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Topic: degrees of freedom
Replies: 0
degrees of freedom
Posted: Jul 28, 2013 11:25 AM
Suppose I calculate a likelihood as a function of M parameters from a
chi-square using the incomplete gamma function. I wish to plot
likelihood contours in 2 dimensions, i.e. "suppress" the other
parameters. Suppose M is 3. If I calculate 3-D contours and project
them onto the 2-D plot, then obviously M is 3. If I hold the third
parameter constant, then obviously M is 2. What about the case when,
for a given point in the 2-D plot, I take the maximum of the likelihood
from the set of values described by the third parameter? When I take
the average? For these 2 cases, would M be 2 or 3?
Obviously, one has to know M before calculating the likelihood. | {"url":"http://mathforum.org/kb/thread.jspa?threadID=2584053","timestamp":"2014-04-17T10:56:14Z","content_type":null,"content_length":"14176","record_id":"<urn:uuid:72435b60-76a7-45df-82bd-3e876ff4f919>","cc-path":"CC-MAIN-2014-15/segments/1397609527423.39/warc/CC-MAIN-20140416005207-00471-ip-10-147-4-33.ec2.internal.warc.gz"} |
Clock and Data Recovery/Structures and types of CDRs/The (slave) CDR based on a second order PLL/2nd order type 1
Generalities of a 2^nd order PLL of type 1Edit
Best fit for a regenerator, but with a slow acquisition.
Good filtering of the incoming signal noise but poor filtering of the VCO noise.
Becomes underdamped if the loop gain increases.
Fundamental equationsEdit
A linear, time-invariant model of the circuit is described here.
The case of a second order loop, fit for a regenerator CDR, is presented.
This architecture is in fact the best for regenerator applications, to recover a clock and to resend the data stream further, as the introductory example has shown.
There are two ways of looking into the model of the 2^nd order CDR loop:
1. to specify the CDR overall characteristics, or to measure its performances. (ω[n2] and $\zeta$).
In these cases of overview approach, the "engineer" will use two variables that characterize the overall performances of the control loop.
The variables are the undamped natural frequency ω[n2] and the damping ratio $\zeta$.
2. to design, verify and validate the CDR, as well as engineer its individual blocks. (G and $\tau_f$).
In these other cases there are again just two variables that correspond to the two basic degrees of freedom of the circuit.
The “designer” variables are the loop gain, G, and the time constant of the loop filter, $\tau_f$.
ω[n2] and ζ as functions of G and τ[f]Edit
It is important to remember that the two sets of parameters are interchangeable. Deciding the two values for a set fixes the values also for the other set, as restated in the table below:
Two degrees of freedom: relations amongst
equivalent quantities
Setting ω[n2] and ζ Setting G and τ[f]
G = ω[n2] / 2ζ ω[n2]^2 = G/τ[f]
τ[f] = 1 / 2ζω[n2] ζ^2 = 1 / 4τ[f]G
The value of ζ (when D[T] is at its minimum expected value) must be close to 1 (0.7 to 1.3), to ensure that the behavior of the loop is neither under-damped nor over-damped.
Saying that the second order loop has one more degree of freedom than the first order loop is therefore an overstatement.
It is more fair to say that in both cases the time (and frequency) response of the loop are primarily set by the gain G. In the case of the second order loop it is possible to have a slightly higher
variability associated with the (limited) range of variation allowed for $\tau$[f].
The main advantage of the second order loop is the sharper filtering action for input frequencies in excess of ω[n2].
The filtering corresponds to attenuating the jitter components beyond ω[n2] with a slope of 40 dB/decade, while a first order loop filters with a slope of just 20 dB/decade beyond its ω[n].
The meaning of the loop parameters (just ω[n] or τ in a first order system, ω[n2] and ζ in a second order system) is slightly different in the two cases:
• ω[n2] is essentially the same as ω[n]: it represents the angular frequency at which the loop changes behavior and a "filterng" action begins (e.g. filtering of the input jitter) or ends (e.g
filtering of the VCO noise).
• τ of a first order system is the time constant of the system unit step response. τ[f] in a second order system can not be seen exactly as the same because the unit step response varies for
different values of the other loop parameter, ζ. In general however a relation exists and the time response of a second order system is close to about 2 times τ[f], as shown in more detail
further on in this book where unit step responses of 1st and 2nd order loops are contrasted, under the condition ω[n2] = ω[n].
The most important functions that describe the relations amongst the variables shown in the figure, expressed for the sinusoidal jitter condition, are described in the following three subpages.
A dedicated section further down in this page gives the equations and the diagram for the time function that describes how this circuit reacts to an abrupt change of phase in the received signal.
ζ is close to 1 (0.7 to 1.3)Edit
From the equations in the figure above, some interesting considerations can be made about PLLs of 2^nd order and type 1 that are part of a CDR.
First, it should be emphasized that the value of ζ shall be set by design close to 1 (0.7 to 1.3).
□ When ζ < 0.707, the jitter at frequencies around ω[n2] will be amplified during the transit through the PLL as shown below, and more so for smaller values of ζ. Amplification of jitter may -
if the input jitter at those frequencies is large enough to start with - accumulate when more identical regenerators are interconnected in a chain along the signal path (a not infrequent case
in geographical networks). This (unfiltered) jitter moves the sampling instant away from its optimum point so that errored bits may ensue: a condition to be avoided at all costs in a CDR!!
□ When ζ >> 1, the loop behaves more and more like a first order loop. The sharp cutoff at about ω[n2] , typical of a good 2nd order loop, smooths out, and the jitter at frequencies around ω
[n2] is less effectively rejected.
τ[f] is the time constant of the added filter and 1/ τ[f] = ω[f] its cut off frequency; G is the open loop DC gain = G[φ] * G[f] * G[VCO].
ω[n2] and ω[f] or ω[z]Edit
In a second order loop, the cut-off frequency of the loop filter block (ω[f] or ω[z], as the case may be) fixes the bandwidth of the closed loop (apart from minor adjustments as ζ varies within its
allowed range).
Anticipating some simple formulas that will be derived later in the book,
the natural frequency of the PLL ω[n2] (which can also be seen as the jitter cutoff frequency of the CDR) is defined by the loop gain and by ω[f] :
ω[n21] = 1 / 2ζ[21]τ[f]
(using the subscripts to identify loop order and type) and can also be expressed as a function of just ζ and ω[f] :
ω[n21] = ω[f] /2ζ[21]
This simple formula tells that (in a 2^nd order PLL of type 1 where ζ[21] must remain close to 1) the natural frequency ω[n21] remains close to half the cutoff frequency of the loop filter!
Similarly, for the model of a 2^nd order type 2 loop (where ζ shall also preferably remain close to 1):
τ[z] = 2 ζ[22]/ ω[n22]
ω[n22] = 2 ζ[22] ω[z]
This simple formula tells that (in a 2^nd order PLL of type 2 where ζ[22] shall remain ≈ 1) the natural undamped frequency ω[n22] remains close to twice the cutoff frequency of the loop filter!
ω[n21] and ω[n1]Edit
In a 1^st order loop, the quantity ω[n] = G tells how fast the loop reacts. The higher ω[n], the faster the loop response.
For the 2nd order loop it is difficult to relate ω[n2] to how fast the loop reacts to a change. In the 2 - 1 loop, for ζ ≈ 1, and setting -for sake of comparison- the gain G equal for the two loops:
ω[n21] = 2ζ[21] G
ω[n21] ≈ 2 ω[n1]
In other words a 2^nd order type 1 PLL (ζ ≈ 1) wold seem twice as “fast” as a 1st order PLL of the same gain!
But it can also be seen that, setting, for sake of comparison, ω[n2] = ω[n], then the 2nd order type 1 loop is not practically slower than the 1st order loop!
“Slow or fast” in the sentences above means both:
- slow or fast to acquire the lock condition
- slow or fast to drift to its free running frequency when the input signal disappears.
The following figure contrasts the Unit Step Responses of the 3 important PLL models:
Note that the USR of the 1 - 1 loop model is plotted for two different values of ω[n1]: ω[n1] = ω[n2] of the other two loops and ω[n1] = ω[n2] / 2 of the other two loops
Jitter transfer functionEdit
The transfer function for a given input X(s) (that is the Jitter Transfer function!) is:
$\tfrac{Y(s)}{X(s)}=\tfrac{1}{\left ( \tfrac{s^2}{\omega_{n2}^2}+\tfrac{2s\zeta}{\omega_{n2}}+1\right ) }$
The magnitude of the jitter transfer frequency response (= absolute value of the jitter transfer, once modified with s replaced by jω) tells, at each frequency f = ω/2π , the ratio of the magnitude
of the output jitter divided by the magnitude of the input jitter.
It can also be seen as the amplitude of the (sinusoidal) output jitter for a sinusoidal input jitter of amplitude = 1 radian ≈ 57.3°.
$\left | \tfrac{Y(j\omega)}{X(j\omega)}\right | =\tfrac{1}{\sqrt{\left ( 1 - \tfrac{\omega^2}{\omega_{n2}^2}\right )^2+\left ( \tfrac{2\omega\zeta}{\omega_{n2}}\right )^2}}$
The following figure is the Bode magnitude plot of the jitter transfer function. Curves for different values of the parameter ζ (damping ratio) are shown:
This PLL is a good low-pass filter [40 dB/decade] for the phase jitter. There is no amplification of the input jitter except for values of the damping ratio smaller than √1/2 ≃ 0.707, at some
frequencies around ω[n2]. The peak amplification occurs at ω = ω[n2] √1 − 2ζ² and is:
$\tfrac{1}{\sqrt{\left ( 2 \left ( \tfrac{\omega_r}{\omega_{n2}}\right ) ^2 \left ( 1- \left ( \tfrac{\omega_r}{\omega_{n2}}\right )^2 \right ) +\left (\left (1-\left (\tfrac{\omega_r}{\omega_{n2}}\
right )^2\right )\right )^2\right ) }}$
The jitter transfer function of a simpler 1^st order type 1 loop (where the signal processing block is just a flat gain block of gain G[f]) can be easily obtained as:
$\tfrac{Y(s)}{X(s)}=\tfrac{1}{\left ( \tfrac{s}{\omega_{n}}+1\right ) }$
Similar to the case above, it filters out all jitter components above ω[n] (ω[n] = G[φ] * G[f] * G[VCO] = G), but without peaking nor smoothing of the corner, and with a single slope of 20 dB/decade
instead of 40 dB/decade, as seen in the previous page.
Important changes in jitter filtering going from the 1 - 1 loop tp the 2 - 1 loopEdit
The most important new feature is the sharper cut off of the input jitter frequencies above ω[n2] , with a slope of -40 dB/decade instead of just -20 dB/decade.
The addition of the 1^st order filter between the phase comparator and the VCO has added one pole to the transfer characteristic of the loop.
All CDRs act as low pass filters on the timing signal, and remove all the (jitter) frequencies above a cut off frequency, called ω[n] in the model of a 1^st order loop and ω[n2] in the model of a 2^
nd order loop.
The figure above contrasts the two cases in a particular example. Also the frequency of the received line pulses, ω[p], is shown, as the vertical marker of the yellow line.
The (CDR) PLL models in general refer to discrete time systems with sample frequency equal to ω[p], and therefore all characteristics above ω[p] are not investigated (they repeat themselves
periodically along the frequency axis).
In the case shown as well as in general, the choice of ω[n2] implicitly defines the input jitter below ω[n2] as useful signal, and the input jitter above ω[n2] as noise to be rejected.
The stronger filtering effect of the 2nd order loop can be reasonably estimated assuming that:
1. the jitter has a flat spectral density from ω = 0 to ω = ω[p],
2. the jitter from ω = 0 to ω = ω[n2] is useful signal and
3. the jitter from ω = ω[n2] to ω = ω[p] is noise to be rejected.
With these assumption, it can be calculated that the 2^nd order loop fiters the noise jitter 17.23 dB more than the 1^st order loop.
Error signalEdit
The previous section about the jitter transfer function has shown that some PLLs may even amplify the input jitter amplitude.
This happens for the frequency components around ω[n2] of the input jitter, if the damping ratio ζ is lower than √1/2 ≃ 0.707.
Another tool to investigate the same critical frequency band is the function that describes the time distance (that is the phase distance if time is divided by the duration of a line pulse, i.e. by
2π/ω[p]) between output and input.
This function is called error signal, indicated respectively as ε(t) or E(s) or E(jω).
The error signal is less important in a pure PLL circuit, whose task is to track an input clock (and to dejitter it, maybe), but is extremely important if the PLL is serving into a CDR circuit.
E(jω) is especially meaningful, because:
• jitter is investigated in "steady state" conditions, i.e. with the Fourier transforms: sinusoidal input, sinusoidal output and sinusoidal waveforms at every inner node of the PLL;
• the PLL output tracks exactly the input with just the difference ε(t);
• ε(t) is exactly the difference between the eye center and the sampling instant: when ε(t) is maximum the lateral eye margin is minimum;
• the maximum value of a sinusoidal deviation is its magnitude: the useful function is therefore |E(jω)|.
Its magnitude tells, at every jitter frequency, the amplitude of the sinusoidal distance between the output and the input phases. It is easy to realize that this function is the loop error function:
Ε(s) = Y(s) − X(s) = X(s) $\tfrac{\left ( \tfrac{s^2}{\omega_{n2}^2}\right)+2\zeta\left ( \tfrac{s}{\omega_{n2}}\right)}{\left( \tfrac{s^2}{\omega_{n2}^2}\right )+2\zeta\left ( \tfrac{s}{\omega_{n2}}
\right )+1}$
$\left |\tfrac{E(j\omega)}{X(j\omega)}\right | = \left |\tfrac{\left ( \tfrac{s^2}{\omega_{n2}^2}\right)+2\zeta\left ( \tfrac{s}{\omega_{n2}}\right)}{\left( \tfrac{s^2}{\omega_{n2}^2}\right )+2\zeta\
left ( \tfrac{s}{\omega_{n2}}\right )+1}\right | = \tfrac{ \sqrt{ \tfrac{\omega^4}{\omega_{n2}^4} + \tfrac{4\zeta^2\omega^2}{\omega_{n2}^2} } } {\sqrt{ \tfrac{\omega^4}{\omega_{n2}^4} - (1-2\zeta^2)\
tfrac{2\omega^2}{\omega_{n2}^2} + 1} }$
The regeneration of the data depends on sampling the received pulses (that have undergone amplification, equalization and filtering of out-of-band noise) close to the time of maximum amplitude of
each pulse. At that time the remaining noise and intersymbol interference still alter the pulse to a certain extent. A significant phase error makes the probability of a wrong detection higher: in
other words a phase error that is not affecting clock tracking may still increase to intolerable levels the bit error rate!
The following figure plots the magnitude (not in dB, but in radian referred to the line pulse duration) of the error signal if the input jitter amplitude at any given frequency is exactly 1 rad (1
radian ≈ 57.3°).
The error signal is, in other words, normalized to the value of the loop input, plotting in fact the error "transfer" function: E(s)/X(s).
At low jitter frequencies there is practically no error, because the tracking is very good.
At very high jitter frequencies the error is practically identical to the input: in fact the PLL is not able to track the jitter and the local clock stays unmoving with respect to it.
At intermediate jitter frequencies, around ω[n2], the error increases with frequency till it is as large as the input jitter itself, with an intermediate region where it can become larger, at
frequencies just above ω[n2] for low values of ζ.
Large values of ζ ( >> 1) involve a large error even at frequencies much lower than $\omega_{n2}$; small values of ζ ( < 1.0) correspond to large overshoots of the phase error just above $\omega_{n2}
Values of ζ between 0.7 and 1.3 are therefore an inevitable design choice. Other considerations that can be drawn from the study of the jitter tolerance function suggest an even tighter range of ζ
values for the regenerator CDR design.
Both the jitter transfer and the jitter error functions are true "transfer functions". They tell the ratio of an output to an input (The function in the figure above represents in fact the magnitude
of the error transfer function, as well as the error magnitude, for an input with a fixed value of 1 radian).
The function in the next Section instead -the jitter tolerance function- is not a transfer function. The aspect of causality (the fact that the input generates the output) is not present there. The
jitter tolerance function describes the values of input jitter that generate a fixed value of the error signal.
The steady state error due to f[p] – f[fr] difference and the steady state error of the sampling timeEdit
The free-running frequency characteristic of the VCO, f[fr], never coincides exactly with f[p].
As introduced earlier in the VCO page,
the VCO control signal -while the PLL is tracking the input signal -
exhibits a d.c. offset from its center value, proportional to the frequency mismatch.
This steady state error (in a control system of type 1 with unity feedback, like the system under consideration here) in presence of a unit ramp applied to the input of the system (as seen earlier in
this book) is also present at the output and has the constant finite value of:
steady state output error = E[s] = 1/G [rad]
If the ramp applied to the input, instead of a slope of 1 rad/sec, has a slope of S rad/sec, E[s] becomes:
Es = S/G
and if, like in most practical cases, the ramp is caused by the frequency difference between the clock embedded in the CDR incoming pulse stream (f[p]) and the free-running frequency of its VCO (f
[fr]), the phase error in those type 1 systems is:
E[s] = (ω[p] – ω[fr]) / G
The frequency offset
This offset (of the drive signal at the node connected to the VCO input), depends on the type of the PLL:
1. may not be a steady state signal because at some other node the signal would ramp until some non-linearity of the loop limits it (and the loop loses its control ability - loops of type 0), or
2. may be present at the other PLL nodes with a proportionality coefficient (loops of type 1, or
3. may be compressed to zero at some other PLL nodes by some integration in the loop (loops of type 2).
Sampling time shifted from the best instant in type 1 PLLs
For the type 1 PLLs the dc offset that reconciles f[p] and f[fr] at the VCO input is also present -with some coefficients of proportionality- at the other nodes.
In particular this offset is present also between the two inputs of the phase comparator, has the value E[s] seen above and represents a deviation of the sampling time from its best (=center)
The VCO centering is the limit to a very narrow jitter bandwidth
In the 2^nd order, type 1, loop, the VCO frequency mismatch f[p] – f[fr] becomes a sampling time error E[s] according to:
E[s] = (ω[p] – ω[fr])/G (G= G[φ]*G[f]*G[VCO])
For fixed open loop dc gain G and filter time constant τ[f], the jitter cut-off frequency ω[n2] is:
$\omega_{n2}$ = G * 2ζ = √G/τ[f]
E[s] = ((ω[p] – ω[fr]) * 2ζ ) / ω[n2]
It is easy to see that, for a 1st order type 1 loop: E[s] = ((ω[p] – ω[fr]) / ω[n1]
The same equation, rearranged, tells that the frequency mismatch and the maximum E[s] define how tight the the loop jitter bandwidth can be:
E[s] / 2ζ = (ω[p] – ω[fr]) / ω[n2] = ((ω[p] – ω[fr])/ω[p]) / (ω[n2]/ω[p])
(ω[n2] / ω[p] ) = ((ω[p] – ω[fr]) /ω[p]) / (E[s] /2ζ ))
It is easy to see that, for a 1st order type 1 loop: (ω[n1]/ω[p] ) = ((ω[p] – ω[fr]) /ω[p]) / E[s] )
For instance, if E[s] is conservatively set as low as = 0.1 rad, then ω[n2] can be : ω[n2] ≃ 20 * (ω[p] – ω[fr]) .
As seen already about CDRs and parts per million of frequency mismatch, the free running frequency of a slave CDR may differ no more than 50 ppm from the frequency of its remote master (very low cost
quartz crystal), or 10000 ppm (monolythic RC oscillator after EWS trimming), or even differ less than 1 ppm, still without big cost concerns (quartz for GPS receivers inside mobile phones). Less than
0.1 ppm is typical of professional equipment.
The four mentioned cases set a respective minimum for the design choice of ω[n2] at: 0.001 ω[p] , 0.2 ω[p] , 2 e-5 ω[p] , 2 ppm ω[p] .
Jitter toleranceEdit
For the slave 2 - 1 loop, the jitter tolerance can be obtained as |X(jω)|[|Ε(jω)| = Φ[leo]], which corresponds to the hypothesis that the tolerance is limited by (and only by) the lateral eye
opening, see also the relevant page. (Φ[leo] is the lateral eye opening expressed in radian).
In other words the mathematical function is obtained as if it was the range of the phase comparator that set the tolerance limit, but reduced to the lower value Φ[leo].
The comparator range corresponds to the ideal width of the received pulse left and right of its mid point ( ±π ).
The function that gives the ratio of the input to the error is:
$\tfrac{X(s)}{\Epsilon(s)}=\tfrac{X(s)}{X(s)-Y(s)}=\tfrac{1}{1-\left ( \tfrac{\omega_n^2}{s^2+2s\zeta\omega_n+\omega_n^2}+\right ) }=\tfrac{s^2+2s\zeta\omega_n+\omega_n^2}{s^2+2s\zeta\omega_n}$
Its magnitude in steady state (s = jω), that represents the magnitude of the sinusoidal input generating an error of magnitude 1 rad ( |Ε(jω)| = 1 rad ), is the normalized jitter tolerance function:
$\left | \tfrac{X(j\omega)}{\Epsilon(j\omega)}\right |$ = $\left |X(j\omega)\right |_{\left |\Epsilon(j\omega)\right | = 1 }$ = $\sqrt{\tfrac{\left ( 1 - \tfrac{\omega^2}{\omega_n^2}\right )^2+\left
( \tfrac{2\omega\zeta}{\omega_n}\right )^2}{\left ( \tfrac{\omega^2}{\omega_n^2}\right )^2+\left ( \tfrac{2\omega\zeta}{\omega_n}\right )^2}}$
The magnitude of the jitter tolerance function (de-normalized) is the same function but taken for |E(jω)| = Φ[leo] :
$\left |X(j\omega)\right |_{\left |\Epsilon(j\omega)\right | = \Phi_{leo} }$ = $\left | \tfrac{X(j\omega) \Phi_{leo}}{\Epsilon(j\omega)}\right | =\Phi_{leo} \sqrt{\tfrac{\left ( 1 - \tfrac{\omega^2}
{\omega_n^2}\right )^2+\left ( \tfrac{2\omega\zeta}{\omega_n}\right )^2}{\left ( \tfrac{\omega^2}{\omega_n^2}\right )^2+\left ( \tfrac{2\omega\zeta}{\omega_n}\right )^2}}$
The Bode magnitude plot is given in the following figure:
If another value $\mathrm{\Psi_{leo}}$ better simulates the circuit tolerance limit, the curve plotted still applies provided it is translated vertically by the amount $\mathrm{20log_{10}(\Psi_{leo}/
At low jitter frequencies there is very good tracking, even if the jitter has a large amplitude. The circuit has time to follow these large but slow variations. It takes an extremely large jitter
amplitude to reach the limit of tolerance.
At very high jitter frequencies the circuit is unable to follow the jitter that varies too fast. The tolerance is in practice exactly the lateral eye opening (or the phase comparator range, whichever
limit is reached first). In fact, the PLL is not able to track the jitter at all and the local clock stays unmoving with respect to it.
At intermediate jitter frequencies, just above ω[n], the circuit is tricked by the jitter into overreaching while the jitter is coming back to its zero value. In the range of such intermediate
frequencies, at and above ω[n], the tolerance is correspondingly reduced below its asymptotic value, and especially so for low values of ζ.
