternatives in order to contest something.
Matthew R Marler”
You say “That is over-wrought. Vaughan Pratt described exactly what he did and found, and published the data that he used and his result.“. Yes it’s true that VP published his data, methods and
result. I congratulated him on publishing all the data and workings, and I truly meant it. If only all climate scientists did that then surely climate science couldn’t have got into its current
mess. But the result was still obtained by circular logic.
38. jack hudson says:
December 13, 2012 at 11:07 am
On statistics –
I have notice that since computers and statistical packages became readily available in the 1980′s there has been a shift away from using a trained statistician to do-it-yourself statistics. ‘Six
Sigma’ in industry is an example.
The statistical training I got from the ‘Six Sigma’ program at work was absolute crap. All they taught was how to use the computer program with not even a basic explanation of different types of
distribution to go with it or even the warning to PLOT THE DATA so you could see the shape of the distribution. They did not even get into attributes vs variables!
It reminds me of the shift from the use of well trained secretaries who would clean up a technonut’s English and pry the need infor out of him to having everyone write their own reports. My plant
manager in desperation insisted EVERYONE in the plant take night courses in English composition.
Too bad Universities do not insist that anyone using statistics must take at least three semesters of Stat.
39. lsvalgaard says:
December 13, 2012 at 12:43 pm
It is OK to derive two time series of the external field driven by the same source, but not to confuse and mix the external and internal fields that have different sources and don’t interact.
As this apparatus does:
records combined solar and Earth’s fields
or as Vukcevic does in here:
calculates combined solar and Earth’s fields
Do you suggest than the combined field curve that happens to match temperature change in the N. Hemisphere is coincidental, it just appeared by chance ?
40. fhhaynie says:
December 13, 2012 at 12:17 pm
Substitute a 1300 year wave length sine with a max at MWP and min at LIA for the AGW function and you will get similar results.
That is pretty much what the Chinese, Liu Y, Cai Q F, Song H M, et al. did. They used 1324 years.
GRAPH: http://jonova.s3.amazonaws.com/graphs/china/liu-2011-cycles-climate-tibet-web.gif
Figure 4 Decomposition of the main cycles of the 2485-year temperature series on the Tibetan Plateau and periodic function simulation. Top: Gray line,original series; red line, 1324 a cycle;
green line, 199 a cycle; blue line, 110 a cycle. Bottom: Three sine functions for different timescales. 1324 a, red dashed line (y = 0.848 sin(0.005 t + 0.23)); 199 a, green line (y = 1.40
sin(0.032 t – 0.369)); 110 a, blue line (y = 1.875 sin(0.057 t + 2.846)); time t is the year from 484 BC to 2000 AD.
41. @Gail. anyone using statistics must take at least three semesters of Stat.
I agree. Three semesters of statistics would be more generally useful throughout adult life than three semesters of Calc. I’m 34 years out of my B.Sc, 31 from my Ph.D. The college texts I return
to most often are my Stat books, Johnson & Leone 1977.
Not that calc isn’t useful. Not that it isn’t required for “Diff_E_Q”. But statistics is the one of the only courses that by design gives you training in uncertainty, to quantify what you don’t
42. vukcevic says:
December 13, 2012 at 1:22 pm
records combined solar and Earth’s fields
It records the external and internal fields superposed by Nature and thus existing in Nature
or as Vukcevic does in here:
calculates combined solar and Earth’s fields
no, you calculate a field that does not exist in Nature by combining two that are not physically related
Do you suggest than the combined field curve that happens to match temperature change in the N. Hemisphere is coincidental, it just appeared by chance ?
I say that the quantity you calculate does not exist in Nature and therefore that any correlation is spurious or worse. But I thought your case was closed. Keep it that way, instead of carpet
bombing every thread on every blog with it.
43. Can’t you just take the fourier transform of the raw data to find the frequency components and phase shifts and whatever else is left over?
I do applaud the guy’s work with sines and exponentials. At least it isn’t the standard linear regression garbage!!!
44. Vaughan Pratt has commented on ClimateEtc that “The game therefore is to come up with two curves we can call NAT and HUM for nature and humans, such that HadCRUT3 = NAT + HUM + shorter-term
variations, where NAT oscillates without trending too far in a few centuries, and some physically justifiable relation between HUM and the CDIAC CO2 data can be demonstrated.
To within MRES I’ve proposed SAW for NAT, AGW for HUM, [...]“.
In order to show that VP’s version of NAT and HUM are unsupported, it is sufficient to show, as I have done, that they are unsupported. It is not necessary to propose a different version.
However, others have looked at NAT. eg, http://wattsupwiththat.com/2010/09/30/amopdo-temperature-variation-one-graph-says-it-all/
I haven’t investigated their workings, so I am not in a position to say whether their graph is worth anything, but at least it is using real data on the PDO and AMO. If they have got it right
(NB. that’s an “If”), then they have nailed NAT, and HUM looks to be around a flat zero.
45. lsvalgaard says:
December 13, 2012 at 1:41 pm
I calculate combined effect of two variables, as you could calculate combined effect of wind and temperature on the evaporation, but in my case it happens that both variables are magnetic fields.
Are you happy now?
I post on other blogs so the readers also should be aware You don’t need to follow me around, if you think it is not worth your attention. Why are you so concerned ?
In a way I am pleasantly surprised that you are devoting all your attention to my ‘nonsense’ rather than the ‘brilliant’ work of your Stanford colleague, discussed above. Either you think Dr.
Pratt’s work is of a superb quality or utter rubbish, in either case no comment of yours is required.
Good night.
46. vukcevic says:
December 13, 2012 at 3:01 pm
I calculate combined effect of two variables, as you could calculate combined effect of wind and temperature on the evaporation, but in my case it happens that both variables are magnetic fields.
Are you happy now?
Wind and temperature and evaporation are physically related. Your inputs are not. that they are both magnetic fields is irrelevant, it makes as much sense to combine them as it would the fields
of the Sun and Sirius.
I post on other blogs so the readers also should be aware
I think the readers are ill served with nonsense.
You don’t need to follow me around, if you think it is not worth your attention. Why are you so concerned ?
Because scientists have an obligation to combat pseudo-science and provide the public with correct scientific information. Even though not all do that.
In a way I am pleasantly surprised that you are devoting all your attention to my ‘nonsense’ rather than the ‘brilliant’ work of your Stanford colleague, discussed above.
You should be ashamed of peddling your nonsense, not pleased when found out.
Either you think Dr. Pratt’s work is of a superb quality or utter rubbish
Curve fitting is what it is. If one believes in it has little bearing on the mathematical validity of the fitting procedure. I asked him to make an experiment for me and the result was that what
he called the ‘solar curve’ was different in solar data and in CET and HadCRUT3 temperature data and between the latter two as well. This settled the matter for me at least.
47. Mike Jonas: But the result was still obtained by circular logic.
In filtering, there is a symmetry: if you know the signal, you can find a filter that will reveal it clearly; if you know the noise, you can design a filter to reveal the signal clearly. Pratt
assumed a functional form for the signal (he said so at ClimateEtc), and worked until he had a filter that revealed it clearly.
The thought process becomes “circular” if you “complete the circle”, so to speak, and conclude that: since he found what he assumed, then it must be true. My only claim is that, given what he
did, the result can be, and should be, tested on future data. I have written about the same regarding the modeling of Vukcevic and Scafetta. I would say the same regarding the curve-fitting of
Liu et al cited by Gail Combs above. Elsewhere I have written the same of the modeling of Latif and Tsonis, and of the GCMs. I do not expect any extant model to survive the next 20 years’ worth
of data collection, but I think that the data collected to date do not clearly rule out very much — though alarmist predictions made in 1988-1990 look less credible year by year.
48. Mike Jonas: However, others have looked at NAT. eg, http://wattsupwiththat.com/2010/09/30/amopdo-temperature-variation-one-graph-says-it-all/
I haven’t investigated their workings, so I am not in a position to say whether their graph is worth anything, but at least it is using real data on the PDO and AMO. If they have got it right
(NB. that’s an “If”), then they have nailed NAT, and HUM looks to be around a flat zero.
Well said.
49. RobertInAz – “As nearly as I can tell, Dr. Pratt has done everything responsible skeptics ask:
- Formulated a hypothesis” – Yes, he did that.
“- Presented all of the supporting data” – Yes, he did that, and I congratulated him for it.
“- Published in an accessible forum” – Yes, he did that.
“- Asked for feedback.” – Yes, he did that.
And just in case you haven’t noticed, I have given him feedback. That feedback stated in no uncertain terms, and with full explanation, using the same data as Dr. Pratt, that he had used circular
logic and that his findings were worthless.
Matthew R Marler – “The thought process becomes “circular” if you “complete the circle”, so to speak, and conclude that: since he found what he assumed, then it must be true. My only claim is
that, given what he did, the result can be, and should be, tested on future data.“.
VP’s claimed results flowed from his initial assumptions. That’s what makes it circular.
And just in case you didn’t notice, I tested the key part of his result (the sawtooth) against existing data (the PDO and AMO) and found that it did not represent the “multidecadal ocean
oscillations” as claimed.
So (a) the logic was circular, and (b) it was tested anyway and found wanting.
50. I submit that a simpler and better fit of the unfiltered data is 0.573-0.973sine(x/608+.96)+0.108sine(x/63+1.21)+0.038sine(x/20+1.46) where x=2PI*year. AGW may be covarient with that 608 year
cycle and contributes a little bit to the magnitude of the coefficient -.973. Most of the residual looks like a three to five year cycle.
51. Marler & Jonas, Vuk, et al. 8<)
The thought process becomes “circular” if you “complete the circle”, so to speak, and conclude that: since he found what he assumed, then it must be true. My only claim is that, given what he
did, the result can be, and should be, tested on future data. I have written about the same regarding the modeling of Vukcevic and Scafetta. I would say the same regarding the curve-fitting of
Liu et al cited by Gail Combs above. Elsewhere I have written the same of the modeling of Latif and Tsonis, and of the GCMs. I do not expect any extant model to survive the next 20 years’ worth
of data collection, but I think that the data collected to date do not clearly rule out very much — though alarmist predictions made in 1988-1990 look less credible year by year.
OK, so try this:
We have a (one) actual real world temperature record. It has a “lot” of noise in it, but it is the only one that has actual data insideits noise and high-frequency (short-range or month-to-month)
variation and its longer-frequency (60 ?? year and (maybe) 800-1000-1200 year variations. Behind the noise of recent changes – somewhere “under” the temperatures between 1945 and 2012 – there
“might be” a CAGW HUM signal that “might be” related to CO2 levels: and there “might be” a non-CO2 signal related to UHI effect starting about 1860 for large cities that tails off to a steady
value , and starts off between 1930 and 1970 for smaller cities and what are now rural areas. Both UHI “signals” would begin at near 0, ramp up as population increases in the area between 5000
and 25,000, then slows as the area saturates with new buildings and people after 25,000 people.
The satellite data must be assumed correct for the whole earth top to bottom.
The satellite data varies randomly month-to-month by 0.20 degrees. So it appears you must first put an “error band” of +/-0.20 degrees around your thermometer record BEFORE looking for any trends
to analyze. Any data within +/- .20 degrees of any running average can be proven with today’s instruments for the entire earth to be just noise.
Then, you try to eliminate the 1000 year longer term cycle – if you see it at all.
Then, after all the “gray band” is eliminated, then you can start eliminating the short cycle term. Or looking at the problem differently, start looking for the potential short cycle.
52. So we take a mathematical process we know can fit any curve (I seem to recall Fourrier first showed that a set of sinusoidal oscillations at various frequencies and amplitudes would fit any curve
to any accuracy given enough different sine waves; in other words any curve can be expressed as a spectrum) and gasp when it only takes 4 curves to fit to a few mK data that are (a) known to
oscillate, so will fit closely with relatively few curves (b) have the non-oscilliatory component arbitrarily removed (see (a)) and (c) don’t actually vary very much, so a few milli Kelvin is
proportionally not as precise a fit as it sounds. Of course we don’t mention that none of the data points can possibly be defined to the level that expressing in milli Kelvin has any meaning.
53. RACookPE1978 – What you say seems to generally make sense. One problem is that every time you enter a new factor (UHI, etc, and there are lots of them) into the equation, you also have to widen
your error band.
What seems reasonable to me is that the ocean oscillations are visible in the last century or so of surface temperature record, and obviously contributed significantly to 20thC warming. UHI is
clearly present too – see Watts 2012, though there is probably more of it in the USA than elsewhere. Other factors such as land clearing, sun + GCRs, volcanoes, etc, etc, need to be added to the
mix. The amount “left over” for CO2-driven AGW ends up being a small amount with large error bands.
The failure of the tropical troposphere to warm faster than the surface does seem to indicate that the IPCC estimate of AGW is way too high.
54. Mike Jonas: VP’s claimed results flowed from his initial assumptions. That’s what makes it circular.
When results follow from assumptions that’s logic or mathematics. It only becomes circular if you then use the result to justify the assumption. You probably recall that Newton showed that
Kepler’s laws followed from Newton’s assumptions. Where the “circle” was broken was in the use of Newton’s laws to derive much more than was known at the time of their creation. In like fashion,
Einstein’s special theory of relativity derived the already known Lorentz-Fitzgerald contractions; that was a really nice modeling result, but the real tests came later.
I tested the key part of his result (the sawtooth) against existing data (the PDO and AMO) and found that it did not represent the “multidecadal ocean oscillations” as claimed.
Pratt claimed that he did not know the exact mechanism generating the sawtooth. You showed that one possible mechanism does not fit.
I think we agree that Pratt’s modeling does not show the hypothesized CO2 effect to be true. At ClimateEtc I wrote that Pratt had “rescued” the hypothesis, not tested it. That’s all.
Most of what you have written is basically true but “over-wrought”. There is no reason to issue a retraction.
55. Just because something can be explained by sine waves proves nothing. In VPmK the anthropogenic component could be replaced by another sine wave of long periodicity to represent the rising AGW
That said, I am of the opinion that much of the change in global temperatures can indeed be explained by AGW and the AMO (plus TSI (sunspots) and Aerosols (volcanoes)). See for example my
discussion of the paper by Zhou and Tung and my own calculations at:
The main conclusion of this work is that AOGCMs have overestimated the AGW component of the temperature increase by a factor of 2 (welcomed by sceptics but not by true believers in AGW) but there
is still a significant AGW component (welcomed by true believers but not by sceptics).
56. Steven Mosher says:
December 13, 2012 at 9:04 am
4. This is basically the same approach that many here praise when scafetta does it.
As usual, Mosher continues in his attempts to mislead people about the real merits of my research. The contorted Mosher’s reasoning is also discussed here:
I do not hope that my explanation will convince him because he clearly does not want to understand.
However, for the general reader this is how the case is.
My research methodology does not have anything to do with the curve fitting exercise implemented in Pratt’s poster.
My logic follows the typical process used in science, which is as follows.
1) preliminary analysis of the patterns of the data without manipulation based on given hypotheses.
2) identification of specific patterns: I found specific frequency peaks in the temperature record .
3) search of possible natural harmonic generators that produce similar harmonics; I found them in major astronomical oscillations.
4) search of whether the astronomical oscillations hindcast data outside the data interval studied in (1): I initially studied the temperature record since 1850, and tested the astronomical model
against the temperature and solar reconstructions during the Holocene!
5) use an high resolution model to hindcast the signal (1): I calibrate the model from 1850 to 1950 and check its performance in hindcasting the oscillations in the period 1950-2012 and
6) the tested harmonic component of the model is used as a first forecast of the future temperature.
7) wait the future to see what happens: for example follow the (at-the moment-very-good) forecasting performance of my model here
There is nothing wrong with the above logic. It is the way science is actually done, although Mosher does not know it.
My modelling methodology equivalent to the way the ocean tidal empirical models (which are the only geophysical models that really work) have been developed.
Pratt’s approach is quite different from mine, it is the opposite.
He does not try to give any physical interpretation to the harmonics but interprets the upward trending at surely due to anthropogenic forcing despite the well known large uncertainty in the
climate sensitivity to radiative forcing. I did the opposite, I interpret the harmonics first and state that the upward trending could have multiple causes that also include the possibility of
secular/millennial natural variability that the decadal/multidecadal oscillations could not predict.
Pratt did not tested his model for hindcasting capabilities, and he cannot do it because he does not have a physical interpretation for the harmonics. I did hindcast tests, because harmonics can
be used for hindcast tests.
Pratt’s model fails to interpret the post 2000 temperature, as all AGW models, which implies that his model is wrong.
My model correctly hindcast the post 2000 temperature: see again
In conclusion, Mosher does not understand science, but I cannot do anything for him because he does not want to understand it.
However, many readers in WUWT may find my explanation useful.
57. Nicola Scafetta says:
December 13, 2012 at 9:32 pm
7) wait the future to see what happens: for example follow the (at-the moment-very-good) forecasting performance of my model here
It fails around 2010 and you need a 0.1 degree AGW to make it fit. I would say that there doesn’t look to be any unique predictability in your model. A constant temperature the past ~20 years
fits even better.
58. anybody know the formula for the sawtooth (which multidecadal series and what factors)as there is certainly no resemblance to the amo… by far the most dominant oscillation. The sawtooth presented
looks well planned to me, must have taken a lot of work to construct to get the desired residuals…
59. lsvalgaard says:
December 13, 2012 at 9:47 pm
It fails around 2010 and you need a 0.1 degree AGW to make it fit.
A model cannot fail to predict what it is not supposed to predict. The model is not supposed to predict the fast ENSO oscillations within the time scale of a few years such as the ElNino peak in
2010. That model uses only the decadal and multidecadal oscillations.
60. Nicola Scafetta says:
December 13, 2012 at 10:07 pm
That model uses only the decadal and multidecadal oscillations.
Does that model predict serious cooling the next 20-50 years?
61. lsvalgaard says:
December 13, 2012 at 10:29 pm
Nicola Scafetta says:
December 13, 2012 at 10:07 pm
That model uses only the decadal and multidecadal oscillations.
Does that model predict serious cooling the next 20-50 years?
Yes it does
see graph 2.
What has that to do with Scafetta? Not much directly, but put in simple terms :
Either the solar and the Earth’s internal variability, which we can only judge by observing the surface effects or changes in the magnetic fields as an internal proxy:
- sun affects the Earth, since the other way around appears to be negligible.
- or caused by common factor, possibly planetary configurations
Surface effects correlation:
Internal variability (as derived from magnetic fields as a proxy) correlation:
As the CET link shows, the best and yhe longest temperature record is not immune to the such sun-Earth link, and neither are relatively reliable shorter recent records from the N. Hemisphere
You and Matthew R Marler call it meaningless curve fitting, but as long as data from which the curves are derived it is foolish to dismiss as nonsense.I admit that I can’t explain above in the
satisfactory terms, if you whish to totally reject it you got your reasons, and you were welcome to say that in the past as you are now and future.
62. The model used here is fine for interpolation, ie to calculate the temperature at time T-t where T is the present and t is positive. So it would be useful to replace the historic temperature
record by a formula. If we need to know temperatures at time T+t then this is an extrapolation which is valid only if the components of the formula represent all the elements of physical reality
that detemine the evolution of the climate. But this is precisely what has not been shown!
There was a transit of Venus earlier this year and we are told thar the next one will be in 2117.
We can be confident in this prediction because we know that the time evolution of the planets is given accurately by the laws of Newton/Einstein.
Climate science contains no equivalent body of knowledge
63. Matthew R Marler – Oh the perils of re-editing a comment in a tearing hurry. You correctly point out that what I said “VP’s claimed results flowed from his initial assumptions. That’s what makes
it circular” was wrong. The correct statement is “VP’s claimed results are his initial assumptions. That’s what makes it circular.“.
What we have is this: He assumed that climate consisted of IPCC-AGW and something else. His finding was that the climate consisted of IPCC-AGW and something else.
Now, if we had learned something of value about the ‘something else’, then there could have been merit in his argument. But we didn’t. The ‘something else’s only characteristic was that it could
be represented by a combination of arbitrary sinewaves and three box filters. The ‘something else’ began its short life as “all the so-called multidecadal ocean oscillations“, but that didn’t
last long because it clearly could not be even remotely matched to the actual ocean oscillations. The ‘something else’ ended its short life as a lame “whatever its origins“. The sum total of VP’s
argument is precisely zero.
On the ‘something else’ you say of me that “You showed that one possible mechanism does not fit.“. Well, actually I tested the one and only mechanism postulated by VP. There wasn’t anything else
that I could test. As I point out above, even VP walked away from that postulated mechanism.
I am bemused by your assertion that VP “had “rescued” the hypothesis” and that “There is no reason to issue a retraction.“. There was no rescue of anything, since no argument was presented other
than the abovementioned assertions and meaningless curve-fitting. Since the poster has absolutely no merit, the retention of its finding is misleading. The only decent thing for VP do now is to
retract it.
64. vukcevic says:
December 14, 2012 at 1:26 am
it is foolish to dismiss as nonsense.
The nonsense part is to make up a data set from two unrelated ones.
65. richardscourtney says:
December 13, 2012 at 11:29 am
Stephen Rasey:
re your post at December 13, 2012 at 11:00 am.
Every now and then one comes across a pearl shining on the sand of WUWT comments. The pearls come in many forms.
Your post is a pearl. Its argument is clear, elegant and cogent. Thankyou.
66. Nicola Scafetta says:
December 13, 2012 at 9:32 pm
My modelling methodology equivalent to the way the ocean tidal empirical models (which are the only geophysical models that really work) have been developed.
Correct. The tidal models are not calculated from first principles in the fashion that climate models try and calculate the climate. The first principles approach has been tried and tried again
and found to be rubbish because of the chaotic behavior of nature.
67. lsvalgaard says:
December 14, 2012 at 5:21 am
vukcevic says:
December 14, 2012 at 1:26 am
it is foolish to dismiss as nonsense.
The nonsense part is to make up a data set from two unrelated ones.
That the data sets are unrelated is an assumption. This can never be know with certainty unless one has infinite knowledge. Something that is impossible for human beings.
The predictive power of the result is the test of the assumption. If there is a predictive power greater than chance, then it is unlikely they are unrelated. Rather, they would simply be related
in a fashion as yet unknown.
68. ferd berple says:
December 14, 2012 at 7:14 am
That the data sets are unrelated is an assumption.
That they are related is the assumption. Their unrelatedness is derived from what we know about how physics works.
69. lsvalgaard says:
December 14, 2012 at 5:21 am
The nonsense part is to make up a data set from two unrelated ones.
All magneto-meters recordings around the world do it and did it from the time of Rudolf wolf as you yourself show here:
to today at Tromso
Unrelated ? Your own data show otherwise
How does it compare with Pratt’s CO2 millKelvin?
More you keep repeating ‘unrelated’ more I think you are trying to suppress this to be more widely known.
70. I have always figured the models were fits to the data. When I took my global warming class at Stanford the head of Lawrence livermores climate modeling team argued at first the models were based
on physical formulas but I argued that they keep modifying the models to match the hind casting they do more and more and all the groups do the same. Studies have shown and he admitted readily
that none of the models predict any better than each other. In fact only the average of all the models did better than any individual model. Such results are what one expects from a bunch of
fits. He acknowledged that they were indeed fits.
If what you are doing is fitting the models then a Fourier analysis of the data would produce a model like vpmk which would be much better fit than the computer models and all that vpmk did was
demonstrate that if you want to fit the data to any particular set of formulas you can do it easier and with much higher accuracy using a conventional mathematical technique than trying to make a
complex iterative computer algorithm with complex formulas match the data. No wonder they need teams of programmers and professors involved since they are trying to make such complex algorithms
match the data. Vpmks approach is simpler and way more accurate.
The problem with all fits however is that since they don’t model the actual processes involved they are no better at predicting the next data point and can’t be called “science” in the sense that
experiments are done and physical facts uncovered and modeled. Instead we have a numerical process of pure mathematics which has no relationship to the actual physics involved. Vpmks “model”
conveniently shows the the cyclical responses fading and the exponential curve overtaking. This gives credence to the underlying assumptions he is trying to promulgate but it is no evidence that
indeed any physical process is occurring so it is as likely as not to predict the next data point. The idea that the effect of all the sun variations and amo/pdo/Enso have faded is counter to all
intuitive and physical evidence. The existence of 16 years of no trend is indicative that whatever effect co2 is having is being completely masked by natural phenomenon vpmk is diminishing yet if
the 16 year trend would show that if anything the natural forcings are much higher than before. Instead vpmk attributes more of the heat currently in the system to co2 and reduces the cooling
that would be expected by the current natural ocean and sun phenomenon. So just as vpmks model shows natural phenomenon decreasing to zero effect the actual world we know that this is not the
case so again another model not corresponding to reality.
71. Steveta_UK “If you can, please present it here”:
Here is FOBS without any Multidecadal removed:
If you squint closely you might see that there are two lines a red one and a blue one, the standard deviation of the residuals over the 100 year period, 1900 to 2000, is 0.79 mK.
For fun, here it is run forward to 2100:
-There is a lot to be said for thinking about sawteeth :-)
72. Gail Combs says:
[IF] “I publish a paper showing that the rise and fall of women’s skirts plus a saw tooth pattern provides a good fit to the curve. Since no one can provide a better ‘fit’ than that the paper has
to stand?”
In the spreadsheet:
AGW = ClimSens * (Log2CO2y – MeanLog2CO2) + MeanDATA
Perhaps I’m a little dense but somebody might have to explain to me the physics behind that formula before I could take any of this seriously.
Also in the spreadsheet is a slider bar for climate sensitivity, given enough time to “play” one should be able to slide that to 1 C per 2x CO2 and adjust the sawtooth to arrive at the same
results, thereby “proving” (not) climate sensitivity is 1 C / 2X CO2. Seems like a lot of time and effort for nothing to me.
73. “The sawtooth which began life as “so-called multidecadal ocean oscillations” later becomes “whatever its origins“.”
Wonderfully scientific approach – we need a fudge factor to save the “AHH theory”, so we’ll simply take the deviation from reality, call it “a sawtooth, whatever its origin” and the theory is
PARDON ME? Don’t we give the warmists billions of Dollars? Can’t we expect at least a little bit of effort from them when they construct their con?
74. Stephen Rasey says:
December 13, 2012 at 11:00 am
“I want to draw people’s attention to the frequency content of VPmK SAW2 and SAW3 wave forms. ”
Very powerful! Dang, I didn’t think of that!
75. Here is a note to Dr. Svalgaard, not for his (he knows it, possibly far better than I do) but for benefit of other readers.
Relatedness of solar and the Earth’s magnetic fields could be considered in three ways
1. Influence of solar on the Earth’s field, well known but short lasted from few hours to few days
2. Common driving force (e.g. planetary ) – considered possible but insignificant forcing factor.
3. Forces of same kind integrated at point of impact by the receptor. Examples of receptors could be: GCR, saline ions in the ocean currents, interactions between ionosphere and equatorial storms
(investigated by NASA), etc
Simple example of relatedness through a receptor:
Daylight and torch light are unrelated by sources and do not interact, but a photocell will happily integrate them, not only that but there is an interesting process of amplification, which I am
very familiar with.
In the older type image tubes, before CCDs era (saticon, ledicon and plumbicon) there is an exponential law of response at low light levels from the projected image, which further up the curve is
steeper and more linear.
A totally ‘unrelated’ light from back of the tube known as ‘backlight or bias light’ is projected at the photosensitive front layer. Effect is a so called ‘black current’ which lifts ‘the image
current’ from low region up the response curve, result is more linear response and surprisingly higher output, since the curve is steeper further away from the zero.
Two light sources are originally totally unrelated, they do not interact with each other in any way, but they are integrated by the receptor, and further more an ‘amplification’ of the output
current from stronger source is achieved by presence of a weaker.
I know that our host worked in TV industry and may be familiar with the above.
So I suggest to Dr. Svalgaard to abondan ‘unrelated’ counterpoint and consider the science in the ‘new light’ of my finding
76. vukcevic: You and Matthew R Marler call it meaningless curve fitting,
I don’t think I said that your modeling was “meaningless”; I have said that that the test of its “truth” will be how well it fits future data.
77. Re: models and harmonics
I was asked by one of WUWT participants (we often correspond by email) would it be possible to extrapolate CET by few decades. I had ago.
The first step was to separate the summer and the winter data (using two months around the two solstices, to see effect of direct TSI input) result:
this graph at a later stage was presented on couple of blogs, but Grant Foster (Tamino), Daniel Bailey (ScSci) and Jan Perlwitz (NASA) fell flat on their faces trying to elucidate, why no
apparent warming in 350 years of the summer CET, but gentle warming in the winters for whole of 3.5 centuries.
Meteorologists knows it well: the Icelandic Low semi-permanent atmospheric pressure system in the North Atlantic. Its footprint is found in the most climatic events of the N. Hemisphere. The
strength of the Icelandic Low is the critical factor in determining path of the polar jet stream over the North Atlantic
In the winter the IL is located at SW of Greenland (driver Subpolar Gyre), but in the summer the IL is to be found much further north (most likely driver the North Icelandic Jet, formed by
complex physical interactions between warm and cold currents), which as graps show had no major ups or downs.
Next step: finding harmonic components separately for the summers and winters, Used one common and one specific to each of the two seasons, all below 90 years. By using the common and two
individual components, I synthesized the CET adding average of two linear trends. Result is nothing special, but did indicate that a much older finding of the ‘apparent correlation ‘ between the
CET and N. Atlantic geological records now made more sense.
I digressed, what about the CET extrapolation?
Well, that suggest return to what we had in the 1970s, speculative. Although the CET is 350 years long, I would advise caution, anything longer than 15-20 years is no more than the ‘blind fate’.
Note: I am not scientist, in no way climate expert, only models I did are the electronic ones, both designed and built working prototypes.
78. Matthew R Marler since I have quoted you wrongly, I do apologise.
79. I mention retraction in my initial post, but I’ll now make it an explicit request:
Vaughan Pratt, please will you issue a formal retraction of your poster “Multidecadal climate to within a millikelvin”.
80. vukcevic says:
December 14, 2012 at 9:45 am
3. Forces of same kind integrated at point of impact by the receptor.
So I suggest to Dr. Svalgaard to abondan ‘unrelated’ counterpoint and consider the science in the ‘new light’ of my finding
There is no integrated effect as the external currents have a short life time and decay rapidly. Your ‘findings’ are not science in any sense of that word. You might try to explain in one
sentence how you make up the ‘data’ you correlate with. Other people have asked for that too, but you have resisted answering [your 'paper' on this is incomprehensible, so a brief, one sentence
summary here might be useful].
81. Ha! Mr.Whack-a-mole must have gone to bed ;-)
Time for a teeny weeny extrapolation, methinks.
The past 1,500 years temperature history, (base data in red)
(The five free phase sine waves, as above)
82. First, let me also congratulate the author for having the courage to provide all of the data and calculations. Such transparency is in the best interests of science.
I also really liked the very first question raised in this thread – “Assume AGW is a flat line and repeat the analysis” and thought that should be a challenge to take up. What I did may be overly
simplistic so please correct my attempt.
I downloaded the Excel spreadsheet and reset cell V26 (ClimSens) to a value of zero. As expected, the red AGW line on the graph dropped to flat. I then set up some links to the green parameters
so they could be dealt with as a single range (a requirement for the Excel Solver Add-in). I played with a few initial parameters to see what they might do, then fired off Solver with the
instruction to modify the parameters below with a goal of maximizing cell U35 (MUL R2). No other constraints were applied.
Converging to the parameters below, Solver returned a MUL R2 of 99.992%, very slightly higher than the downloaded result. The gray MRES line in the chart shows very flat. (I think it needs one
more constant to bring the two flat lines together but couldn’t find that on the spreadsheet.) Have I successfully fit the elephant? Does this result answer Steveta_uk’s challenge above (Dec 13
at 8:55 am)? Or have I missed something here?
Cell name value
D26 ToothW 2156.84…
G23 Shift 1 1928.48…
G26 Scale 1 1489.03…
H23 Shift 2 3686.05…
H26 Scale 2 1386.24…
I23 Shift 3 4356.71…
I26 Scale 3 2238.07…
J23 Shift 4 3468.56…
J26 Scale 4 0
K23 Shift 5 2982.83…
K26 Scale 5 781.58…
M26 Amp 2235.58…
83. Update: Might have found that constant. Setting cell D32 to a value of -0.1325 roughly centers the MRES line around zero and makes the gray detail chart visible.
84. Chas says:
December 14, 2012 at 12:45 pm
“Ha! Mr.Whack-a-mole must have gone to bed ;-)
Time for a teeny weeny extrapolation, methinks.
The past 1,500 years temperature history, (base data in red)”
Exactly like in the history books! /sarc
Thanks, Chas. Beautiful.
85. Mike, I guess that because you are maximising the R2 you are ending up with two parellel but offset fits (by about 32mK). If you minimised the sum of the residuals you would kill two birds with
one stone.This would have to be the sum of the absolute values of the residuals or the sum of the squared residuals to stop the negative residuals cancelling out the positive ones. I get the
standard deviation ALL of your residuals to be about 2mK; PV selected his SD from the best 100year period to get the ‘less than a millikelvin’ bit, I think.
-I notice that the residuals seem to have a clear sine wave in them with an amplitude of about 5mK whilst at the same time you have SAW4 with an amplitude of zero -I wonder if Solver hasnt
This all is on the basis that I have entered your solutions correctly!
In some ways your fit ought to make more sense to VP than his fit; you have the first sine wave and he is left wondering why his first wave doesnt exist.
86. Leif Svalgaard says:
December 14, 2012 at 12:22 pm
how do you make up the ‘data’
‘Phil Jones from CRU’ syndrome, unable to read the Excel file?
How to calculate spectrum of the changes in the Earth’s magnetic field see pages 13 & 14, you can repeat the calculations.
Since you are so infuriated by ‘unrelated’ magnetic fields, and my ‘art of making-up the data’ you should closely examine Fig. 26, in case you did miss it. That should make it even more
See you.
87. Mike Rossander – Thanks for doing that. Please can you put the result graph online.
Regarding my request to Vaughan Pratt to retract. I made the same request on ClimateEtc, to which he replied:
“The “emperor has no clothes” gambit. Oh, well played, Mike. Mate in one, how could I have overlooked that? ;)
Mike, I would agree with your “simple summary” with two small modifications: replace the first F by F(v) and the second by F(V). My clothes will remain invisible to the denizens of Bill Gates’
case-insensitive world, but Linux hackers will continue to admire my magnificent clothing.
Here F is a function of a 9-tuple v of variables, or free parameters v_1 through v_9, while V is a 9-tuple of reals, or values V_1 through V_9 for those parameters (a valuation in the terminology
of logic).
F(v) is a smooth 9-dimensional space whose points are curves expressible as analytic functions of y (smooth because F is an analytic function of the variables and therefore changes smoothly when
the variables do). F(V) is one of those curves.
To summarize:
1. I assumed F(v).
2. I found F(V) (just as you said, modulo case)
3. at the surface of F(v) very near F3(HadCRUT3).
That’s all I did. As you say, very simple.
If needed we can always make the simple difficult as follows. With the additional requirement that “near” is defined by the Euclidean or L2 metric (as opposed to say the L1 or taxicab metric),
“near” means “least squares.” The least squares approach to estimation is perhaps the most basic of the wide range of techniques treated by estimation theory, on which there is a vast literature.
Least-squares fitting has the downside of exaggerating outliers and the advantage of Euclidean geometry, whose metric is the appropriate one for pre-urban or nontaxicab geometry. Euclidean
geometry works just as nicely in higher dimensions as it does in three, thereby leveraging the spatial intuitions of those who think visually rather than verbally.
We picture F(v) as a 9-dimensional manifold (i.e. locally Euclidean) embedded in the higher-dimensional manifold of all possible time series for 1850-2010 inclusive. Without F3 the latter would
be a 161-dimensional space. F3 cuts this very high-dimensional space down to a mere 7*2 = 14 dimensions, on the premise that 161/7 = 23 years is the shortest period still barely visible above the
prevailing noise despite losing 20 dB. F3(HadCRUT3), H for short, is a point in this 14-dimensional space. The geometrical intuition here is that F3(HadCRUT3) is way closer to F(V) than HadCRUT3,
not because F3 moved it anywhere but merely because the dimension decreased. Given two points near and far from a castle wall, the nearest point on the wall to either can be estimated much more
accurately for the point near the wall than for the one far away. Whence F3. Isn’t geometry wonderful?
(The factor of two in 7*2 comes from the fact that the space of all sine waves of a given period 161/n years, for some n from 1 to 7, is two-dimensional, having as its two unit vectors sin and
cos for that frequency (as first noticed by Fourier?). Letting our imagination run riot, the same factor of 2 is arrived at via De Moivre’s theorem exp(ix) = cos(x) + i sin(x) but that might be
too complex for this blog—when I wrote to Martin Gardner in the mid-1970s to complain that his Scientific American column neglected complex numbers he wrote back to say they were a tad too
complex for his Scientific American readers.)
We’d like F(V) to be the nearest point of F(v) to H in F(v), i.e the global minimum, though this may require a big search and we often settle for a local minimum, namely a point F(V) in F(v) that
is nearest H among points in the neighborhood of F(V).
In either case MRES is the vector from F(V) to H, that is, H – F(V). Since manifolds are smooth, MRES is normal to (the surface of) F(v) at F(V). Hence very small adjustments to V will not change
the length of MRES appreciably, as one would hope with a local minimum.
Hmm, better stop before I spout even more abstract nonsense. ;)“.
As a very capable and experienced physicist once said to me: Nonsense dressed up in complicated technical language is still nonsense.
The “simple summary” to which he referred is on WUWT here:
Vaughan Pratt then added a further shot, it seems it was aimed as much at WUWT as at me:
“Mike, you can find my “formal retraction” here (right under the post you just linked to). I wrote “The ‘emperor has no clothes’ gambit. Oh, well played, Mike. Mate in one, how could I have
overlooked that? ;)”
Feel free to announce on WUWT that I “formally retracted” with those words. In the spirit of full disclosure do please include the point that only Windows fans would consider that a retraction,
not Linux hackers. WUWT readers won’t have a clue what that means and will simply assume I retracted, whereas those at RealClimate will have no difficulty understanding my meaning. Climate Etc.
may be more evenly split.“.
88. vukcevic says:
December 14, 2012 at 2:52 pm
my ‘art of making-up the data’ you should closely examine Fig. 26, in case you did miss it.
You are ducking the question again.
89. lsvalgaard says:
December 14, 2012 at 10:11 pm
Let’s summaries:
Subject of my article is calculation which shows that the natural temperature variability in the N. Hemisphere is closely correlated to the geomagnetic variability with no particular mechanism is
1. You objected: the data was artificially ‘made up’
- this was rebutted by showing that the ‘new data’ is simple arithmetic sum of two magnetic fields.
2. You said: this is not valid since the fields do not interact.
- this was rebutted by showing that interaction is the property of receptor, e.g. magneto-meters do react to both combined fields. Secondary interaction is also recognized via induction of
electric currents.
3. You said: currents are of short duration from few hours to up to a few days, therefore effect is insignificant.
- this happens on a regular bases and it may be sufficient to alter the average temperature of about 290K by + or – 0.4K.
4. You are returning to the starting point: ‘made up’ data (see item 1)
- It is not my intention to go forever in circles.
You made more than 20 posts, here and elsewhere, regarding my finding, with a very little or no success to invalidate it.
My intention is to get more scientific appraisal, as a next step I emailed Dr. J. Haig (from my old university), her interests include solar contribution to the climate change.
She is a firm supporter of the AGW theory, you can contact and join forces, if you whish to do so. Content of the email is posted here:
90. vukcevic says:
December 15, 2012 at 4:01 am
1. You objected: the data was artificially ‘made up’
- this was rebutted by showing that the ‘new data’ is simple arithmetic sum of two magnetic fields.
Repeating an error is not a rebuttal, and a simple inspection of your graph shows that your made up data is not the sum of two ‘magnetic fields’. So, again, how exactly is the data made up?
91. This is in a way of explanation to Dr. Svalgaards statement ‘you made up data’ you are pseudo scientist ,some would think even possibly fraudster, but I am certain that Dr.S didn’t imply it.
Here we go: Since the AMO is trend-less time function (oscillating around zero value) than it is assumed that the signed and normalized SSN (to the AMO values) is an adequate representative of
the heliospheric magnetic field at the Earth’s orbit. It is equally possible to use McCracken, Lockwood or Svalgaard & Cliver data, but these do not contain either sufficient resolution and
mutually disagree, so it is considered that the SSN, as most familiar and internationally accepted data set is the best for the purpose. Earth Magnetic field has number of strong spectral
components. One of them is exactly same as the Hale cycle period (as calculated from the SSN). I could have used it as an non-dumped oscillator (clean Cos wave) , but match to the AMO is not as
good as using the signed SSN. This points to the fact that the SSN is more likely factor, unless of course the Earth harmonic has same annual ‘modulation’ in manner of the SSN, which would be an
extraordinary finding. Such possibility is considered on page 14 Fig. 25 curve dF(t). For purpose of comparison to the AMO from the Earth spectrum is then taken second component and employed as
clean Cos wave. I suspect that this component is due to the feedback ‘bounce’ due to propagation delay in the Earth’s interior (see link to Hide and Dickey paper) but this is speculative.It is
huge puzzle why the Earth’s magnetic field oscillation component should have as its main period one which is exactly same as the SSN derived Hale cycle, but of much stronger intensity than the
heliospheric field . I do not think, but many solar scientist (including yourself) postulate that the solar dynamo has amplification property. Would something of a kind exist within the Earth’s
dynamo than it would explain the strong Earth’s component as well as Antarctic field http://www.vukcevic.talktalk.net/TMC.htm .How could this occur: I speculate that since dept of Earth’s crust
is 20-40 km, and the geomagnetic storm induced currents reach down to 100km (Svalgaard). It is possible that a magnetized bubble of liquid metal is formed than amplified by the field of the
Earth’s dynamo, in a manner of the solar dynamo amplification. Although highly speculative, and despite you promoting solar dynamo amplification you will reject the geo dynamo amplification, but
it would explain a lot. Mathematics of periodic oscillations ‘amplification’ is dead simple Cos A + Cos B = 2 Cos (A+B)/2 x Cos (A–B)/2, and vice versa, result: one short and one long period of
oscillation, giving rise to the two AMO’s characteristic periods of 9 and 64 years (see pages 5 & 6).Where this process occurs it is not known (it could be in the magnetic field itself or in the
oceans as receptor of the two oscillation. Now to the Excel file: Word sum is used with its more general meaning, to described any of the four arithmetic operations as used in the Excel file, but
here is a list:
Column1: Year; Column2: SSN; Column3: (+ & – 1 to sign SSN); Column4: Hale cycle- SSN with sign (times); Column5: Normalized SSN to AMO (divide); Column6: – Earth field oscillator (Cos, times,
minus, divide); Column7: Geosolar oscillator (times); Column8: Geosolar oscillator moved forward by 15 years; Column9: AMO; Column10: AMO 3yma (plus & divide). So what is all this about: http://
www.vukcevic.talktalk.net/GSC1.htm Annoying fact is that you know all of the above, why do you want it all spelt out god only knows. I am not answering any more questions, have go at your
Stanford colleague and his milliKelvins. , instead I shall refer you to appropriate page in my article, Excel file you have and this post.