It may be noted that the weak spot just above ω[n] when ζ is low, corresponds to an asymptote towards low frequencies (-20 dB/decade typical of all type 1 loops) that is correspondignly higher
[translated vertically by -20log[10](2ζ) with respect to the asymptote going through (ω[n] , Φ[leo])].
This does not compensate, in CDR applications, the reduction of the tolerance margin in the critical frequency band, and the 2 - 1 loop should never be significantly under-damped.
The following table gives the values of the maximum tolerance reduction for different values of ζ:
Values of ζ between 1.0 and 1.5 are therefore an inevitable design choice. The designer shall pay attention to the fabrication tolerances of the CDR blocks (that may often be as large as ± 20 %) as
well as to the dependence of them from operating conditions like the power supply variations or the density of transitions inside the incoming signal.
It is therefore important to check the behaviour of the CDR circuit (whatever its actual implemetation is, analog, digital or ..) in the frequency range around the transition from tracking (typical
at low jitter frequencies) to inability to track (typical at very high jitter frequencies): an undershoot of the jitter tolerance may be present, and go unnoticed otherwise!.
The normalized jitter tolerance for all three important loop models used in (slave) CDRs is shown in the following figure.
under the hypothesis that the tolerance is limited by (and only by) the lateral eye opening, which is reasonable in most practical cases of SLAVE CDRs, see also the relevant page
It may be noted that the Bode plot of the curve of normalized jitter tolerance is the mirrored image (across the x axis) of the normalized error curve.
The asintote towards low frequencies (in the log-log plot) has a slope of 20 dB/ decade for the type 1 systems, and of 40 dB/ decade for the type 2 system.
The rejection of generated (VCO) noise…..Edit
Noise injected into the different nodes of the PLL loop reaches the output modified by the loop operation.
[See also the page Clock and Data Recovery/Noise is shaped by the PLL structure].
The description in the s domain of such modifications for a 2^nd order type 1 PLL is:
The noise generated by the filter N[f] is modified by the transfer equation TF[Nf] = (TF[VCO]) / (1 + TF[PhComp]*TF[filter][*]TF[VCO]))
The noise generated by the VCO N[o] is modified by the transfer equation TF[No] = 1/(1 + TF[PhComp]*TF[filter][*]TF[VCO])
The 2 - 1 magnitude is: |Y(jω)/N[o](jω)| = 20log10(((ω^2τ[z])^2+ω^2)^1/2 / ( (G-ω^2τ[z])^2 + ω^2 )^1/2)
The 2 - 1 loop attenuates the VCO noise at all frequencies below $\omega_{21}$, with a single slope (20 dB/decade) and with a gain overshoot that - whenever ζ > 0 , i.e. always - peaks at $\tfrac{\
omega^2}{\omega_{n21}^2} = \tfrac{1 + \sqrt{1+8\zeta_{21}^{2}}}{2}$ reaching a peak value that can represent a problem whenever ζ[21] is lower than 0.5.
The following figure illustrates the behavior of the 1-1 and 2-1 loops (using on purpose for the 2-1 a damping ratio that is unacceptably low) and gives the relevant formulas.
The 2-1 loop is normally used in regenerator applications (when $\omega_{21}$ is low): a low noise VCO must be used.
It also needs very good control of ζ[21] that must be kept close to 1.0 (this rules out the use of blocks with hard non-linearities and largely variable gain, like bang-bang phase detectors).
Unit step responseEdit
The analysis in the frequency domain (jitter transfer, error function, jitter tolerance, noise transfer) models the circuit behavior in the "small signal" conditions.
The Unit Step Response is part of the same mathematical model, and gives a time domain perspective of the circuit,
that complements the frequency domain perspective offered by the jω functions.
The Unit Step Response can be obtained from the jitter transfer function, multiplying it by 1/s and inverse transforming.
In fact the (unit) step response shows the response of the PLL output phase to an abrupt variation of the input phase.
The (unit) step response is used primarily to see how the acquisition phase evolves towards phase lock.
The step response of this type of loop is slower than that of the 1^st order (and than that of the 2^nd order type 2), because the loop filter does attenuates the high frequencies (while there is no
attenuation of high frequencies in the case of the other two loops).
At the beginning of an acquisition the signal pops up with some timing difference from the phase of the -free-running until that moment- local oscillator, and this can be well represented by a step
function, with a value of the step that may vary randomly, from acquisition to acquisition, in the range of -π to +π rad.
It is however important to be cautious when using the step response to model the system in case of large abrupt variations of the input phase.
Care is necessary because the boundaries of linear operation of the system (the range of the phase comparator, the swing of the filter output, the control range of the VCO, ..) can easily be reached
and exceeded during a (relatively) large transient.
The CDR will -very likely- still operate correctly but its performances may not be well described by this linear model during such transient.
The response when left free-runningEdit
The unit step response of the loop filter can give some understanding also on the behavior of the CDR when the latter loses lock and becomes free-running.
(The CDR is left free-running, for instance, when the incoming signal has no level transitions for the phase comparator to detect its phase. At that moment the phase comparator output steps abruptly
to its neutral -i.e. no phase difference- level.)
1. In a 1^st order loop, the comparator output waveform is applied directly to the VCO input (apart from some flat amplification). The VCO abruptly changes its frequency to its free-running
frequency. The sampling point drifts from its lock-in point following a linear (phase) ramp. The slope of the ramp is the frequency difference between the free-running frequency of the local
oscillator and the frequency of the remote transmit oscillator (Δf = f[fr] -f[s]).
2. In a 2^nd order type 1 loop, the loop filter smooths down the frequency step. The phase ramp does not start immediately, but gradually. The graduality corresponds to the time constant of the loop
filter, τ.
The following figure contrasts the behavior of the two loops, assuming that both have the same f[p] and the same VCO gain.
This helps understand the difference in the architectural sense, but may be misleading in practice, because the two loops, for the same f[p] , would be used in different applications with widely
different values of G[VCO].
G[VCO] would be much larger in a 1^st order loop, in consideration of the fast acquisition expected, and much smaller in the 2^nd order loop, in consideration of the tight jitter filtering expected.
The (1^st order type 1) loop drifts 1 rad exactly after a time equal to: f[p] / |f[p] – f[fr]| periods of received pulses.
But errored bits might appear even before, if the lateral eye opening was less than 1 rad (1 rad is an improbable value, 2 rad is a more likely threshold for the onset of error-ed bits).
In the case of a 2^nd order type 1 loop, (for the property of the 1^st order USR -of the filter inside the 2^nd order PLL- that the area lost versus the unit step is exactly τ = 1/ω) :
The filter of 2^nd order type 1 loop delays the onset of errored bits by about τ seconds ( by τ once settled, by somewhat less before )
Last modified on 28 March 2014, at 08:26 | {"url":"http://en.m.wikibooks.org/wiki/Clock_and_Data_Recovery/Structures_and_types_of_CDRs/The_(slave)_CDR_based_on_a_second_order_PLL/2nd_order_type_1","timestamp":"2014-04-17T15:37:14Z","content_type":null,"content_length":"74470","record_id":"<urn:uuid:edd12c4f-2ade-46be-8135-77ea07a47770>","cc-path":"CC-MAIN-2014-15/segments/1398223204388.12/warc/CC-MAIN-20140423032004-00383-ip-10-147-4-33.ec2.internal.warc.gz"} |
│ Symmetrical components │ Power invariance │ Impedance transformation │ Sequence networks │
│ Fault analysis │ Causes of system faults │ Symmetrical faults │ Sudden 3-ph short-circuit of unloaded generator-Definition of reactances │
│ System representation during short circuit │ Thevenin's equivalent │ Three-phase fault │ Single-line to ground fault │
│ │ │ │ │
│ │ │ │ OBJECTIVE TYPE QUESTIONS │
│ │ │ Internet sites: │ │
HOME takes you to the start page after you have read these Topics. Start page has links to other topics.
Internet sites:
ABC012: Handy phase sequence calculator. Microsoft Fortran Power Station 4.0 source and executable are included. http://www.ece.utexas.edu/~grady/PCFLO.html ( contains PCFLOW - short circuit, load
flow and harmonics program)
1. If a positive sequence current passes through a transformer and its phase shift is 30 degrees, the negative sequence current flowing through the transformer will have a phase shift of
a. 30 deg.
b. -30deg
c. 120 deg.
Ans: (b)
2. The zero sequence impedances of an ideal star-delta connected transformer (star-grounded)
a. looking from star side is zero and looking from delta side is infinite
b. looking from star side is zero and looking from delta side is also zero
c. looking from star side is infinite and looking from delta side is zero
Ans: (a)
2. The positive, negative and zero sequence impedances of a transmission line are 0.5,0.5 and 1.1 pu respectively. The self (Zs) and mutual (Zm) impedances of the line will be given by
(a) Zs = 0.7 pu, Zm =0.2 pu
(b) Zs = 0.5 pu, Zm =0.6 pu
(c)Zs = 1 pu, Zm =0.6 pu
Ans: (a)
3. Symmetrical component method of analysis is more useful when
a. system has unsymmetrical fault and the network is otherwise balanced
b. system has symmetrical fault and the network is otherwise unbalanced
c. system has unsymmetrical fault and the network is unbalanced.
Ans: (a)
2. For a line to line fault analysis using symmetrical components,
(a) the positive and negative sequence networks at the fault point are
connected in series
(b) the positive and negative sequence networks at the fault point are
connected in parallel
(c) the positive, negative, and zero sequence networks at the fault point are
connected in parallel
Ans: (b)
3. The machine reactances used for computation of short circuit current ratings of a circuit breaker are
a. synchronous reactance
b. transient reactance
c. sub-transient reactance
Ans: (c)
2. The load currents in short-circuit calculations are neglected because
i. short-circuit currents are much lager than load currents
ii. short -circuit currents are greatly out of phase with load currents
The correct alternative is
a. both (i) and (ii) are wrong
b. (i) is wrong and (ii) is correct
c. both (i) and (ii) are correct.
Ans: (c)
2. The positive sequence network of a sample power system is shown in Fig.2 and the primitive reactance of each element is marked in ohms. The element Zcc of the bus impedance matrix Zbus will be
a. 1.0
b. 0.1
c. 0.21
Ans: (c)
2. Which one of the following statements is not true?
a. Fault levels in an all a. c system are less than in an a. c system with a D.C. link operating
b. Large systems may be interconnected with d. c link of small capacity
c. Limitation on the critical length of underground cables for use in A. C no longer exists if D.C transmission by cables is used.
d. Corona loss and radio and TV interference with D.C. transmission is less
Ans. a
2. Advantage of power system interconnection is
a. Large size circuit breakers are required because of large short-circuit currents
b. Machines of one system remain in step with machines of another system
c. Fewer machines are required as reserve for operation at peak loads
Ans. c
2. The interenal voltages of a 3-phase synchronous generator correspond to
a. Positive sequence
b. Negative sequence
c. Zero sequence
These were developed to ease the calculations for unbalanced 3-phase systems and as a help to numerical solution using network analyzers. Even with present day digital computation, the symmetrical
components help in solution of unbalanced systems, besides explaining many phenomena such as rotor heating in machines, neutral current etc.
The basic concept is to convert a set of three phasors into another set of three phasors with certain desirable properties. The symmetrical components (introduced by Fortescue) is only one such set,
the other set is the Kimbark/Clarke components.
The unique property of symmetrical components is that they retain the concept of 3-phase system associated with each component phasor. Thus the positive sequence retains the concept of 3 balanced
phasors having the same phase sequence as the original phasors whereas the negative sequence component retains the concept of 3 balanced phasors but rotating in opposite direction of rotation. The
zero sequence component is balanced set of 3- coincident phasors but rotating in the same direction as the original unbalanced phasors.
The basic definition for a set of unbalanced three-phase system in terms of the sequence components is
I[a] = I[0a] + I[1a] +I[2a]
I[b] = I[0b] + I[1b] + I[2b] + I[0 a] + a^2 I[1 a] +a I[2 a]
I[c ]= I[0 c] +I[1 c] +I[2 c] = I[0 a] + a I[1 a] + a ^2 I[2 a]
These equations may be written in matrix form as
I ^a,b,c = [Ts] I ^0,1,2
Where Ts is called the symmetrical component transformation matrix.
I ^0,1,2 = [Ts]^-1 I ^a,b,c
Similar transformation may be applied for unbalanced voltages also.
Illustrate these transformations by phasor additions.
If the above transformation Ts is used simultaneously on the voltage and current values of three phase network elements, then
Spq ^abc = P pq + j Qpq =[( I pq^ abc )*]^t e pq^ abc
Spq^012 = [( I pq^012)*]^t e pq^012
Note that Spq^abc is not equal to Spq^012.
Spq^abc = 3Spq^012 since Spq^012 refers to only phase "a" power , and similar amount of power is in phases "b" and "c" also. Thus all three phases of symmetrical components must be used. Often for
computer work, the symmetrical transformation given by
T[si ] = (1/Ö 3) T[s]
is use where T[si]^*t . T[si] = unity matrix. This is a property of orthogonal matrices. Further since T[si]^* = T[si]^-1 we can show that Spq^abc = Spq^012. However, in earlier works, and even now,
the originally defined non-power invariant transformation is being used
To solve the network in terms of the sequence components, sequence components of impedance are required. These are obtained from their corresponding three phase values.
E^abc = Z^abc I ^abc
Ts E^012 = Z ^abc Ts I ^012
E^012 = T[si]^-1 Z ^abc Ts I ^012
= Z^012 I ^012
Thus Z^012 = T[s]^-1 Z ^abc Ts
Z^abc = T[s ]Z ^012 Ts^-1
Forms of Z^012 for balanced stationary and rotating elements should be known. Decoupling of sequence components and its limitations are to be stressed.
Sequence generated voltages in a balanced network are:
E^1 = Ea, E^2 = 0 , E^0 =0
Sequence networks have the advantage since for balanced network there is no mutual coupling between sequence component elements unlike what happens in balanced 3-phase components. Thus balanced
3-phase network can be assembled component for component in three separate sequence networks. Sequence networks of generator, transmission line, transformer, and loads should be known. Zero sequence
networks for different transformer connections should also be known. The ideal earth and the neutral point should be distinguished. Relative magnitudes of sequence impedances of generator and
transmission lines should be known.
• Find Z^012 for^ rotating and non-rotating elements in terms of corresponding phase quantities
• Study the measurements of zero sequence impedances of transformer for different connections
• Find phase shifts in sequence components in D U transformation
• If the positive, negative and zero sequence impedances of a transmission line are 0.3,0.3, and 0.5 respectively, find the self and mutual impedances between the phases.
An essential part of the design of a power system is the calculation of currents which flow in the components and the resulting voltages , when the faults of various type occur. Common faults on a
transmission system are;
1. LG
2. LL
3. LLG
4. LLL or LLLG
5. Open conductor
6. Simultaneous faults that may be any combination of the above five.
Faults may also occur in switchgear, transformers and machines but their nature may be different than those in transmission lines.
Faults are mostly due to lightning and switching. Most of them are temporary. LG fault is most common and LLL fault is least common. Relays provided on the system detect these faults and produce a
trip signal for the circuit breaker to isolate the faulty portion form the system. Thus to determine the circuit breaker and relay operating times, fault currents & voltages have to be calculated and
for many other applications.
Except for the three phase faults ,all other types of faults cause unbalanced operation, an fault currents & voltages under such conditions are required to be obtained using symmetrical components,
or phase components ( the latter analysis is much more difficult even with digital computers.
The calculation of 3-phase balanced faults is relatively simple but forms the basis of determining the circuit breaker ratings.
When a sudden short circuit occurs on the electric supply system, the currents & voltages are of transient nature before they settle down to steady state values.
The fault current at any time consists of
1. Dc transient , also called the DC offset, arising out of the terms of the type
2. Ac transients consisting of the terms of the type
Ae^-Rt/L Sin(w t + f ) where w = supply frequency
3. Steady state value of the type A Sin(w t + f ).
For a single machine system, the maximum value of the Dc transient could be equal to the peak value of the total Ac component.
Fig.5 shows the symmetrical current in one phase with no DC offset.( Other phases will have DC offset). The cause of subtransient/transient current could be explained on the basis of the theorem of
constant flux linkages associated with the field winding. The damper winding affects the initial value of the fault current. The initial value decays faster(region PQ in Fig.5) because of the very
low time constant of this circuit. Later, the closed field circuit effects dominate (region QR). The region QR may be extended up to the origin as the dotted line bQ.
Definition of reactances
X[d]" = Ö 2 .E. /Oa
X[d]'= Ö 2 .E. /Ob
X[d]= Ö 2 .E. /Oc
What are the typical values of these reactances for turbo and hydro generators?
For the interruption current rating of a circuit breaker (time involved being 2-5 cycles), the subtransient current is important. The transient current is important for transient stability studies
lasting from 1-2 seconds to 10 seconds.
Short circuit volt-amperes =Ö 3. (short-circuit current). [Nominal Voltage (line value)]
System representation during short circuit
Simplifying assumptions
1. Each machine is represented by a constant voltage behind the machine reactance, subtransient or transient depending on the situation.
2. Shunt connections i.e.; loads , line charging etc. are neglected.
3. All transformers are assumed to be at nominal tap settings.
4. The above three assumptions do not imply that the system is unloaded before the fault.. However , it is assumed that the network is unloaded and that all the generator voltages are at 1 pu. This
assumption is justified because fault currents in the network are much larger than the load currents.
If assumptions 2-4 are not taken into account, pre-fault currents are added to the fault currents obtained by using the Thevenin's equivalent of the network.
The Thevenin's equivalent of the network at the point of the fault location (either in phase or sequence components) is obtained for calculating the fault currents. If the network is balanced before
the fault, then at the fault point , Thevenin's sequence voltages become
E[a1]= E[a]
E[a2]= 0
E[a0]= 0.
If the network is not balanced , the above voltages may have finite values and the sequence component approach may not be useful. The performance equation for the balanced network in terms of
sequence components is :
[Vs] = [Eph]- [Zs][Is],
[Eph] = [0,E1a,0]
[Zs] = diagonal matrix of [Z0,Z1,Z2]
[Is] = [I0a,I1a,I2a]
The network at the fault point F appears as shown in
Fig.6 where Z1,Z2 and Z0 are the lumped values of the Thevenin's impedances between the fault point and the neutral.
Depending upon the type of fault, the sequence components of currents and voltages are constrained leading to particular connections of sequence network. After the fault currents coming out of the
network at the fault point are determined, the current & voltage distributions inside the network are found out . If originally, there are load currents , these are restricted to the positive
sequence network , and they are superposed on the fault currents for accurate results, but this is often not done.
Here ,
I[a1 ]= E[a]/(Z[1] +Z[F])
I[a2 ]= 0
I[a0]= 0
Single-line to ground (SLG)fault
Fault is assumed in phase "a". If it is in any other phase, symmetrical shifting will be required.
The connections at the fault point are given in
FIG.7 for various types of faults. The connections of the sequence networks for SLG fault are given in FIG.8 .
Study the network connections for L-L and DLG faults.
• How do you distinguish switching transient currents from sub-transient currents?
• Sketch the subtransient/transient current wave-shapes in phase b and c respectively for a synchronous generator when the current in phase a is symmetrical.
• Under what conditions can you expect a negative or zero sequence Thevenin's voltage at the fault point?
• Determine and draw the sequence network connection diagrams for
1. Single conductor opening at a point
2. Two conductors opening at a point. | {"url":"http://cindulkar.tripod.com/notes9.htm","timestamp":"2014-04-19T04:19:17Z","content_type":null,"content_length":"42203","record_id":"<urn:uuid:63768f03-c713-4c0a-8988-1854b3e16118>","cc-path":"CC-MAIN-2014-15/segments/1397609535775.35/warc/CC-MAIN-20140416005215-00223-ip-10-147-4-33.ec2.internal.warc.gz"} |
When Shape Really Matters
Volume 10
Story 2 - 27/7/2010
When Shape Really Matters
Particles in the nanoworld have mostly been modeled on unrealistically ideal shapes. A recent research takes advantage of a more complex and realistic model of the complex 3D shape of a nanoparticle.
3D tomography of a nanoparticle. The 3D shape of a nanoparticle can be reconstructed from many 2D SEM images of the same. Once the 3D morphology of the nanoparticle is known, its optical response can
be calculated with the algorithm developed by the researchers at the National University of Córdoba.
The nanoworld features a very complex reality. In order to have a more thorough insight into this reality we need to employ the use of models – be that as it may, every model has its own limitations.
The geometrical modeling of nanostructures, for example, has mostly been done through the use of ideal shapes. A new technique makes it possible to do away with any approximation in shape, thus
bringing the model even closer to reality. This work is the product of the collaboration between researchers at the National University of Córdoba (Córdoba, Argentina), at the Bariloche Atomic Centre
(San Carlos de Bariloche, Argentina), and at the University of Cambridge (Cambridge, UK).
Nanotechnologies are already permeating many areas in science and technology, where they help interpret, and take advantage of, the world of the ultra-small. The design of materials at the nanometer
scale in particular is becoming a routine process in many scientific disciplines. For this reason, even higher accuracy, reliability and reproducibility in the manufacturing of nanomaterials is of
utmost importance. Furthermore, there is a need for models that can help interpret, predict and design the properties of such nanomaterials.
Nanostructures are usually molded into shapes such as cubes, prisms or rods. This begs the question:
to which extend do the objects of the nanoworld reproduce such ideal shapes?
At a closer look, we will probably start noticing all kinds of surface defects and irregularities; this means that the particles are uglier than the actual ideal shape itself. We can get a vivid
picture of this by making a parallel with the everyday world around us: in the middle of the Egyptian desert, for example, the Pyramids, from afar, look exactly like the shape they take their name
from; a closer look, however, will reveal their rounded corners, the roughness of their walls, and quite a few more irregularities, undetected at first glance.
The optical properties of metallic nanostructures are largely dependent on their shape. Very intense electromagnetic fields are generated on the surface of the metal when hit by light of the right
color — color that can be tuned precisely by changing the shape of the nanostructure itself. In order to predict the behavior of metallic nanostructures, simulations have been performed mostly by
approximating their shape with ideal geometries. There is, however, a problem with this approach! In practice, every single deviation from the ideal shape can substantially affect the final optical
properties of the real nanostructure. "I started getting interested in this research topic," says Eduardo A. Coronado at the National University of Córdoba, "when trying to understand the factors
that control the sensing capabilities of metallic probes used in many techniques, such as TERS and MEF. In order to understand how these probes work it is of fundamental importance to be able to
correlate theory with experiments."
The work of Coronado and colleagues combines an established technique for the reconstruction of the 3D shape of a nanoparticle (3D tomography) with a novel algorithm to calculate its optical
properties. The new algorithm, unlike the previous ones, takes into account all the information of the 3D morphology of the nanoparticle. As Coronado says, "this achievement has only been possible
thanks to the joint effort of the researchers at Cambridge University, who offered their expertise in 3D tomography, and the group here in Córdoba, who developed a suitable algorithm to simulate the
optical response of the complex 3D nanoparticle."
How to know the exact 3D shape of a nanostructure? The method that makes it possible is called 3D tomography and already finds application in many fields, such as radiology, archeology, material
sciences and so on. The basic idea is that an image can only offer a 2D representation of an object, but many images, acquired from different perspectives, can be used to extract the 3D shape of the
same object. The greater the number of images, the better the reconstruction. The first step of Coronado and colleagues’ proposal generates, in fact, an exact 3D geometrical model of the
nanostructure from several SEM images.