Thank you and good bye.
92. I suggest that Vaughan Pratt and some commentators here read up a bit on Fourier theory. Any (yes ANY) periodic or aperiodic continuous function can be decomposed into sine waves to any precision
wanted. So it follows that you can also subtract any arbitrary quantity (for example an IPCC estimate of global warming) from that continuous function and it can still be decomposed into sine
waves just as well as before, though they would be slightly different sine waves.
However note that there is absolutely no requirement that those sine waves have any physical reason or explanation.
93. vukcevic says:
December 15, 2012 at 10:06 am
why do you want it all spelt out god only knows.
What you describe is a perfect example of fake data, selected and made up to fit the best, based on invalid physics. That you call the ‘data’ is deceptive in the extreme. The one deceived in in
first line yourself. What happened to your grandiose plan of sending your stuff to all geophysics departments [before AGU] at all major Universities?
94. vukcevic says:
December 15, 2012 at 10:06 am
why do you want it all spelt out god only knows
What you describe is a perfect example of fake, selected, tortured, and made-up stuff twisted to fit an idea. Calling the result ‘data’ is deceptive in the extreme; the one most deceived being
yourself, belying your claim that “ the ‘new data’ is simple arithmetic sum of two magnetic fields.
BTW, what happened to your grandiose plan of carpet-bombing [before AGU] geophysics departments at all major universities to drum up support for your ideas?
95. Mike Jonas: Regarding my request to Vaughan Pratt to retract. I made the same request on ClimateEtc, to which he replied:
I thought I would mention again that Einstein’s 1905 paper on special relativity showed that: by assuming the speed of light to be constant he could derive the already well-known
Lorentz-Fitzgerald contraction, an exercise you would regard as circular because the Lorentz-Fitzgerald contraction was already known, and the mechanism by which the speed of light can be
independent of the relative motions of source and receiver (whereas the frequency and wavelength are not so independent) is a mystery.
I don’t mean to embarrass Dr Pratt by elevating him into the Pantheon with Einstein, but the logic of the two theoretical derivations is the same in the two cases: the result is known, and the
procedure produces it. The suspicion surrounding Einstein’s result was such that the Swedish Academy awarded him the Nobel Prize for a different 1905 paper, and the general theory did not begin
to gain widespread acceptance until the Eddington expedition in 1919, and that was the subject of acrimonious debate.
There is no more reason for Pratt to withdraw this paper than there would have been for Einstein to withdraw his first paper on relativity.
96. Dr. Svalgaard said:
What you describe is a perfect example of fake data, selected and made up to fit the best, based on invalid physics. That you call the ‘data’ is deceptive in the extreme. The one deceived in in
first line yourself.
Calling the result ‘data’ is deceptive in the extreme; the one most deceived being yourself, belying your claim that “ the ‘new data’ is simple arithmetic sum of two magnetic fields.
No need to be so furious, sun matters, you know.
Hey, not just the ‘ordinary garden nonsense’ this time, something far more valuable.
‘would need to distill the argument into relatively simple points, show a few key figs’ as another university professor said, and than I’ll dispatch few emails.
Have a happy Xmas and N. Year.
p.s. Apparently a new fable by Hans Christian Anderson is discovered; try to get a first print copy for your grandchildren.
97. vukcevic says:
December 15, 2012 at 12:05 pm
No need to be so furious, sun matters, you know.
No need to be so evasive. Honesty matters, you know.
Hey, not just the ‘ordinary garden nonsense’ this time, something far more valuable
D-K effect again. There is nothing valuable at all in your stuff.
98. lsvalgaard says:
Nicola Scafetta says: “7) wait the future to see what happens: for example follow the (at-the moment-very-good) forecasting performance of my model here”
It fails around 2010 and you need a 0.1 degree AGW to make it fit. I would say that there doesn’t look to be any unique predictability in your model. A constant temperature the past ~20 years
fits even better.
I recently read an article by a professor at Stanford, one of the top universities in the U.S. , that claimed at 3 deg. / 2XCO2 model was accurate to within one thousandth of a degree. But I’m a
bit concerned because he doesn’t know how to do a running mean.
Do you think it matters ?
99. lsvalgaard says: A constant temperature the past ~20 years fits even better.
The same could be said of 3K per doubling, it sure as hell isn’t within a 1/1000 degree whatever way you spin it.
100. Matthew R Marler says “I thought I would mention again that Einstein’s 1905 paper on special relativity showed that: by assuming the speed of light to be constant he could derive the already
well-known Lorentz-Fitzgerald contraction, an exercise you would regard as circular because the Lorentz-Fitzgerald contraction was already known[...]“.
That doesn’t look at all logical to me. Circular logic is where your finding is what you assumed in the first place. The fact that you can derive A from B when A is already known doesn’t make the
logic circular. Circular is deriving A from A.
101. Mike Rossander and tty – Thanks for your comments. I refer to them on ClimateEtc -
- as follows:
Thanks, tty. It explains neatly why VP’s finding is completely meaningless without any physical reason or explanation for the sinewaves.
Thanks also to Mike Rossander for repeating VP’s exact process but with slightly different parameters. These are the sinewaves generated using VP’s spreadsheet and Mike Rossander’s parameters:
and this is the result:
Every single argument of VP’s in support of the process that he used with IPCC-AGW applies absolutely equally to the exact same process with AGW = zero.
Vaughan Pratt – Now do you see that your argument has no merit?
102. The updated ‘AGW = zero” spreadsheet is here:
103. Mike Rossander says:
December 14, 2012 at 1:18 pm
Mike Rossander is my hero!
104. By showing that a climate sensitivity to CO2 of 0 C/W/m^2 gives as good (or better) fit than the consensus climate sensitivity, Mike Rossander and Mike Jonas have completely rubbished the
Thanks for your efforts!
Now assume a climate sensitivity to CO2 of -3 C/W/m^2. That should be fun! I predict an almost perfect fit once again can be achieved with the correct weighting of suitable sinusoids.
105. Scafetta doesnt share his code or his data. he is a fraud
106. Vukcevic says that:
he has discovered/invented following formula
AMO = SSN x Fmf
AMO = Atlantic Multidecadal Oscillation (or de-trended N.H. temp)
SSN = Sunspot number with polarity
Fmf = frequency of the Earth’s magnetic field ripple (undulation).
He calls above arithmetic sum ‘Geo-Solar Cycle’
calculations are accurate, but he is unable to provide valid physical mechanism.
Svalgaard says that:
Vukcevic – writes nonsense, pseudo science, suffers from Denning-Kruger mental aberration, making up data, deceptive in the extreme (implies fraud), honesty matters (implies dishonest)
In the years gone by, where I come from, the above attributes had to be earned. Fortunately that is not the case any more. Similar pronouncements were often repeated by the self-appointed
‘guardians of eternal truth’ regardless of geography or historic epoch.
(@ mosher forwarded the relevant Excel file to you too)
107. Mike Jonas: Every single argument of VP’s in support of the process that he used with IPCC-AGW applies absolutely equally to the exact same process with AGW = zero.
On that we agree. The test between the two models will be made with the next 20 years’ worth of data. Having found filters and estimated coefficients, they are not free to modify those
coefficients willy-nilly to get good fits each time the data disconfirm their model forecast.
108. Matthew R Marler – Neither of the “sawtooths” bears any relationship to the real world. There is no point in testing them.
109. vukcevic says:
December 15, 2012 at 10:47 pm
AMO = SSN x Fmf
AMO = Atlantic Multidecadal Oscillation (or de-trended N.H. temp)
SSN = Sunspot number with polarity
Fmf = frequency of the Earth’s magnetic field ripple (undulation).
He calls above arithmetic sum ‘Geo-Solar Cycle’
Zeroth: Is it AMO or [manipulated] temps? Not the same.
First, the formula uses multiplication [I assume that is what the 'x' stands for], so is not a sum.
Second, which SSN is used? The International [Zurich] SSN or the Group SSN?
Third, the ‘polarity’ of the SSN is nonsense. You might talk about the polarity of the HMF, but then that should go from maximum to maximum [when the polar fields change]. In any case, the
polarity that is important for the interaction with the Earth is the North-South polarity which changes from hour to hour.
Fourth, ‘frequency’ is not a magnetic field [you said that you added two magnetic fields].
Fifth, ‘ripple’ is what? and undulation?
Sixth, ‘Earth’s field’, measured where? And why there?
In the years gone by, where I come from, the above attributes had to be earned.
With the expertise you perfected back then, you are still earning it in earnest now.
Similar pronouncements were often repeated by the self-appointed ‘guardians of eternal truth’ regardless of geography or historic epoch.
Nonsense is nonsense no matter where and when.
110. lsvalgaard says:
December 16, 2012 at 6:21 am
Some of the points you raise are explained beforehand, see my post above
The above formula is a summary in its most abstract form. I have also made it clear that all four basic arithmetic calculations for simplicity I always address as ‘sum’, but for your benefit (see
the link above) each of the arithmetic calculations has been specifically itemized in the Excel file description.
You also have access to my article, which is over 20 pages long, contains 39 illustrations of which 35 are my own product, and many of the questions you pose are elaborated in detail in the
article. You also have the Excel file with further information.
You will appreciate that the blog is not capable of furnishing full reproduction, therefore reading of the article is a prerequisite, which of course you are welcome to do.
Thank you for the note, once the final publication (this is the first draft) is composed the points you made will be fully considered.
For time being you may consider snippets of information as ‘unofficial leaks’ from a future publication, which currently is a ‘high vogue’ on the fringes of climate science, and should be treated
as such, or else ignored.
Thanks again for your attention.
111. lsvalgaard says:
December 16, 2012 at 6:21 am
May I add, I feel highly privileged and grateful that most if not all, of your attention and time during the last few days, on this blog and elsewhere, was devoted to the ‘ironing out’ of any
inadvertent inconsistencies, that may be found in the draft of my article, to the preference of the Dr. Pratt’s paper, which surely must be this year’s the most important contribution to
understanding of the anthropogenic warming.
Thank you again, sir.
112. vukcevic says:
December 16, 2012 at 11:09 am
May I add, I feel highly privileged and grateful that most if not all, of your attention and time during the last few days, on this blog and elsewhere, was devoted to the ‘ironing out’ of any
inadvertent inconsistencies, that may be found in the draft of my article
As I said, your descriptions here on WUWT have been deceptive and your ‘paper’ is incomprehensible. You cannot presume that anybody would try to decipher what you actually mean. The purpose of
publication is to make the paper comprehensible to a [scientifically literate] reader with only cursory knowledge of the subject to understand your claims by simply skimming the paper [your
version is much too dense with details and loose ends]. You could start by responding to my seven points, right here on WUWT. As things stand, the paper is still nonsense, and you commit the
deadly sin of arguing with the referee instead of responding concisely to the points raised.
113. Mike Rossander is the first, and so far only, person to respond to my challenge to provide an alternative description of modern secular (> decadal) climate to the one I presented at the AGU Fall
Meeting 12 days ago. I only wish there were more Mike Rossanders so as to increase the chances of obtaining a meaningful such description. Thank you, Mike!
Although Mike’s magic number of 99.992% may seem impressive, it’s unclear to me how a single number addresses the conclusion of my poster, which starts as follows.
“We infer from this analysis that the only significant contributors to modern multidecadal climate are SAW and AGW, plus a miniscule amount from MRES after 1950.”
If one views MRES as merely the “unexplained” portion of modern secular climate then 99.992% does indeed beat 99.99%.
However a cursory glance at these charts reveals two clear differences between the upper and lower charts, respectively mine and Mike’s.
1. On the left, the upper chart is “within a millikelvin” (as measured by standard deviation) for the “quiet” century from 1860-1960. It then moves up until 1968, quietens back down to 1990, then
moves up again. At no point does it go below the century-long quiet period. (Ignore the decade at each end, secular or multidecadal climate can’t be measured accurately to within a single
I justify my “within a millikelvin” title by claiming that, although the fluctuations from 1860 to 1960 are certainly “unexplained” (R2 is defined as unity minus the “unexplained variance”
relative to the total variance), the non-negative bumps thereafter admit explanations, namely non-greenhouse impacts of growing population and technology in conjunction with the adoption of
emissions controls. Their clear pattern therefore makes it unreasonable to count them as part of the unexplained variance.
The lower chart on the other hand simply wiggles up and down throughout the entire period, and moreover with a standard deviation three times as large. It is just as happy to go negative as
positive, and it draws absolutely no distinction between the low-tech low-population 19th century and the next century. WW1 consisted largely of firing Howitzers, killing many millions of
soldiers with bayonets and machine guns, and dropping bricks from planes, while WW2 consisted largely of blowing cities and dams partially or in some cases completely to smithereens with
conventional and nuclear weapons of devastating power. The clear consequences of this trend led to the cancellation of WW3 by mutual agreement.
There is not a trace of this progression in Mike’s chart, just oscillations both above and below the line. Moreover they even die down a little near the end—what clearer proof could you ask for
that the increasing human population and its technology cannot be having any impact whatsoever on the climate?
2. On the right, the upper chart shows in orange the Hale or magnetic cycle of the Sun. The lower one does the same except that it is much messier and gives no reason to suppose that a cleaner
picture is possible.
Now let’s look at how far one needs to bend over in order to cope with zero AGW. Here are the ten coefficients Mike and I are using to express the sawtooth shape, expressed in natural units
rather than the incomprehensible slider units. For shifts this is the fraction of a sawtooth period t to be shifted by, so for example 0.37t means a shift of 37% of the sawtooth period. For scale
it is the attenuation of the harmonics, so for example 0.37X means an attenuation down to 37% of full strength for that harmonic in the case of a perfect sawtooth. 0 and X are synonymous with 0X
and 1X respectively.
Shifts: 0.092848t 0.268605t 0.335671t 0.246856t 0.198283t
Scales: 1.48903X 1.38624X 2.23807X 0 0.78158X
(The 0 for Scale4 is clearly a bug in how the problem was presented to Solver.)
Shifts: 0 0 0 t/40 t/40
Scales: 0 X X X/8 X/2
If we take the number of bits needed to represent these coefficients as a measure of how much information each of us has to pump into the formulas to force them to fit the data, the difference
should be clear to those who can count bits. Also note that all of my shifts are way smaller than all of Mike’s shifts. His five sine waves bear no resemblance whatsoever to the harmonics of a
My question to Mike, and to anyone else who shares Mike’s and my interest in modern secular climate analysis, is, can one create a more plausible MRES, a cleaner HALE cycle, and a less obviously
contrived collection of coefficients, while still setting climate sensitivity to zero?
If this can be done we would have a much stronger case against global warming.
(Incidentally the reason the logic looks circular to MIke Jonas is that iterative least-squares fitting is circular: it entails a loop and loops are circular. The correct question is not whether
the loop is circular but whether it converges.)
114. Good afternoon, Vaughan. I think you may have misunderstood the intent of my comment. And perhaps of the original criticism in the post above. My analysis was trivial, limited and deeply flawed.
It had to be so because it was based on no underlying hypothesis about the factors being modeled (other than the ClimSens control value). It was an exercise that took about 15 min and was
intended only to illustrate the dangers of over-fitting one’s data.
For example, you argue above that the fact that the shifts in your scenario are smaller is an element in favor of that scenario. Unless there is a physical reason why small shifts should be
preferred, that claim is unjustifiable. A shift of zero may be perfectly legitimate – or totally irrelevant. That statistics are only useful to the extent that they illuminate some underlying
physical process.
By the same token, you can’t say that the coefficients are “contrived” unless you have a physical understanding that indicates what they SHOULD be.
The closeness of R2 is also essentially irrelevant. Minor changes to the parameters drove that value off the 0.99x values quite easily. We can reasonably interpret an R2 of 0.2 as “bad” and an R2
of “0.8″ good but the statement that “R2 of 0.99992 is better than 0.9990″ is well beyond the reliability of the statistics to honestly say. On the contrary, given everything we know about the
randomness of the input data (and about the known uncertainties of the measurement techniques), an R2 that high is almost certainly evidence that we have gone beyond the underlying data. There’s
too little noise left in the solution. I say that because my physical model includes assumptions about the existance and magnitude of human error in data collection, transcription, etc. and that
those errors should be random, not systemic.
What you need (and what neither of us have done) is a formal test of overfitting. Unfortunately, without an assumed physical model, I don’t know of any reliable way to structure that test. A
common approach would be to run the Student’s t Test on each parameter in isolation, comparing the results of the model at the set parameter value vs the hypothesized null value. But your model
does not make apparent what the null value ought to be. As noted above, we can not blindly assume that it is zero.
An alternate approach would be to restructure the model so you can feed an element of noise into all your data and rerun the analysis a few thousand times. Parameters which remain relevant across
the noisy samples are probably more reliable. That’s not an approach that can be easily built in Excel, however.
Having said all that, you are certainly more familiar with your model than I ever will be. (The organization of your workbook is well above average but reverse-engineering someone else’s excel
spreadsheet is almost always an exercise in futility.) Maybe you see a way to run a proper test against the parameters that I don’t.
One last thought. I ran my trivial test with the hypothesis that ClimSens should equal zero. There is no reason why that is necessarily the appropriate null, either. ClimSens really should be its
own parameter, also included in the statistical tests of overfitting.
115. Vaughan.
I have used a statistical curve fitting approach in testing one component of the AGW model that you assume to be valid.That component is the assumption that anthropogenic emissions has caused all
of the atmospheric increase in CO2. Read http://www.retiredresearcher.wordpress.com.
116. Hi Mike. All your points are eminently sensible, particularly as regards the dangers of over-fitting (your main concern). That was also a concern of mine, and is why I locked down 5 degrees of
freedom in SAW.
One way to structure a test of overfitting is to compare the dimensions of the data and model spaces. To my mind the best protection against overfitting is to keep the latter smaller than the
former. When it can be made a lot smaller one can claim to have a genuine theory. When only a little smaller, or barely at all, it is not much better than a mere description from some point of
view (namely that of a choice of basis). I would say my Figure 10 was more the latter, argued as follows.
For this data, 161 years of annualized HadCRUT3 anomalies is a point in the 160-dimensional space of all anomaly time series centered on the same interval, one dimension being lost to the
definition of anomaly. My F3 filter projects this onto the 14-dimensional space of the first seven harmonics of a 161-year period. However it attenuates harmonics 6 and 7 down into the noise,
making 10 dimensions (two per harmonic) a more reasonable estimate, perhaps 12 if you can crank up the R2 really high to bring up the signal-to-noise ratio.)
In principal my model has 14 dimensions, of which I lock down 5 leaving 9. You locked down the 3 AGW dimensions leaving the 11 SAW dimensions. However two of these only partially benefited you
because Solver wanted to drive Scale4 negative. I’m guessing you left the box checked that told Solver not to use negative coefficients, so it stopped moving Scale4 when it hit 0, which in turn
made tShift4 ineffective. The Evolutionary solver might have been able to add 1250 (half a period) to tShift4 discontinuously to simulate Scale4 going negative.
9 dimensions for the model space is dangerously close to the 10 dimensions of the model space, so I’m within a dimension of being as guilty of overfitting as you. The real difference is not
dimensional however but choice of basis: sine waves in your case, a more mixed basis in mine including 3 dimensions for the evident rise that I’ve modeled as log(1+exp(t)) per the known physics
of radiative forcing and increasing CO2.
Rather than using a t-test I’d be inclined to move the data and model dimensions apart by dropping the 4th and 5th harmonics altogether (since they aren’t really carrying their weight in
increasing R2) while halving the period of F3. The data space would then have 20-24 dimensions and the model space 6.
But I would then spend 5-6 dimensions in describing HALE as a smoothly varying sine wave. That would greatly reduce the contribution of HALE to the unexplained variance, at the price of reducing
the 6:20 gap to 11-12:20-24. That’s still an 8-13 dimensional gap between the data and model spaces, which I would interpret as not being at serious risk of overfitting. 3 of the HALE dimensions
describe the basic 20-year oscillation, which the remaining 2-3 modulate.
As Max Manacker insightfully remarked at Climate Etc. a day or so ago, it’s not the number of parameters that counts so much as whether they’re the right ones. Whether parameters are meaningful
depends heavily on the choice of basis for the space. Do they have any physical relevance?
Physics aside, your suggesting of feeding noise in would be a straightforward way of quantifying the dimension gap empirically that also took into account the attenuation by F3 of harmonics 6 and
7 as well as the role of R2, all in the one test so that would be very nice. I have this on my list of things to look into, hopefully the other things won’t push it down too far.
I would have replied sooner except that I spent today following up on the suggestion to try other ClimSens values besides 0 and 2.83, made by both you and “Bill” on CE. Very interesting results,
more on this later as this comment is already so deep into tl;dr territory that I ought to submit it for the 2013 literature Nobel.
117. Hi fhaynie. I’m having trouble reading your first figure about dependence on latitude. When I click on it I get only an arctic plot. Is there some way I can blow it up to a readable size?
Your emphasis on carbon 13 and 14 is commendable, but I must confess I have thought less about them than the raw CDIAC emissions and land-use-change data since 1750. These can be converted to
ppmv contributions using 5140 teratonnes as the mass of the atmosphere and 28.97 as its average molecular weight. One GtC of emitted CO2 therefore contributes 28.97/12/5.14 = 0.47 ppmv CO2 to the
CDIAC says that in 2010 the anthropogenic contribution including land use changes was 10.6 GtC. For that year Mauna Loa recorded an increase of 2.13 ppmv. The former translates to a contribution
of 10.6 * 0.47 = 4.98 ppmv.. Hence 2.13/4.98 = 42.7% percent of our contribution was retained in the atmosphere in 2010, with the remaining 57.3% being presumably taken up by the ocean, plants,
soil, etc.
It would be very interesting to compare this with your analysis based on molecular species.of CO2. Do you have an estimate of the robustness of this sort of analysis?
I very much like this direction you’re pursuing, it could lead to useful insights. What feedback have you had from those knowledgeable about this sort of approach? And are there more detailed
papers I can read on this?
118. Vaughan,
I am in the process of using similar techniques doing mass balances dividing the earth’s surface into five regions. If you are interested, you can find my email address at http://
I will gladly share what I am doing as well as thoughts on the mistakes we can make using curve fitting programs. I have had some long email conversations with two individuals that promote the
global mass balance you cite. They have a strong vested interest in being right. Most of the favorable comments did not go into any technical detail.
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Swan like theorem and covering spaces
up vote 7 down vote favorite
Let $X$ be a finite CW complex. Swan's theorem provide an equivalence \[ Vec(X)~\xrightarrow{\sim} ~ProjMod(hom_{Top}(X,\mathbb{R})) \] between the category of finite dimensional vector bundles over
$X$ and the category of finitely generated projective modules over the ring of continuous functions from $X$ to the reals. This isomorphism behaves well with the monoidal structures $\oplus$.
There is an intermediate step in this construction: The category $Vec(X)$ of finite dimensional vector bundles over $X$ is equivalent to locally free modules of finite rank over the sheaf $C_X(-)=
hom_{Top}(-,\mathbb{R})$ on $X$.
The category $Cov(X)$ of covers of $X$ is equivalent to the category of locally constant sheaves of sets on $X$. Is it possible to formulate this analogously to the above correspondence? So maybe
locally constant sheaves are somehow special modules over $C_X(-)$ and this category possibly corresponds to some special modules over $C_X(X)$. Maybe, this is also compatible with disjunct unions of
coverings and sums of the corresponding modules. Maybe it is also necessary to require, that the covering is regular.
(The fat things are edits made, partially based on the answers below.)
at.algebraic-topology vector-bundles gt.geometric-topology
It might be worth exploring the connection between vector bundles with flat connections and covering spaces via representations of the fundamental groupoid. It then remains to think about what
vector bundles with flat connections are modules for, but my guess is that stacks would come in and/or differential cohomology. – David Roberts Oct 27 '10 at 22:31
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2 Answers
active oldest votes
If I can take only the finite covers, then yes, I think. (After all, Swan's theorem is a characterization of finite-dimensional vector bundles, not all vector bundles.) This is easier
to do over $\mathbb{C}$ than over $\mathbb{R}$. In addition to the entire sheaf $C_X(-)$, let $C(X) = C_X(X)$ be the algebra of global continuous functions.
Since the module $M$ is locally free, what you want to do is to choose a basis for $C(U) = C_X(U)$ for enough open sets $U$, and such that the bases agree when you restrict to smaller
up vote 5 open sets. You could just ask for this directly, but there is an indirect algebraic condition that comes to the same thing. Namely, you can ask for $M$ to not only be finitely generated
down vote and projective, but also a semisimple commutative algebra over $C(X)$. This gives you the unordered basis in each fiber.
Over $\mathbb{R}$, it's not quite enough to require that $M$ be a semisimple algebra, because you could end up creating $C(Y,\mathbb{C})$ for a finite cover $Y$ of $X$. So, you could
also impose the condition that $f^2 + g^2 = 0$ has no non-trivial solutions in $M$.
Unfortunately I do not understand your answer. What is $M$ here? If the cover of $X$ is regular, the fibers are discrete groups and in particular normal subgroups of $\pi_1(X)$. I
wonder what the action of $C(X)$ is. – roger123 Oct 28 '10 at 9:32
3 In the end $M = C(Y)$, where $Y$ is the covering space of $X$. The projection from $Y$ to $X$ induces a ring homomorphism from $C(X)$ to $C(Y)$, and then $C(X)$ acts on $M$ via this
ring homomorphism. – Greg Kuperberg Oct 28 '10 at 10:05
1 What I'm saying is, if $M$ is a $C(X)$-algebra which (a) satisfies Swan's conditions as a module, (b) is a semisimple algebra, and (c) has no solutions to $f^2 + g^2 = 0$, then,
conclusion, it is isomorphic to $C(Y)$ for a finite cover $Y$. – Greg Kuperberg Oct 28 '10 at 17:40
1 @roger123 You can construct a locally constant set-theoretic sheaf directly this $C(X)$-algebra $M$. You can first make a sheaf version of $M$, and then you can look at the
corresponding sheaf of minimal idempotents. (Recall that an idempotent in a ring is an element $a$ such that $a^2 = 0$. An idempotent is minimal if it is not the sum of two
[commuting] idempotents.) Or, if you like, the minimal idempotents are a canonical basis of $M(U)$ for an open set $U$ on which $M$ is trivial. – Greg Kuperberg Oct 29 '10 at 17:57
@roger123 If $Y$ is a finite-sheeted covering of $X$, then the gluing maps between local trivializations of $Y$ are functions from $U \subset X$ to permutations. You can get a vector
1 bundle if you replace these permutations by the same permutation matrices. Or, in the present context, $C(Y)$ satisfies Swan's theorem as a module over $C(X)$ and thus gives you a
vector bundle, exactly the same bundle as you would get from using permutation matrices. – Greg Kuperberg Oct 30 '10 at 6:03
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A theorem I found in a paper of Horst Madsen:
For every locally connected space $X$ there exists a category equivalence between the category of regular covering spaces of $X$ and the category of Galois extensions of $C(X)
up vote 1 down vote
I've just found this on the web and don't know how this fits to your question. I would also like to see if there is such a relation as you mentioned.
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How to mathematically calculate the accuracy and resolution of weighing system
If you are only measuring after you remove weight from your system in batch format, you should use the hysteresis specification of your load cells combined with the accuracy of your electronics to
determine accuracy.
If you measure after adding AND removing weight from your system, you can use the combined error (linearity and hysteresis) of your load cells combined with the accuracy of your electronics to
determine system accuracy. You have to take into account the error of your controller, load cells, gravity correction, and verify that there are no other issues that may affect your accuracy reading
(like EMI/RFI noise, scale binding, correct load cell placement, etc) and environmental variations (such as humidity, temperature and wind). For example, using a 16,500lb Hardy load cell: Linearity
0.012% = 1.98lb or approximately 1 in 10,000 Hysteresis 0.025% = 4.125lb or approximately 1 in 3000 Combined error 0.02% = 3.3lb or approximately 1 in 5000 The above does not including mechanical,
electronic or electrically induced errors. Before designing a system, an engineer should consider carefully what he or she expects from it and then relate this to the component accuracies making up
the system. No physical measuring system can be completely accurate. An error band must be defined for a system which gives an indication of any expected deviations from true value. The parameters
under which this applies must also be clear and concise. Accuracy terms such as "1 part in 3000" are commonly used. Calculating true weigh system accuracy is very difficult, and many customers do not
know what they really require from their system.
They often request a "system to be as accurate as possible". Proper installation is critical to maximize a systems accuracy, but other considerations such as connecting pipes and conduits must be
taken into account. One thing is certain, good load cells do not make a poor system good, but poor load cells can make a good system poor. Hardy Instruments and load cells are considered by many to
be the most accurate available in today’s process weighing market.
In the vast majority of applications, weighing occurs in only a small portion of the load cell’s range. Thus non-repeatability is the most important specification for most system designers. LOAD
SENSOR NON-REPEATABILITY: A standard non-repeatability for a typical load sensor is 0.01% of Full Scale. This is the equivalent of one part in 10,000 of total system load cell capacity. In a 300
pound example the non-repeatability would be + or - 0.03 pounds. The simple definition for non-repeatability is: the maximum error seen if the same amount of material was repeatedly add to or removed
from the same vessel, under the same environmental conditions. This situation is often encountered in batching applications. SYSTEM STATISTICAL ERROR: In a multiple load cell system the “combined
error” parameters are not added together but are combined using the following formula: The square root of the sum of load cell number one’s “combined error” squared, plus load cell number two’s
“combined error” squared, plus load cell number three’s “combined error” squared, etc. In a system using three 10,000 lb load cells, the “combined error” of each load cell is 3 pounds. Entering this
data in the formula: Square root of (32 + 32 + 32) or Square root of (9 + 9 + 9) or Square root of 27 or 5.196 lbs (the square root of 27). SYSTEM RESOLUTION: The stability of the weight reading
(often referred to as useful resolution) is affected by electrical noise in the form of both Radio Frequency Interference (RFI), and Electro Magnetic Interference (EMI).
These sources of interference affect the signal to noise ration of the load censor input to the weight controller. Following standard electrical wiring procedures a weight stable to 0.3 micro volt is
typical for Hardy controllers. Utilizing 2 mV/V load cells and a 5 volt excitation this translates to one part in 30,000. In a 300 lb example, a stable weight reading of + or - 0.01 pound would be
obtained. To insure good results attention must be paid to shielding, grounding and cable routing. EQUIPMENT DISPLAYED RESOLUTION: The high internal and displayed resolution of the Hardy Instrument
weight controller line allows precise mathematical computations. This resolution yields good results for such features as WAVERSAVER and development of various digital and analog outputs without
introducing errors in either displayed or transmitted data. The Internal Resolution is one part in 654,000 for a 2 mV/V load cell and one part in 985,000 for a 3 mV/V load point. SYSTEM ACCURACY: All
of the above terms relate only to the electrical specifications of the system. Mechanical errors often introduce system errors. Mechanical errors can at times be difficult to identify. Proper system
performance requires that the mechanical system be properly designed. In a properly designed system all of the weight will be vertically applied to the load points. In addition, there will be no
redundant load paths from non-flexible connections, such as piping, ducting, tubing, etc. As you can see the proper installation of load cells is critical to a scale systems accuracy. Mechanical
errors caused by binding is the number one cause of inaccuracy in a scale system. By using the above terms and formula’s you can calculate your weighing system’s accuracy.
Respective topics:
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Re: Cochlear travelling wave. An epiphenomenon?
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Re: Cochlear travelling wave. An epiphenomenon?
Dear Antony and List:
I agree that there is more mathematical backing for the TW concept, but
that's because coherent alternatives have just not existed! And it is worth
while remembering that mathematical models are only as good as the physical
insights they represent.
As auditory scientists, we would do well not to let the TW theory become a
dogma that permits no other hypothesis to be
considered. My modeling has got to the point of being able to explain the
shape of the partition's tuning curve, but it would be nice to do more
elaborate modeling. I hope that additional mathematical analysis will be
developed once the initial concepts have been clarified. Patience is needed,
for it took many decades for sophisticated mathematical models of the TW to
be developed after Bekesy's initial model was put forward.
I am trained as a physicist, not a mathematical modeler, but if anyone out
there has a particular interest in exploring the mathematical properties of
a resonance theory of the cochlea, I would welcome their interest.
Thankyou for your good wishes.
Andrew Bell.
-----Original Message-----
From: AUDITORY Research in Auditory Perception
[mailto:AUDITORY@LISTS.MCGILL.CA]On Behalf Of Antony Locke
Sent: Thursday, 29 June 2000 8:39
To: AUDITORY@LISTS.MCGILL.CA
Subject: Re: Cochlear travelling wave. An epiphenomenon?
Dear Andrew
Many thanks for your reply that attempted to address the most basic 'level
of description' of energy flow within the cochlea; i.e. how stapes
vibrational energy is transferred to the characteristic place. Many thanks
also to Jont Allen's comments, all of which I agree with.
In my opinion, the travelling wave model captures the qualitative and, to a
large extent, quantitative features of the cochlear response to sound at
this level of description. The model is based on established anatomical,
physiological and mechanical observations within a rigorous mathematical
framework. Any model that counters the travelling wave model must refer to
the same observations and be developed within an equally valid mathematical
In conclusion, I reiterate Jont's suggestion: I look forward to reading
about your model in a quality peer reviewed journal.
Best wishes
Antony Locke | {"url":"http://www.auditory.org/mhonarc/2000/msg00220.html","timestamp":"2014-04-17T21:26:19Z","content_type":null,"content_length":"5736","record_id":"<urn:uuid:bba26d74-cebd-458d-bca4-6f1706075357>","cc-path":"CC-MAIN-2014-15/segments/1397609532128.44/warc/CC-MAIN-20140416005212-00254-ip-10-147-4-33.ec2.internal.warc.gz"} |
Lagrange Multipliers
April 5th 2009, 09:11 AM #1
Jan 2009
Lagrange Multipliers
Find the maximum value of log(x) + log(y) + 3log(z) on the octant of the sphere x^2 + y^2 + z^2 = r^2.
Deduce that if a,b and c are real numebrs, abc^3<=27[(a+b+c)/(5)]^5
I've done the first part (I think). Not suire at all about the second though.. I guess it uses Lagrange multipliers but cant see what to you as the constraints etc
many thanks
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What is the function for this graph? (picture inside)
September 2nd 2010, 11:54 AM #1
Aug 2010
what type of function is the attached graph? why??
thank you!
p.s: it isn't a parabola.... i was thinking some sort of sin or cosine? i dont know....
also, here's the equation :
Last edited by mr fantastic; September 2nd 2010 at 03:53 PM. Reason: Re-titled in lower case.
If this is $4x^3-32x^2+64x$
then it's called a cubic equation, a polynomial of degree 3.
As Archie Meade indicates, it's a cubic function. Does this answer your question?
September 2nd 2010, 12:02 PM #2
MHF Contributor
Dec 2009
September 2nd 2010, 02:02 PM #3
Oct 2009 | {"url":"http://mathhelpforum.com/algebra/155047-what-function-graph-picture-inside.html","timestamp":"2014-04-18T13:24:32Z","content_type":null,"content_length":"35120","record_id":"<urn:uuid:ca12f0be-a9a8-46c2-abcc-34844c7ad87e>","cc-path":"CC-MAIN-2014-15/segments/1397609533689.29/warc/CC-MAIN-20140416005213-00508-ip-10-147-4-33.ec2.internal.warc.gz"} |
Quantum Mechanics/The Hydrogen Atom
The Hydrogen AtomEdit
By now you're probably familiar with the Bohr model of the atom, which was a great help in classifying the position of fundamental atomic specta lines. However, Bohr lucked out in more ways than one.
The hydrogen atom turns out to be one of the few systems in Quantum Mechanics that we are able to solve almost precisely. This has made it tremendously useful as a model for other Quantum Mechanical
systems, and as a model for the behavior of atoms themselves.
We can assume that the hydrogen atom is governed by the Coulomb potential, namely:
$V(r) = -\frac{e^2}{4 \pi \epsilon _0} \frac{1}{r}$
such that, $H\Psi = -\frac{\hbar ^2}{2m}\frac{d^2\Psi}{dx^2} - \frac{e^2}{4 \pi \epsilon _0} \frac{1}{r}$
Obviously, simply by inspection, we can see that the Hydrogen Atom is a spherical system. Hence it makes more sense to deal with the Hydrogen atom in spherical coordinates. One should remember at
this point that, via Separation of variables, you can obtain the solution to the spherical Laplacian in three-dimensional space:
$abla ^2f = {1 \over r^2} {\partial \over \partial r} \left( r^2 {\partial f \over \partial r} \right) + {1 \over r^2 \sin \theta} {\partial \over \partial \theta} \left( \sin \theta {\partial f \
over \partial \theta} \right) + {1 \over r^2 \sin^2 \theta} {\partial^2 f \over \partial \phi^2}$
the solutions to this function when we use separation of variables inside a Hamiltonian gives us two different functions, the Radial Wave functions (not useful now, but good to know):
$A \cdot J_l(Z_{nl}\frac{r}{a})$
where $J_l$ are the spherical Bessel functions of type l, and $Z_{nl}$ are the zeroes of said Bessel functions.
The other component, the angular component are the Spherical harmonics which are explored in detail on Wikipedia.
Essentially, we're ahead of the game at this point. We already know the answers. The hydrogen wave function will have to involve the spherical solutions to the Laplacian and will be related to both
the angular and radial components. This is most fortunate; for us to attempt to solve the Laplacian while doing the hydrogen atom would be difficult. However, we have some tasks left. The situation
must be normalized, and we must deal with the fact that our potential has a pesky r dependence.
We end up with the results that you knew we were going to. We can write down the hydrogen wave equation as:
$\Psi _{nlm}(r, \phi, \theta) = R_{nl}(r)Y^l_m(\theta, \phi)$
where $Y^l_m(\theta, \phi)$ are the spherical harmonics.
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Solving trig
January 7th 2010, 02:07 AM
Solving trig
Need help on how solve the following equation for 0 <= x <= $2\pi$:
This is what i have done, but it is incorrect.
$tan2x=0$which gets 0
January 7th 2010, 02:22 AM
January 7th 2010, 02:38 AM
Hello Paymemoney
Noting that $\sin2x = 2\sin x \cos x$, we get:
$2\sin x \cos x = \cos x$
You need to be a little careful now. Don't just divide both sides by $\cos x$, because it might be zero, and the first commandment of arithmetic is 'Thou shalt not divide by zero'. So you need to
$2\sin x \cos x = \cos x$
$\Rightarrow \cos x = 0$ or $2\sin x = 1$
Can you go on to complete this now?
January 7th 2010, 02:45 AM
thanks i get how to do it now | {"url":"http://mathhelpforum.com/trigonometry/122745-solving-trig-print.html","timestamp":"2014-04-18T18:38:22Z","content_type":null,"content_length":"10188","record_id":"<urn:uuid:3576b701-8ebe-4f8b-b1c4-2375ffff6c81>","cc-path":"CC-MAIN-2014-15/segments/1397609535095.7/warc/CC-MAIN-20140416005215-00233-ip-10-147-4-33.ec2.internal.warc.gz"} |
Strengths of R (was Re: [SciPy-dev] IPython updated (Emacs works now))
Anthony Rossini rossini at u.washington.edu
Wed Feb 20 11:39:42 CST 2002
On Tue, 19 Feb 2002, eric wrote:
> Hey Tony,
> > Note that one thing I'm working on doing is extending ESS (an Emacs mode for
> > data analysis, usually R or SAS, but also for other "interactive" data analysis
> > stuff) to accomodate iPython, in the hopes of (very slowly) moving towards using
> > R and SciPy for most of my work.
> What are the benefits of R over Python/SciPy. Is there a philosophical
> difference that makes it better suited for statistics (and other things for that
> matter), or is it simply that it has more stats functionality and is much more
> mature? If there is a different philosophy behind it, can you summarize it?
> Maybe we can incorporate some of its strong points into SciPy's stats module.
> Travis Oliphant is working on it as we speak. We could definitely do with some
> of you stats guys input!
John Barnard, who is on this list, should speak up, being one of the few other people that I know of (Doug Bates and some of his students at U Wisc being another exception) that actually use Python for "work" (database or computation).
R (and the language it implements, S) is a language intended for primarily interactive data analysis. So, a good bit of thought has gone into data structures (lists/dataframes in R parlance, which are a bit like arrays with row/column labels which can be used interchangeably instead of row/column numbers), data types such as factors (and factor coding -- some analytic approaches are not robust to choice of coding style for nominal (categorical, non-ordered) data. It has a means for handling missing data (similar to the MA extension for Numeric), and it also has a strong modeling structure, i.e.
fitting linear models (using least squares or weighted least squares) is done in a language which "looks right", i.e.
lm(y ~ x)
fits a model which looks like
y = b x + e, e following the usual linear models assumptions.
as well as smoothing methods (splines, kernels) done in similar fashion. Models are a data object as well, which means that you can act on it appropriately, comparing 2 fitted models, etc, etc.