The challenge for the researchers was to develop an
ad hoc
method to calculate the optical response of such a complex 3D nanostructure. "The challenge for our group in Córdoba," Coronado explains, "was to develop a numerical algorithm, based on the Discrete
Dipole Approximation (DDA) approach, which would be able to take into account all the features of the 3D shape of the nanostructure. So far all the simulations using this approach have been performed
for nanoparticles whose shape was approximated by regular shapes, such as spheres, spheroids, cylinders, prisms or truncated prisms." George C. Schatz at the Northwestern University (Illinois, USA)
points out that "what is novel here is the combination of the 3D tomography technique for measuring the shape of a complex nanoparticle with the DDA approach for calculating its optical properties.
Prior to this work, most of the optical property modeling was done using ideal particle shapes and, only in a few simple cases, was the shape of the particle derived by electron microscopy
"The gathered 3D information has allowed us to correlate the structural and morphological characteristics of the nanostructure under study with their physico-chemical properties," Coronado explains.
"There is no doubt that this correlation will also help redirect or optimize the present processes of fabrication in any nanotechnology field." The most straightforward application of these results
is the molecular sensing mediated by nanostructures, where the gathered information could help detect the most suited particle for specific applications. Schatz confirms that "this work shows that
nanoparticles are not the smooth objects we once thought them to be; they can have lumpy protrusions and irregular shapes. However, even these
particles might be effective for sensing applications. The results in the paper provide a simple way to determine when
particles might in fact have beautiful capabilities for applications in a broad range of technologies."
Giorgio Volpe
2010 © Optics & Photonics Focus
GV is currently working on his doctoral thesis at ICFO - The Institute of Photonic Sciences, Barcelona (Spain).
Eduardo M. Perassi, Juan. C. Hernandez-Garrido, M. Sergio Moreno, Ezequiel R. Encina, Eduardo A. Coronado & Paul A. Midgley, Using Highly Accurate 3D Nanometrology to Model the Optical Properties of
Highly Irregular Nanoparticles: A Powerful Tool for Rational Design of Plasmonic Devices, Nanoletters (2010) 10, 2097–2104 (link). | {"url":"http://www.opfocus.org/index.php?topic=story&v=10&s=2","timestamp":"2014-04-16T19:32:52Z","content_type":null,"content_length":"16270","record_id":"<urn:uuid:96493a85-29ee-4269-8732-45a66a689fdb>","cc-path":"CC-MAIN-2014-15/segments/1398223207985.17/warc/CC-MAIN-20140423032007-00492-ip-10-147-4-33.ec2.internal.warc.gz"} |
Please explainhow did he simplified this given?
January 31st 2010, 02:14 AM #1
Sep 2009
Please explainhow did he simplified this given?
Hi, Can you please show me the step-by-step process of this example. Especially the second and third line?
Here is the given:
Any help will be appreciated, thanks
x is a common factor of xy and x, so $xy-x=x(y-1)$
$x^2$ is a common factor of $x^2y^2$ and $x^2$, so $x^2y^2-x^2=x^2(y^2-1)$
Now $y^2-1$ is a common factor, so $x^2(y^2-1)+(y^2-1)=(y^2-1)(x^2+1)$
Finally, $y^2-1=(y+1)(y-1)$
hence y-1 is common to both sides and can be eliminated to obtain the last line.
January 31st 2010, 04:11 AM #2
MHF Contributor
Dec 2009 | {"url":"http://mathhelpforum.com/algebra/126398-please-explainhow-did-he-simplified-given.html","timestamp":"2014-04-16T11:47:18Z","content_type":null,"content_length":"36147","record_id":"<urn:uuid:a181c91f-1f65-448e-b9e6-0be37951c695>","cc-path":"CC-MAIN-2014-15/segments/1397609532128.44/warc/CC-MAIN-20140416005212-00580-ip-10-147-4-33.ec2.internal.warc.gz"} |
Tip: Use AS3′s .Histogram() for color averaging
Tip: Use AS3′s .Histogram() for color averaging
By LABS — February 28, 2011 - 7:11 pm
For several of my projects, I’ve been reusing a simple method of finding the average color of a region, based on an often-used technique of looping over each pixel, grabbing the pixel’s color using
BitmapData.getPixel(x,y), and bit-shifting the results to arrive at the individual color sums.
Great, I thought. No need to ever revisit this.
Then, once I got started exploring computer vision and signal processing, I started needing color averages for live video, pumping in at 30 frames per second, on top of a screaming mountain of other
intense math, and every millisecond started to matter.
So I followed up on a hunch I had about Actionscript’s BitmapData.Histogram() function. There’s a lot of uses for this function, but even after looking around, I haven’t found anyone who uses it for
finding the average color values. I gave it a shot and was very surprised at the result.
What the Histogram() method does is read over a BitmapData object and return a Vector.<Vector.<Number>> (a vector of vectors of numbers, if that makes sense) object with 4 rows–one each for red,
green, blue, and alpha–and 256 columns storing the total number of pixels matching each of the available brightness levels, 0-256, for that channel. With this information at our fingertips, shouldn’t
we be able to sum up the brightness levels and divide by the total number of pixels? Let’s try it out.
The old way: Bitshifting color values
function averageColor(source:BitmapData):uint {
var pixel:uint;
var red:Number = 0;
var green:Number = 0;
var blue:Number = 0;
var w = source.width;
var h = source.height;
var countInverse:Number = 1 / (w*h);
for (var i:int = 0; i < h; i++) {
for (var j:int = 0; j < w; j++) {
red += pixel >> 16
green += pixel >> 8 & 0xFF;
blue += pixel & 0xFF;
red *= countInverse;
green *= countInverse;
blue *= countInverse;
return red << 16 | green << 8 | blue;
Take a look at what’s going on. First I set up the variable holders for the red, green, and blue sums. Then I set up holders for the width, the height, and the inverse of the total pixel count
(Note: I use the inverse so that I only have to perform 1 division operation at the beginning of the function instead of 3 at the end).
Then I loop through each and every pixel in the image, store the pixel’s color, decompose it using the standard bit-shifting method, and add it to each sum. Assuming even a small image of 100 x 100
px, this result in 10,000 getPixel() calls.
The new way: Using the Histogram() function
function histogramTest(source:BitmapData):uint {
var histogram:Vector.<Vector.<Number>> = source.histogram();
var red:Number = 0;
var green:Number = 0;
var blue:Number = 0;
var w = source.width;
var h = source.height;
var countInverse:Number = 1 / (w*h);
for (var i:int = 0; i < 256; ++i) {
red += i * histogram[0][i];
green += i * histogram[1][i];
blue += i * histogram[2][i];
red *= countInverse;
green *= countInverse;
blue *= countInverse;
return red << 16 | green << 8 | blue;
See what I did there? Since I know each row in the Histogram() function’s result is a color channel, and each column is the count of how many pixels match each brightness level from 0 to 256, I can
do some quick math to come up with the exact same color average results in a fraction of the time.
2x – 10x Performance increase
My test image was reasonably large, at 675 x 644 px (a total of 434,700 pixels), and the speed difference was staggering. The Histogram method consistently booked a 2x speed increase in the browser
(10ms vs 5ms) and a 10x speed increase in Flash player (50ms vs 5 ms – for some reason the .getPixel() method is consistently 5x faster in the browser).
Give it a shot on your own projects and let me know if you see the same results.
UPDATE: A new method was recommended to me, using the BitmapData.draw() method to resize the image down to 1px x 1px and then doing a single BitmapData.getPixel() call to read it. While is is, like,
crazy fast (100 trials averaged 0.6 ms), the result wasn’t accurate enough for what I need it for. Who knows what perceptual color algorithms Actionscript uses, but it definitely wasn’t a pure
average function. For another project, maybe, but not when I need pixel-perfect accuracy.
actionscript, as3, development, performance, speed
One Response to “Tip: Use AS3′s .Histogram() for color averaging”
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Learning Materials
Data.LearningMaterials History
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[INS:*[[http://dataninja.wordpress.com | Dataninja ]] - tips, cheatsheets, etc. on data analysis, Stata, SAS, R.:INS]
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*[[http://www-stat.wharton.upenn.edu/~buja/sci.html | The Science of Scientific Writing]] by Gopen & Swan
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*[[http://www.ats.ucla.edu/stat/ | UCLA ATS Statistical Computing Resources]] - Extensive instructional materials on statistical analysis and software applications.
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-[DEL:<:DEL]Interesting list of statistical analysis books from the political blog site, Daily Kos. Short synopses plus comments from readers.
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[DEL:- Interesting entry from the political blog site, Daily Kos. Short synopses plus comments from readers. Here's the list.
!!!!Basic and Introductory and General and Miscellaneous
* Statistics as principled argument, by Robert Abelson.
* The elements of statistical learning by Hastie et al.
* An introduction to the bootstrap by Efron and Tibshirani.
* Classification and regression trees by Breiman et al.
!!!!Regression methods
* Regression modelling strategies, by Frank Harrell.
* Regression models for Categorical and limited dependent variables by J. Scott Long.
* Applied logistic regression by Hosmer and Lemeshow.
* Regression models for discrete longitudinal data by Molenberghs and Verbeke.
* Linear mixed models for longitudinal data by Verbeke and Mollenberghs.
* Applied longitudinal analysis by Fitzmaurice, Laird, and Ware.
* Visualizing data by William S. Cleveland
* The elements of graphing data
* All of Tufte's work
!!!!Categorical data
* Categorical data analysis, by Alan Agresti
* An introduction to categorical data analysis by Alan Agresti.
* Applied categorical data analaysis, by Chap Le.:DEL]
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!!!Found on the web:
[[http://www.dailykos.com/story/2006/12/3/62238/9507 | Statistics 101 Book Recommendations]]
- Interesting entry on the political blog site, Daily Kos. Short synopses plus comments from readers. Here's the list.
!!!!Basic and Introductory and General and Miscellaneous
* Statistics as principled argument, by Robert Abelson.
* The elements of statistical learning by Hastie et al.
* An introduction to the bootstrap by Efron and Tibshirani.
* Classification and regression trees by Breiman et al.
!!!!Regression methods
* Regression modelling strategies, by Frank Harrell.
* Regression models for Categorical and limited dependent variables by J. Scott Long.
* Applied logistic regression by Hosmer and Lemeshow.
* Regression models for discrete longitudinal data by Molenberghs and Verbeke.
* Linear mixed models for longitudinal data by Verbeke and Mollenberghs.
* Applied longitudinal analysis by Fitzmaurice, Laird, and Ware.
* Visualizing data by William S. Cleveland
* The elements of graphing data
* All of Tufte's work
!!!!Categorical data
* Categorical data analysis, by Alan Agresti
* An introduction to categorical data analysis by Alan Agresti.
* Applied categorical data analaysis, by Chap Le. :INS] | {"url":"http://lib.berkeley.edu/wikis/datalab/Data/LearningMaterials?action=diff","timestamp":"2014-04-18T18:23:19Z","content_type":null,"content_length":"34051","record_id":"<urn:uuid:431224df-5f4a-4f61-ac32-e973c669ac9c>","cc-path":"CC-MAIN-2014-15/segments/1397609535095.7/warc/CC-MAIN-20140416005215-00204-ip-10-147-4-33.ec2.internal.warc.gz"} |
Julian Barbour on does time exist
I need to plug ahead with how time (as a flow on the observable algebra) emerges. For continuity, here are the essentials of the last post:
Given an algebra
of observables and a state of the world ω it is possible to derive a unique flow α
on the algebra. Taking each observable A into a progression of "later" evolved observables α
A, for every timeparameter number t.
A nice thing is that this "thermal time" construct RECOVERS ordinary time when we start with a conventional Hamiltonian H and hilbertspace
. this is what Connes Rovelli show on pages 16 and 17 of their paper. See link:
pages 16,17
WikiP: "Gelfand-Naimark-Segal construction"
WikiP: "KMS state"
WikiP: "Tomita-Takesaki theory" (not so good I think, but at least article exists)
WikiP: "Polar decomposition" (article exists, I haven't used or evaluated it)
The basic situation that general covariant quantum physics deals with is an algebra
of observables. That's the world. After all QM is about making measurements/observations. And a temporal flow α
is a oneparameter group of automorphisms of that algebra.
automorphism means it maps an observable A onto another observable α
A which you can think of as making the same observation but "t timeunits later".
oneparameter group means that doing α
and then doing α
has the same flow effect as doing α
. the parameter t is a real number.
And automorphism means it preserves the algebra operations, it is linear etc etc.
Observables are in fact an algebra because you can add and multiply observables together to predict other observables or to find how they correlate with each other.
The statistical quantum state of the world is represented by a positive functional on the algebra which we can think of as a density matrix ω and its value on an observable A can be written either as
ω(A) or as trace(Aω). The state ω is what gives the observables their expectation values and their correlations.
A nice thing about a density matrix ω is that it has a square root ω
. Think of writing it as a diagonal matrix with all positive entries down the diagonal, and taking the square root of each entry.
The observable algebra (think matrices) IS a vector space. You can add matrices entry-wise and so on. The celebrated GNS construction makes a vectorspace out of |ω
⟩ together with all the other density matrices and their like which you can get by applying elements A of the algebra to that root vector. that is called the FOLIUM of ω
⟩ for all A in
It is a hilbertspace. The special good things about this hilbertspace (they give it a name,
) is that the algebra acts on it, after all it was MADE by having the algebra act on the single root vector |ω
⟩ and seeing what you get, and the other thing is just that: it has what is called a "cyclic vector", a root or generator: the whole hilbertspace is made by having the algebra of operators act on
that one |ω
⟩, as we have seen.
ω(A) = ⟨ω
Now what C&R do is they construct an operator, by giving its polar decomposition. This is what happens on page 17. And the operator obtained by putting the polar decomp. together has the effect of
doing a matrix transpose, or mapping A → A*. They call this operator S.
SA |ω
⟩ = A* |ω
There is some intuition behind this (there is already something about it on page 7 but I'm looking at page 17). It is like swapping creation and annihilation operators. Undoing whatever an operator
does. Partly it is like getting your hands on what is implicitly an infinitesimal time-step, except there is no time yet. More importantly, transpose is tantamount to commuting
(AB)* = B*A*
So if we can just take the anti-unitary part out of the picture it's almost like swapping order: AB → BA. Yes very handwavy, but there is some underlying intuition, will get back to this.
We are going to build from that swapping or reversal operator S. In particular we will use the positive self-adjoint part of the polar decomposition. More about this later. | {"url":"http://www.physicsforums.com/showthread.php?p=4174050","timestamp":"2014-04-17T21:41:44Z","content_type":null,"content_length":"104810","record_id":"<urn:uuid:b7c01f7b-8984-44fb-aa31-74e6a4027834>","cc-path":"CC-MAIN-2014-15/segments/1397609532128.44/warc/CC-MAIN-20140416005212-00505-ip-10-147-4-33.ec2.internal.warc.gz"} |
Newest 'lie-algebras' Questions - Page 5
Lie algebras are algebraic structures which were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" (after Sophus Lie) was introduced by Hermann Weyl in the
1930s. In older texts, the name "infinitesimal group" is used. Related mathematical concepts include Lie ...
learn more… | top users | synonyms | {"url":"http://mathoverflow.net/questions/tagged/lie-algebras?page=5&sort=newest&pagesize=15","timestamp":"2014-04-21T15:16:08Z","content_type":null,"content_length":"172602","record_id":"<urn:uuid:3776c2c5-2e32-4f65-a401-0110d24cb54f>","cc-path":"CC-MAIN-2014-15/segments/1397609540626.47/warc/CC-MAIN-20140416005220-00340-ip-10-147-4-33.ec2.internal.warc.gz"} |
Re: Re: (OT) Where is programming headed?
in reply to Re: (OT) Where is programming headed?
in thread (OT) Where is programming headed?
> Perhaps I misunderstand your meaning, but it seems to me that Gödel's
> first incompleteness theorem says exactly that algebra cannot be
> reduced to a finite set of postulates.
Godel proved that the axiom of choice can't be disproved using the other axioms of Cantor set theory. Cohen followed that up by proving that the axiom of choice is independent of the remaining
axioms. In other words, instead of a single concept of set theory, we have two options: one where we assume that the axiom of choice is true, and one where we assume that it's false. Cantor's
postulates don't give us enough information to choose one over the other, and that makes them incomplete.
You can't apply Godel's theorem the opposite way, though. In either case, the system you choose is completely generated from its particular basis.
> If "brute-force examination" were sufficient, one would write a
> computer program to perform the brute-force examination and come up
> with the answer. But this is precisely what we know is impossible.
We're missing each other slightly on this one.. you're talking about a finite, algorithmic solution, and I'm talking about the infinite, theoretical one. You're saying the halting problem can't be
solved by a Turing machine, and I'm saying it can, but only in infinite time. We're don't disagree, we're just saying the same thing in different ways.
By contrast.. and I'll refer back to this in a minute.. some problems can't be solved at all, even in infinite time. Random number prediction is an example. It doesn't matter what equipment we use,
or how much information we gather, we'll never be able to predict the next value of a true random sequence.
> I won't address your assertion that a solution to the halting problem
> will provide a solution to "the automatic programming problem",
> because you still haven't said what "the automatic programming
> problem" is, and I'm starting to suspect that you don't have a very
> clear idea of it yourself.
Well, I certainly haven't gotten it across so far.. ;-)
Let's try this: In your original response, you said that gcc "takes an input specification and writes a machine language program to implement that specification".
Now let me ask this: where does that input specification come from? Could you write a program to generate it? If so, what would you have to feed into that program?
Compilers are translators. You don't get more out of them than you put in.
Theorem provers, OTOH, can start from a set of basic axioms and discover arithmetic and higher mathematics on their own. They can even solve problems that humans haven't.. In 1933, Herbert Robbins
came up with a set of axioms that he thought were a basis for Boolean algebra, but nobody could prove it. In 1996, a program called EQP crunched out the proof. Nobody gave EQP an input specification
that it translated into a proof, it came up with the proof on its own. We got something out that we didn't put in.
Genetic algorithms also produce solutions to problems we can't specify, but they do it through random perturbation. In effect, they assemble jigsaw puzzles by shaking the box for a while, opening it,
gluing any interlocked pieces together, then closing the box and shaking it again. Even so, we get something out that we didn't put in.
If we put theorem provers on one side, and genetic algorithms on the other, automatic programming falls somewhere between.
If programming can be reduced to a finite set of generative postulates, automatic programming will collapse to theorem proving. We'll be able to write programs that derive other programs from the
core set of postulates, and they'll eventually write code faster and better than humans can.
If programming can't be reduced to a finite set of generative postulates, it means that writing software is like trying to guess random numbers. No algorithmic solution is possible, and genetic
algorithms hooked to expert systems are the closest we'll ever get to programs that generate software without requiring an input specification that completely defines the output.
> Are you now saying that code to compute the square root of 2 will fail
> "as often as not"? That isn't true either.
Uh, no.. I'm saying that computers represent floats as N-bit binary logarithms, and that logs are notorious for producing microscopic variations when you truncate them. See Question 14.4 from the C
FAQ, for instance. Subtracting one root of 2 from another might give you 1e-37, which is awfully close to zero, but still different enough to fail a straight equality test. That goes back to my
original point regarding computers being finicky about precision.. as opposed to humans. ;-) | {"url":"http://www.perlmonks.org/?node_id=132326","timestamp":"2014-04-23T23:32:10Z","content_type":null,"content_length":"40271","record_id":"<urn:uuid:8e31ae3b-83b0-4994-b6a8-57609cca593a>","cc-path":"CC-MAIN-2014-15/segments/1398223203841.5/warc/CC-MAIN-20140423032003-00527-ip-10-147-4-33.ec2.internal.warc.gz"} |
Finding the radius of an inscribed circle - Math Central
Maria, we have two responses for you:
Hi Maria.
Draw the radii to each of the three points of tangency and connect the vertices of the triangle to the center of the circle. Do you see that you have three pairs of congruent triangles? They are
congruent because they are right triangles whose hypotenuses is shared and they have the same length of a leg (the radius).
Now there are three new variables to calculate (actually, just getting one of them is sufficient for your goal):
Since these are congruent triangles, you know that angle C was divided exactly in half, so you know the measures of all the angles here. If you know the length y, then you can use the Tangent
function to find the radius r. So now the problem is: what is y?
Actually, you can find that quickly by noticing that there are three equations and three variables:
x + z = 21
x + y = 51
y + z = 34.
Solve these simultaneous equations (using either the substitution or the elimination method) for y. Then use it in the Tangent function to find r.
Hope this helps,
Stephen La Rocque.
Hi Maria.
Stephen's answer overlooked a small problem: The angles cannot be very accurate -- they do not sum to 180 degrees.
Here is a formula in terms of the three sides:
If the sides have length a, b, c, we define the semiperimeter s to be half their sum, so s = (a+b+c)/2. Given this, the radius is given using the following:
r^2 = (s - a)*(s - b)*(s - c) / s.
Take the square root of this expression to find r.
Prof. J. Chris Fisher. | {"url":"http://mathcentral.uregina.ca/QQ/database/QQ.09.06/s/maria2.html","timestamp":"2014-04-21T09:37:06Z","content_type":null,"content_length":"9077","record_id":"<urn:uuid:ce01891d-b9ae-4869-a96c-5d35acc44077>","cc-path":"CC-MAIN-2014-15/segments/1397609539705.42/warc/CC-MAIN-20140416005219-00618-ip-10-147-4-33.ec2.internal.warc.gz"} |
Computing efficiently
September 3rd 2010, 02:02 AM #1
What is gained when $x^y$ is computed as $2^{y\log_2 x}$ (for $x>0$) rather than the "direct" (but not naive) way?
You get a smaller base (not sure if that helps, but I'll leave it to be corrected ... on the other hand, $\log$ introduces (additional) error.
I don't think I'll have the answer in any case, but you should probably specify whether x,y are integers/floats/other.
This is from a chapter on IEEE-754 standard, so call $x$ and $y$ "floats". Though, at the level of 23 (or 52 for double precision) bits, they're all really just integers, aren't they?
The textbook uses the above (questioned) fact just for showing, that even 52 bits (double precision) aren't good enough for IEEE and we need 64 bits (Extended precision). But it doesn't show why
$x^y$ is better computed as $2^{y\log_2 x}$.
Last edited by courteous; September 3rd 2010 at 08:12 AM. Reason: [math]
This is from a chapter on IEEE-754 standard, so call $x$ and $y$ "floats". Though, at the level of 23 (or 52 for double precision) bits, they're all really just integers, aren't they?
The textbook uses the above (questioned) fact just for showing, that even 52 bits (double precision) aren't good enough for IEEE and we need 64 bits (Extended precision). But it doesn't show why
$x^y$ is better computed as $2^{y\log_2 x}$.
Perhaps this will be of some use?
Info: (gmp) Normal Powering Algorithm
There might be further notes in the source code of GMP.
Couldn't help myself with GMP documentation.
Should've looked at Wikipedia first, though this paragraph is still unclear (to me):
Moreover, $cd = b^{(\log_b c)d}$ reduces the exponentiation to looking up the logarithm of c, multiplying it with $d$ (possibly using the previous method*) and looking up the antilogarithm of the
Looking up, as in "looking up in the tables"? Is this then what makes it efficient, the fact that computer has those tables already stored?