R, as opposed to the commercial version S or S-PLUS, also has a flexible packaging scheme for add-on packages (for things like Expression Array analysis, spatial-temporal data analysis, graphics, and generalized linear models, marginal models, and it seems like hundreds more. It also can call out to C, Fortran, Java, Python, and Perl (and C++, but that's recent, in the last year or so). Database work is simple, as well, though not up to Perl (or Python) interfaces.
It also has lexical scoping, and is based originally on scheme (though the syntax is like python).
However, it's not a true OO language like python, and some things seem to be hacks. This is mostly an aesthetic problem, not a functional problem.
It's worth a look if you do data analysis. In many ways, the strength is in the ease of programming good graphics, analyses, etc, with output which is easily read and intelligible.
It has problems, in terms of scope of data and speed. It's not as clean to read as python (i.e. I _LIKE_ meaningful indentation, which makes me weird :-), and
isn't as generally glexible (it took me twice as long to write a reader for Flow Cytometry standard data file formats in R than in Python) but annotation of the resulting data is much easier in R than in Python (and default summary statistics, both numerical and graphical, are easier to work with).
So, I don't think I'll be giving up R, but I am looking forward to SciPy (esp things like the sparse array work, which is much more difficult to handle in R, in a nice format).
One thing that I did write was RPVM, for using PVM (LAN cluster library) with R; and patched up PyPVM so that the previous authors work actually worked :-); that is how I'm thinking of doing the interfacing.
In general, R is great for pre- and post-processing small and medium sized datasets, as well as for descriptive and inferential statistics, but for custom analyses, one would still go to C, Fortran, or C++ after prototyping (much like Python).
I can try to say more, but it's hard to describe a full language quickly. See http://www.r-project.org/ for more details (and for ESS, http://software.biostat.washington.edu/statsoft/ess/, if anyone is interested).
HOWEVER, it doesn't have
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The n-Category Café
Brown representability
Posted by Mike Shulman
Time for more debunking of pervasive mathematical myths. Quick, state the Brown representability theorem!
Here’s (roughly) the original version of the theorem that Brown proved:
Theorem. Let $\mathcal{H}$ denote the homotopy category of pointed connected CW complexes, and let $F:\mathcal{H}^{op}\to Set_*$ be a functor to pointed sets which takes coproducts to products, and
homotopy pushouts to weak pullbacks (i.e. spans with the existence, but not the uniqueness, property of a pullback). Then $F$ is a representable functor.
(The restriction to CW complexes is, of course, equivalent to saying we invert weak homotopy equivalences among all spaces.) As a graduate student, it was a revelation to me to realize that Brown
representability is closely related, at least in spirit, to the Adjoint Functor Theorem. In fact, although this isn’t always made explicit, an essential step in the usual proof of the (special, dual)
AFT is the following:
Lemma. Let $\mathcal{C}$ be cocomplete, locally small, and well-copowered, with a small generating set, and let $F:\mathcal{C}^{op}\to Set$ be a functor that takes colimits in $\mathcal{C}$ to
limits. Then $F$ is representable.
Clearly there is a family resemblance. But how does Brown’s theorem get away with such weaker assumptions on limits? The category $\mathcal{H}$ is certainly not cocomplete — it only has weak colimits
— but we still get a strong representability theorem, not merely a “weak” one of some sort. The answer is that Brown uses a stronger generation property, as can be seen from the following abstract
version of Brown’s original theorem (also due to Brown):
Theorem. Let $\mathcal{K}$ be a category with coproducts, weak pushouts, and weak sequential colimits, which admits a strongly generating set $\mathcal{G}$, closed under finite coproducts and weak
pushouts, and such that for $X\in\mathcal{G}$ the hom-functor $\mathcal{K}(X,-)$ takes the weak sequential colimits to actual colimits. Then if $F:\mathcal{K}^{op}\to Set$ takes coproducts to
products and weak pushouts to weak pullbacks, it is representable.
Recall that $\mathcal{G}$ is a generating set if the hom-functors $\mathcal{K}(X,-)$ are jointly faithful, i.e. detect equality of parallel arrows, and a strongly generating set if these functors are
moreover jointly conservative, i.e. detect invertibility of arrows. In the homotopy category of pointed connected spaces, the pointed spheres $\{S^n | n \ge 1 \}$ are of course strongly generating,
since by definition, mapping out of them detects homotopy groups.
Now in surprisingly many places, you may find Brown’s original representability theorem quoted without the adjective “connected”. You may even be tempted to conjecture that it should also still be
true without the adjective “pointed”. However, there is no obvious strongly generating set in either of these cases! If we add $S^0$, then we can detect $\pi_0$-isomorphisms of pointed spaces, but
the other pointed spheres only detect higher homotopy groups of the basepoint component. On the other hand, mapping out of something like $(S^n)_+$ doesn’t detect homotopy groups of other components
either, since in that case the loops are “free” rather than based — and the unpointed spheres $S^n$ have the same problem in the unpointed homotopy category.
(This should, of course, be contrasted with the fact that the $(\infty,1)$-category of unpointed spaces does have a strong generator in the $(\infty,1)$-categorical sense: namely, the point.
Strong-generation is not preserved on passage to homotopy categories!)
In fact, as pointed out recently on MO by Karol Szumiło (thanks Karol!), Brown representability for non-connected and unpointed spaces is false! This was proven back in 1993 by Peter Freyd and Alex
Heller. Their construction begins with the group $G$ that is freely generated by an endo-homomorphism $\phi:G\to G$ and an element $b\in G$ such that $\phi(\phi(x)) = b \cdot \phi(x)\cdot b^{-1}$ for
all $x\in G$.
(Aside: This is a slightly unusual sort of “free generation”. Exercise for homotopy type theorists in the audience: define this group $G$ as a higher inductive type.)
Now $\phi$ induces an endomorphism $B \phi :B G \to B G$, and since free homotopies of continuous maps between classifying spaces of groups correspond to conjugations, $\phi$ is idempotent in the
homotopy category of unpointed spaces. Freyd and Heller proved that this idempotent does not split. (I have not attempted to understand their proof of this, but I find it believable.) Now, however, $
\phi$ induces an idempotent of the representable functor $[-,B G]$, which does split (since idempotents split in $Set$); let $F$ be the splitting. Since $F$ is a retract of a representable, which
takes (weak) colimits to (weak) limits, it does the same; but it is not itself representable because $\phi$ does not split.
Finally, $B \phi _+ :B G_+ \to B G_+$ gives a similar counterexample in pointed, non-connected spaces.
Now the original, and probably still most common, use of the Brown representability theorem is to produce spectra which represent cohomology theories. In that case, the suspension axiom $H^n(X) \cong
H^{n+1}(\Sigma X)$ (for reduced cohomology) means that the whole thing is determined by its behavior on pointed connected spaces (since $\Sigma X$ is connected), so this is not a problem. There are
also different tricks that appear to work for any single functor as long as it lands in (abelian) groups rather than sets. Note also that the collection of spheres $\{S^n | n \in \mathbb{Z}\}$ is
strongly generating in the category of all spectra. But outside of those domains, Brown representability is subtler than I used to think.
(I was first made aware of this issue several years ago, when Peter May and Johann Sigurdsson ran into it while trying to use the abstract version cited above to produce right adjoints to derived
pullback and smash product functors in parametrized homotopy theory. However, I didn’t realize until now that the nonconnected and unpointed versions were actually false even in the classical case of
ordinary spaces.)
Posted at August 24, 2012 5:55 PM UTC
Re: Brown representability
Their construction begins with the group $G$ that is freely generated by an endo-homomorphism $ϕ\colon G \to G$ and an element $b \in G$ such that $\phi(\phi(x)) = b \cdot \phi(x) \cdot b^
{-1}$ for all $x \in G$.
…and that group is, if I’m not mistaken, Thompson’s group $F$.
(Freyd seems to prefer not to call it Thompson’s group, because he believes he discovered it first, and maybe he did. But everyone else calls it Thompson’s group.)
Posted by: Tom Leinster on August 25, 2012 1:41 AM | Permalink | Reply to this
Re: Brown representability
Hah, I was going to point out the appearance of F, but Tom has (fittingly) beaten me to it.
Posted by: Yemon Choi on August 25, 2012 9:21 AM | Permalink | Reply to this
Re: Brown representability
Posted by: David Corfield on August 25, 2012 10:38 PM | Permalink | Reply to this
Re: Brown representability
Ah, excellent! I wasn’t sure whether this post was worth writing, but you’ve proven that it absolutely was, because otherwise I might never have realized that.
Posted by: Mike Shulman on August 26, 2012 1:57 AM | Permalink | Reply to this
Re: Brown representability
Posted by: David Corfield on August 25, 2012 10:23 PM | Permalink | Reply to this
Re: Brown representability
Well, there are variants of the classical AFT. E.g. it’s enough for the category to be total, rather than cocomplete with a generating set. And there are versions for indexed categories and enriched
categories, and more generally for objects in a Yoneda structure.
Posted by: Mike Shulman on August 26, 2012 4:37 AM | Permalink | Reply to this
Re: Brown representability
That’s a very nice summary of the whole story. However, it occurred to me that the notion of strongly generating set is not the right one here. Namely, the functors represented by spheres are not
jointly faithful in the homotopy category of based connected spaces. Think about any nontrivial cohomology operation of nonzero degree, it is represented by a map $K(A, m) \to K(B, n)$ with $m <
n$. It induces zero on homotopy groups just like the trivial map but is nontrivial. On the other hand the functors represented by spheres are jointly conservative and it seems to be all that is
relevant to the Representability Theorem.
I also want to point out that an earlier paper of Heller that I mentioned in the comment to your MO question also gives a counterexample in the non-based case which is even more striking since it is
a half-exact functor which is not even a retract of a representable functor. I admit that I don’t really understand the construction, but it seems that the problem is that the fundamental group of
the representing space would have to be proper class.
Posted by: Karol Szumiło on August 25, 2012 10:41 PM | Permalink | Reply to this
Re: Brown representability
the notion of strongly generating set is not the right one here
An excellent point.
It may be worth mentioning at this point, in case some reader is unaware, that there is no unanimity in the literature on the meaning of “generating set”. I like the definitions I gave in the post
above, because they are systematic: $\mathcal{G}$ is generating if for all $X$, the family of all morphisms $G\to X$ for $G\in\mathcal{G}$ is jointly epimorphic, and strongly generating if that
family is jointly strong-epimorphic (or, more precisely, extremal-epimorphic — so “extremally generating” would be an even better name — but strong and extremal epimorphisms are the same in any
category with pullbacks, so people are often sloppy about the difference, and I’ve decided not to try to fight that battle). However, some people say “separating set” and “generating set” where I say
“generating set” and “strongly generating set”, and I don’t know whether all of these people require a generating set (in their terminology) to also be separating. (In a category with equalizers, the
implication is automatic — but of course the homotopy category lacks equalizers.)
It would be nice to have a word for morphisms which satisfy the property that makes an epimorphism strong, but are not necessarily themselves epic, and similarly for the sort of “strongly
not-generating sets” that we see here.
a counterexample in the non-based case which is even more striking since it is a half-exact functor which is not even a retract of a representable functor. … it seems that the problem is that the
fundamental group of the representing space would have to be proper class.
Thanks for pointing that out here! Although I’m not sure whether I find it more striking to have a functor that fails to be representable because an idempotent fails to split, or a functor that fails
to be representable for cardinality reasons. I think both of them seem like rather “boring” reasons. (-: But I suppose the second example implies that Brown representability also fails in the Cauchy
completion of the unbased homotopy category, which is not obvious if you’ve only seen the first example.
Posted by: Mike Shulman on August 26, 2012 4:51 AM | Permalink | Reply to this
Re: Brown representability
It may be worth mentioning at this point, in case some reader is unaware, that there is no unanimity in the literature on the meaning of “generating set”.
There is one such notion that is especially relevant to this discussion, namely generating sets in triangulated categories. A set of objects $\mathcal{G}$ of a triangulated category $\mathcal{T}$ is
generating if given an object $X$ such that for all $G \in \mathcal{G}$ and $m \in \mathbb{Z}$ we have $\mathcal{T}(\Sigma^m G, X) = 0$, then it follows that $X = 0$. In the presence of coproducts
this is (somewhat non-trivially) equivalent to $\mathcal{G}$ generating all objects $\mathcal{T}$ under shifts, cones and coproducts.
For example in the stable homotopy category the sphere spectrum $\mathbb{S}$ is generating even though $\{\Sigma^m \mathbb{S}\}$ is not a generating set in the sense of your post because of the
existence of non-trivial stable cohomology operations.
The relevance is that if $\mathcal{T}$ has coproducts and a set of compact generators, then every cohomological functor $\mathcal{T}^\mathrm{op} \to \mathbf{Ab}$ is representable. The proof is a
variation on the usual proof, but it works even more neatly because of the stability.
Posted by: Karol Szumiło on August 27, 2012 11:08 AM | Permalink | Reply to this
Re: Brown representability
Isn’t that the same as this sort of “strong non-generation”, though, because a morphism in a triangulated category is an isomorphism iff its cone is zero, and hom-functors are exact?
Posted by: Mike Shulman on August 27, 2012 6:24 PM | Permalink | Reply to this
Re: Brown representability
You are right of course. I noticed that this material is not very well covered in nLab. Maybe I will try to add something.
Posted by: Karol Szumiło on August 28, 2012 10:25 AM | Permalink | Reply to this
Re: Brown representability
Maybe it’s worth adding another “issue” that disappears when you go stable, say the smallness of the generators. I learned that from the paper “Representability theorems for presheaves of spectra” by
Jardine. It contains a version of Brown representability for any compactly generated model category (an object $x$ is compact in $C$ if $colim_i[x,y_i]\to [x,colim_i y_i]$ is a bijection in $Ho(C)$
for every (countable) sequential colimit). However, Brown representability seems to be unknown if the category is $\alpha$-compactly generated for a regular cardinal $\alpha$ (change sequential
colimits for $\alpha$-sequential colimits). In the case $C$ is stable, its homotopy category will be well generated and you can proof Brown representability even without any model structure (as done
in Neeman’s book). Jardine also shows that this problem disappears if we “enrich” the problem, i.e. if we start with a simplicial $\alpha$-compactly generated (cofibrantly generated pointed) model
category $C$ and a simplicial functor $F\colon C^{op}\to sSet_*$ (sending w.e. to w.e and homotopy colimits to homotopy limits), then $F(-)\simeq Map(-,X)$.
Posted by: Oriol Raventós on September 5, 2012 4:59 PM | Permalink | Reply to this
Re: Brown representability
I’d just like to explicitly cite Heller’s paper “On the representability of homotopy functors” J. London Math. Soc. (2) 23 (1981), no. 3, 551–562, which Karol already mentioned on math overflow, as
his Theorem 1.3 is the most general positive result that I know, but I don’t think the paper is as well known as it should be.
I should also mention that the representability of homology functors (which isn’t representability in the sense of category theory, since it involves a symmetric monoidal structure), is much more
Posted by: Dan Christensen on September 26, 2012 5:18 PM | Permalink | Reply to this
Re: Brown representability
Alex Heller’s papers from that time were full of insights that perhaps now are better appreciated, yet you rarely see references to them. He did a lot of work on homotopy Kan extensions, for
Posted by: Tim Porter on September 26, 2012 8:32 PM | Permalink | Reply to this
Re: Brown representability
Indeed they are! I’ve been discovering that his monograph “Homotopy theories” and later paper on stable homotopy theories are full of useful stuff about what we now call derivators.
Posted by: Mike Shulman on September 26, 2012 10:36 PM | Permalink | Reply to this
Re: Brown representability
Yes, in that memoir Alex had some comment about Grothendieck doing something but it seems to be going in a different direction! I seem to remember telling both of them about the others work way back
in 1984 or shortly after, but I do not know if the message was understood. They thought of themselves doing different things.
Jean-Marc Cordier and I worked on trying to extend his proofs (including that of Brown representability) to the simplicially enriched context (as we called it then). That part of our results were
never published but some of the more basic stuff did appear in the paper (Homotopy Coherent Category Theory, Trans. Amer. Math. Soc. 349 (1997) 1-54.).
Posted by: Tim Porter on September 27, 2012 6:45 AM | Permalink | Reply to this
Re: Brown representability
How about a link to `Alex Heller’ if there is such an entry?
And if it’s there, does it due justice to his work?
Posted by: jim stasheff on September 27, 2012 1:37 PM | Permalink | Reply to this
Re: Brown representability
There is now, but it does not do justice to his contribution to the subject. I have made some links but not enough.
Posted by: Tim Porter on September 28, 2012 7:56 AM | Permalink | Reply to this | {"url":"http://golem.ph.utexas.edu/category/2012/08/brown_representability.html","timestamp":"2014-04-16T07:57:07Z","content_type":null,"content_length":"60484","record_id":"<urn:uuid:5caed21b-9d51-40d0-8b6a-015a47b648f8>","cc-path":"CC-MAIN-2014-15/segments/1398223206118.10/warc/CC-MAIN-20140423032006-00030-ip-10-147-4-33.ec2.internal.warc.gz"} |
Finding a point on a line, given another point on the same line, and knowing its dist
June 8th 2010, 03:51 PM #1
May 2010
Finding a point on a line, given another point on the same line, and knowing its dist
Hi there. I'm tryin to find a point, lets call it C. I'm working on a Rē. What I know is that the point belongs to the line L: $y=\displaystyle\frac{x}{2}+\displaystyle\frac{1}{2 }$ And that the
distance to the point B(1,1), that belongs to L is $\sqrt[ ]{20}$.
How can I find it? I know there are two points, cause of the distance over the line.
Hi there. I'm tryin to find a point, lets call it C. I'm working on a Rē. What I know is that the point belongs to the line L: $y=\displaystyle\frac{x}{2}+\displaystyle\frac{1}{2 }$ And that the
distance to the point B(1,1), that belongs to L is $\sqrt[ ]{20}$.
How can I find it? I know there are two points, cause of the distance over the line.
Hi Ulysses,
A straightforward way to solve is to consider the slope of the line, 1/2, which means that for a rise of 1, we have a run of 2.
The hypotenuse length was obtained using the Pythagorean theorem. Then notice that $2 * \sqrt{5} = \sqrt{4*5} = \sqrt{20}$. That means that for a run of 4 (or -4) and rise of 2 (or -2) starting
from B we get to C.
So C = (5, 3) or C = (-3, -1).
Thanks undefined!
June 8th 2010, 04:16 PM #2
June 8th 2010, 04:59 PM #3
May 2010 | {"url":"http://mathhelpforum.com/algebra/148310-finding-point-line-given-another-point-same-line-knowing-its-dist.html","timestamp":"2014-04-17T19:39:46Z","content_type":null,"content_length":"38036","record_id":"<urn:uuid:e6bfb12a-b0d1-40a8-9d5c-eeb7a3f89d46>","cc-path":"CC-MAIN-2014-15/segments/1397609539230.18/warc/CC-MAIN-20140416005219-00547-ip-10-147-4-33.ec2.internal.warc.gz"} |
Factors and Multiples - Hamiltonian Path
Date: 11/02/98 at 22:29:02
From: Joey Dawson
Subject: Factors and Multiples
We have to make a sequence of numbers, all different, each of which is
a factor or a multiple of the one preceding it. The numbers in this
sequence have to be from 1-100, 1 being the last number in the
sequence. An example of this is: 3-15-30-10-5-35-7-21-3. We have to
use as many numbers as possible. Everyone in my class has figured out
the most number of numbers there has to be in the sequence. Can you
help me figure out how many numbers is the most numbers there can be
in the sequence? I would appreciate it very much if you could help me.
Thank you very much.
Your mathematician friend,
Joey Dawson
Date: 11/04/98 at 12:01:56
From: Doctor Peterson
Subject: Re: Factors and Multiples
Hi, Joey. This is a pretty big problem. I haven't found any easy
solution. I'll just tell you how I've been looking at it and maybe you
will be able to finish it up.
I see this in terms of what we call graph theory, which has to do with
things like maps that show relations between objects. To keep things
simple, let's just work with numbers up to 12, rather than all the way
to 100. I have drawn a diagram that shows which numbers could be
adjacent in your sequence, by drawing a line between any pair of
numbers one of which is a divisor of the other:
/ \ / \ / \
9 6----2---8
This says that 2 divides 4, 6, 8, 10, and 12; 3 divides 6, 9, and 12;
4 divides 8 and 12 and is a multiple of 2; 5 divides 10; 6 divides 12
and is a multiple of 2 and 3; 8, 9, 10, and 12 don't divide anything
but are multiples of other numbers. 7 and 11 divide nothing in this
range, so they are disconnected from the rest of this "graph." Of
course 1 could be connected to every number, but since it has to be
last number in the sequence (the answer would be a little different if
that weren't required), we don't need to show it.
You can see that 7 and 11 won't be included in any sequence you can
make, because there is no number other than 1 that they could be next
to. In general, any prime bigger than half the limit (100 in your
problem) will be neither a multiple nor a factor of any number in the
The problem now is just to draw the longest path that goes through
number only once. That's not hard. There's one path that will go
through the whole "connected" part of the graph:
3 12---4
/ \ / \
9 6 2---8
The longest sequence, then, is:
5, 10, 2, 8, 4, 12, 6, 3, 9, 1
(or the reverse), and includes all the numbers that could possibly
been in such a sequence. (This is called a "Hamiltonian path.")