Without this stored table, this method isn't any more efficient, because antilogarithm is exponentiation, right (or is there some subtle difference (like between antiderivative and integral, I
think ... )?
* $cd = b^{\log_b c}*b^{\log_b d} = b^{\log_b c + \log_b d}$
Couldn't help myself with GMP documentation.
Should've looked at Wikipedia first, though this paragraph is still unclear (to me):
Looking up, as in "looking up in the tables"? Is this then what makes it efficient, the fact that computer has those tables already stored?
Without this stored table, this method isn't any more efficient, because antilogarithm is exponentiation, right (or is there some subtle difference (like between antiderivative and integral, I
think ... )?
* $cd = b^{\log_b c}*b^{\log_b d} = b^{\log_b c + \log_b d}$
Isn't exponentiation with base 2 just a bitshift?
2^0 -> 1
2^1 -> 10
2^2 -> 100
Or am I missing something? I haven't really worked with floating point in this way.
It is (or rather can be) more efficient because you are using base 2 and a binary float representation, so logs are only ever required for numbers in the range 1 to 2, and the main part of a log
can be read straight out of the exponent (-ish if we ignore the normalisation/non-normalisation complication) and the same sort of tricks can be played in the reverse direction.
September 3rd 2010, 06:53 AM #2
September 3rd 2010, 08:11 AM #3
September 3rd 2010, 09:44 AM #4
September 4th 2010, 06:31 AM #5
September 4th 2010, 06:50 AM #6
September 4th 2010, 08:17 AM #7
Grand Panjandrum
Nov 2005
September 4th 2010, 11:09 PM #8 | {"url":"http://mathhelpforum.com/math-topics/155098-computing-efficiently.html","timestamp":"2014-04-18T04:36:47Z","content_type":null,"content_length":"63283","record_id":"<urn:uuid:769d008d-a393-46b0-9def-a1fec78d8109>","cc-path":"CC-MAIN-2014-15/segments/1397609532480.36/warc/CC-MAIN-20140416005212-00184-ip-10-147-4-33.ec2.internal.warc.gz"} |
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A few proofs in ranks and null
April 25th 2008, 06:11 AM #1
Super Member
Mar 2006
A few proofs in ranks and null
Let U be a linear operator on a finite dimensional vector space V, prove that:
a) $N(U) \subseteq N(U^2) \subseteq N(U^3) \subseteq ... \subseteq N(U^k) \subseteq N(U^k+1) \subseteq . . .$
b) If $rank (U^m) = rank (U^{m+1})$ , then $N(U^m) = N(U^k) \ \ \ \forall k \geq m$
c) Let T be linear operator on V whose characteristic polynomial splits, and let $\lambda _1 \lambda _2 , . . . , \lambda _k$ be distinct eigenvalues of T. Then T is diagonalizable if and only if
$rank (T- \lambda _i I ) = rank ((T- \lambda _i I )^2 )$ for $1 \leq i \leq k$
I start you off. Let $x\in N(U)$ then by definition (I assume it is the nullspace) $U(x) = \bold{0}\implies U(U(x)) = U(\bold{0}) = \bold{0}$. Thus, $x\in N(U^2)$. Now generalize the argument to
prove that $N(U^{k}) \subseteq N(U^{k+1})$.
b) If $rank (U^m) = rank (U^{m+1})$ , then $N(U^m) = N(U^k) \ \ \ \forall k \geq m$
Using the rank nullity theorem we find that $\text{rank}(U^m) + \text{nullity}(U^m) = \text{rank}(U^{m+1})+\text{nullity}(U^{m+1})$ thus, $\text{nullity}(U^m) = \text{nullity}(U^{m+1})$. Thus, $\
text{dim}N(U^m) = \text{dim}N(U^{m+1})$ since $N(U^m)$ is a subspace of $N(U^{m+1})$ it actually means $N(U^m) = N(U^{m+1})$ (because their dimensions are the same). Now generalize this argument
for $m+2,m+3,...$.
Sorry I cannot help with (c), my linear algebra is not that great.
April 25th 2008, 08:37 AM #2
Global Moderator
Nov 2005
New York City | {"url":"http://mathhelpforum.com/advanced-algebra/35950-few-proofs-ranks-null.html","timestamp":"2014-04-17T08:39:05Z","content_type":null,"content_length":"38510","record_id":"<urn:uuid:6c879ffc-7d09-42ef-946b-072c82a833d7>","cc-path":"CC-MAIN-2014-15/segments/1397609526311.33/warc/CC-MAIN-20140416005206-00499-ip-10-147-4-33.ec2.internal.warc.gz"} |
Math help needed ASAP please
February 18th 2010, 11:07 AM #1
Feb 2010
Can anyone help me understand how to do these questions?
I f the number of meals served in a hospital is a normal distribution with amean of 250 and a standard deviation of 37
a) What is the probability that more than 300 meals are served?
B) For the middle 50% of the distribution, what is the range of the number of meals served?
a) Hint: $300$ is $\frac{50}{37} = 1.35135...$ standard deviations away from the mean.
b) There is a number $a$ such that the interval $(250 - a, 250 + a)$ comprises 50% of the distribution. Determine a in terms of the standard deviation.
Using this information, you can find both answers with a table.
I still don't understand, what do I use for a table?
What is a probability density function for the normal distribution? Sorry, I have no background with this stuff, and I really don't have a clue how to do it? Can you help me further?
In general, a probability density function is a function of x. If the probability density function for a random variable x is given by p(x),
then $\int _a ^b p(x) \cdot dx$ gives you the probability that x is between a and b. Have you never seen this before?
No, never. I am not able to complete this problem...I am so lost.
Suffice it to say that the probability density function for the normal distribution is a specific probability density function that describes the normal distribution in particular. If you don't
know what that function is, and you don't have a table, then I'm not sure how you're supposed to solve the problem either.
Suffice it to say that the probability density function for the normal distribution is a specific probability density function that describes the normal distribution in particular. If you don't
know what that function is, and you don't have a table, then I'm not sure how you're supposed to solve the problem either.
Where do you get a table? If I had a table how do I use it?
Most statistics books have tables in the back that give you the information you need to solve problems relating to specific types of distributions, the normal distribution being the most common.
For instance, I have a table in my statistics book that gives me the probability of x being farther than 1.35 standard deviations from the mean in the positive direction, which is the answer to
Most statistics books have tables in the back that give you the information you need to solve problems relating to specific types of distributions, the normal distribution being the most common.
For instance, I have a table in my statistics book that gives me the probability of x being farther than 1.35 standard deviations from the mean in the positive direction, which is the answer to
Ok, so is a 0.4115?
February 18th 2010, 11:17 AM #2
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Feb 2010 | {"url":"http://mathhelpforum.com/statistics/129486-math-help-needed-asap-please.html","timestamp":"2014-04-19T10:03:44Z","content_type":null,"content_length":"73328","record_id":"<urn:uuid:9b59ca81-af5d-40d5-8ee1-32065c8a2f90>","cc-path":"CC-MAIN-2014-15/segments/1397609537097.26/warc/CC-MAIN-20140416005217-00632-ip-10-147-4-33.ec2.internal.warc.gz"} |
The String Coffee Table
More Vietri Talks
Posted by Urs Schreiber
I’ll very quickly mention a few aspects of some further talks that we heard ($\to$) on Thursday. A collection of slides for all talks should be available online soon.
There was a talk on topological membranes by Allesandro Tanzini
Giulio Bonelli, Alessandro Tanzini, Maxim Zabzine
On topological M-theory
As you all know, the idea of topological M-theory is to realize topological string theories as limits of topological membrane theories, in analogy to how phyiscal superstrings are realized as limits
of physical membranes.
Instead of saying anything about this particular work, I’ll restate a question which was raised, but not fully answered after the talk:
The topological membrane couples to the supergravity 3-form just like the topological string couples to the Kalb-Ramond 2-form. We know that this 3-form is best thought of as the Chern-Simons 3-form
of an ${E}_{8}$ connection on spacetime. Hence, if we forget about the other couplings of the toplogical membrane for a moment, its coupling to the SUGRA 3-form makes it look a loot like ${E}_{8}$
Chern-Simons theory on its 3-dimensional worldvolume.
Not quite of course, because what we vary is the membrane “embedding” into spacetime, not the ${E}_{8}$ connection, which is merely pulled back from spacetime to the membrane worldvolume using this
embedding. But anyway, it seems suggestive that, (according to CMP 121, 3 (1989), 351-399 ) on the boundary of an ${E}_{8}$ Chern-Simons theory there lives a 2D conformal field theory with ${E}_{8}$
current algebra.
Hence it looks like the boundary theory of topological membranes coupled to just the SUGRA 3-form should be closely related to ${E}_{8}$WZW models. But a similar picture is of course implied by
Hořava-Witten theory, where the boundary theory of the membranes is the heterotic string carrying ${E}_{8}$ current algebras.
Ok, so the question is: Does anyone know if there is a precise way in which the ${E}_{8}$ current algebra of the heterotic string can be understood as the boundary theory of an ${E}_{8}$-Chern-Simons
theory living on a membrane, induced from its coupling to the SUGRA 3-form?
A few notes from another talk: Jarah Evslin sketched work in progress which is supposed to illuminate the true nature of RR-fields, clarifying in particular the behaviour under S-duality ($\to$). In
the available 45 minutes he could only sketch the most basic setup, which, even further condensed, is the following:
Let ${G}_{p}$ be the $p$-form field strength of an RR-$\left(p-1\right)$-form in either IIA or IIB. Let $H$ be the 3-form field-strength of the Kalb-Ramond 3-form. (This induces the “twist” in the
twisted K-theory classification of the RR-fields.) The equations of motion say that
(1)$H\wedge {G}_{p}=d{G}_{p+2}\phantom{\rule{thinmathspace}{0ex}}.$
Jarah’s idea is to use this equation in order to regard the operator of left wedge multiplication with $H$ as a “BRST operator” in the following sense.
The ${G}_{p}$ are the “physical fields”.
Gauge transformation parameters $\Lambda$ of these fields are one type of “ghosts” and D-branes are another type of “ghosts”.
This makes sense in that $d{G}_{p}$ is proportional to magnetic “brane density”, so that $H\wedge$ can be regarded as mapping “fields” (${G}_{p}$) to “ghosts” (brane current densities). Hence we have
a complex looking something like this:
(2)$\left\{\Lambda \right\}\stackrel{H\wedge }{\to }\left\{{G}_{p}\right\}\stackrel{H\wedge }{\to }\text{D-branes}\phantom{\rule{thinmathspace}{0ex}}.$
Think of this as ${ℤ}_{2}$-graded with the central term living in grade 0.
Now, physical fields, namely gauge orbits of field strenghts, do live in the ghost 0 cohomology of $Q=H\wedge$, which corresponds to the K-theory ${K}_{H}^{0}\left(M\right)$, with $M$ being spacetime
and $H$ indicating twisted K-theory.
Similarly, the D-branes themselves live in ghost-number 1 cohomology of $Q=H\wedge$, corresponding to ${K}_{H}^{1}\left(M\right)$.
I can sort of see how this should work, but if you find any of this suspicious we will have to ask Jarah to help us with the details…
Jarah said this way of looking at things should eventually be useful for understanding S-duality covariant K-theory. Namely, in order to obtain something of this sort, he expects it to be sufficient
to enlarge the above described $H\wedge$-cohomology by similar cohomologies which include also the gauge transformations of the KR-field itself.
He would probably have needed 450 instead of 45 minutes to get the details across. But a pdf with more notes is supposed to appear in the online proceedings of the conference.
Finally, I should mention that Brano Jurco gave nice overview talk about the latest on 1- and 2-gerbes with structure group the string group for $\mathrm{Spin}\left(n\right)$ and for ${E}_{8}$, and
how this is related to the Green-Schwarz anomaly and possibly to M5-brane anomalies. If I find the time, I’ll report on more details of this talk. But right now I have absolutely no time left.
Posted at April 16, 2006 5:29 PM UTC
Re: More Vietri Talks
The PDF that you mentioned with the details on the isomorphism between twisted K-theory and a BRST cohomology has now appeared, it is at http://paft06.sa.infn.it/contributi13/Evslin.pdf.
Posted by: Jarah on April 22, 2006 4:56 PM | Permalink | Reply to this
Re: More Vietri Talks
Ah, now the manuscripts are all online.
Hm, looking at the photographs one gets the impression that I never did anything else than talking to Brano. ;-)
Posted by: urs on April 22, 2006 5:23 PM | Permalink | Reply to this | {"url":"http://golem.ph.utexas.edu/string/archives/000791.html","timestamp":"2014-04-19T19:33:35Z","content_type":null,"content_length":"20626","record_id":"<urn:uuid:c1e3cb13-7ef0-450d-aff1-8485ada3dc15>","cc-path":"CC-MAIN-2014-15/segments/1398223206647.11/warc/CC-MAIN-20140423032006-00538-ip-10-147-4-33.ec2.internal.warc.gz"} |
Equation for a quadratic curve in a three-dimensional space?
July 24th 2010, 07:31 PM
Equation for a quadratic curve in a three-dimensional space?
I'm not really sure if this is the right area BUT I did similar things such as this in trig so I thought this might be a good place to ask.
I'm working within a three dimensional space and I have three points that I must create a quadratic curve out of. The curve must touch all three points. In 2d the process for solving this is easy
enough but in three dimensions I begin to have trouble. I built a script that can find and output the X,Y,and Z coordinates and their distance in relation to each other but I need to know a
formula (or possibly two combining formulas. I'm not really sure how using two would work for programming) to plug in the values. Anyone know a formula like this?
July 24th 2010, 07:37 PM
Do you think this is really a question for the trigonometry sub-forum?
Best ask a mod to move it to improve the chanes of getting a proper reply.
July 25th 2010, 09:11 AM
A curve is one-dimensional so to write a curve in three dimensions, you need to reduce from 3 variables to one. you can do that by either using two equations in the three coordinates (so that the
two equations can be solved for two of them in terms of the other one) or three equations for x, y, and z in terms of a parameter, t, say. That is, something like:
$x= at^2+ bt+ c$, $y= dt^2+ et+ f$, and $z= gt^2+ ht+ i$.
You are free to choose what values t takes at each point, although choosing values reflecting the distances between the points might simplify things. Putting the t values and the x, y, z values
of the points in the parametric equations will give 9 linear equations for the nine coefficients, a, b, c, d, e, f, g, h, and i.
July 25th 2010, 09:59 AM
Hello, Korinkite!
I don't agree with your intial statements.
I'm working within a three-dimensional space
and I have three points that I must create a quadratic curve out of.
In 2-D the process for solving this is easy enough. . really?
Three points do not determine a quadratic curve.
General quadratic function: . $Ax^2 + Bxy + Cy^2 + Dx + Ey + F \:=\:0$
. . which has six coefficients to be determined.
We have only three points: . $(x_1,y_1),\;(x_2,y_2),\;(x_3,y_3)$
They produce this system of equations:
. . $\begin{Bmatrix}Ax_1^2 + Bx_1y_1 + Cy_1^2 + Dx_1 + Ey_1 + F &=& 0 \\ \\[-3mm]<br /> Ax_2^2 + Bx_2y_2 + Cy_2^2 + Dx_2 + Ey_2 + F &=& 0 \\ \\[-3mm]<br /> Ax_3^2 + Bx_3y_3 + Cy_3^2 + Dx_3 + Ey_3
+ F &=& 0 <br /> \end{Bmatrix}$
. . which obviously does not have a unique solution.
July 25th 2010, 04:52 PM
Got it!
Sorry for the late reply. First to Soroban, I apologize if I wasn't being entirely clear. I didn't mean to suggest that there could be one perfect equation for a quadratic curve with three
points. In the end, I did get Hallsofivy's method to work so a few screenshots might save the trouble of explaining.
Quadratic Curve :: Quadratic Curve Finished! picture by Korinkite - Photobucket
As you can see, the result isn't perfect (or at least not entirely what I had intended. Originally I wanted the second point to always be the origin but this works just fine for my purposes I
In those two pictures, I demonstrate how I can move any point in any direction and as far out as necessary and the curve will compensate!
Thank you Hallsofivy and all that commented for your time and assistance ^_^. | {"url":"http://mathhelpforum.com/trigonometry/151899-equation-quadratic-curve-three-dimensional-space-print.html","timestamp":"2014-04-17T11:02:56Z","content_type":null,"content_length":"9471","record_id":"<urn:uuid:96faba2f-10cc-4b04-aeaa-7b5106129045>","cc-path":"CC-MAIN-2014-15/segments/1397609527423.39/warc/CC-MAIN-20140416005207-00419-ip-10-147-4-33.ec2.internal.warc.gz"} |
Punch Line Algebra Book A 7 11
What are all the answers for punchline algebra book a page, Answers to page 17.14 punchline algebra book b? she was having contractions. what is the answer to page 8.3 punchline algebra book a? i
just did it and im pretty sure. What is the answer to page 7.8 in punchline algebra book a, What is the answer to page 7.8 in punchline algebra book a what is used to repair big brass band
instruments?. Classzone - algebra 1, Welcome to algebra 1. this course will make math come alive with its many intriguing examples of algebra in the world around you, from baseball to theater
lighting to.
Algebra - wikipedia, the free encyclopedia, The start of algebra as an area of mathematics may be dated to the end of 16th century, with françois viète's work. until the 19th century, algebra
consisted. Punch line math worksheet answers at askives, Marcy math works ( punchline algebra) punchline algebra set • books a & b punchline bridge to algebra punchline problem solving
mathimagination more info: about us. Algebra survival guide: a conversational guide for the, Josh rappaport is the author of the parents' choice award-winning algebra survival guide, and co-author of
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game roundup books. josh. Untitled document [www.marcymathworks.com], The punchline algebra set consists of two binders, each containing 192 pages. book a includes topics often taught in the first
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X-bit’s Guide: Contemporary LCD Monitor Parameters and Characteristics. Page 21
First, the matrix has viewing angles. The problem of the early 16msec matrices was in awfully small viewing angles, which prevented the user from working normally. Even sitting motionlessly and
looking at the center of the screen perpendicularly you could see that the top of the screen was much darker than its bottom, and colors were yellowish at the sides. By the way, this thing – a strong
darkening of the image when it is viewed from below – unmistakably distinguishes a TN matrix from the other types, which have no such defect.
Of course, some improvements have been made since. The horizontal viewing angles are now wide enough for you to sit before the screen with your friend without complaining at the “impure” white color;
the vertical angles aren’t provoking any big discomfort too, although the vertical irregularity of the screen brightness is still visible even in the best samples. Unfortunately, the manufacturers of
monitors on TN matrices, trying to match the competitive matrix types in this parameter at least on paper now often declare the viewing angles measured by the fall of the contrast ratio to 5:1 rather
than to 10:1. This way TN matrices acquired 160-degree angles into their specs, without any real improvements. I want to warn you once again and remind you the above-described method of measuring the
viewing angles. A true specified angle of 140 degrees doesn’t mean that its defects are only visible “when you are looking at it from under the desk” or “dance before it when working” as some users
think, because visible distortions of the image appear much sooner before you reach those specified angles. The number “140 degrees” means you’ll see strong image distortion when looking at the
screen at such an angle. For example, the vertical irregularity of brightness is visible in a TN matrix whatever position you choose before the monitor, so if this matters to you, a monitor on a TN
matrix would be the worst choice you could make.
Then, the contrast ratio of TN matrices is not ideal, either. Although many manufactures declare a contrast ratio of about 500:1, the real contrast ratio seldom exceeds 300:1, some rare samples
getting as high as 400:1. In practice this means that you can’t get a good black color on a TN-matrix monitor, and the black background on the screen will look highlighted if you’re in a dim room
(for example, when you’re watching a movie). I should note, however, that the contrast ratio of matrices depends heavily on the manufacturer. For example, the latest matrices from Samsung have a
standard contrast ratio of 300…400:1, while matrices from Chunghwa Picture Tubes (CPT) often have a deplorable contrast, so monitors on them can only be recommended as inexpensive office devices.
By the way, another drawback of TN matrices is that if a thin-film transistor crashes, there appears a bright dot on the screen, as pixels in TN matrices let light pass through in their inactive
state. Such bright dots are more annoying than black pixels, especially if you’re intending to use the monitor at home, i.e. in the evening, for watching movies or playing games.
Next, the color reproduction of matrices of this type is far from perfect, too. All fast matrices are 18-bit, i.e. they employ Frame Rate Control to display all 16.2 millions of colors, but the
colors of TN matrices are anyway bad – they are faded, unimpressive, rather far from natural ones, which makes TN matrices a bad choice for working with colors even amateurishly.
Thus, the small response time turns to be not just the main, but the only advantage of TN matrices. All of their other parameters are rather average. Monitors with matrices of that type suit for
playing games and watching movies and for routine office work, but professionals may want to consider other types of the matrix. Unfortunately, this means limiting yourself with monitors with a
diagonal of 19” and bigger, since the majority of 17” models have TN+Film matrices now. | {"url":"http://www.xbitlabs.com/articles/monitors/display/lcd-guide_21.html","timestamp":"2014-04-19T02:22:15Z","content_type":null,"content_length":"36452","record_id":"<urn:uuid:e9540523-63d9-43f9-8118-7c5c35201da3>","cc-path":"CC-MAIN-2014-15/segments/1398223205375.6/warc/CC-MAIN-20140423032005-00509-ip-10-147-4-33.ec2.internal.warc.gz"} |
Simon's Graphics Blog
Silhouettes are commonly used for real-time shadowing algorithms. Usually these are generated from the existing edges of a mesh using the face normals. Since shading is usually interpolated over the
triangle from the vertex normals, this can introduce shading artifacts where the vertex and face normals do not agree. In addition, these silhouettes move discontinuously when the light or mesh is in
motion, which can cause nasty popping artifacts when using penumbra wedge soft shadows, since the projected penumbra volumes are very sensitive to the distance from the silhouette edge to the light
This post describes an idea of how to generate silhouette geometry using the vertex normals of a mesh. These silhouettes match the vertex lighting exactly, and also move continuously under smooth
lighting or geometry changes.
Note: since writing this article I’ve found a SIGGRAPH 2000 paper that uses a similar isocontour technique for NPR shading. The paper is by Hertzmann and Zorin on Illustrating Smooth Surfaces. I
haven’t yet found anyone doing any work related to stencil volumes using this technique, so I’ll attempt to put a demo together when I have the time and see what it looks like.
Algorithm Overview
Consider the gouraud-shaded triangle in the diagram below:
The attributes are linearly interpolated between the vertices using barycentric coordinates, and in particular they are linearly interpolated across the edges. Therefore it is possible to efficiently
cut a triangle along a zero by testing the sign of the attribute at each vertex. If one vertex has a different sign to the other two, then the two edges that meet at this vertex contain the
end-points of the cut. The end-points can trivially be found by solving (1 – t)a + tb = 0 for t on each edge.
For the case that the mesh is globally continuous, these cut vertices will always agree, since the vertex attributes agree for each face that shares a given edge. If a given edge is discontinuous,
you will need to generate linking cuts to ensure that you generate closed silhouette loops.
The above algorithm has been implemented using OpenGL and compared with the silhouette edges generated using face normals.
The yellow lines are the silhouette edges generated by this algorithm. The blue lines are the silhouette edges generated using the traditional face-normal method. Note that the yellow silhouette
matches the vertex lighting much more closely.
Future Work
The main applications for this technique are:
• Hard Stencil Shadows. Using continuous silhouette edges should eliminate the popping artifacts caused by the disagreement of vertex-lighting and face-shadowing.
• Penumbra Wedge Soft Shadows. Using continuous silhouette edges as penumbra wedge sources should eliminate the animation artifacts caused by depth discontinuities in traditional silhouette edges. | {"url":"http://www.sjbrown.co.uk/2004/05/11/continuous-silhouettes/","timestamp":"2014-04-20T10:51:13Z","content_type":null,"content_length":"17925","record_id":"<urn:uuid:0dba8f90-812e-4dd1-9e12-3752626f1501>","cc-path":"CC-MAIN-2014-15/segments/1397609538423.10/warc/CC-MAIN-20140416005218-00150-ip-10-147-4-33.ec2.internal.warc.gz"} |
[Numpy-discussion] matrix wart
Konrad Hinsen konrad.hinsen@laposte....
Fri Feb 22 09:14:48 CST 2008
On Feb 22, 2008, at 15:55, Travis E. Oliphant wrote:
>> ColumnVector, and RowVector. It would work like this:
>> m = Matrix([[1, 2], [3, 4]])
>> m[0, :] --> ColumnVector([1, 3])
>> m[:, 0] --> RowVector([1, 2])
> These seem backward to me. I would think that m[0,:] would be the
> RowVector([1,2]) and m[:,0] be the ColumnVector([1,3]).
> What is m[0] in this case? The same as m[0, :]?
>> However, the matrix implementation in Numeric was inspired by Matlab,
>> where everything is a matrix. But as I said before, Python is not
>> Matlab.
> It should be kept in mind, however, that Matlab's matrix object is
> used
> successfully by a lot of people and should not be dismissed as
> irrelevant.
Matlab's approach is fine for Matlab, of course. All I am saying is
that it is a misfit for Python. Just like 1-based indexing is used
successfully by lots of Fortran programmers, but would be an eternal
source of confusion if it were introduced for a specific container
object in Python.
More information about the Numpy-discussion mailing list | {"url":"http://mail.scipy.org/pipermail/numpy-discussion/2008-February/031544.html","timestamp":"2014-04-21T05:03:55Z","content_type":null,"content_length":"3627","record_id":"<urn:uuid:6aeadfe9-89b8-4949-8e0d-8b08687fa7b6>","cc-path":"CC-MAIN-2014-15/segments/1397609540626.47/warc/CC-MAIN-20140416005220-00148-ip-10-147-4-33.ec2.internal.warc.gz"} |
Total # Posts: 1,121
I would go with marginal utility per dollar as it can be used to guide the consumer to allocate is dollars of income to maximize utility. Although, truth be said, whenever I think of "marginal
utility", I presume it means "marginal utility per dollar"
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is it possible that 0.10 is the depreciation rate and 1/E-1 is the foreign int...
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function from a). If so, the MPCH=.75-.25 = .50 c)correct d) D=C+I+G+...
Try this for starters. http://en.wikipedia.org/wiki/Trade_bloc
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International Economics
This is the easy part of your question. Draw initial world-market supply and demand curves for tobacco; and another for domestic food items. If the US subsidises foreign tobacco farmers, foreign
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initiall thought false; of course investors care about the real interest rates. Howev...
International Economics
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An EXCEL spreadsheet is very helpful for solving these type of problems. write out spaces for 7 numbers. 1) the Median is 16, so put 16 in the 4-spot. 2) the sum of the first two spots (two lowest)
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National output, aka GNP, is the sum of C+I+G+(X-M) (X-M is net exports). So, for each scenario, decide how one (or more) of the above factors would change. A fall in interest rates should spur
investments (I), an appreciation of the home currency should make imports cheaper a...
do the algebra. Team X is +3, then -5, ten -2, for a net -4, So.....
You probably have a demand function or table, which shows price (P) and the quantity demanded (Q). Total revenue is simply P*Q. You also probably have some supply function or table, which shows the
average and/or marginal costs of producing each level of Q. Average cost*Q = to...
a) If Y goes up by a dollar, how much would C go up by? MPC is: (change-C)/(change-Y) b) disposable income is Y-T, which can either be consumed or saved. Calculate C when Y-T=100 c) what if taxes
suddenly changed? d)if we taxed everybody by $100 to fund the transfer program an...
Draw standard supply and demand graphs for both markets. In the beachfront market, supply is inelastic; so make the supply curve nearly vertical. In the auto market, supply is elastic, so make the
supply curve nearly horizontal. Now shift the demand curves in each market. You ...
ok, the party of 4 can sit on either the left side or the right side -- 2 possibilities. Once it chooses a side, there are 8*7*6*5=1680 ways they could be seated on that side of the table. There are
8*7=56 ways for the party of two to sit on the other side. With these two par...
With respect to the convergence hypothesis: http://econc10.bu.edu/economic_systems/Lecture_notes/Introduction/intro_types_convergence_lg.htm with respect to S=I, GNP is the value of everything we
produce GNP=C+I. (Ignore government expenditures or net exports for now). GNP is ...
Computers, faster telecommunications, and the reduction of several trade barriers (NAFTA and GAPP) to name 3.
Think about why similar goods cost different amounts. -- Is there some level of service or services involved? -- Is location a factor? (e.g., one seller is next to a major highway or railroad) -- Is
financing typically involved? -- What about legal protections? (e.g., is one s...
Economic Question
Sorry, international macro is not my area. I too am lost on this one.
economics Need help immediatly
I would go with b) if it causes an increase in nominal wages BY THE SAME PERCENTAGE AMOUNT.
Take a shot, what do you think? This is an easy question, just calculate the means of for each of the three vehicles.
Take a shot, what do you think?
Economics Questions.
first off, international macro is not my area, That said.... 1) you have given 50$=100L (in jeans). So, under PPP, 0.5$=1L (so true). (Note: in the real world, the exchange rate is closer to 1$=.5L)
2) I think True. If the euro depreciates against the $, then money investors w...
Take a shot, what do you think. Hint. Draw a supply and demand graph for loanable funds. The price of loanable funds is the interest rate. Now, with each of your situations, decide if the supply
curve would shift or the demand curve (or neither).
Take a shot, what do you think? Hint: As I understand, the initial endowments of persons 1 and 2 are both utility maximizing. For person 1, he will maximize when x1=y1. For person 2, he is
indifferent between getting an extra x2 or a y2. Hint 2: On question 2, I dont believe a...
No, see my post below.
micro economics
Huh? Producer surplus is represented by the area above supply and below price. Consumer surplus in represented by the area below demand and above price. Under what conditions can this be true?
First, lets break apart the Utility function. We know Aneeta consumes 15 x and 10 y. Her utility is therefore min(4*15, 2*15+10) = min(60, 40) = 40. So, for here relevant consumption pattern, her
utility function is simply U=2x+y. Utility will be maximized when MUx/MUy = Px/Py...
Take a shot, what do you think?
something is not right, either your M is too high or your SS is too low
tough question. Here is a link that summarizes the myriad of types of auctions. http://en.wikipedia.org/wiki/Auction For a) my knee jerk reaction was a common value auction (but I could be pursuaded
otherwise). For b) definately not an English auction. The FCC should assume al...
Take a shot, what do you think? Hint: draw supply and demand graphs, compare a graphs where demand is highly elastic (almost flat) vs highly inelastic (almost vertical) Hint: question has nothing do
do with the incomes of consumers; eliminate C.
Managerial Economics
Take a shot, what do you think? Hint: economies of scale imply a declining average cost curve
Managerial Economics
Take a shot, what do you think? Hint: if advertising is used to discurage entry, then definately advertise before new firms actually enter. Hint2: think about the capital investment needed for a new
day-care center?
Managerial Economics
R+D is, in general, expensive and the returns are very uncertain. One purpose of R+D is to lower production costs. However, knowledge is very difficult to contain, despite the best patent laws. So,
it is difficult for a firm to capture the total economic benefits from R+D. Tha...
Managerial Economics
a) you could say the nightclub had zero or negitive economic profits. b) The rate of return on capital in the nightclub business is 10%
econ help
Let me make a correction, we can approximate MR from an elasticity by the following formula MR=P*(1+1/e), where e is the elasticity. So, we have, MC=125=MR=P*(1 - 1/1.8) for full timers. Solve for P.
I get 281.25 Repeat for part-timers. (I guess I would assume the school could...
econ help
Im not sure I understand your question. In general, you want to set Marginal cost = Marginal revenue. You are given marginal cost. I dont think you can derive a marginal revenue from just a known
demand elasticity. Further, its not clear from your question whether the school c...
As MC=MR, the monopolist is at its optimal position in the short run. However, as average total cost are above $4, the firm is losing money. So, long run, either the firm shuts down, or figures out a
way to cut costs.
Econ T/F
I think false. The exchange rate generally equalizes purchasing power. That said, let me give an example. Say the exchange rate is $2=L1. (L for pounds). Now say a burger in the U.S. costs $2 and a
similar burger in the U.K. costs L1. Here purchasing powers are the same. But t...
econ (economyst)
Right. I was a bit confuse by your terminology. I missed the word "real" While the exchange rate may not be unity (one-to-one), the REAL exchange rate would be unity.
Curious, all the recent econ questions have been about exchange rates. I think false. See my post to "..." above.
Take a shot, what do you think? Hint. The government spending multiplier in this example is 1/MPS = 1/.4 = 2.5. The taxation multiplier is the government multiplier minus one = 1.5
Take a shot, what do you think. Hint, the government spending multiplier is 1/mps = 1/.33333 = 3.
econ (economyst please help)
I am mucho confused by your given: A "basket" costs $120 in the UK. Given the current exchange rate of $/L=2, does that mean the basket in the UK costs L60. Your question hinges on this
interpretation. (Note that I'm using L for British Pounds). a) I agree b) Ass...
False I found the following power point on the web which explains. (Broken Link Removed)
False. exporters love when the home currency depreciates and hate when it appreciates. Take Ms Sue's example. Say a firm has a good which it sells for $1. Initially, europeans could take 1 euro,
exchange for $1, and buy the good. Today, a european could take .64 euros, exc...
False. Think it through. You have $/euros = 3/2 which means $3 = 2E. Also, you have 2E=1L. Ergo, $3=1L.
Peter Pundit is full of S. a) First, what is causing the super-increase in productivity, and why is that reason unique to the EU? Unless there is a learning divide, information leading to
productivity increases is very difficult to keep secret. Second, even if the EU is becomi...
For convenience, let c= $15 of consumption. (Dont worry, you would get the same answer if c=$1 of consumption) So, your equations are: Max(U)=2*ln(c)+ln(L) subject to: c+L=120 To get one L,the person
must give up one c. So the price of L in terms of C is 1. Soooooo, we want th...
Ta Da. Yes.
The price in NY is lower than Tokyo. An investor could buy euros in NY, wire the euros to Tokyo, then sell the euros in Tokyo, making a gross profit (before expenses) of $.05 per euro. But such an
arbitrager is increasing the demand for euros in NY and increasing the supply of...
Somewhat of an open ended question. A direct piece of information: "does the business have competitors"? A bit indirect, but, "are there barriers to entry in the firm's industry or product line".
Take it from here
Probability. NEED HELP!!!
The number of possible ways to deal 5 cards from a standard deck is 52-choose-5 = 52!/5!(52-5)! where ! means factorial. This will be the denominator in your probablity calcuator. Now for the
numerator. There are 13 possible choices for your 4-of-a-kind (A,2,3,..Q,K). That tak...
Think it through. 1oz = $35 and $2=L1, so 1oz = L17.5 Now then, if 1oz = $40 and 1oz remains at L17.5 then $40 = L17.5 or 40/17.5 = L1 or $2.286 = L1
Think it through. $1.25 = E1 30-day forward. So to buy 10 million E 30 days from now, to hedge it will cost $12.5 million, profit=$0.5 million b) absolutely c) This call option says that 30 days from
now, you have the option of buying euros at $1.27. If the spot market is abov...
I need HELP bad! Macroeconomics!
1) Nominal gains are 220-200 = $20, with a 50% tax rate, tax=$10 Real gains are 220 - (1.05*200) = $10, tax on real gains, if real gains is the base, is $5 2) Nominal gains are 400-200=$200, real
gains are 400-1.5*200 = $100. Tax on nominal gains is $100. So after tax real cap...
Think it through. What do you think. Take a shot.
see my response above
Sorry. See my respose to Sarah above.
See my response to Sally above.
Sally had a nominal capital gain of $200. With a 50% tax rate, she has a $100 liability. I believe the inflation rate is a red herring in this problem
a) could only work if the Home country was extremely large, relative to the sum of all the other countries. Otherwise you would have a knife-edge problem; either all the other countries would hold
the Home contries currency or none of them would. For more info, google "Wa...
I presume you gave the probability of the biased die, where there is a 33.33% chance of rolling a 1 and a 13.33% chance of rolling a 2,3,4,5 or 6. For a normal die, the probability of rolling any
given number is 1/6 = 16.67%. There is one way to roll a 2, both dies are 1. so, ...
Sounds right to me.
I too have never heard the term "the economy is isolating:
As you may already know, economies e.g., go through "business" cycles, which results in boom periods and bust periods. In booms, unemployment is low, but there are inflationary pressures. In busts,
inflation is low, but unemployment is high. Stabilization policy is d...
Wow. thats really an open ended question. I'm not really sure what to suggest. However, if you have some ideas, I or others will be happy to critique.
Based the information given there are: 960 A-R 640 A-S 192 B-R 608 B-S 48 C-R 552 C-S so, c) = 552/3000 di) = (1.-552/3000) * (1.-551/2999) dii) = 1.-(di) diii) = (552/3000) * (48/2999) div) = 1. - (
Boston has an absolute advantage in producing both red socks and white socks. For red, a worker in Boston can produce 3 for every 2 a Chicago worker can (3>2). For white, a Boston worker can produce
3 for every 1 a Chicago worker can (3>1). Chicago has a comparitive adva...
use algebra and plug in known values. You have MC=MR=(1-1/e)*P where Mc=MR=27 and P=45. Solve: 27=(1-1/e)*45 27/45 = (1-1/e) -.4 = -1/e e = 2.5 -- elastic If P were higher, then e would have to be
lower (more inelastic)
see my response to Tiffany above.
Widespread use of computers and computer technology. Improved communication technologies. Improved health-related technologies. More open trade, etc.
economic theory
You could short sell a portion of one of the currencies. You could use a portion of the currencies to buy another currency or another asset.
Do you mean America - USA, if so, go to the economic report of the president. http://www.gpoaccess.gov/eop/download.html If you mean latin America, I too had lots of trouble. I found plenty of sites
that likely had the information. However, the sites were either pay-for or had...
Data Management, Math
in week 1 you have a 1 in 10 chance to win, or 10%. In week 2 you have a 2 in 20 chance to win or 10% .... In week 5 you have a 5 in 50 chance to win, or 10%. a) The probability that you always lose
is .9^5 = .59049 -- so the probability that you win at least once is 1-.59049 ...
Macro Help
Take a shot. what do you think. Hint, S=I, where I is composed of intended or autonomous investment Ia and unintended investment Ib (e.g., change in inventories). Hint 2. Multiplier is 1/MPS
Think it through. GDP per capita is simply GDP/Pop. Let a be the growth in real GDP, and let b be the growth in the population. Then the growth in per captita real GDP is (a*GDP)/(b*Pop). Take in
from here.
I dont know what your "Grapher tool" is. That said, if the economy has zero inflation and 5% unemployment for several years, that sounds like equilibrium to me. Any attempts to move to an alternative
position is, very likely, doomed to fail.
Hummm. As you can see, graphing in Jiskha doesnt really work. That said, producer surplus is represented by the area below price and above supply. If this is some triangle, use the geometry formula
for calculating the area of a triangle.
Very creative way to learn to use EXCEL for typical financial calculations. This is more of a "how to use" EXCEL problem than a finance problem. And would be quite difficult to to here in Jishka.
Repost if you have a specific question. Hint: put your SUN in A1 In B1 ...
1a) this is just algebra. Plug in the known values for Y, Pc, and Pf. Then set Qd=Qs and solve for Pb 1b) Raise price by 1%, calculate the %change in Q. Elasticity is %changeQ/%changeP 1c) demand
curve shifts out. 1d) raise price of chicken by 1%, then calculate the %change in...
They are nearly all applicable. You and I will likely work more with higher wage rates (micro). A country's quantity of labor supplied will increase with higher wage rates (macro). I receive some
consumer surplus when I buy jelly beans (micro). The consumers of a country c...
I presume that the numbers mean some sort of production in a given period (e.g., "Ben-cones 8" means the number of cones Ben can produce in a time period) So then, the opportunity cost for Ben to
produce 8 cones is 2 ice cream, or 4 cones=1 ice cream. For jerry 1.5 c...
I would pick B.
A good programming forum???
Here is a link to the forum I use. Although, I would not recommend it as a basic programming homework help site; most of the blogs are high-level intensive stuff. http://softwarecommunity.intel.com/
I'd say about all of them.
Here is a link to an explanation of the Laffer curve. Google Laffer Curve for more info. http://en.wikipedia.org/wiki/Laffer_curve
Take a shot, what do you think? Hint: does an author have marginal costs? Does a publisher have marginal costs?
This is the notation used for combinations. In particular, its top-choose-bottom. In your example, you have 15 items (top) and you want to determine the number of ways of picking 15 of them (bottom).
BTW, the answer in this example is 1.
1a(i) there are two O's, so P=(2/8) 1b(i) 4+ points are received with F1,F2,O1,O2,T1, and T2, so P=(6/8) 1b(ii) first toss O1 and O2. 3+ points are receive with F1,F2,T1,T2 and E. so P=(5/6) In 2,
put your information into a spreadsheet or table, and take it from there
Economics 101
Take a shot, what do you think? Hint, (T,T,T,T)
Take a shot, what do you think? Hint: no.
See my later post
What do you mean? Do you mean "what alternatives are there to Acxiom" e.g, who are Acxiom's competitors. Or do you mean, "what alternative business markets could Acxiom get into?" Or something else?
I believe you have supply and demand equations; Qd on the left, Qs on the right. Normally, we put P on the y-axis, and Q on the x-axis. So rewrite each equation in terms of P. 800 - 100P = Qd is
transformed into P=8-Qd/100. Do the same for Supply. As for a graph, each equation...
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Sierpitski-Zygmund Functions and other Problems on Lineability
Gámez Merino, José Luis and Muñoz Fernández, Gustavo Adolfo and Sánchez, V.M. and Seoane Sepúlveda, Juan Benigno (2010) Sierpitski-Zygmund Functions and other Problems on Lineability. Proceedings of
the American Mathematical Society, 138 (11). pp. 3863-3876. ISSN 0002-9939
We find large algebraic structures inside the following sets of pathological functions: (i) perfectly everywhere surjective functions, (ii) differentiable functions with almost nowhere continuous
derivatives, (iii) differentiable nowhere monotone functions, and (iv) Sierpinski-Zygmund functions. The conclusions obtained on (i) and (iii) are improvements of some already known results.
Repository Staff Only: item control page | {"url":"http://eprints.ucm.es/16916/","timestamp":"2014-04-21T14:51:19Z","content_type":null,"content_length":"23095","record_id":"<urn:uuid:0c1b55f1-5889-4dd1-a883-145120a92230>","cc-path":"CC-MAIN-2014-15/segments/1397609540626.47/warc/CC-MAIN-20140416005220-00332-ip-10-147-4-33.ec2.internal.warc.gz"} |
Are these terms consisting of logarithms of primes rationally independent?
up vote 4 down vote favorite
I expected it to be basic, but seem unable to find a proof of the following:
Let $p_0, p_1, .., p_m$ be distinct primes. Then the $m+1$ terms $\dfrac{\log p_0}{\log p_j}$, are rationally independent.
diophantine-approximation nt.number-theory
2 Perhaps you've already observed this, but it follows from Schanuel's conjecture that in fact the logarithms of the primes are algebraically independent. – Qiaochu Yuan Sep 25 '11 at 23:50
Sorry, I voted to close because I misread the question as asking about $\log p_j / \log p_0$ which is trivial. – Felipe Voloch Sep 26 '11 at 10:41
@Felipe, no problem. Although, now that I think about it, I would have preferred it to be closed because of triviality, because I need this result :D @Qiaochu, thanks. Actually, I don't really
mind if the part of the paper I'm writing in which I use this result, was based on a conjecture like Schanuel's. But if that should happen, should I then use the 'easiest' conjecture from which
this follows, or the most well-known, or some other criterium? – Woett Sep 26 '11 at 12:45
@Qiaochu, I probably misread your observation. I now believe that what you mean is that IF my question has an affirmative answer, then Schanuel's conjecture shows that the terms are algebraically
independent, right? – Woett Sep 26 '11 at 13:18
I think that Qiaochu is saying that because $\log p_0, \dots, \log p_m$ are linearly independent over $\mathbb{Q}$, Schanuel's conjecture would imply that they are algebraically independent over $
\overline{\mathbb{Q}}$. This then would imply the linear independence result you want. See Chapter 1 of Waldschmidt's book "Diophantine approximation on linear algebraic groups" for more
information on algebraic independence of logarithms. Conjecture 1.15 there is all that you would need, but it's just a special case of Schaneul's conjecture which is more widely known. – Matt
Papanikolas Sep 30 '11 at 4:20
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QMW Hyperspace - A set of hypertext based services for general relativity research provided by the QMW Relativity group.
Relativity bookmarks - Rob Salgado's bookmarks
General Relativity Simulation Contest - The purpose of this Contest is to prove General Relativity using a (simple) algorithm.
Ricci - A Mathematica package for doing tensor calculations in differential geometry and general relativity.
NOVA Online/Einstein Revealed - Profile of Albert Einstein, with additional teaching resources, Shockwave demonstrations, and animations of relativity concepts.
Gravitational Lensing - A popular site with interactive demonstrations of gravitational lensing. The background material is at a graduate level
Shapiro Radar Bounce Test - A two part overview of the Shapiro radar bounce test of general relativity. (The two parts consist of a section for normal people, and one for nerds)
Hyperspace at the University of British Columbia - Includes links to online journals, news and preprint archives, as well as lists of faculty at this university.
Solving Einstein's Equations in Three Dimensions - This is a Mathematica notebook showing how to plot light cones in two plus one spacetime dimensions with a point mass.
Interactive Experiments in Gravity - Try an experiment that illustrates the gravitational attraction between two objects or use a Java applet to understand how orbits work in strongly curved
Gravity Probe B - Gravity Probe B is the relativity gyroscope experiment being developed by NASA and Stanford University to test two extraordinary, unverified predictions of Albert Einstein's
general theory of relativity.
Physics Bookshelf - Relativity - A collection of articles about relativity
Gravity of Gravity - An experiment at the University of Washington seeks to determine whether the gravitational binding energy of an object generates gravity of its own.
The 5D Spacetime Consortium - We are a group of physicists and astronomers working on a 5-dimensional version of general relativity.
Derivation of gravitational waves in Einstein, Brans-Dicke and Rosen theories of gravity. - Mathematical derivation of gravitational waves to quadrupole order in the Brans-Dicke scalar-tensor and
Rosen bi-metric theories of gravity.
Gravitational Waves: An Introduction - This paper presents an elementary introduction to the theory of gravitational waves. This article is meant for students who have had an exposure to general
relativity, but results from general relativity have been derived in the appendices.