I expanded this to include the numbers 1-25, and found no Hamiltonian
path through all the numbers. I had to leave out three numbers in
addition to 13, 17, 19, and 23 that are not connected to the rest. But
I'm not sure I could prove I found the longest path, without trying
possible paths.
With 100 numbers, it would be even harder. But at least we know a
maximum size for the path (that is, which numbers _have_ to be left
out), although we don't know whether we can actually make a path that
size until we try.
Keep playing with this, and let me know if anyone finds a way other
than brute force to find the longest sequence or to prove that you can
find a Hamiltonian path in the graph for 1-100. You may find some
feature of these particular graphs that makes it easy to find a path.
- Doctor Peterson, The Math Forum | {"url":"http://mathforum.org/library/drmath/view/54255.html","timestamp":"2014-04-16T22:13:08Z","content_type":null,"content_length":"8704","record_id":"<urn:uuid:caa33db1-e92d-4d38-bf9b-0414a831866a>","cc-path":"CC-MAIN-2014-15/segments/1397609525991.2/warc/CC-MAIN-20140416005205-00548-ip-10-147-4-33.ec2.internal.warc.gz"} |
Causal inference using the algorithmic Markov condition
, 2008
"... We describe eight data sets that together formed the CauseEffectPairs task in the Causality Challenge #2: Pot-Luck competition. Each set consists of a sample of a pair of statistically dependent
random variables. One variable is known to cause the other one, but this information was hidden from the ..."
Cited by 8 (7 self)
Add to MetaCart
We describe eight data sets that together formed the CauseEffectPairs task in the Causality Challenge #2: Pot-Luck competition. Each set consists of a sample of a pair of statistically dependent
random variables. One variable is known to cause the other one, but this information was hidden from the participants; the task was to identify which of the two variables was the cause and which one
the effect, based upon the observed sample. The data sets were chosen such that we expect common agreement on the ground truth. Even though part of the statistical dependences may also be due to
hidden common causes, common sense tells us that there is a significant cause-effect relation between the two variables in each pair. We also present baseline results using three different causal
inference methods.
, 2007
"... and their relation to thermodynamics ..."
"... We consider two variables that are related to each other by an invertible function. While it has previously been shown that the dependence structure of the noise can provide hints to determine
which of the two variables is the cause, we presently show that even in the deterministic (noise-free) case ..."
Cited by 6 (2 self)
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We consider two variables that are related to each other by an invertible function. While it has previously been shown that the dependence structure of the noise can provide hints to determine which
of the two variables is the cause, we presently show that even in the deterministic (noise-free) case, there are asymmetries that can be exploited for causal inference. Our method is based on the
idea that if the function and the probability density of the cause are chosen independently, then the distribution of the effect will, in a certain sense, depend on the function. We provide a
theoretical analysis of this method, showing that it also works in the low noise regime, and link it to information geometry. We report strong empirical results on various real-world data sets from
different domains. 1
- In UAI , 2011
"... This work addresses the following question: Under what assumptions on the data generating process can one infer the causal graph from the joint distribution? The approach taken by conditional
independencebased causal discovery methods is based on two assumptions: the Markov condition and faithfulnes ..."
Cited by 3 (1 self)
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This work addresses the following question: Under what assumptions on the data generating process can one infer the causal graph from the joint distribution? The approach taken by conditional
independencebased causal discovery methods is based on two assumptions: the Markov condition and faithfulness. It has been shown that under these assumptions the causal graph can be identified up to
Markov equivalence (some arrows remain undirected) using methods like the PC algorithm. In this work we propose an alternative by defining Identifiable Functional Model Classes (IFMOCs). As our main
theorem we prove that if the data generating process belongs to an IFMOC, one can identify the complete causal graph. To the best of our knowledge this is the first identifiability result of this
kind that is not limited to linear functional relationships. We discuss how the IFMOC assumption and the Markov and faithfulness assumptions relate to each other and explain why we believe that the
IFMOC assumption can be tested more easily on given data. We further provide a practical algorithm that recovers the causal graph from finitely many data; experiments on simulated data support the
theoretical findings. 1
- In NIPS 2008 workshop on causality, volume 7. JMLR W&CP, in press, 2009a
"... The NIPS 2008 workshop on causality provided a forum for researchers from different horizons to share their view on causal modeling and address the difficult question of assessing causal models.
There has been a vivid debate on properly separating the notion of causality from particular models such ..."
Cited by 3 (3 self)
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The NIPS 2008 workshop on causality provided a forum for researchers from different horizons to share their view on causal modeling and address the difficult question of assessing causal models.
There has been a vivid debate on properly separating the notion of causality from particular models such as graphical models, which have been dominating the field in the past few years. Part of the
workshop was dedicated to discussing the results of a challenge, which offered a wide variety of applications of causal modeling. We have regrouped in these proceedings the best papers presented.
Most lectures were videotaped or recorded. All information regarding the challenge and the lectures are found at
"... The causal Markov condition (CMC) is a postulate that links observations to causality. It describes the conditional independences among the observations that are entailed by a causal hypothesis
in terms of a directed acyclic graph. In the conventional setting, the observations are random variables a ..."
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The causal Markov condition (CMC) is a postulate that links observations to causality. It describes the conditional independences among the observations that are entailed by a causal hypothesis in
terms of a directed acyclic graph. In the conventional setting, the observations are random variables and the independence is a statistical one, i.e., the information content of observations is
measured in terms of Shannon entropy. We formulate a generalized CMC for any kind of observations on which independence is defined via an arbitrary submodular information measure. Recently, this has
been discussed for observations in terms of binary strings where information is understood in the sense of Kolmogorov complexity. Our approach enables us to find computable alternatives to Kolmogorov
complexity, e.g., the length of a text after applying existing data compression schemes. We show that our CMC is justified if one restricts the attention to a class of causal mechanisms that is
adapted to the respective information measure. Our justification is similar to deriving the statistical CMC from functional models of causality, where every variable is a deterministic function of
its observed causes and an unobserved noise term. Our experiments on real data demonstrate the performance of compression based causal inference. 1
, 2009
"... Telling cause from effect based on high-dimensional observations ..."
"... Directed acyclic graph (DAG) models are popular tools for describing causal relationships and for guiding attempts to learn them from data. They appear to supply a means of extracting causal
conclusions from probabilistic conditional independence properties inferred from purely observational data. I ..."
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Directed acyclic graph (DAG) models are popular tools for describing causal relationships and for guiding attempts to learn them from data. They appear to supply a means of extracting causal
conclusions from probabilistic conditional independence properties inferred from purely observational data. I take a critical look at this enterprise, and suggest that it is in need of more, and more
explicit, methodological and philosophical justification than it typically receives. In particular, I argue for the value of a clean separation between formal causal language and intuitive causal | {"url":"http://citeseerx.ist.psu.edu/showciting?cid=10431007","timestamp":"2014-04-16T10:54:58Z","content_type":null,"content_length":"33968","record_id":"<urn:uuid:19b42a46-9b9a-4077-8d8c-47c5b7708e88>","cc-path":"CC-MAIN-2014-15/segments/1397609523265.25/warc/CC-MAIN-20140416005203-00515-ip-10-147-4-33.ec2.internal.warc.gz"} |
MathGroup Archive: June 2010 [00017]
[Date Index] [Thread Index] [Author Index]
Re: Expanding Integrals with constants and 'unknown' functions
• To: mathgroup at smc.vnet.net
• Subject: [mg110068] Re: Expanding Integrals with constants and 'unknown' functions
• From: "David Park" <djmpark at comcast.net>
• Date: Tue, 1 Jun 2010 04:20:45 -0400 (EDT)
The Presentations package ($50) from my web site has functionality for this.
The Student's Integral section allows you to write integrate[...] (small i)
instead of Integrate[...] and obtain the integral in a held form. There are
then commands to manipulate the integral (operating on the integrand, change
of variable, integration by parts, trigonometric substitution, and breaking
the integral out) and then evaluate using Integrate, NIntegrate, or a custom
integral table. Also, you can pass any assumptions at the time you use
Integrate, instead of when the integral is initially written, so it always
formats nicely.
The following code also uses another Presentations routine, MapLevelParts,
that allows you to apply an operation to a subset of level parts in an
expression - here the two integrals that can be evaluated without asking
Mathematica to evaluate the integral with the undefined function. I copied
the output as text, with box structures, so if you copy from the email back
into a notebook is should format as a regular expression.
integrate[a + z + s[z], {z, clow, chigh}]
% // BreakoutIntegral
% // MapLevelParts[UseIntegrate[], {{1, 2}}]
\*SubsuperscriptBox[\(\[Integral]\), \(clow\), \(chigh\)]\(\((a + z +
s[z])\) \[DifferentialD]z\)\)
a \!\(
\*SubsuperscriptBox[\(\[Integral]\), \(clow\), \(chigh\)]\(1
\*SubsuperscriptBox[\(\[Integral]\), \(clow\), \(chigh\)]\(z
\*SubsuperscriptBox[\(\[Integral]\), \(clow\), \(chigh\)]\(s[z]
chigh^2/2+a (chigh-clow)-clow^2/2+\!\(
\*SubsuperscriptBox[\(\[Integral]\), \(clow\), \(chigh\)]\(s[z]
David Park
djmpark at comcast.net
From: Jesse Perla [mailto:jesseperla at gmail.com]
I have an integral involving constants and an 'unknown' function. I
would like to expand it out to solve for the constants and keep the
integrals of the unknown function as expected.
Integrate[a + z + s[z], {z, clow, chigh}]
I want to get out:
(a*chigh + chigh^2/2 - a*clow - clow^2/2) + Integrate[s[z], {z, clow,
However, FullSimplify, etc. don't seem to do anything with this. Any | {"url":"http://forums.wolfram.com/mathgroup/archive/2010/Jun/msg00017.html","timestamp":"2014-04-19T02:09:45Z","content_type":null,"content_length":"27397","record_id":"<urn:uuid:9f491cf3-69a1-4b71-b129-9a03883d5be4>","cc-path":"CC-MAIN-2014-15/segments/1397609535745.0/warc/CC-MAIN-20140416005215-00150-ip-10-147-4-33.ec2.internal.warc.gz"} |
Fuzzy logics as the logics of chains
From Mathfuzzlog
Title: Fuzzy logics as the logics of chains
Journal: Fuzzy Sets and Systems
Volume 157
Number 5
Pages: 604-610
Year: 2006
Download from the publisher
Post-publication comments
Even though we still consider our arguments valid, it seems that the proposed delimitation of the class of fuzzy logics (in the technical sense) has not become widely accepted in the community of
mathematical fuzzy logic. Perhaps a more neutral term for Cintula's class of weakly implicative fuzzy logics would be more acceptable to other researchers---e.g., semilinear logics (Nick Galatos has
pointed out to us that semi-X is often used in universal algebra for the property that all irreducibles are X, like in semi-simple).
The best behaved among "semilinear weakly implicative logics" seem to be those which are substructural in Ono's sense (i.e., logics of residuated lattices),^[1] as they also internalize the deductive
role of conjunction and implication. Since the name substructural fuzzy logics has already been taken for a slightly different concept by Metcalfe and Montagna in their JSL paper, a suitable name for
this class of logics (also called deductive fuzzy logics in the paper On the difference between traditional and deductive fuzzy logic) could be "semilinear substructural logics", i.e., the logics of
semilinear residuated lattices.
These proposals, however, are not intended to suggest replacing the name "mathematical fuzzy logic" for the discipline of logic, and only regard the name of the formally defined class of logics
(which is not regarded as coinciding with the agenda of mathematical fuzzy logic by many researchers anyway).
-- LBehounek 20:10, 2 September 2008 (CEST)
(The comment was inspired by a discussion with Petr Cintula, Nick Galatos, Rosťa Horčík, Petr Hájek, and Carles Noguera in Prague, August 2008. However, it only expresses my own opinion
immediately after the discussion; the opinions of other participants of the discussion may differ, and my own opinion may easily change in the future.)
References for this page
1. ↑ Ono, H., 2003, "Substructural logics and residuated lattices — an introduction". In F.V. Hendricks, J. Malinowski (eds.): Trends in Logic: 50 Years of Studia Logica, Trends in Logic 20: | {"url":"http://www.mathfuzzlog.org/index.php/Fuzzy_logics_as_the_logics_of_chains","timestamp":"2014-04-21T15:00:31Z","content_type":null,"content_length":"19351","record_id":"<urn:uuid:a2b382ee-0f6f-426a-bbcc-06ff94bab60a>","cc-path":"CC-MAIN-2014-15/segments/1398223211700.16/warc/CC-MAIN-20140423032011-00613-ip-10-147-4-33.ec2.internal.warc.gz"} |
MathGroup Archive: January 2010 [00082]
[Date Index] [Thread Index] [Author Index]
Re: Difficulty with NDSolve (and DSolve)
• To: mathgroup at smc.vnet.net
• Subject: [mg106167] Re: Difficulty with NDSolve (and DSolve)
• From: Mark McClure <mcmcclur at unca.edu>
• Date: Sun, 3 Jan 2010 03:43:31 -0500 (EST)
• References: <201001021004.FAA07409@smc.vnet.net>
On Sat, Jan 2, 2010 at 5:04 AM, KK <kknatarajan at yahoo.com> wrote:
> I am trying to numerically solve a differential equation. However, I
> am encountering difficulty with Mathematica in generating valid
> numerical solutions even for a special case of that equation.
> The differential equation for the special case is:
> F'[x] == - (2-F[x])^2/(1-2 x + x F[x]) and
> F[1]==1.
> These equations are defined for x in (0,1). Moreover, for my context,
> I am only interested in solutions with F[x] in the range <1.
Of course, the basic problem is that the solution is not one-to-one in
a neighborhood of your initial condition. Thus, let's set up an
initial condition y0 at x0=1/2 and try to choose y0 so that the
solution passes through (1,1) - kind of like the shooting method for
solving boundary value problems. We can set up a function that tells
us the value of the solution at x=1 as a function of the initial
condition at x0=1/2 like as follows:
In[1]:= valAt1[y0_?NumericQ] := Quiet[Check[(F[x] /.
NDSolve[{F'[x] == -(2 - F[x])^2/(1 - 2 x + x F[x]),
F[1/2] == y0}, F[x], {x, 1/2, 1}][[1, 1]]) /. x -> 1,
In[2]:= {valAt1[-4], valAt1[-3]}
Out[2]= {0.208967, bad}
Note that we use Check to check for error messages in case that x=1 is
outside the range of the solution. The computation shows that this
singularity between y0=-4 and y0=-3. We can use a bisection method to
find more precisely where this happens like so:
In[3]:= a = -4;
In[4]:= b = -3;
In[5]:= Do[
c = (a + b)/2;
If[valAt1[c] === bad, b = c, a = c],
In[6]:= N[a]
Out[6]= -3.35669
Now, we find the solution taking y0=a.
In[7]:= Clear[F];
In[8]:= F[x_] = F[x] /.
NDSolve[{F'[x] == -(2 - F[x])^2/(1 - 2 x + x F[x]),
F[1/2] == a}, F[x], {x, 0.001, 1}][[1]];
In[9]:= F[1]
Out[9]:= 1.
The computation shows that it worked.
Hope that helps,
Mark McClure
• References: | {"url":"http://forums.wolfram.com/mathgroup/archive/2010/Jan/msg00082.html","timestamp":"2014-04-19T04:42:25Z","content_type":null,"content_length":"27250","record_id":"<urn:uuid:c36da26e-7b06-4399-b0ed-b2500ba9d6c6>","cc-path":"CC-MAIN-2014-15/segments/1398223211700.16/warc/CC-MAIN-20140423032011-00254-ip-10-147-4-33.ec2.internal.warc.gz"} |
Monte Carlo Inference is a graduate level course taught in serial with Time Series (last 16 lectures). This year it is being offered in the Lent term. It covers topics like random number generation,
importance sampling, the bootstrap and jacknife for estimation and hypothesis tests, MCMC, sequential importance sampling, simulated annealing, and the EM algorithm.
Course syllabus (including recommended texts)
• The revision examples class will be held in MR4 on 18 May, from 4-6pm. The last three years worth of exam questions (except on uniform pseudo-random number generation) will be covered
Example sheets:
• Example Sheet 1 (12 Feb 2010)
Examples class 24 Feb, 4-6pm in MR9 (CMS) -- Solutions
• Example Sheet 2 (22 Feb 2010), requires the data file ar3.txt and optionally the code rmultnorm.R
Examples class 9 March, 4-6pm in MR13 (CMS) -- Solutions
• Example Sheet 3 (8 March 2010)
Examples class 4 May, 4-6pm in MR11 (CMS) -- Solutions
□ R code to accompany #3: boh.R
• reject.R: Rejection sampling from Lecture 2
• rou.R: Ratio of Uniforms sampling from Lecture 3
• is.R: Importance Sampling from Lectures 4 and 5
• resample.R: Resampling methods (Jackknife and Bootstrap) from Lecture 8
• gibbs.R which requires gibbs_norm.R: Gibbs sampling from Lecture 9
• da.R: Data augmentation from Lecture 9
• mh.R which requires mh_norm.R: the Metropolis Hastings algorithm from Lecture 10
• ess.R which requires gibbs_norm.R and mh_norm.R: Effective sample size from Lecture 11
• rjmcmc.R: Reversible Jump MCMC from Lecture 12
• sa.R: Simulated Annealing from Lecture 14
• em.R: Expectation Maximisation from Lecture 16
Robert B. Gramacy -- 2010
straight to the top | {"url":"http://www.statslab.cam.ac.uk/~bobby/teaching.html","timestamp":"2014-04-16T10:16:44Z","content_type":null,"content_length":"17765","record_id":"<urn:uuid:e4e4a3ae-bdf8-4c69-b177-23c841007c29>","cc-path":"CC-MAIN-2014-15/segments/1397609523265.25/warc/CC-MAIN-20140416005203-00143-ip-10-147-4-33.ec2.internal.warc.gz"} |
Prime Curios!: 5
Curios Search:
5 = 12 in base 3. There is no other prime p that can be expressed as p = 12...n in base n+1, because p has n as a factor if n is odd, or n/2, if n is even. [Necula]
Submitted: 2004-02-03 17:12:04; Last Modified: 2008-01-30 11:28:00. | {"url":"http://primes.utm.edu/curios/cpage/8188.html","timestamp":"2014-04-19T11:56:49Z","content_type":null,"content_length":"4616","record_id":"<urn:uuid:3c5dee9d-cfdd-419a-84d3-24c5c5325416>","cc-path":"CC-MAIN-2014-15/segments/1397609537186.46/warc/CC-MAIN-20140416005217-00083-ip-10-147-4-33.ec2.internal.warc.gz"} |
The Prime Pages: Paul Underwood
A titan, as defined by Samuel Yates, is anyone who has found a titanic prime. This page provides data on those that have found these primes. The data below only reflect on the primes currently on
the list. (Many of the terms that are used here are explained on another page.)
Proof-code(s): p6, x43, p98, p102, p62, L97, L158, L256, p308, p367, p372, c70
E-mail address: paulunderwood@mindless.com
Web page: http://www.mersenneforum.org/showpost.php?p=298027&postcount=44
Username: Underwood (entry created on 01/18/2000)
Database id: 181 (entry last modified on 04/18/2014)
Active primes: on current list: 7.5 (unweighted total: 13), rank by number 112
Total primes: number ever on any list: 1520.78 (unweighted total: 2977)
Production score: for current list 46 (normalized: 32), total 46.8134, rank by score 103
Largest prime: 3 · 2^3136255 - 1 (944108 digits) via code L256 on 03/08/2007
Most recent: V(89849) (18778 digits) via code c70 on 01/09/2014
Entrance Rank: mean 78386.92 (minimum 12, maximum 103515)
Surname: Underwood (used for alphabetizing and in codes)
Unverified primes are omitted from counts and lists until verification completed. | {"url":"http://primes.utm.edu/bios/page.php?id=181","timestamp":"2014-04-18T20:44:19Z","content_type":null,"content_length":"18379","record_id":"<urn:uuid:7b18d7d0-d759-49e6-a888-2433e79b2cf7>","cc-path":"CC-MAIN-2014-15/segments/1398223202457.0/warc/CC-MAIN-20140423032002-00636-ip-10-147-4-33.ec2.internal.warc.gz"} |
Need help differentiating an expression. Answer is given just unsure of the steps.
November 11th 2011, 03:00 PM #1
Nov 2009
Need help differentiating an expression. Answer is given just unsure of the steps.
Hello, this problem states to differentiate the expression with respect to q. I am not sure of the steps taken to get the final result.
This is an economics problem differentiating the variable cost function to get the marginal cost function.
Re: Need help differentiating an expression. Answer is given just unsure of the steps
So you want to differentiate:
$6(\frac{q}{10k^{0.4}})^{1.67}$ with respect to q?