Are There Pictorial Examples of Covariant and Contravariant Vectors - We present pictorial means of distinguishing contravariant vectors (or simply vectors) from covariant vectors (or linear
Handbook of Space Astronomy and Astrophysics - Relativity - A collection of equations important to both special relativity and general relativity.
An Essay on General Relativity - This site contains a comprehensive introduction to the basic ideas and tests of general relativity
The Special Theory of Relativity - Self-tutorial with short essays, questions and answers.
The Cosmological Constant - An overview of why Einstein added an extra term in general relativity, and why it is still examined.
Experimental Gravitational Physics Using Drag-Free Satellites - Covers current and past projects in experimental gravitational physics using Drag-Free Satellites including the Relativity Gyro,
Equivalence Principle, and LISA.
Publications in Physics and Mathematics by Walter Pfeifer - A textbook for relativistic quantum mechanics is presented. The Dirac equation is dealt with in detail.
Why Time is Absolute, and Relative, But Never Universal - An article by Vincent Sauvé elaborating upon a materialist view of the absolute and relative nature of time.
Relativity - Provides information on the history, experiments and paradoxes of relativity.
Lecture Notes on General Relativity - Sean M. Carroll's lecture notes from a one-semester graduate-level course he taught at MIT in 1996. Notes are in postscript format.
Dulkyn: Scientific Center for Gravitational-Wave Research - Research includes detection of periodic gravitational radiation from relativistic astrophysical objects, investigation of
infra-low-frequency variations of Terrestrial gravity field in the role of crystal growing and compound formation, and investigation of time variations of Terrestrial inertial field and the
geophysical consequences. | {"url":"http://www.bazsites.com/Science/Physics/Relativity/","timestamp":"2014-04-17T03:49:27Z","content_type":null,"content_length":"12023","record_id":"<urn:uuid:2433f5c0-901b-4a3e-af89-b1d5456ef5af>","cc-path":"CC-MAIN-2014-15/segments/1398223205137.4/warc/CC-MAIN-20140423032005-00148-ip-10-147-4-33.ec2.internal.warc.gz"} |
Post a reply
Sketching sine and cosine curves are relatively easy. You just need the vertical translation, the magnitude, the period, and the horizontal translation.
Sketching tangent curves are slightly more difficult.
But when it comes to sketching the arc forms of trigonometric functions, I don't really know what to do. I could just study the patern that develops with each one, but that teaches me little. I could
plot the points one by one in a few spots by just inserting values into the function, but I think this is cheating. I'm not really sure what else I'm supposed to do. | {"url":"http://www.mathisfunforum.com/post.php?tid=1592&qid=14420","timestamp":"2014-04-20T16:46:19Z","content_type":null,"content_length":"18112","record_id":"<urn:uuid:610285a6-619c-4572-b88e-2629e2189b1f>","cc-path":"CC-MAIN-2014-15/segments/1397609538824.34/warc/CC-MAIN-20140416005218-00640-ip-10-147-4-33.ec2.internal.warc.gz"} |
Ratio to Percent Calculator
Percent means "for each hundred" and is denoted as " % ". Percent is a way to express ratio with respect to hundred. Percent is a way of saying that one number is some percent of another.
Ratio to percent calculator is used to determine the percent of a given ratio. Ratio to percent calculator contains two input fields where we have to give the ratio to get the corresponding
percentage. This calculator which is in the form of ratio a : b, converts ratio to percent. | {"url":"http://calculator.mathcaptain.com/ratio-to-percent-calculator.html","timestamp":"2014-04-16T16:29:10Z","content_type":null,"content_length":"39815","record_id":"<urn:uuid:ff7d5ecb-6447-497f-b623-26056cbd32c2>","cc-path":"CC-MAIN-2014-15/segments/1398223206118.10/warc/CC-MAIN-20140423032006-00082-ip-10-147-4-33.ec2.internal.warc.gz"} |
[with patch] Symbolic substitution infinite recursion (was "NumberField infinite recursion")
Reported by: phatsphere Owned by: mhansen
Priority: critical Milestone: sage-2.9.1
Component: calculus Keywords:
Cc: Merged in:
Authors: Reviewers:
Report Upstream: Work issues:
Branch: Commit:
Dependencies: Stopgaps:
I think the input is wrong, but it should not loop forever and throw an error.
Exception (click to the left for traceback):
RuntimeError: maximum recursion depth exceeded in cmp
here the infinite traceback:
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
File "/home/server4/sage_notebook/worksheets/phatsphere/0/code/123.py", line 4, in <module>
exec compile(ur'K3=NumberField([x**Integer(2)+Integer(1),sqrt(x**Integer(3))+Integer(1)],names=(\u0027a\u0027,)); (a,) = K3._first_ngens(Integer(1))' + '\n', '', 'single')
File "/usr/local/sage-2.6/local/lib/python2.5/site-packages/sympy/plotting/", line 1, in <module>
File "/usr/local/sage-2.6/local/lib/python2.5/site-packages/sage/rings/number_field/number_field.py", line 245, in NumberField
return NumberFieldTower(polynomial, name)
File "/usr/local/sage-2.6/local/lib/python2.5/site-packages/sage/rings/number_field/number_field.py", line 389, in NumberFieldTower
w = NumberFieldTower(v[1:], names=names[1:])
File "/usr/local/sage-2.6/local/lib/python2.5/site-packages/sage/rings/number_field/number_field.py", line 387, in NumberFieldTower
return NumberField(v[0], names=names)
File "/usr/local/sage-2.6/local/lib/python2.5/site-packages/sage/rings/number_field/number_field.py", line 251, in NumberField
polynomial = polynomial.polynomial(QQ)
File "/usr/local/sage-2.6/local/lib/python2.5/site-packages/sage/calculus/calculus.py", line 1344, in polynomial
dict([(var(V[i]),G[i]) for i in range(len(G))]), ring=R)
File "/usr/local/sage-2.6/local/lib/python2.5/site-packages/sage/calculus/calculus.py", line 2825, in substitute_over_ring
return X._recursive_sub_over_ring(kwds, ring)
File "/usr/local/sage-2.6/local/lib/python2.5/site-packages/sage/calculus/calculus.py", line 3763, in _recursive_sub_over_ring
new_ops = [op._recursive_sub_over_ring(kwds, ring=ring) for op in ops]
File "/usr/local/sage-2.6/local/lib/python2.5/site-packages/sage/calculus/calculus.py", line 4842, in _recursive_sub_over_ring
return ring(ops[0](ops[1]._recursive_sub_over_ring(kwds, ring=ring)))
File "/usr/local/sage-2.6/local/lib/python2.5/site-packages/sage/rings/polynomial/polynomial_ring.py", line 218, in __call__
return x._polynomial_(self)
File "/usr/local/sage-2.6/local/lib/python2.5/site-packages/sage/calculus/calculus.py", line 1409, in _polynomial_
return self.substitute_over_ring(dict(sub), ring=R)
File "/usr/local/sage-2.6/local/lib/python2.5/site-packages/sage/calculus/calculus.py", line 2825, in substitute_over_ring
return X._recursive_sub_over_ring(kwds, ring)
File "/usr/local/sage-2.6/local/lib/python2.5/site-packages/sage/calculus/calculus.py", line 4842, in _recursive_sub_over_ring
return ring(ops[0](ops[1]._recursive_sub_over_ring(kwds, ring=ring)))
File "/usr/local/sage-2.6/local/lib/python2.5/site-packages/sage/rings/polynomial/polynomial_ring.py", line 218, in __call__
return x._polynomial_(self)
File "/usr/local/sage-2.6/local/lib/python2.5/site-packages/sage/calculus/calculus.py", line 1409, in _polynomial_
return self.substitute_over_ring(dict(sub), ring=R)
File "/usr/local/sage-2.6/local/lib/python2.5/site-packages/sage/calculus/calculus.py", line 2825, in substitute_over_ring
return X._recursive_sub_over_ring(kwds, ring)
File "/usr/local/sage-2.6/local/lib/python2.5/site-packages/sage/calculus/calculus.py", line 4842, in _recursive_sub_over_ring
return ring(ops[0](ops[1]._recursive_sub_over_ring(kwds, ring=ring)))
File "/usr/local/sage-2.6/local/lib/python2.5/site-packages/sage/calculus/calculus.py", line 1409, in _polynomial_
return self.substitute_over_ring(dict(sub), ring=R)
File "/usr/local/sage-2.6/local/lib/python2.5/site-packages/sage/calculus/calculus.py", line 2825, in substitute_over_ring
return X._recursive_sub_over_ring(kwds, ring)
File "/usr/local/sage-2.6/local/lib/python2.5/site-packages/sage/calculus/calculus.py", line 4842, in _recursive_sub_over_ring
return ring(ops[0](ops[1]._recursive_sub_over_ring(kwds, ring=ring)))
File "/usr/local/sage-2.6/local/lib/python2.5/site-packages/sage/rings/polynomial/polynomial_ring.py", line 218, in __call__
return x._polynomial_(self)
File "/usr/local/sage-2.6/local/lib/python2.5/site-packages/sage/calculus/calculus.py", line 1387, in _polynomial_
vars = self.variables()
File "/usr/local/sage-2.6/local/lib/python2.5/site-packages/sage/calculus/calculus.py", line 3524, in variables
return self.simplify().variables(vars)
File "/usr/local/sage-2.6/local/lib/python2.5/site-packages/sage/calculus/calculus.py", line 2555, in simplify
S = evaled_symbolic_expression_from_maxima_string(self._maxima_init_())
File "/usr/local/sage-2.6/local/lib/python2.5/site-packages/sage/calculus/calculus.py", line 6466, in evaled_symbolic_expression_from_maxima_string
return symbolic_expression_from_maxima_string(maxima.eval(x))
File "/usr/local/sage-2.6/local/lib/python2.5/site-packages/sage/interfaces/expect.py", line 707, in eval
return '\n'.join([self._eval_line(L, **kwds) for L in code.split('\n') if L != ''])
File "/usr/local/sage-2.6/local/lib/python2.5/site-packages/sage/interfaces/maxima.py", line 540, in _eval_line
File "/usr/local/sage-2.6/local/lib/python2.5/site-packages/sage/interfaces/maxima.py", line 600, in _synchronize
File "/usr/local/sage-2.6/local/lib/python2.5/site-packages/sage/interfaces/maxima.py", line 453, in _expect_expr
i = self._expect.expect(expr,timeout=timeout)
File "/usr/local/sage-2.6/local/lib/python2.5/site-packages/pexpect.py", line 911, in expect
compiled_pattern_list = self.compile_pattern_list(pattern)
File "/usr/local/sage-2.6/local/lib/python2.5/site-packages/pexpect.py", line 843, in compile_pattern_list
compiled_pattern_list.append(re.compile(p, re.DOTALL))
File "/usr/local/sage-2.6/local/lib/python2.5/re.py", line 180, in compile
return _compile(pattern, flags)
File "/usr/local/sage-2.6/local/lib/python2.5/re.py", line 222, in _compile
p = _cache.get(cachekey)
RuntimeError: maximum recursion depth exceeded in cmp
Attachments (1)
Change History (9)
• Milestone set to sage-2.9.1
• Priority changed from major to critical
I'm taking a look a this....
here is a simpler example that causes the same problem:
sage: K3.<a> = NumberField(sqrt(x))
aha it has nothing to do with number fields:
sage: f = sqrt(x)
sage: g = f.polynomial(QQ)
ok it's some nastiness to do with recursive substitution in symbolic expressions. The same thing happens with cos(x).polynomial(QQ) etc. I will leave this voodoo to someone else who understands the
symbolic calculus package.
• Component changed from algebraic geometry to calculus
• Summary changed from NumberField infinite recursion to Symbolic substitution infinite recursion (was "NumberField infinite recursion")
• Owner changed from was to mhansen
• Status changed from new to assigned
• Summary changed from Symbolic substitution infinite recursion (was "NumberField infinite recursion") to [with patch] Symbolic substitution infinite recursion (was "NumberField infinite
• Resolution set to fixed
• Status changed from assigned to closed | {"url":"http://trac.sagemath.org/ticket/1566","timestamp":"2014-04-20T08:14:12Z","content_type":null,"content_length":"31697","record_id":"<urn:uuid:0fa6875e-1c02-46b5-8e51-a91e11087de3>","cc-path":"CC-MAIN-2014-15/segments/1397609538110.1/warc/CC-MAIN-20140416005218-00519-ip-10-147-4-33.ec2.internal.warc.gz"} |
Floating-point exceptions and the CLR
I’ve decided to share some of our experiences during the development of our math and statistics libraries in the hope that they may contribute to improvements in the .NET platform as the next version
is being designed.
The CLR is a general-purpose runtime environment, and cannot be expected to support every application at the fastest possible speed. However, I do expect it to perform reasonably well, and if a
performance hit can be avoided, then it should be. The absence of any floating-point exception mechanism incurs such a performance hit in some fairly common situations.
As an example, let’s take an implementation of complex numbers. This is a type for general use, and has to give accurate results whenever possible. For obvious reasons, we want the core operations to
be as fast as possible. This means we want to inline when we can, and make our code fast, too. Most operations are fairly straightforward, but division isn’t. Let’s start with the ‘naïve’
struct Complex
private re, im;
public static operator/(Complex z1, Complex z2)
double d = z2.re*z2.re + z2.im*z2.im;
double resultRe = z1.re * z2.re + z1.im * z2.im;
double resultIm = z1.im * z2.re – z1.re * z2.im;
return new Complex(resultRe / d, resultIm / d);
If any of the values d, resultRe, and resultIm underflow, the result will lose accuracy, because subnormal numbers by definition don’t have the full 52 bit precision. The CLR also offers no
indication that underflow has occurred. This can be fixed, mostly, by modifying the above to:
public static operator/(Complex z1, Complex z2)
if (Math.Abs(z2.re) > Math.Abs(z2.im)
double t = z2.im / z2.re;
double d = z2.re + t * z2.im;
double resultRe = (z1.re + t * z1.im);
double resultIm = (z1.im – t * z1.re);
return new Complex(resultRe / d, resultIm / d);
double t = z2.re / z2.im;
double d = t * z2.re + z2.im;
double resultRe = (t * z1.re + z1.im);
double resultIm = (t * z1.im – z1.re);
return new Complex(resultRe / d, resultIm / d);
This will give accurate results in a larger domain, but is slower because of the extra division. Worse still, some operations that one would expect to give exact results now aren’t exact. For
example, if z1 = 27-21i and z2 = 9-7i, the exact result is 3, but the round-off in the division by 9 destroys the exact result.
IEEE-754 exceptions would come to the rescue here – if they were available. Exceptions (a term with a specific meaning in the IEEE-754 standard – not to be confused with CLR exceptions) raise a flag
in the FPU’s status register, and can also be trapped by the operating system. We don’t need a trap here. We can do what we need to do with a flag. The code would look something like:
public static operator/(Complex z1, Complex z2)
double d = z2.re*z2.re + z2.im*z2.im;
double resultRe = z1.re * z2.re + z1.im * z2.im;
double resultIm = z1.im * z2.re – z1.re * z2.im;
if (FloatingPoint.IsExceptionFlagRaised(
// Code for the special cases.
return new Complex(resultRe / d, resultIm / d);
Note that the CLR strategy to “continue with default values” won’t work here, because complete underflow is defaulted to 0, which can not be distinguished from the common case when the result is
exactly zero (and therefore no special action is required). The only way to do it right would be a whole series of ugly comparisons, which would make the code slower and harder to read/maintain. Even
if a language supported a mechanism to check for underflow (by inserting comparisons to a suitable small value before and after storing the value), this would bloat the IL, introduce unnecessary
round-off (by forcing a conversion from extended to double on each operation), and slow things down unnecessarily.
This type of scenario occurs many times in numerical calculations. You perform a calculation the quick and dirty way, and if it turns out you made a mess, you try again but you’re more cautious. The
complex number example is the most significant one I have come across while developing our numerical libraries.
Nearly all hardware that the CLR (or its clones) run on, supports this floating-point exception mechanism. I found it somewhat surprising that a ‘standard’ virtual execution environment would not
adopt another and well-established standard (IEEE-754) for a specific subset of its functionality.
For more on the math jargon used in this post, see my article Floating-Point in .NET Part 1: Concepts and Formats.
One thought on “Floating-point exceptions and the CLR” | {"url":"http://www.extremeoptimization.com/Blog/index.php/2006/01/floating-point-exceptions-and-the-clr/","timestamp":"2014-04-18T13:09:17Z","content_type":null,"content_length":"23922","record_id":"<urn:uuid:abe2d6bb-8853-42ce-8699-18cfd6251126>","cc-path":"CC-MAIN-2014-15/segments/1398223205375.6/warc/CC-MAIN-20140423032005-00069-ip-10-147-4-33.ec2.internal.warc.gz"} |
Describing 2-D and 3-D Shapes
Unit Plan
Describing 2-D and 3-D Shapes
Grade Levels
Related Academic Standards
Assessment Anchors
Eligible Content
Big Ideas
Students will identify, describe, and analyze two- and three-dimensional shapes. Students will:
• identify and name familiar shapes in the classroom.
• learn key geometry vocabulary.
• describe key attributes of two- and three-dimensional shapes.
• sort shapes based on attributes.
• create shapes by folding and cutting paper.
• identify and draw lines of symmetry.
Essential Questions
• None listed in the SAS materials.
Related Unit and Lesson Plans
Related Materials & Resources
The possible inclusion of commercial websites below is not an implied endorsement of their products, which are not free, and are not required for this unit.
• National Council of Teachers of Mathematics (NCTM) Principles and Standards 2000.
Formative Assessment
I like how this unit plan offers a varied approach to learning. The numerous activities and games are sure to engage students throughout the lessons."Tangram Puzzles" in materials and resources would
be a great way to enrich this unit.
Posted more than a year ago
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Under Construction
Thank you for your patience. Not all features of the site are currently available. | {"url":"http://pdesas.org/module/content/resources/5438/view.ashx","timestamp":"2014-04-18T23:14:57Z","content_type":null,"content_length":"41831","record_id":"<urn:uuid:344e2b51-d824-47e3-b1e2-f3c3201e041c>","cc-path":"CC-MAIN-2014-15/segments/1398223202548.14/warc/CC-MAIN-20140423032002-00231-ip-10-147-4-33.ec2.internal.warc.gz"} |
Probability-One Homotopy Algorithms for Robust Controller Analysis and Synthesis with Fixed-Structure Multipliers
Probability-One Homotopy Algorithms for Robust Controller Analysis and Synthesis with Fixed-Structure Multipliers
(1996) Probability-One Homotopy Algorithms for Robust Controller Analysis and Synthesis with Fixed-Structure Multipliers. Technical Report ncstrl.vatech_cs//TR-96-05, Computer Science, Virginia
Polytechnic Institute and State University.
Full text available as:
Postscript - Requires a viewer, such as GhostView
TR-96-05.ps (669808)
To enable the development of M-K (i.e., multiplier-controller) iteration schemes that do not require (suboptimal) curve fitting, mixed structured singular value analysis tests that allow the
structure of the multipliers to a priori be specified, have been developed. These tests have recently been formulated as linear matrix inequality (LMI) feasibility problems. The least conservative of
these tests always results in unstable multipliers and hence requires a stable coprime factorization of the multiplier before the control synthesis phase of the M-K iteration. This paper first
reviews the LMI formulations of robustness analysis. It then develops alternative formulations that directly synthesize the stable factorizations and are based on the existence of positive definite
solutions to certain Riccati equations. These problems, unlike the LMI problems, are not convex. The feasibility problem is approached by posing an associated optimization problem that cannot be
solved using standard descent methods. Hence, we develop probability-one homotopy algorithms to find a solution. These results easily extend to provide computationally tractable algorithms for
fixed-architecture, robust control design, which appear to have some advantages over the bilinear matrix inequality (BMI) approaches resulting from extensions of the LMI framework for robustness | {"url":"http://eprints.cs.vt.edu/archive/00000444/","timestamp":"2014-04-19T12:36:10Z","content_type":null,"content_length":"8219","record_id":"<urn:uuid:5f95ec72-4735-4590-b075-c658ce56290f>","cc-path":"CC-MAIN-2014-15/segments/1398223206770.7/warc/CC-MAIN-20140423032006-00143-ip-10-147-4-33.ec2.internal.warc.gz"} |
College, NY Algebra 2 Tutor
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Posts by
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5th grade
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In what distant deeps or skies Burnt the fire of thine eyes? On what wings dare he aspire? What the hand dare seize the fire? and Tyger! Tyger! burning bright In the forests of the night, What
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Math 111
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Math 111
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College Chemistry
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Organic Chemistry
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Which of the following elements has an ionic radius greater than its atomic radius a) Ca c) Fe b) Li d) N
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so that means the ion is smaller because your removing the 2 outside electrons ?
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write a balanced chemical reaction for the reaction of aqueous magnesium chloride with aqueous potassium phosphate to form solid magnesium phosphate (Mg3(PO4)2) and aqueous potassium chloride (KCl).
Indicate the physical states of the reactants and products
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Probability - CHECK ANSWERS
Thanks a bunch!
Probability - CHECK ANSWERS
I have an eight pack of crayons. Colors red, green, blue, orange, black, gray, pink, and purple. What is the probability of choosing the green crayon? 12.5 percent What is the probability of choosing
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Home work for Gym
Thank You soo much!! ;D
Home work for Gym
To join a gym, your friend pays a one-time fee of $75 and $45 per month for the duration of the membership. Your friend has paid a total of $345. How long has your friend been a member of the gym?
HELP ME1 :D
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A few Precalculus function problems
April 16th 2011, 11:58 AM #1
Apr 2011
A few Precalculus function problems
Hey guys. Here are some math problem I'm having trouble with. Try and answer any one and I'll be very very thankful.
1. Describe in general terms how to find the domain of the composite function f(g(x)) then illustrate with these functions f(x)=1/x^2-1 g(x)=sqroot(x+1)
2. There's a 2 cylinder water tank with a 4 ft diameter thats 1 ft. tall. When there's a hole in the bottom it drains at 5ft/sec^2. Write a function for the rate water drains as a function of t
3. The length "C" of the hypotenuse of a right triangle is a function of the lengths "a" and "b" of the legs.
a) state a rule for the function [a,b]
b) find c(3,4) and c(12,5)
First thing to do is find ther domains of f ang g. What do you get?
This sounds like a calculus question to me.
What is the formula for a cylinder?
Think about pythagora's theorem here.
response pickslides
1. So the domain of of f(x) is basically all real numbers except x= 1,-1 and the domain of g(x) is (-1,infinity) but what i do not understand is what the domain of f(g(x) is.
2.So the formula for the volume of a cylinder is pir^2h. But how can this help??
and for 3....so c^2=a^2+b^2
the rule would be C(a,b)= a^2+b^2?
so would this mean that C(3,4) = 5? and C(12,5) = 13?
Your best approach with (1) in order to get understanding is to draw pictures.
This might help:
Definition:Composition of Mappings - ProofWiki
Work out what the domain of g is. Bear in mind f(g(x)) is "g first, then do f of what you get when you've done g."
Then work out what the image of g is (some call it "range", it's the set of numbers that can appear as the output of g).
Then the only numbers available for f to work on are the image of g.
But then there are numbers (-1 and 1) which can not be in the domain of f.