In which case, we have:
= $\frac{6q^{1.67}}{(10k^{0.4})^{1.67}}$
But the $\frac{6}{(10k^{0.4})^{1.67}}$(everything except the $q^{1.67}$) is just constant. It's a stable value, like 4 or 7 - it isn't a variable.
For the sake of simplicity, I'm going to let $\frac{6}{(10k^{0.4})^{1.67}}=c$
When we differentiate $4x^n$ with respect to $x$, we get: $4nx^{n-1}$.
When we differentiate $ax^n$ with respect to $x$, we get: $anx^{n-1}$
We have: $c\times~q^{1.67}$
So, differentiating, we get: $1.67c\times~q^{0.67}$
Then, slipping c back in as $\frac{6}{(10k^{0.4})^{1.67}}$, we have:
$\frac{1.67\times~6\times~q^{0.67}}{(10k^{0.4})^{1. 67}}$
and $1.67\times~6=10.02$ which is where that number arises.
November 11th 2011, 03:31 PM #2 | {"url":"http://mathhelpforum.com/calculus/191672-need-help-differentiating-expression-answer-given-just-unsure-steps.html","timestamp":"2014-04-16T11:03:34Z","content_type":null,"content_length":"37387","record_id":"<urn:uuid:baa9c500-7e21-4fbe-ae72-72c113a4954b>","cc-path":"CC-MAIN-2014-15/segments/1397609523265.25/warc/CC-MAIN-20140416005203-00546-ip-10-147-4-33.ec2.internal.warc.gz"} |
The graduate student seminar meets from 3:30 to 4:30 pm Thursdays in Goldsmith 317. Want to give a talk? Email Steve Hermes at srhermes@etc... for details.
January 17
Organizational Meeting
January 24
A Gentle Introduction to A[∞]-Algebras
Stephen Hermes
In topology, the singular cochain complex of a space comes with some extra structure which allows us to define an associative product called the cup product. This structure of a chain complex with an
associative product appears throughout various branches of mathematics, leading us to the notion of a differential graded algebra (dga). Unfortunately, these associative products are not well-behaved
with respect to homotopy: a homotopy equivalent chain complex may not inherit an associative product from one that does.
In this talk we will discuss how we can alleviate this problem by introducing the notion of A[∞]-algebras. Along the way we will see how to cure all sorts of non-ideal behaviour of dgas.
January 31
Ramsey Theory
Nick Stevenson
By the pigeonhole principle of Dirchlet, if we partition 'many' objects into a 'small' number of classes, there will be 'many' objects contained in a single class. Suppose that these objects are now
interrelated in a 'small' number of ways. Given a set of 'many' such objects, will there be 'many' objects all of which are related in a single way? Ramsey Theory studies this question. I will begin
by surveying the Ramsey theory of the edge colourings of finite graphs and hypergraphs, then moving on to partition calculus (the Ramsey theory of infinite sets), an important branch of set theory.
February 7
Kedlaya's Point Counting Algorithm
Andrija Peruničić
Let X be a hyperelliptic curve of genus g defined over a finite field. Kedlaya developed an algorithm for (quickly) counting the number of points on this curve. It can also be used for computing its
zeta function. The goal of my talk will be to describe this algorithm.
First, I will describe how to lift objects associated to the curve to a p-adic setting. The advantage of doing so is that there is a well-behaved cohomology theory, called Monsky-Washnitzer
cohomology, which allows us to use the Weil conjectures. With their help, we will be able to explicitly compute the zeta function of the curve under consideration.
You will be able to follow the algorithm even if you've never dealt with the p-adic numbers before, and are only now learning about cohomology!
February 14
CAT(0) Cube Complexes and Hyperplanes
Mike Carr
Department policy states that graduate students must see the definitions of the CAT(0) condition and of Right-Angled Artin Groups at least once per semester. In addition to fulfilling our quota, we
will describe the relationship between a cube complex and its hyperplanes. Finally, we'll outline an application: a "geometric" proof of a theorem from combinatorial group theory.
February 28
No Seminar
No Seminar due to Brandeis-Harvard-MIT-Northeastern Colloquium
March 7
No Seminar
March 14
π Day
Chris Ohrt
March 21
No Seminar
No Seminar due to Brandeis-Harvard-MIT-Northeastern Colloquium
April 4
Lower Bounds for Knot Genus via Contact Topology
Katherine Raoux
Computing the genus of an arbitrary knot can be difficult. Adding a contact structure to R^3, allows us to compute a lower bound for the genus of an arbitrary Seifert surface, of any Legendrian knot.
Since any knot can be nicely approximated by a Legendrian knot, this is also a lower bound for the genus of the knot. I will show that the trefoil is not the unknot.
April 11
I Got 99 Problems but a Word Ain't One:
A Geometric Method to Solve the Word Problem in Right-Angled Artin Groups
Matt Cordes
Given a finitely generated group, the word problem asks if there is an algorithm to determine if a word in the generators of the group is the identity. The word problem is not solvable in general,
but it is in the case of right-angled Artin groups. I will give a geometrically flavored proof of this fact.
April 18
Alyson Burchardt
April 25
John Bergdall
Previous Semesters | {"url":"http://people.brandeis.edu/~srhermes/gss.html","timestamp":"2014-04-19T04:51:42Z","content_type":null,"content_length":"5723","record_id":"<urn:uuid:c01c5caa-98a2-4d1c-9779-e67c0dca6a0a>","cc-path":"CC-MAIN-2014-15/segments/1398223206672.15/warc/CC-MAIN-20140423032006-00455-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Topic: Re: Needing a Random List of Non-repeating Values whose Range and Length
Replies: 0
Re: Needing a Random List of Non-repeating Values whose Range and Length
Posted: Jan 19, 2013 1:13 AM
On 1/18/13 at 12:54 AM, brtubb090530@cox.net wrote:
>Consider the following:
>ril[range_Integer]:= RandomInteger[{1,range},range]
>This function is almost what I want; but I need one which doesn't
>include any repeated values. This is intended for use, for example,
>for a card deck, or dice, etc.
Perhaps you should look at RandomSample. For example, if you
associate the integers from 1 to 52 with the 52 possible cards
(excluding jokers) in a deck
would return 5 values representing a random 5 card draw from the deck | {"url":"http://mathforum.org/kb/thread.jspa?threadID=2430024&messageID=8109711","timestamp":"2014-04-16T11:27:08Z","content_type":null,"content_length":"14228","record_id":"<urn:uuid:59e5ac27-59c5-48b1-9ba4-d023bc6ee767>","cc-path":"CC-MAIN-2014-15/segments/1397609523265.25/warc/CC-MAIN-20140416005203-00159-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Dynamics of Nonholonomic Systems
             
Translations of The goal of this book is to give a comprehensive and systematic exposition of the mechanics of nonholonomic systems, including the kinematics and dynamics of nonholonomic systems
Mathematical with classical nonholonomic constraints, the theory of stability of nonholonomic systems, technical problems of the directional stability of rolling systems, and the general
Monographs theory of electrical machines. The book contains a large number of examples and illustrations.
1972; 518 pp;
Volume: 33
reprinted 2004
List Price: US$135
Member Price:
Order Code: MMONO/ | {"url":"http://ams.org/bookstore?fn=20&arg1=mmonoseries&ikey=MMONO-33-S","timestamp":"2014-04-20T16:08:46Z","content_type":null,"content_length":"13976","record_id":"<urn:uuid:49adb929-2677-4bcf-be45-45d110bfefa3>","cc-path":"CC-MAIN-2014-15/segments/1397609538824.34/warc/CC-MAIN-20140416005218-00614-ip-10-147-4-33.ec2.internal.warc.gz"} |
Index of /sandbox/latex_directive
System Message: ERROR/3 (../sandbox/latex_directive/README.txt, line 5)
Unknown interpreted text role "latex".
.. role:: m(latex)
Role and directive
There is a role called latex that can be used for inline latex expressions: :latex:`$\psi(r) = \exp(-2r)$` will produce :m:`$\psi(r)=\exp(-2r)$`. Inside the back-tics you can write anything you would
write in a LaTeX ducument.
System Message: ERROR/3 (../sandbox/latex_directive/README.txt, line 16); backlink
Unknown interpreted text role "m".
For producing displayed math (like an equation* environment in a LaTeX document) there is a latex role. If you write:
.. latex-math::
$\psi(r) = e^{-2r}$
you will get:
System Message: ERROR/3 (../sandbox/latex_directive/README.txt, line 30)
Unknown directive type "latex".
.. latex::
$\psi(r) = e^{-2r}$ | {"url":"http://docutils.sourceforge.net/sandbox/latex_directive/","timestamp":"2014-04-17T03:56:07Z","content_type":null,"content_length":"5519","record_id":"<urn:uuid:d077fd93-be9e-47ce-b6da-e35281092375>","cc-path":"CC-MAIN-2014-15/segments/1397609537754.12/warc/CC-MAIN-20140416005217-00573-ip-10-147-4-33.ec2.internal.warc.gz"} |
e^(i*pi) = -1: pi = 0 ?
Date: 10/17/97 at 11:29:15
From: John K. Koehler
Subject: Weird tricks
Dr. Math,
I know that from a certain trig identity we get the equation
e^(i*pi) = -1. I have been playing around with this equation and have
found some disturbing things. I hope that you can help me.
e^(2*i*pi) = 1
e^(-2*pi) = 1 (raised both sides to i and 1^n is 1)
-2*pi = 0 (took the ln of both sides and ln(1) = 0)
pi = 0 !
i*pi = ln(-1) {1}
ln(-n) = ln(-1) + ln(n)
therefore by {1}
ln(-n) = i*pi + ln(n)
i^2 = -1
e^(i*pi) = i^2
e^(-pi) = i^(2i)
e^(-pi/2) = i^i
e^(-pi/2) is a real number and therefore so is i^i
Can you please tell me why I get such ridiculus results?
I asked my calc2 teacher last year - but all he could tell me
was that imaginary numbers don't work like reals, and I still
don't see how that would change anything.
Thanks in advance,
Date: 10/17/97 at 13:08:36
From: Doctor Bombelli
Subject: Re: Weird tricks
Well, John, not all the results you have are ridiculous! In fact, you
and your teacher are both right; complex numbers don't always work
like real numbers, but sometimes they do.
Complex numbers can be written in many equivalent ways:
z = x+iy
z = re^*(it) where t is the angle the point (x,y) makes with the
positive x axis.
t is called the argument of z--arg(z)
z = r[cos(t)+i sin(t)]
In fact, the definition of e^(it) is cos(t)+i sin(t).
This is why we have e^(-ipi) = cos(pi)+i sin(pi)= -1.
We would like w = log(z) and z = e^w to match up, just as for real
Let z = re^(it) and w = u+iv.
z = e^w = e^(u+iv) = e^u*e^(iv) and z is also re(^it)
Match the real part and the "i part" (the imaginary part).
r = e^u and e^(it) = e^(iv)
So u = ln(r) and v can be any value with cos(v) = cos(t)
[since e^(it) is cos(t)+i sin(t)].
So we say, for complex numbers, ln(z)=ln(r)+i arg(z)+i2kpi
(k an integer). In fact, ln(z) has many values.
We also define, in this way, that
z^a = e^[a ln(z)] when a is complex.
(Check that this works for real numbers a, also.)
Note that 1^a = e^[ln(1)] = e^[0+i2kpi], so only one of the answers
is e^0 = 1.
Here is your problem:
e^(2*i*pi) = 1
e^(-2*pi) = 1 (raised both sides to i and 1^n is 1)
-2*pi = 0 (took the ln of both sides and ln(1) = 0)
pi = 0
In steps 2 and 3 you don't have one-to-one functions any more.
(The reason e^x = 1 implies x = 0 is because the real logarithm
function is one-to-one. It is kind of like why x^3 = 1 gives x = 1
and x^2 = 1 gives x = 1 or -1. The cube root is one-to-one, but
the square root isn't.)
Now in your second experiment:
i*pi = ln(-1) {1}
ln(-n) = ln(-1) + ln(n)
therefore by {1}
ln(-n) = i*pi + ln(n)
This is okay.
ln(-n) = ln(n) + i pi + i 2kpi from the definition above.
In your third experiment:
i^2 = -1
e^(i*pi) = i^2
e^(-pi) = i^(2i)
e^(-pi/2) = i^i
e^(-pi/2) is a real number and therefore so is i^i
i^i is e^[i ln(i)], and ln(i) is ln(1) + i pi/2 + i 2kpi
So... i^i = e^[0 + i^2 pi/2+ i^2 2kpi] = e^[- pi/2 - 2kpi]
which is a real number.
I commend you on playing around with this stuff. That is how new
mathematics is learned, on all levels.
-Doctor Bombelli, The Math Forum
Check out our web site! http://mathforum.org/dr.math/ | {"url":"http://mathforum.org/library/drmath/view/53907.html","timestamp":"2014-04-16T17:22:39Z","content_type":null,"content_length":"8387","record_id":"<urn:uuid:4fc767a4-418e-4d6c-8678-c3d9113e656a>","cc-path":"CC-MAIN-2014-15/segments/1397609538423.10/warc/CC-MAIN-20140416005218-00381-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Abstract: FUZZYSTUDIOTM1.0 - W.A.R.P.2.0 using FUZZYSTUDIOTM2.0 - MATLAB - C Language - Fu.L.L. (Fuzzy
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, 17. Solar Energy Fuzzy Control System Structure Fuzzy Logic Controller + Fuzzy Knowledge Base , C ds 30 with 8051 Solar Charge Controller driver circuits Solar Sensor Light Intensity
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Abstract: desired height, is transmitted to the Levitating a Beach Ball Using Fuzzy Logic Wanting a
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Braingle: 'Consecutive numbers' Brain Teaser
Consecutive numbers
Math brain teasers require computations to solve.
Puzzle ID: #14553
Category: Math
Submitted By: Gerd
Between 1000 and 2000 you can get each integer as the sum of nonnegative consecutive integers. For example,
147+148+149+150+151+152+153 = 1050
There is only one number that you cannot get.
What is this number?
Show Hint
Show Answer
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Programming project help?
10-08-2008 #1
Registered User
Join Date
Oct 2008
Programming project help?
Hello, I am really confused as to how to start my project that I was given for my C++ course.
The instructions are:
Write a function declaration for a function that computes interest on a credit card account balance. The function takes arguments for the initial balance, the monthly interest rate, and the
number of months for which interest must be paid. The value returned is the interest due. Do not forgetto compound the interest- that is, to charge interest on the interest due. This interest due
is added into the balance due, and the interest for the next month is computed using this larger balance. Use a while loop. Embed the function in a program that reads the values for the interest
rate, initial account balance, and number of months, then outputs ths interest due. Embed your function definition in a program that lets thge user compute the interest due on the credit account
balance. The program should allow the user to repeat the calculation until the user said he or she wants to end the program.
The information that I am given through my instructor is:
Read the description carefully, insure that you allow the user a choice to run the program or exit.
The function prototype is:
double interest (double initBalance, double rate, int months);
Your initial output to the user should be something like this:
Credit card interest
Enter: initial balance, monthly interest rate as
a decimal fraction, e.g. for 1.5% per month write 0.015
and the number of months the bill has run.
I will give you the interest that has accumulated.
100 .1 2
Interest accumulated = $21.00
Y or y repeats, any other character quits
Credit card interest
Enter: initial balance, monthly interest rate as
a decimal fraction, e.g. for 1.5% per month write 0.015
and months the bill has run.
I will give you the interest that has accumulated.
100 .1 3
Interest accumulated = $33.10
Y or y repeats, any other character quits
Therefore, I am confused as to how to start with the function prototype that he gave me to start with...
double interest (double initBalance, double rate, int months);
Is there a possibility that anyone can lead me in the right direction to start with?
Thank you very much,
Debbie Bremer
The prototype looks good. Surely you can program a function that compounds interest.
I dont know how to type in the code for coompound interest. Basically right now, I have the program to display the instructions...but as far as the math goes for interest...im confused. Can
anyone help?
Google for "compound interest calculation", perhaps?
Compilers can produce warnings - make the compiler programmers happy: Use them!
Please don't PM me for help - and no, I don't do help over instant messengers.
How would you calculate it by hand? Write it out in non-code, then put it in code.
#include <cmath>
double interest (double initBalance, double rate, int months)
return initBalance * pow(M_E, rate*months);
This will do continuously compounded interest... but I am no economics expert so its the formula I know better. Of course this is a start, and just a mere example since this is NOT the formula
you were asked to use.
For some reason I cannot get my math to work.
This is what I have so far:
#include <iostream>
using namespace std;
int main ( )
int months, count = 1;
double init_Balance, rate, interest = 0, new_balance, total_Interest = 0, int_Accumulated;
char repeats;
total_Interest = 0;
cout << " Credit card interest\n ";
cout << "Enter: Initial balance, monthly interest rate as a decimal fraction, e.g. for 1.5% per month write 0.015, and the number of months the bill has run.\n ";
cout << "I will give you the interest that has accumulated.\n ";
cin >> init_Balance, rate, months;
for ( int count; count <= months; count++)
interest = ( rate * init_Balance );
new_balance = ( init_Balance + interest );
total_Interest = ( interest + );
cout << count ++;
cout << "Interest accumulated = $\n";
cin >> int_Accumulated;
cout << "Y or y repeats, any other character quits. ";
}while ( repeats != 'Y' && repeats != 'y' );
return 0;
Please indent your code. You cannot possible ask us to read your unreadable code mess.
For information on how to enable C++11 on your compiler, look here.
よく聞くがいい！私は天才だからね！ ^_^
ok sorry about that.
Well? Edit your original post and indent.
All the buzzt!
"There is not now, nor has there ever been, nor will there ever be, any programming language in which it is the least bit difficult to write bad code."
- Flon's Law
#include <iostream>
using namespace std;
int main ( )
int months, count = 1;
double init_Balance, rate, interest = 0, new_balance, total_Interest = 0, int_Accumulated;
char repeats;
total_Interest = 0;
cout << " Credit card interest\n ";
cout << "Enter: Initial balance, monthly interest rate as a decimal fraction, e.g. for 1.5% per month write 0.015, and the number of months the bill has run.\n ";
cout << "I will give you the interest that has accumulated.\n ";
cin >> init_Balance >> rate >> months;
for ( int count; count <= months; count++)
interest = ( rate * init_Balance );
new_balance = ( init_Balance + interest );
total_Interest = ( interest + );
cout << count ++;
cout << "Interest accumulated = $\n";
cin >> int_Accumulated;
cout << "Y or y repeats, any other character quits. ";
}while ( repeats != 'Y' && repeats != 'y' );
return 0;
The code I just posted will not compute my math in the program. can anyone help me fix my problem?
You don't initialize count in the for-loop, which means it has a pretty much random initial value - very likely more than months, so the loop never runs.
All the buzzt!
"There is not now, nor has there ever been, nor will there ever be, any programming language in which it is the least bit difficult to write bad code."
- Flon's Law
im a little confused by what you mean. does that mean i usea different variable there?
also, im not even sure if my math equations are right?
for ( int count; count <= months; count++)
Should be:
for ( int count = 0; count <= months; count++)
Also, if you want to execute the loop once for every month, you need to compare with <, not <=.
total_Interest = ( interest + );
This line is meaningless and syntactically incorrect.
new_balance = ( init_Balance + interest );
You overwrite new_balance every iteration without ever doing anything to the value.
All the buzzt!
"There is not now, nor has there ever been, nor will there ever be, any programming language in which it is the least bit difficult to write bad code."
- Flon's Law
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Quantitative Methods Optimal Solution a-j
1. 422410
Quantitative Methods Optimal Solution a-j
Quality Air conditioning manufactures three hoe air conditioners: an economy model, a standard model, and a deluxe model. The profits per unit are $63, $95, and $135, respectively. The production
requirements per unit are as follows:
Number of Fans Number of Cooling Coils Manufacturing Time
Economy 1 1 8 hours
Standard 1 2 12 hours
Deluxe 1 4 14 hours
For the coming production period, the company has 200 fan motors, 320 cooling coils, and 2400 hours of manufacturing time available. How many economy models (E), standard models (S), and deluxe
models (D) should the company produce in order to maximize profit? The linear programming model for the problem is as follows:
Max 63E + 95S + 135D
1E + 1S + 1D < 200 Fan motors
1E + 2S + 4D < 320 Cooling coils
8E + 12 S + 14 D < 2400 Manufacturing Time
E, S, D, > 0
Use the 100% rule for objective function coefficients and right hand side ranges where appropriate. Do not run the changed model. Assume that any changes given in a of the problem are the only
changes being made in the model
a) What is the optimal solution? What values of the decision variables lead to it? (3 points) I believe this is the right answer for this first question, the rest I am not sure of.