So these numbers must be removed from the image of g.
So any numbers in the domain of g which map to 1 and -1 must be removed for the domain of f and g for f(g(x)) to be a mapping.
So the domain of f(g(x)) is: all the numbers in the domain of g, removing the numbers that map to 1 and -1 (whatever they are, I haven't looked at what f and g actually are).
Does this help? It is understood that the concepts behind function theory are more complicated and fiddly than they look on the surface. Take time to study this stuff because it will save hours
of confusion in the future.
The thing that bothers me about this problem is the information that "it drains at 5 ft/sec^2". The rate at which water drains from at tank should have units of "volume per time" or "ft^3/s".
With a tank of uniform cross section, you could give the rate at which the height of water goes down, calling it (with "abuse of notation") the rate at which the water drains, but that would
still be "ft/sec", not "ft/sec^2".
April 16th 2011, 01:48 PM #2
April 17th 2011, 12:31 AM #3
Apr 2011
April 17th 2011, 01:15 AM #4
April 17th 2011, 11:58 AM #5
MHF Contributor
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$10 A Day Equals 3/4 Of A Million Dollars - No Credit Needed
$10 A Day Equals 3/4 Of A Million Dollars
Thursday evening, my son and I went to a local restaurant for supper. After eating, my son asked for some chewing gum – you know, the little round gum that you get for a quarter from one of the
machines at the front of the store. He’s three, so he wanted gum for our whole family – Mommy, Daddy, Sister, and himself. I had four quarters, so we loaded up on gum and headed home.
When we got home, I opened up a spreadsheet and began playing around with a few scenarios. What if, instead of spending that dollar on gum, I had saved it? And, what if I saved a dollar, every day,
for a year? for a decade? for 50 years? Also, what if I save more than a dollar, say, two dollars, or five dollars or even 10 dollars, per day?
Check out this chart I created -
Note, the following numbers represent the total amounts contributed to savings (w/ no accounting for interest earned).
│1 Day│1 Month│1 Year│ 10 Years│ 50 Years│
│ │ │ │ │ │
│ $1│ $30│ $360│ $3,600│ $18,000│
│ $2│ $60│ $720│ $7,200│ $36,000│
│ $3│ $90│$1,080│ $10,800│ $54,000│
│ $4│ $120│$1,440│ $14,400│ $72,000│
│ $5│ $150│$1,800│ $18,000│ $90,000│
│ $6│ $180│$2,160│ $21,600│ $108,000│
│ $7│ $210│$2,520│ $25,200│ $126,000│
│ $8│ $240│$2,880│ $28,800│ $144,000│
│ $9│ $270│$3,240│ $32,400│ $162,000│
│ $10│ $300│$3,600│ $36,000│ $180,000│
If I save $1 per day, for 50 years, I’ll have $18,000.
If I save $10 a day, for 50 years, I’ll have $180,000.
But, as I mentioned before, this chart represents contributions – and doesn’t calculate interest earned.
So, let’s kick it up a notch and do some calculations, assuming a 5% annual return.
│Yearly│10 Years│20 Years│30 Years│40 Years│50 Years│
│ │ │ │ │ │ │
│ $360│ $4,754│ $12,499│ $25,114│ $45,662│ $79,134│
│ $720│ $9,509│ $24,998│ $50,228│ $91,325│$158,267│
│$1,080│ $14,263│ $37,497│ $75,342│$136,987│$237,401│
│$1,440│ $19,018│ $49,996│$100,456│$182,649│$316,534│
│$1,800│ $23,772│ $62,495│$125,569│$228,312│$395,668│
│$2,160│ $28,527│ $74,994│$150,683│$273,974│$474,801│
│$2,520│ $33,281│ $87,493│$175,797│$319,636│$553,935│
│$2,880│ $38,036│ $99,991│$200,911│$365,299│$633,068│
│$3,240│ $42,790│$112,490│$226,025│$410,961│$712,202│
│$3,600│ $47,544│$124,989│$251,139│$456,623│$791,335│
If I contribute $1 per day ($360 per year) for 50 years, at 5%, I’ll have more than $79,000!
If I contribute $10 per day ($3600 per year) for 50 years at 5%, I’ll have more than $790,000!
(Please note, I used a very simplified formula to calculate interest, based on annual contributions. In reality, if you made deposits throughout a calendar year, you would earn even more interest!)
Isn’t that cool? I like to look at charts like these. I am motivated by “what if”.
As a side note, my son will only be 4 once, so I’m still going to buy him the chewing gum! But, I’m also going to pocket some additional savings, so that when the time comes, I can buy my grand-kids
a warehouse full!
If you want to get really “pumped” about saving money – checkout what happens if you max our your 401k and Roth IRA contributions.
49 Responses to $10 A Day Equals 3/4 Of A Million Dollars
1. Holy crap!!
2. If you want to really blow your mind, calculate what you would have if you invested that same $10 in a growth stock mutual fund making the market average of 10-12%.
3. Are you me? Mythbusters, Car Talk, sports radio talk, getting out of debt…I think I have a twin!!!
4. So is your son three or four years old?
5. If you compute the numbers using daily compounding at 5%, your balance would be over $816,000 in 50 years. If you can manage a 10% rate of return, you balance goes to…
What a difference 5% makes!
6. I second the mutual fund idea. Compound interest is an amazing thing.
Also, not to rain on anyone’s parade, but just remember that a quarter million in 50 years isn’t worth the same as a quarter million today – it’s worth *substantially* less due to inflation.
7. Andrea,
My son turns 4 in January – and the new baby is due in April…
I’ve spent so much time telling folks, “he’ll be 4 when the baby gets here” that sometimes I call him 4, sometimes I call him 3…
8. I don’t have any money. Compound interest of 0 is still 0! I’m still motivated and will try $1/day
Wish me Luck!
9. Great post! It really makes you stop and think do I need to purchase this candy bar…
10. Pingback: Want an Extra $750,000? « Friendzy
11. Sometimes it’s hard to think that far into the future. Hard, but necessary!
12. What about inflation? How much would you end up in 50 years really.
13. For those asking about inflation -
The purpose of this post was to show how quickly things “add” up, if you keep at it -
In the “real world” – one would have to contribute the equivalent of 1 per day – if you wanted to keep up with inflation…
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16. You should keep this in mind with the 3 (going to be 4) year old and the newborn. Just think about the $10 per day, in 18 years plus 5% interest is just above $100,000. For what? For COLLEGE,
what else? You can use this as a quick and simple (theoretically) method of saving for college.
17. Wow, so $10 a day for a year is $3,600 not $3,650. When did the length of a year get shortened by 5 days? I bet banks did this just to save money on interest.
18. well, though a bit unrealistic, considering the fact that in most case it is hard to keep away from your saving account and let them untouch for 50 years,your post is worth to read. At least it
give more consideration before we investing somewhere
19. @bobimmy – I simply used 360 for the sake of simplification…
@pipholic – it’s not really that unrealistic…
my wife and I plan to do this for our newborn…
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22. That’s amazing!
I’m glad I’m not the only one out there running ‘what if’s on a regular basis.
23. Great example for your son!
- Steven Burda
24. My husband thinks I am mad saving $10 a week I know it isn’t a lot but I am the only one working in the house and I only work 25 hours a week. I find that $10.00 per week add’s up at the end of
the year with my interest that is paid from a saving’s account. I put in extra where i can and I get maintaince for my son which is just under $40 per month so in effect I save $80 approximatley
per month. what am I going to do with the money is my most asked question…my response is just leave it there you never know when you may need it!
25. Pingback: Carnival of Personal Finance:Naughty or Nice Edition
26. Intersting. I did the same calculations… but with 10,000 dollars a year being added and I was comparing percentages. I wanted to see the difference of 1% interest rate. The difference between 4%
and 5% is over 610,000. The difference between 6% and 5% is ver 879,000.
So saving isn’t the only difference. If you bank somewhere simply for conveniece, this is something to look at.
27. OOOOooo that is sooo neat. How much will you have at 100 years? Hello, when you’re ass is 80 years old, you won’t give a shit if you have $80 or $8,000,000
28. Aard-dude, the only way you will not care how much money you have at 80 is if you have Alzheimer’s. You don’t stop being a human being who needs money at eighty. My grandmother is almost eighty,
and it sure would not hurt her feelings to have $80k in the bank considering all the meds she has to be on and the property taxes she has to pay.
29. where can i find a free download to figure out
what the balance would be if i had 100,000 dollars
to put into a safe fund for 40years,as if i did not
waste my time&money going to college and just put
the same amount in a fund that earned interest
then go to work at a fastfood resturent and not
have worry about my retirement,thank,s
30. I just want to add a comment what will happen to that same $1 a day if I put it under my mattress and do not allow the gov to tax me on my savings? If you haven’t noticed the quote green backs
are not going to be tangible some day all of our money will just be on paper we are almost there now the only people who have real cash are the poor and drug dealers even our casinos are cards no
more clink clink clink when we win
31. That’s why you put it in a ROTH IRA and it’s tax free when you pull it back out.
Great post!
32. Look, this is the same strategy as buy and hold for the long run in stocks. Over a 50 year period just about any type of investment looks good on paper. Projections don’t meana hill of ………….. Ask
the Japanese how much they’ve made on thier money over the past 20 years. They lost in the stock market big time. Next to nothing on their treasuries, and lost huge in real estate. No one can
project that far out, because quite frankly what’s 750k going to be worth in 50 years? How will it stack up against inflation? Well not to be a poop but you catch my drift. However you have to
save something for a rainy day.
33. To all the nay-sayers who diss the numbers by reason of inflation / uncertainty and so on. $791,000 buys a lot more than zero. Which, if you don’t get busy socking some away, is what -you- will
have to buy your off-brand dog food with.
34. Saving money is great and it $10 a day sure adds up but honestly, one of my worries if what if I die before I get to spend all that money? I don’t plan to have kids and have no one to leave it to
so why would I want to accumulate millions when I’m dead. I want to spend the money while I am still on this earth. That doesn’t mean I want to be in debt of spend foolishly, I just don’t see the
point the point of being 80 and having 10 mil in the bank since I will be too old to enjoy it.
35. one cent doubled a day for 30 days equals over 1 million dollars. it is true.
36. It’s amazing what the small stuff can add up to!!
37. This was absolutely inspiring! I will definitely put $10 aside per day!, Thank you so much for sharing. Your graphics made it incredibly clear to see that $1 actually has great value, and that if
we were to skip that “daily coffee from whatever coffee shop” and save those $10 instead, we could certainly retire with a smile on our faces and with enough financial peace!
38. I did a search to find out how much money would represent all the money people put into Medicare and Social Security (SS) and that is how I got to this website. People pay Medicare till they die
even if they get SS the government deducts Medicare from SS. I did this because most people do not think how much money the government is stealing from people every year. Imagine if people start
saving at 10-12% compound interest for SS and Medicare which is very feasible. Then at age 60 they start drawing monthly payments from the account for retirement and leave the account for Medical
Care untouched unless there is a need due to illness. Those monies will never be depleted and when the person dies their family inherits whatever amount is left in the accounts. Now, people do
not want for Congress to put 500 billion dollars back for Medicare and that is peanuts if you do the numbers. I have always been of the thinking that SS and Medicare should be privatize and let
people save the money for their future. So you know, from the time Linden B. Johnson changed SS from being a non-public fund to a public fund in 1967 presidents have taken more than 5 trillion
dollars from SS alone…Now, do the math and see how much money that would generate at a compound interest for 50 years.
39. There are 365 days in a year
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the encyclopedic entry of computer circuit
Cube is a 1997 Canadian psychological thriller/horror/science fiction movie directed by Vincenzo Natali. The film was a very successful product of the Canadian Film Centre's First Feature Project.
Despite its low budget, the film achieved moderate commercial success and has acquired cult status as a niche science-fiction title.
Plot summary
Much of the film's appeal lies in its surreal,
settings — no extensive attempt is made to explain what the cube in which the characters are confined is, why it is created, or how the "inmates" were selected. Although the world "outside" is
referred to, it is presented in an extremely abstract fashion - either a dark void or a bright white light. Its appeal is also its complex mathematics, which is valid save for minor plot holes.
The film opens with a man named Alderson waking up in a strange, cube-shaped room with glowing, computer circuit-like walls and six doors, one at the center of each wall, including the ceiling and
floor. After recovering from his confusion, he opens two of the doors and looks into them to find rooms that differ to the one he is in only by color. He then opens and goes through a third door. He
looks around and then takes a step, but is suddenly cut into large cubes. He falls apart and the rack of crosshatched wires which diced him moves into view. It folds up and retracts.
In another room, several people find each other: Quentin the cop, Worth the office worker, Holloway the doctor, Rennes the escape artist, and Leaven the college student. None of them know where they
are, how they got there, or why they are there. Quentin, however, knows that there are traps, as he had looked into a room and nearly got his head cut off. The five decide to stay together and look
for the way out. Rennes, who had escaped from at least seven prisons, takes the lead. He shows them how to test for traps by tossing a boot into the rooms while holding onto the laces, to trigger
potential traps, figuring that the trapped room contain motion detectors. At one point he throws the boot in and comes with nothing, but figures out from smell that there are sensors rigged to detect
the chemicals that come off skin. Not long later, Rennes jumps into a room tested with a boot, and is sprayed in the face with acid. The others pull him back, but he dies as the acid corrodes his
face and the inside of his head. The group deduces that the floor must have pressure or thermal sensors, and decide that they need a better way to test for traps.
Quentin asks everyone about their occupations. He is a cop, Holloway is a free clinic doctor, and Worth works "in an office building, doing office building stuff". Leaven claims to do nothing but
"hang out" with her friends. Quentin believes that nothing is a coincidence, that each of them has a purpose in the cube. He asks why Leaven has her glasses, while Holloway has had her jewellery
taken away. Leaven reveals herself to excel at mathematics, and recalls that each room had a set of numbers engraved in the crawlspace between the doors. She theorizes that when one of those numbers
is prime, the room is trapped.
Leaven's purpose becomes attempting to "crack the cube's code", and they progress through the cubes. When they find themselves in a room with trapped rooms all around and below, Quentin checks the
door in the ceiling, through which falls a seventh person: Kazan. He appears to be mentally handicapped. At least two of the others see him as a burden, but Holloway decides to bring him along.
The group starts speculating about their surroundings, which leads to a conflict between Quentin and Holloway. Quentin dismisses Holloway's ideas as conspiracy theories, and Holloway thinks that
Quentin is naive. They also fight over Holloway's sympathy for Kazan, whom she believes to be autistic, as Quentin is angered by him and wants to leave him behind.
Quentin enters a cube without prime numbers and narrowly avoids death. Leaven's theory that non-prime-numbered rooms are safe is shown to be incorrect, and the group rests. Worth and Quentin get into
a fight, and it is revealed that Worth is one of the architects who designed the massive cube-shaped shell which contains the cube-shaped rooms. Although the others begin to distrust Worth, he is
able to give them information about the dimensions of the outer cube: it is 434 feet on each side. Leaven then realizes that the numbers between the cubes represent encoded Cartesian coordinates,
which show the starting location of the rooms. With this information, she can now guess at the size of the maze. She paces 14 feet, and deduces that the maze could be at most, 26 cubes by 26 cubes by
26 cubes, or 17,576 rooms. (Although not stated in the film, her calculations seem to be based on the correct assumption that the outer dimensions of the rooms are one and a half feet longer than
their inner dimensions, so that each room measures 15.5 feet on an external side. One could calculate the external dimensions from the length of tunnels between rooms.) Using the encoded coordinates,
Leaven claims to be able to navigate the maze, but could not do so without knowing what to use as the origin (0,0,0), and also without knowing that the rooms are moving which they discover later, or
how fast the rooms are moving around (Leaven estimates this later).
As they make their way there, hunger and dehydration take hold. Each person becomes on edge and paranoid. Quentin is gradually driven mad by his conflicts with Worth and Holloway, and his annoyance
with Kazan. He begins to seem cold and malicious. The group reaches a room near one face of the cube, but it is trapped with spears that come out of the walls when activated by sounds other than
those of the doors opening and closing. Quentin refuses to backtrack, insisting that they can pass through this room if they are very quiet. He tries to leave Kazan behind, but Holloway and Worth get
Kazan to cover his mouth and follow. As the last person, Quentin, is passing through, Kazan makes a sound. Quentin is nearly killed and is extremely angered, calling Kazan "a fucking fuck." Everyone
begins to argue.
The group finally reaches the last room in that direction, but discovers that there is a gap between the door and the outer shell. They fashion a rope from their clothes, and Holloway volunteers to
swing out on the rope to investigate. As she is suspended outside the room, the cube shakes and Holloway nearly falls. Quentin catches her, but then lets her fall to her death. He tells the others
that she slipped.
As they sleep, Quentin carries Leaven into another room. He tells her that they will "be going to the bottom. It will be quiet there". As he tries to convince Leaven to abandon the others, he also
makes sexual advances at her. Worth and Kazan wake up to save Leaven. Quentin says that he did not trust Holloway, and the group guesses that Holloway's death was not an accident. Enraged, Quentin
throws Worth through a door in the floor. Worth begins to laugh hysterically at what he sees in that room: Rennes's corpse. They think that they have been going in circles, but then Worth notices
that the "acid room" which killed Rennes is no longer adjacent to that room. He and Leaven realize that the rooms change locations by moving inside and outside the cube (a full cube of 26x26x26 rooms
could not not shift without moving a room outside of the cube).
Leaven deduces that the sets of numbers connected to a room represent something other than its traps and initial location: permutations obtained by subtracting the digits that are claimed to be able
to tell them how the rooms are moving. After some calculation, Leaven discovers that a room they have been in would be between the cubed rooms and the outer cube - a bridge to the outside world. If
they had stayed in the room they started in, they would have eventually been linked to the bridge cube and thus the exit (though this means that they may not have found Kazan). Leaven also discovers
that the trapped rooms are rooms in which any of the number of prime factors of the marked numbers is not a prime power. The prisoners then face the task of performing prime factorizations of
three-digit numbers – a task Leaven claims that she requires a computer for to do quickly. Fortunately, Kazan appears to be an autistic savant and performs the factorizations easily. He utters the
number of distinct prime factors each number has, as the room numbers are read to him.
They make their way towards the exit safely with Kazan's help. Worth devises a plan to incapacitate Quentin, who has gone completely mad. Leaven, Worth, and Kazan fight Quentin into a room below them
and leave him to die. They proceed and reach the bridge cube. When they open its door, bright light pours into the room. Worth announces that he will not go, as there is nothing for him in the world
outside. As he and Leaven share a moment, Quentin appears having somehow managed to catch up with the trio, and kills Leaven by stabbing her with a door handle he somehow broke off a door. He stabs
Worth as well, and grabs Kazan, who is climbing out. Worth grabs Quentin's leg with the last of his strength, and Quentin is crushed in the crawlspace between the cubes when the cubes realign. Having
saved Kazan, Worth lies down next to Leaven and dies.
In the final shot, Kazan is seen walking slowly into a bright light, outside which is only explained in Cube Zero
Power struggles and character development
The director and writers state (in the
Commentary) that each character in the film was designed to play through a certain arc of
character development
. This is presented through the plot twists, changes in party leader, and who the audience hopes will escape.
is the character who thrusts himself into the position of leader and appears to be the
main character
when the group first assembles. He claims to be a
and is strong and level headed. He takes on most of the dangerous tasks and claims to look for "practical solutions". However, it is soon revealed (mainly through Quentin's confrontation with Worth
in the Red Room) that he is violent,
- especially to Worth - and slightly unhinged, with a possible "thing for young girls". As the film progresses, he tries to take control of Leaven for her mathematical abilities, and becomes a
villain. He is responsible for tearing the group apart, and is himself literally torn apart at the end.
Dr Helen Holloway
is the elder woman of the group and a
free clinic doctor
. She is shown at the start to be bitter,
, and melodramatic. She spouts
conspiracy theories
and believes that the
U.S Government
is responsible for the Cube. She becomes more human, however, and tends to Quentin's wounds. She looks after Kazan with patience and gentleness. She shows that she can be calm when necessary, when
she explains to Quentin why they need Worth. She attempts to connect with Worth before Quentin kills her. In the film, she changes from being the most unstable member of the group to being the calm
opposition to Quentin.
Joan Leaven
begins the film as a
damsel in distress
. Rather than exploring her surroundings, she screams for help until she attracts Quentin, Holloway and Worth. She is the only member of the group to have personal belongings (her glasses). She is
modest in claiming she's nothing special, because she has expert mathematical skills (especially on little sleep) for her level of education which aid the group (much of the math in cube baffles even
college students). She prompts Worth to keep going, and is invaluable to the group for much of the film. She is killed by Quentin while trying to escape the cube.
David Worth's transformation
begins with him lying on the ground, looking injured and grim. He maintains a doomed outlook throughout the first part of the film, and mocks Quentin's attempts at escape. He is asked why he even
follows the others, when he claims to have no reason to live. He does not contribute much to the group, but sometimes leads Kazan and "boots" rooms (throwing boots into them to test for traps). When
the group reaches the Red Room, Quentin confronts Worth. It becomes known that Worth worked on the design of the outer shell or cube. He claims not to know about the purpose, construction, or traps
of the rooms, but knows that people were being put in for a few months. Quentin reacts in anger to Worth's story, and Worth gives a long, lucid speech about the futility of leadership: "The cube's a
headless blunder operating under the
of a master plan. Can you grasp that, Holloway?
Big Brother
is not watching you." Quentin says that his function in the group is "The Poison", another obstacle for them to overcome, with no function but the cause of conflict. However, he later becomes the
replacement hero. He continues to fight Quentin, and eventually tries to disable him. He is beaten by Quentin but luckily avoids being killed by traps in the rooms that Quentin throws him into. He
rescues Kazan and Leaven at different points in the film. At the end, he decides that he has nothing to live for and chooses to die by the exit. When Leaven asks him what is out there, he answers:
"Boundless human stupidity". He is then stabbed in the stomach by Quentin, but manages to both kill Quentin and aid in Kazan's escape. He then crawls over to Leaven's corpse and passes away.
, also known as "The
", first appears to the group as the most knowledgeable of their surroundings, and the reluctant leader. He is an
escape artist
who "flew the coop" from seven major
, and comes up with the method of using boots to test for traps. He has a facial
, but seems physically fit for an older man. He detects a trap that the boot does not, and uses it to urge the others to "concentrate on what's in front of you". Despite his calm skillfulness and
experience, he is killed by a trap. He changes from a specialist and central character to an early victim.
Kazan is introduced as the autistic man who seems only to be dead weight. He is immediately distrusted by Quentin and Leaven, who believe that he may be put into the cubes to slow them down or kill
them off. He later becomes a pivotal part of their escape, as the only person who can perform the calculations needed for the group to move safely. In the end, he is the only prisoner who makes it
out of the Cube alive.