E = 80, S = 120, D = 0. Profit= $16,440
b) Suppose the coefficient of E is changed to 67. Will the optimal solution change? Why or why not?(3 points)
c) Suppose the coefficient of D is changed to 167 . Will the optimal solution change? Why or why not? (3 points)
d) Suppose the coefficient of E is changed to 70.5, that of S to 90.5 and that of D to 130. Will the optimal solution change? Why or why not? (6 points)
e) Suppose the coefficient of E is changed to 60.25, that of S to 97 and that of D to 142. Will the optimal solution change? Why or why not? (6 points)
f) Which resource is not completely used? (3 points)
g) Suppose the number of available fan motors is increased to 225. By how much will Quality AirĂ¢??s profit increase? (5 points)
h) Suppose the number of available cooling coils is increased to 375. By how much will Quality AirĂ¢??s profits increase? (5 points)
i) Suppose the number of available fan motors is changed to 215, that of cooling coils to 310 and the number of available manufacturing hours to 2350. Will the dual prices of those constraints
change? Why or why not? (7 points)
j) Suppose number of available fan motors is changed to 185, the number of cooling coils to 345 and the number of available manufacturing hours to 2275. Will the dual prices of those constraints
change? Why or why not? (7 points)
Quantitative Methods Optimal Solution a-j | {"url":"https://brainmass.com/math/linear-programming/422410","timestamp":"2014-04-17T16:19:47Z","content_type":null,"content_length":"31846","record_id":"<urn:uuid:23dd2d0c-ed88-43e9-8fa9-7a677186b484>","cc-path":"CC-MAIN-2014-15/segments/1398223205375.6/warc/CC-MAIN-20140423032005-00600-ip-10-147-4-33.ec2.internal.warc.gz"} |
Seeding random number generators
Java tutorials home java.util.Random Random number generators XORShift High quality random Seeding generators Entropy SecureRandom Random sampling Random simulations and nextGaussian()
Seeding random number generators
In our discussion of random number generators so far, we've seen the generator's period as a major limiting factor in "how much randomness" it can generate. But an equally important— and sometimes
more important— issue is how we seed or initialise a random number generator.
Let's take, for example, Java's standard generator java.util.Random, which has a period of 2^48. We can imagine this as a huge wheel with 2^48 randomly-numbered notches. Whenever we create an
instance of java.util.Random, the wheel starts in a "random" place, and moves round by one notch every time we generate a number from that instance. Wherever we start from, we'll end up back at the
same place after generating 2^48 numbers. If the place where we start is "truly random", then a sequence length of 2^48 may be sufficient for our application. But how do we pick a random place on the
wheel to start from?
The random place that we start from is in effect the seed or initial state of the generator. For this, we ideally want to pick a number (or some sequence of bits) that is "truly unpredictable". Or
put another way, we want to find some source of entropy (or "true unpredictability") available to the program.
System clocks and System.nanoTime()
The traditional solution used by java.lang.Random is to use one of the system clocks that the computer makes available. In recent versions of the JVM, the measure taken is that reported by
System.nanoTime(), which typically gives the number of nanoseconds since the computer was switched on. (In older versions of the JVM, or possibly as a fallback position on some platforms,
System.currentTimeMillis() is used, which reports the current "wall clock" time in milliseconds since 1 Jan 1970.)
So how good is System.nanoTime()? Well, on the face of it, it's a 64-bit value, so ample for generating starting points for all 2^48 possible sequences. But now consider that there are "only" 10^9
nanoseconds every second. So in the first five minutes that the computer is switched on, "only" 3x10^11— or about 2^38— possible values could be returned by System.nanoTime(). In other words, in the
first 5 minutes of the computer being on, only about one thousandth of the possible series of java.lang.Random will ever be generated. (Of course, if Windows takes three of those five minutes to boot
up, and/or if the system cannot actually report timings with nanosecond granularity, that reduces the number of possibilities even further...) For many casual applications, 2^38 possibilities or
something in that order is still plenty. But we certainly wouldn't want to use System.nanoTime() to seed, say, a random encryption key, or a "serious" game or gambling application where a user's
ability to guess the random sequence could result in financial loss.
Of course, for such applications, we also shouldn't be using java.lang.Random. But the problem of seed selection still applies. Even if we have a high-quality random number generator with a period
of, say, 2^160, that period doesn't really buy us "extra randomness" if we are only able to generate, say, 2^38 distinct seeds and/or an adversary can make further predictions about the seed.
Looking for other sources of entropy
To get more randomness in the choice of the generator's "initial position", we need to look for more sources of entropy on the local machine. On the next page, we discuss approaches to finding
entropy for seeding a random number generator.
Written by Neil Coffey. Copyright © Javamex UK 2013. All rights reserved. | {"url":"http://javamex.com/tutorials/random_numbers/seeding.shtml","timestamp":"2014-04-18T20:46:41Z","content_type":null,"content_length":"8975","record_id":"<urn:uuid:cf8efe02-708e-461d-9a0c-638c5fd5132e>","cc-path":"CC-MAIN-2014-15/segments/1398223206647.11/warc/CC-MAIN-20140423032006-00113-ip-10-147-4-33.ec2.internal.warc.gz"} |
Wolfram Demonstrations Project
Second Droz-Farny Circle
Let ABC be a triangle. Draw the circles about the midpoints of AB, AC, and BC that pass through the orthocenter (the point of intersection of the triangle's altitudes). These three circles intersect
the sides of ABC or their extensions in six points. Then those points lie on a circle (the second Droz-Farny circle). | {"url":"http://demonstrations.wolfram.com/SecondDrozFarnyCircle/","timestamp":"2014-04-19T15:17:56Z","content_type":null,"content_length":"42139","record_id":"<urn:uuid:3c0de290-738a-47b2-9b3a-d8483200de03>","cc-path":"CC-MAIN-2014-15/segments/1398223205137.4/warc/CC-MAIN-20140423032005-00563-ip-10-147-4-33.ec2.internal.warc.gz"} |
Results 1 - 10 of 12
, 1998
"... . Rational terms (possibly infinite terms with finitely many subterms) can be represented in a finite way via -terms, that is, terms over a signature extended with self-instantiation operators.
For example, f ! = f(f(f(: : :))) can be represented as x :f(x) (or also as x :f(f(x)), f(x :f(x)), ..."
Cited by 21 (12 self)
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. Rational terms (possibly infinite terms with finitely many subterms) can be represented in a finite way via -terms, that is, terms over a signature extended with self-instantiation operators. For
example, f ! = f(f(f(: : :))) can be represented as x :f(x) (or also as x :f(f(x)), f(x :f(x)), . . . ). Now, if we reduce a -term t to s via a rewriting rule using standard notions of the theory of
Term Rewriting Systems, how are the rational terms corresponding to t and to s related? We answer to this question in a satisfactory way, resorting to the definition of infinite parallel rewriting
proposed in [7]. We also provide a simple, algebraic description of -term rewriting through a variation of Meseguer's Rewriting Logic formalism. 1 Introduction Rational terms are possibly infinite
terms with a finite set of subterms. They show up in a natural way in Theoretical Computer Science whenever some finite cyclic structures are of concern (for example data flow diagrams, cyclic te...
- CGH , 1997
"... Acyclic Term Graphs are able to represent terms with sharing, and the relationship between Term Graph Rewriting (TGR) and Term Rewrtiting (TR) is now well understood [BvEG + 87, HP91]. During
the last years, some researchers considered the extension of TGR to possibly cyclic term graphs, which ..."
Cited by 20 (6 self)
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Acyclic Term Graphs are able to represent terms with sharing, and the relationship between Term Graph Rewriting (TGR) and Term Rewrtiting (TR) is now well understood [BvEG + 87, HP91]. During the
last years, some researchers considered the extension of TGR to possibly cyclic term graphs, which can represent possibly infinite, rational terms. In [KKSdV94] the authors formalize the classical
relationship between TGR and TR as an "adequate mapping" between rewriting systems, and extend it by proving that unraveling is an adequate mapping from cyclic TGR to rational, infinitary term
rewriting: In fact, a single graph reduction may correspond to an infinite sequence of term reductions. Using the same notions, we propose a different adequacy result, showing that unraveling is an
adequate mapping from cyclic TGR to rational parallel term rewriting, where at each reduction infinitely many rules can be applied in parallel. We also argue that our adequacy result is more
, 1999
"... . We present a categorical formulation of the rewriting of possibly cyclic term graphs, based on a variation of algebraic 2-theories. We show that this presentation is equivalent to the
well-accepted operational definition proposed by Barendregt et alii---but for the case of circular redexes, fo ..."
Cited by 12 (6 self)
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. We present a categorical formulation of the rewriting of possibly cyclic term graphs, based on a variation of algebraic 2-theories. We show that this presentation is equivalent to the well-accepted
operational definition proposed by Barendregt et alii---but for the case of circular redexes, for which we propose (and justify formally) a different treatment. The categorical framework allows us to
model in a concise way also automatic garbage collection and rules for sharing/unsharing and folding/unfolding of structures, and to relate term graph rewriting to other rewriting formalisms.
R'esum'e. Nous pr'esentons une formulation cat'egorique de la r'e'ecriture des graphes cycliques des termes, bas'ee sur une variante de 2-theorie alg'ebrique. Nous prouvons que cette pr'esentation
est 'equivalente `a la d'efinition op'erationnelle propos'ee par Barendregt et d'autres auteurs, mais pas dons le cas des radicaux circulaires, pour lesquels nous proposons (et justifions
- in Engineering, Communication and Computing 7 , 1993
"... . Dealing properly with sharing is important for expressing some of the common compiler optimizations, such as common subexpressions elimination, lifting of free expressions and removal of
invariants from a loop, as source-to-source transformations. Graph rewriting is a suitable vehicle to accommoda ..."
Cited by 8 (4 self)
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. Dealing properly with sharing is important for expressing some of the common compiler optimizations, such as common subexpressions elimination, lifting of free expressions and removal of invariants
from a loop, as source-to-source transformations. Graph rewriting is a suitable vehicle to accommodate these concerns. In [4] we have presented a term model for graph rewriting systems (GRSs) without
interfering rules, and shown the partial correctness of the aforementioned optimizations. In this paper we define a different model for GRSs, which allows us to prove total correctness of those
optimizations. Differently from [4] we will discard sharing from our observations and introduce more restrictions on the rules. We will introduce the notion of Bohm tree for GRSs, and show that in a
system without interfering and non-left linear rules (orthogonal GRSs), Bohm tree equivalence defines a congruence. Total correctness then follows in a straightforward way from showing that if a
program M co...
, 1989
"... .We study properties of rewrite systems that are not necessarily terminating, but allow instead for trans#nite derivations that have a limit. In particular, we give conditions for the existence
of a limit and for its uniqueness and relate the operational and algebraic semantics of in#nitary theories ..."
Cited by 8 (1 self)
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.We study properties of rewrite systems that are not necessarily terminating, but allow instead for trans#nite derivations that have a limit. In particular, we give conditions for the existence of a
limit and for its uniqueness and relate the operational and algebraic semantics of in#nitary theories. We also consider su#cient completeness of hierarchical systems. Is there no limit? ---Job 16:3
1. Introduction Rewrite systems are sets of directed equations used to compute by repeatedly replacing equal terms in a given formula, as long as possible. For one approach to their use in computing,
see #23#. The theory of rewriting is an outgrowth of the study of the lambda calculus and combinatory logic, and # Preliminary versions #6, 7# of ideas in this paper were presented at the Sixteenth
ACM Symposium on Principles of Programming Languages, Austin, TX #January 1989# and at the Sixteenth EATCS International Colloquium on Automata, Languages and Programming, Stresa, Italy #July 1989#.
, 1992
"... A methodology for polymorphic type inference for general term graph rewriting systems is presented. This requires modified notions of type and of type inference due to the absence of structural
induction over graphs. Induction over terms is replaced by dataflow analysis. 1 Introduction Term graphs ..."
Cited by 5 (1 self)
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A methodology for polymorphic type inference for general term graph rewriting systems is presented. This requires modified notions of type and of type inference due to the absence of structural
induction over graphs. Induction over terms is replaced by dataflow analysis. 1 Introduction Term graphs are objects that locally look like terms, but globally have a general directed graph
structure. Since their introduction in Barendregt et al. (1987), they have served the purpose of defining a rigorous framework for graph reduction implementations of functional languages
(Peyton-Jones (1987)). This was the original intention. However the rewriting of term graphs defined in the operational semantics of the model, makes term graph rewriting systems (TGRSs) interesting
models of computation in their own right. One can thus study all sorts of issues in the specific TGRS context. Typically one might be interested in how close TGRSs are to TRSs and this problem is
examined in Barendregt et al. (19...
, 1995
"... We define the notion of transfinite term rewriting: rewriting in which terms may be infinitely large and rewrite sequences may be of any ordinal length. For orthogonal rewrite systems, some
fundamental properties known in the finite case are extended to the transfinite case. Among these are the Pa ..."
Cited by 4 (1 self)
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We define the notion of transfinite term rewriting: rewriting in which terms may be infinitely large and rewrite sequences may be of any ordinal length. For orthogonal rewrite systems, some
fundamental properties known in the finite case are extended to the transfinite case. Among these are the Parallel Moves lemma and the Unique Normal Form property. The transfinite Church-Rosser
property (CR 1 ) fails in general, even for orthogonal systems, including such well-known systems as Combinatory Logic. Syntactic characterisations are given of some classes of orthogonal TRSs which
do satisfy CR 1 . We also prove a weakening of CR 1 for all orthogonal systems, in which the property is only required to hold up to a certain equivalence relation on terms. Finally, we extend the
theory of needed reduction from the finite to the transfinite case. The reduction strategy of needed reduction is normalising in the finite case, but not in the transfinite case. To obtain a
normalising str...
- Electronic Notes in Theoretical Computer Science , 1995
"... Infinitary rewriting allows infinitely large terms and infinitely long reduction sequences. There are two computational motivations for studying these: the infinite data structures implicit in
lazy functional programming, and the use of rewriting of possibly cyclic graphs as an implementation techni ..."
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Infinitary rewriting allows infinitely large terms and infinitely long reduction sequences. There are two computational motivations for studying these: the infinite data structures implicit in lazy
functional programming, and the use of rewriting of possibly cyclic graphs as an implementation technique for functional languages. We survey the fundamental properties of infinitary rewriting in
orthogonal term rewrite systems, and its relation to cyclic graph rewriting. 1 Introduction Our interest in term and graph rewriting arises from functional languages and their implementation.
Functional programs can be seen as term rewrite systems. 2 Terms can be thought of as trees. Representing these trees as graphs allows repeated subterms to be replaced by multiple pointers to the
same subgraph. This optimisation has a dramatic effect when rewrite steps are performed. Whenever a variable appears more than once on the right-hand side of a rule, when that rule is applied to a
graph multiple poi...
"... Term graph rewriting provides a simple mechanism to finitely represent restricted forms of infinitary term rewriting. The correspondence between infinitary term rewriting and term graph
rewriting has been studied to some extent. However, this endeavour is impaired by the lack of an appropriate count ..."
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Term graph rewriting provides a simple mechanism to finitely represent restricted forms of infinitary term rewriting. The correspondence between infinitary term rewriting and term graph rewriting has
been studied to some extent. However, this endeavour is impaired by the lack of an appropriate counterpart of infinitary rewriting on the side of term graphs. We aim to fill this gap by devising two
modes of convergence based on a partial order resp. a metric on term graphs. The thus obtained structures generalise corresponding modes of convergence that are usually studied in infinitary term
rewriting. We argue that this yields a common framework in which both term rewriting and term graph rewriting can be studied. In order to substantiate our claim, we compare convergence on term graphs
and on terms. In particular, we show that the resulting infinitary calculi of term graph rewriting exhibit the same correspondence as we know it from term rewriting: Convergence via the partial order
is a conservative extension of the metric convergence.
, 1996
"... We will present a syntactical proof of correctness and completeness of shared reduction. This work is an application of type theory extended with infinite objects and co-induction. ..."
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We will present a syntactical proof of correctness and completeness of shared reduction. This work is an application of type theory extended with infinite objects and co-induction. | {"url":"http://citeseerx.ist.psu.edu/showciting?cid=2590162","timestamp":"2014-04-20T14:08:14Z","content_type":null,"content_length":"38669","record_id":"<urn:uuid:994ec4c4-b2ab-4fe4-ae76-4a05a2f44fd2>","cc-path":"CC-MAIN-2014-15/segments/1397609538787.31/warc/CC-MAIN-20140416005218-00572-ip-10-147-4-33.ec2.internal.warc.gz"} |
[FOM] About Paradox Theory
Taylor Dupuy taylor.dupuy at gmail.com
Fri Sep 23 16:20:46 EDT 2011
There is an interesting way to view contradictions in Propositional logic
via arithmetization:
1. Convert propositions, with propositional variables X1,X2,... into
polynomials over FF_2=ZZ/ 2 ZZ with indeterminates x1, x2, ...
X & Y <-> xy
X or Y <-> xy + x +y
!X <-> x+1
True <-> 1
False <-> 0
2. Satisfiability of a proposition is equivalent existence of solutions of
polynomial equations in FF_2. If P(X1,X2,...,Xn) is a proposition and
p(x1,x2,...,xn) is its associated polynomial the P is satisfiable if and
only if p=1 admits a solution over FF_2. For example the proposition X & !X
is clearly not satisfiable which corresponds to the fact that x(x+1)=1 or
x^2+x +1 =0 has no solutions over FF_2.
One would think then that a canonical way to extend truth values in
propostional logic would be to allow values in FF_2[x]/(x^2+x+1) (or the
relevant extension for an unsatisfiable proposition) and carry the truth
tables back over to logic via (1). I've never heard of a theory but have
always wondered if its already been developed and what it would look like.
-Does a theory using this idea in propositional logic exist? Can one
make proofs make sense with these truth values?
-Does a first order theory using ideas like this make sense?
I could never get past what the methods of proof should be. Thoughts?
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X-Rays, Neutrons and Muons
ISBN: 978-3-527-30774-6
248 pages
September 2012
Read an Excerpt
Spectroscopy is a versatile tool for the characterization of materials, and photons in the visible frequency range of the electromagnetic spectrum have been used successfully for more than a century
now. But other elementary particles such as neutrons, muons and x-ray photons have been proven to be useful probes as well and are routinely generated in modern cyclotrons and synchrotrons. They
offer attractive
alternative ways of probing condensed matter in order to better understand its properties and to correlate material behavior with its structure. In particular, the combination of these different
spectroscopic probes yields rich information on the material samples, thereby allowing for a systematic investigation down to atomic resolutions.
This book gives a practical account of how well they complement each other for 21st century material characterization, and provides the basis for a detailed understanding of the scattering processes
and the knowledge of the relevant microscopic interactions necessary for the correct interpretation of the experimentally obtained spectroscopic data.
See More
Foreword V
Preface XI
About the Book XIII
1 Introduction 1
1.1 Some Historical Remarks 1
1.2 The Experimental Methods 2
1.3 The Solid as a Many Body 6
1.4 Survey over the Spectral Region of a Solid 9
References 11
2 The Probes, their Origin and Properties 13
2.1 Origin 13
2.1.1 The Photon 13
2.2 Properties 15
2.3 Magnetic Field of the Probing Particles 25
3 Interaction of the Probes with the Constituents of Matter 27
3.1 The Nuclear Interaction of Neutrons 27
3.2 Interaction of X-Rays with Atomic Constituents 36
3.3 Magnetic Interaction 49
3.4 Corollar 55
References 58
4 Scattering on (Bulk-)Samples 59
4.1 Introduction 59
4.2 The Sample as a Thermodynamic System 59
4.3 The Scattering Experiment 66
4.4 Properties of the Scattering and Correlation Function 83
4.5 General Form of Spin-Dependent Cross-Sections 88
4.6 Summary and Conclusions 123
References 125
5 General Theoretical Framework 127
5.1 Time Development of the Density Operator 127
5.2 Generalized Suspectibility 142
5.3 Dielectric Response Function and Sum Rules 157
References 173
Appendix A: Principles of Scattering Theory 175
A.1 Potential Scattering (Supporting Section 3.1.1) 175
A.2 Two Particle Scattering (Supporting Section 3.1.2) 179
A.3 Abstract Scattering Theory (Supporting Section 3.2) 180
A.4 Time-Dependent Perturbation 183
A.5 Scattering of Light on Atoms 186
A.6 Polarization and its Analysis 188
References 190
Appendix B: Form Factors 191
References 195
Appendix C: Reminder on Statistical Mechanics 197
C.1 The Statistical Operator P 197
C.2 The Equation of Motion 198
C.3 Entropy 198
C.4 Thermal Equilibrium – The Canonical Distribution 199
C.5 Thermodynamics 200
Appendix D: The Magnetic Matrix-Elements 203
D.1 The Trammell Expansion 203
D.2 The Matrix Elements 205
D.3 Conclusion 210
References 211
Appendix E: The Principle of a mSR-Experiment 213
Appendix F: Reflection Symmetry and Time-Reversal Invariance 217
F.1 Invariance Under Space Inversion Q 217
F.2 Invariance Under Time Reversal 218
Appendix G: Phonon Coupling to Heat Bath 221
References 223
Further Reading 225
Index 227
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Walter E. Fischer (1939-2008) was the former head of the Department of Condensed Matter Research with Neutrons and Muons (NUM) at the Paul Scherrer Institute (PSI) in Villigen, Switzerland. He
pioneered in establishing the spallation neutron source SINQ at PSI which went into operation in the mid-1990s. Later he foundes a condensed matter theory group to complement the experimental work at
the neutron source.
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X-Rays, Neutrons and Muons (US $85.00)
-and- Elements of Modern X-ray Physics, 2nd Edition (US $68.00)
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