There is speculation that Kazan and Eric Wynn (from Cube Zero) are the same person. Eric was lobotomized at the end of the film and put back into the Cube behaving and speaking in the same way as
Kazan. Eric was a child prodigy, able to recognize and solve patterns with ease; at one point, he tells the others to read the codes at the doors to him so he can tell them the correct route, like
how Kazan helps his group. The director's commentary for Cube Zero states: "Maybe this is Kazan or how Kazan came to be, making this much more of a prequel than a sequel."
appears at the opening of the film (and on the DVD packaging) but does not meet the others. He is killed within minutes, but is first made to appear to be a main character. The fact that the actor
who played him was fitted with a headpiece to look as if he has a shaven head, and that the group lacks a moral or spiritual authority, have led to the notion that Alderson was a
Catholic monk
. Due to his lack of contact with the other characters, Alderson might have been part of another group placed in the cube earlier, as Worth mentioned that he knew people were being put in the cube
for a few months. (This theory does not fit with
Cube Zero
, in which it is revealed that the cube is "swept" periodically. However, the two films were written by different people, and the sweep might not apply to
Character Names
All the characters are named after
. Quentin is named after
San Quentin State Prison
, Holloway after the
Holloway Prison
, Kazan after the prison in
. Rennes is named after a prison in
, Alderson after
the prison
Alderson, West Virginia
, and Leaven and Worth after
the prison
Leavenworth, Kansas
The characters themselves reflect the prisons in their traits. Kazan (the autistic man) is a disorganized prison. Rennes (the "mentor") pioneered many of today's prison policies. Quentin (the
policeman) is known for brutality. Holloway is a women's prison. Alderson is a prison where isolation is a common punishment. Leavenworth runs on a rigid set of rules (Leaven's mathematics), and the
new prison is corporately owned and built (Worth, hired as an architect).
Most character's first names are either not given or not revealed until later in the film.
The various traps in the Cube:
• The mesh trap - A hallmark trap of the Cube series as the first to be shown in the first movie, with a brief appearance in Cube Zero prequel. It is a rack of razor-sharp wire which unfurls and
falls from the ceiling onto the victim who walks into the room. It slices the victim's body into cube-shaped chunks, which then fall apart. The wire rack then folds up, and returns to its
original position. Alderson is killed at the beginning of the film by this trap.
• Flamethrower - A simple trap, this is simply a flame thrower that comes out of the walls and shoots fire when a victim is detected in the room. This trap can be detected by throwing a boot into
the room to set off the motion sensors. The phrase "Boot it!" is used primarily by Quentin as a command to the others to test a room.
• Acid sprayer - A small hole in the wall opens to spray a strong acid into a victim's face, quickly dissolving it and killing the person. Unlike the flamethrower, this trap is not detectable by
boot and appears to be activated by the chemical detector. Rennes is killed by this trap.
• Spear trap - This trap is sound activated (though built to ignore the noisy opening of cube doors). Metal spears extend out of the walls and across the room to skewer those within. The characters
are still able to pass through a room with this trap by being silent and careful. Production designs in the DVD special features show that the spikes are constructed so that they retract into a
coil, thus explaining how such long spikes can fit in the walls.
• The wire twist trap - Lines of razorwire extend from ceiling to floor, to form a round cage around a target. They twist inwards like a spiral to cut the victim into pieces, much like the swinging
dicer trap. Quentin gets caught in this trap but rolls away before it closes completely. It is dubbed the "Sushi Trap" or "Sushi Machine" by Quentin.
• Unknown trap - Little is known about this trap. Rennes discovers it before entering, due to the dryness of the air in the room. It reacts to the hydrogen sulfide gas emitted by the occupant. This
trap shows that there are several types of sensors used throughout the cube that are immune to the boot test.
• Cube transitions - The movement of the cubes themselves could also be considered a trap. Anyone who is in the crawlspace between the cubes during the shuffle is torn in half as the rooms move
apart. Quentin is killed by this "trap".
Mathematics in Cube
The mathematics in Cube has intrigued and baffled many.
hired a professor to help with the math and it is valid except for a couple of plot holes. Each room is marked by a nine-digit number, or three 3-digit numbers. Leaven discovers that the
Cartesian coordinates
of the original position of each room inside the cube are obtained by adding the digits of each number. So a room marked "123 456 789" is located at x=1+2+3, y=4+5+6, z=7+8+9, or [6,15,24], and
figures the rooms are moving around by moving close to each other, allowing the small space in between the cubes to add up and create enough space for the cubes to move around. This could easily be
possible with anti-gravity, but is difficult (but perhaps possible) to fathom otherwise since we are never shown any kind of engineering that would shift around the cubes.
She also theorizes that the trapped rooms are the rooms in which at least one of these numbers is a prime number, and then later revises this to the theory that the trapped rooms are the rooms where
the number of prime factors of any of the three coordinates are not a prime power. For instance, the first trapped room they encounter since discovering this is one where the number of prime factors
of one of the marked numbers is one, while any room where the number of prime factors of the marked numbers consist of 2, 3, 4, or 5, or any prime power, are safe. This is also why the first theory
worked for them up to a point, because a room with a prime number does mark a trapped room, because the number of prime factors of a prime is one, which is not a prime power, but it is not the only
thing which marks the room. The number of prime factors could be any other non-prime-power number.
This may all be possible, but this is not enough to allow them to navigate. Leaven has no way to know which of the eight corners is the origin, the co-ordinate [1,1,1] (it could not be 0,0,0 because
the maximum co-ordinate is 26, where it would be 25 if the origin were 0,0,0 because that would count as a room). In addition, she could not navigate to begin with because they later find the rooms
are moving around which they did not know at start. Later, they claim to know what pattern they are moving in based on subtracting the digits rather than adding, and Leaven estimates how fast the
rooms are moving based on having heard the cubes shift.
There is also one "bridge", a 17,577th room which begins outside the cube and moves its way through it, which they finally reach but again could not do so without knowing the origin. Kazaan finally
enters it, which leads him to whatever lies outside. It is only explained later in Cube Zero what lies outside.
Production details
The movie was shot on a
soundstage. Only one "cube", measuring 14 by 14 by 14 feet, was built. The colour of the room was changed by sliding panels. Since this task was a time-consuming procedure, the movie was not shot in
sequence; all shots taking place in rooms of a specific colour were shot one at a time.
Another partial "cube" was made for shots from a different room.
There was only one working door which could support the weight of the actors.
It was intended that there would be six different colours of rooms to match the recurring theme of six throughout the movie - five sets of gel panels plus pure white. However, the budget did not
stretch to the sixth gel panel and so there are only five different colours of room in the movie.
is followed by the
sequel Cube 2: Hypercube
(2002) and the
prequel Cube Zero
a conceptual prequel from 1969 is "The Cube" written and directed by Jim Henson
• During Alderson's falling to pieces, you can see what looks like the letters "ED" in the middle slice of his torso.
• Leaven claims that it's impossible for a human being to compute the factors of a 3 digit number because it would be "astronomical". This is highly incorrect. There are numerous tricks and methods
for factoring numbers that would make the job fairly quick.
• Quentin states Rennes escaped from six prisons (before being quickly corrected), this further hints at the connection between the characters' names and six major prisons
• As Leaven discovers trapped rooms are marked with prime numbers, she uses the same amount of time figuring out that 645 and 372 are not prime as she does figuring out that 649 is not a prime.
This makes no sense as numbers ending in 5 or an even number are not primes (except for 2 and 5), as they are divisible by 5 or 2.
• There are no three-digit prime powers numbers that add up to have a Cartesian coordinates of 1, 6, 12, 15, 18, 21, 24, 26.
• There are 192 numbers that are prime powers between 001 and 999.
• If the numbers for the cubes are randomly distributed, there would be a 52.76% chance that one of the three digit numbers will contain a prime power, and hence contain a trap.
See also
External links | {"url":"http://www.reference.com/browse/computer+circuit","timestamp":"2014-04-18T21:45:02Z","content_type":null,"content_length":"106134","record_id":"<urn:uuid:aa552901-5faa-4ae5-94bb-5b824c1f7854>","cc-path":"CC-MAIN-2014-15/segments/1397609535095.9/warc/CC-MAIN-20140416005215-00229-ip-10-147-4-33.ec2.internal.warc.gz"} |
Problems from Another Time
Individual problems from throughout mathematics history, as well as articles that include problem sets for students.
Find the greatest value of y in the equation a4 x2= (x2 + y2)3.
A square walled city measures 10 li on each side. At the center of each side is a gate. Two persons start walking from the center of the city.
Prove that if the sums of the square opposite sides of any quadrilateral are equal, its diagonals interect at right angles.
The authors recount the 'great tale' of Napier's and Burgi's parallel development of logarithms and urge you to use it in class.
A powerful, unvanquished, excellent black snake, 80 angulas in length, enters into a hole at the rate of 7 1/2 angulas in 5/14 of a day, and in the course of a day its tail grows 11/4 of an angula.
A lady being asked how old she was at the time of her marriage replied that the age of her oldest son was 13; that he was born 2 years after her marriage...
A series of circles have their centers on an equilateral hyperbola and pass through its center. Show that their envelope is a lemniscate.
How a translation of Peano's counterexample to the 'theorem' that a zero Wronskian implies linear dependence can help your differential equations students
Given a wooden log of diameter 2 feet 5 inches from which a 7 inch thick board is to be cut, what is the maximum possible width of the board?
There is a log 18 feet long, the diameter of the extremities being 1 ft and 2.6 ft respectively... | {"url":"http://www.maa.org/publications/periodicals/convergence/problems-another-time?page=6&device=mobile","timestamp":"2014-04-16T11:05:22Z","content_type":null,"content_length":"25969","record_id":"<urn:uuid:19f9df6d-5d52-48a2-9fe4-ec55694168d7>","cc-path":"CC-MAIN-2014-15/segments/1397609539705.42/warc/CC-MAIN-20140416005219-00654-ip-10-147-4-33.ec2.internal.warc.gz"} |
Some experimental support for OpenLists, which are streams in the Par monad that support constant-time append.
data OpenList a Source
Show a => Show (OpenList a) OpenLists can only be printed properly in the Par monad. show on an open list will only give a hint -- what the first and last elements of the openlist are.
NFData a => NFData (OpenList a)
cons :: NFData a => a -> OpenList a -> Par (OpenList a)Source
Add an element to the front of an OpenList. Works irrespective | of whether the input is closed.
tail :: OpenList a -> Par (OpenList a)Source
Tail of an OpenList. Beware, if the list contains only one element (e.g. the result of tail will be null), it must be CLOSED for tail to work.
data IList a Source
An IList is the equivalent of a lazy list in the Par monad. The tail of the list is an IVar, which allows the list to be produced and consumed in parallel.
NFData a => NFData (IList a) To fully evaluate an IList means to evaluate both the head and tail. This does not evaluate the entire spine of the list of course, because the tail is an IVar. | {"url":"http://hackage.haskell.org/package/monad-par-0.1/docs/Control-Monad-Par-OpenList.html","timestamp":"2014-04-20T01:01:53Z","content_type":null,"content_length":"19955","record_id":"<urn:uuid:5ec63e08-d159-4ebc-8e4f-849b3f2b10f0>","cc-path":"CC-MAIN-2014-15/segments/1397609537804.4/warc/CC-MAIN-20140416005217-00504-ip-10-147-4-33.ec2.internal.warc.gz"} |
Riverside, MA Math Tutor
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Almost-Kahler Einstein four manifolds
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Are the odd Betti numbers of an Almost-Kahler Einstein four manifolds necessarily even ?
If some of Betti numbers were odd this would obviously contradict to Goldberg's conjecture, stating that Almost-Kahler Einstein manifolds are Kahler. I assume Goldberg's conjecture is open in all
Dmitri Mar 5 '11 at 10:19
Sorry for the double-retag. I think these tags should work.
Elizabeth S. Q. Goodman Mar 5 '11 at 21:19
add comment | {"url":"http://mathoverflow.net/questions/57442/almost-kahler-einstein-four-manifolds","timestamp":"2014-04-19T04:52:25Z","content_type":null,"content_length":"48540","record_id":"<urn:uuid:b08e6e2b-c9e6-45e9-8362-86c9b21596a8>","cc-path":"CC-MAIN-2014-15/segments/1397609535775.35/warc/CC-MAIN-20140416005215-00485-ip-10-147-4-33.ec2.internal.warc.gz"} |
proof of irrationality of sqrt(2)
Here'a a really simple and intuitive proof that sqrt(2) is irrational, which extends immediately to the nth root of any rational number that isn't a perfect nth power.
It goes like this. Let r be a rational number, and let a, b be integers such that r=a/b and (a,b)=1 (so that a/b is r in lowest terms). Then clearly r^2=a^2/b^2, and it is clear that (a^2,b^2)=1, so
that this is r^2 in lowest terms. Thus we see that the square of any rational number, written in lowest terms, has a perfect square in both the numerator and denominator. So we can conclude that if a
rational number does not have this property, it is not the square of a rational number, and so its square root is irrational. Since 2 is not a perfect square, sqrt(2) is irrational.
This seems a lot more straightforward than the standard proof by contradiction. Is this well known, and why isn't it more popular? | {"url":"http://www.physicsforums.com/showthread.php?t=129729","timestamp":"2014-04-17T09:54:10Z","content_type":null,"content_length":"58881","record_id":"<urn:uuid:2df03d56-115e-4fad-9f71-3d619f099e0b>","cc-path":"CC-MAIN-2014-15/segments/1398223206647.11/warc/CC-MAIN-20140423032006-00618-ip-10-147-4-33.ec2.internal.warc.gz"} |
Orthogonality in non-inner product spaces
up vote 9 down vote favorite
I have come across a notion of orthogonality of two vectors in a normed space not necessarily inner product space. Two vectors $x$ and $y$ in a normed space are said to be orthogonal (represented $x\
perp y$) if $||x||\leq ||x+\alpha y||,$ for every $\alpha,$ a scalar.
1) What is the rational behind the definition above? I guess, it has got something to do with minimum overlap between $x$ and $y$.
2) Is this unique generalization of the concept of orthogonality from inner product spaces?
Thank you.
Here's another paper (seems not to be referenced in the ones mentioned in Valerio's answer): projecteuclid.org/… – Ralph Mar 4 '12 at 22:31
add comment
4 Answers
active oldest votes
Concerning question 1: The rational is that in an inner product space $$x\perp y \Leftrightarrow \forall \alpha \in K: ||x||\leq ||x+\alpha y|| \qquad(K = \mathbb{R} \text{ or } K = \
mathbb{C})$$ Now, if no inner product is available (but a norm), the idea is, to just take the right hand side as definition of orthogonality (call it $\perp_1$).
Concerning question 2: No, there are other -non-equivalent - generalizations as well. As an example, note that in an inner product space over the reals $$\langle x,y \rangle = \frac
{1}{4}( ||x+y||^2 - ||x-y||^2).$$ Hence $x\perp y \Leftrightarrow ||x+y|| = ||x-y||$. So the definition $$ x\perp_{\scriptstyle 2}\; y : \Leftrightarrow ||x+y|| = ||x-y||$$
generalizes the orthogonality from an inner product space to any normed space (over the reals).
up vote 12 down
vote accepted Now let's show that $\perp_1, \perp_2$ aren't equivalent. Let $E = \mathbb{R}^2$ with norm $||(a,b)|| = \max(|a|, |b|)$. Then
$\qquad (0,1) \perp_2 (2,1)$ but not $(0,1) \perp_1 (2,1)\quad$ (take $t=-1/4$)
$\qquad (1,1) \perp_1 (2,0)$ but not $(1,1) \perp_2 (2,0).$
Follow up questions: (i) Are these two notions of orthogonality symmetric? $\perp_2$ certainly is, but what about $\perp_1$? (ii) Are these notions invariant under scaling? For
example, if $x \perp_1 y$, then $x \perp_1 \beta y$ for any $\beta \in K$. Most importantly, (iii): In which contexts are these two notions of orthogonality useful? – Martin Mar 4
'12 at 1:43
Another follow up question I have is: Since $\perp_{1}$ and $\perp_{2}$ are equivalent in inner-product spaces, in what way normed spaces are deficient so that these orthogonality
concepts don't agree? – Uday Mar 4 '12 at 2:07
Another question: Are there any more non-equivalent orthogonality definitions? – Uday Mar 4 '12 at 2:16
1 @Martin: (i) $\perp_1$ is not symmetric: From above $(1,1) \perp_1 (2,0)$, but not $(2,0) \perp_1 (1,1)$ (take $t = -1$). (ii) $\perp_1$ is invariant under scalar multiplication,
as is aparant from the definition. $\perp_2$ ist not: Again from above, $(0,1) \perp_2 (2,1)$ but not $3 \cdot (0,1) \perp_2 (2,1)$. – Ralph Mar 4 '12 at 10:00
1 @Uday: Yes, there are more notions of orthogonality in normed spaces - see the survey article quoted by Valerio. In principle you can take any norm-expression that is equivalent to
orthogonality in an inner product space and use it as definition of orthogonality in a normed space. – Ralph Mar 4 '12 at 10:04
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The definition you gave is called Birkhoff-James orthogonality and the intuition is the following: suppose you have $x,y\in\mathbb R^2$ and construct a triangle with sides $x$ and $y$. Now
let $x$ be fixed and consider the same triangle with $-\alpha y$ instead of $y$. Observe that $||x+\alpha y||$ is the length of the third side of this triangle. If you try to write down a
picture, you figure out in a moment that the condition $||x||\leq||x+\alpha y||$ can be true for all $\alpha$ iff $x$ and $y$ are orthogonal (looking at the picture, if they are not
orthogonal and the inequality is true for some $\alpha$, then it is false for $-\alpha$). Birkhoff-James' orthogonality is a tentative to capture orthogonality through this geometric
Birkhoff-James' orthogonality is not the unique notion of orthogonality for normed space.
Some references:
In the following paper
up vote
6 down http://arxiv.org/pdf/0907.1813.pdf
you can find some recent very easy application of BJ's orthogonality, as well, if you go through the bibliography, some references about other notions of orthogonality are given. In
particular I suggest the paper of Diminnie
Diminnie, C.R. A new orthogonality relation for normed linear spaces, Math. Nachr. 114 (1983), 192-203
and the survey by Alonso and Benitez http://dmle.cindoc.csic.es/pdf/EXTRACTAMATHEMATICAE_1989_04_03_03.pdf
P.s. Bikhoff-James orthogonality is not symmetric in general. Some interesting remarks about symmetric orthogonalities can be found in the paper(s) by Partington in the bibliography of the
arxiv paper cited above.
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Concerning your follow-up question (iii) there is the following very nice result: For Birkhoff-James orthogonality it is easy to find examples where $y\perp x$ but $\left\|x\right\|/\left\|
x+\alpha y\right\| > 1$ for some real $\alpha$, and so natural to investigate the largest such value $\left\|x\right\|/\left\|x+\alpha y\right\|$ over $X$. In "R. L. Thele, Some results on
up vote 4 the radial projection in Banach spaces. Proc. Amer. Math. Soc., 42(2):484--486", it is it is shown that this quantity is exactly the Lipshitz constant for the radial projection onto the
down vote unit ball in this norm.
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Well, it depends what do you need it for. You may also have a look at semi-inner-product spaces, which are natural generalizations of inner product spaces.
up vote 3 down vote
add comment
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Cardinality of Equivalence Relation of Eventually Sublinear Functions
up vote 4 down vote favorite
Let $\Bbb{R}^{+}\_{0}$ be the set of non-negative real numbers and $\Bbb{R}^{+}$be the set of positive reals. Let us say that a function $f \colon \Bbb{R}^{+}\_{0} \to \Bbb{R}^{+}\_{0}$ is eventually
sublinear if $\ \forall r \in \Bbb{R}^{+} \ \exists x_0 \in \Bbb{R}^{+} \colon \forall x \geq x_0, f(x) < rx$. Let $S$ be the set of non-decreasing, eventually sublinear functions (we require no
continuity). We define an equivalence relation on $S$: $f \sim g$ if and only if $f$ and $g$ are eventually boundedly close, i.e., $\exists K>0\ \exists x_0 \in \Bbb{R}^{+} \colon \forall x \geq x_0,
|f(x)-g(x)| < K$. Denote the set of equivalence classes of $S$ under $\sim$ by $\ T$.
Question 1: What is the cardinality of $\ T$?
Question 2: Does this cardinality change if we drop the requirement that the functions are non-decreasing? What if the co-domain becomes $\Bbb{R}$?
Looking at the set of pairwise non-equivalent sublinear functions $\{f(x) = r*ln(x) \mid r \in \Bbb{R}\}$ shows that $|T|$ must be at least as big as the cardinality of the reals, $\beth_1$. Also, $|
T|$ can be no bigger than the cardinality of all functions from $\Bbb{R}$ to $\Bbb{R}$, which is $\beth_2$, the cardinality of the power set of the reals. So there seem to be three options:
Option 1: ZFC proves $|T| = \beth_1$.
Option 2: ZFC proves $|T| = \beth_2$.
Option 3: This question is independent from ZFC (but perhaps not independent from ZFC + Generalized CH?).
Different proof ideas involving explicit bounds or forcing are failing me. So does anyone know of an answer and a proof? Thanks.
Could you give some motivation for your question? Note that there are only continuum many nondecreasing functions. (Also, note that a priori there are many other possibilities for a cardinal number
to be between $\aleph_0$ and $\beth_2$, such as $\aleph_1$; however, experience shows that the answer to naive questions about cardinality is almost always $\aleph_0$ or $2^{\aleph_0}$, when
separable spaces are involved. – Goldstern Dec 7 '12 at 11:26
I don't really remember the "motivation" right now. It came from a fellow graduate student's research in geometry group theory, but she is the sort to walk into a room, drop this sort of problem
down without context, and then leave. I was interested in it independently as a toy problem (and because I like infinite cardinality counting problems). If this doesn't answer her original
question, I'll come back! Thanks again! – Charlie Cunningham Dec 7 '12 at 19:41
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2 Answers
active oldest votes
There are only $\beth_1$ non-decreasing functions from $\mathbb R_0^+$ to itself, because such a function is determined by its values on a dense set plus information about its countably
many discontinuities. So the answer to Question 1 is $\beth_1$. For Question 2, the answer is $\beth_2$. Take any proper nonempty subset $A$ of the unit interval (there are $\beth_2$ of
up vote 7 these), and consider the function $f_A$ that agrees with the natural logarithm at points whose fractional part is in $A$ and is identically 0 at all other points. No two of these
down vote functions are boundedly close.
Amazing... When I wrote my answer, I thought that this particular family is very uncanonical. – Goldstern Dec 7 '12 at 11:46
2 It becomes a little more canonical because the OP already mentioned logarithms. – Andreas Blass Dec 7 '12 at 11:53
This is perfect! (Especially with Goldstern's reminder below containing some elementary real analysis.) Thank you! – Charlie Cunningham Dec 7 '12 at 19:39
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(EDIT: I was busy writing this answer when Andreas Blass already submitted his answer. No need to read this answer, it is surprisingly similar to Andreas' answer.)
Every function $f: \mathbb R_0^+\to \mathbb R_0^+$ is completely determined by its restriction to $\mathbb Q \cup D(f)$, where $D(f)$ is the set of points where $f$ is not continuous. If
$f$ satisfies $x\le y \Rightarrow f(x) \le f(y)$, then $D(f)$ is countable since the intervals $(\lim_{x\to a-} f(x), \lim_{x\to a+} f(x))$ are disjoint for $a\in D(f)$. So the number of
nondecreasing functions is $\beth_1=2^{\aleph_0}$.
up vote 7
down vote If the functions are not required to be nondecreasing, then there are $\beth_2 = 2^{\beth_1}$ equivalence classes, which can be seen as follows:
For every set $A \subseteq (0,1)$ define $f_A$ by $f_A(n+x) = \log(n)$ if $x\in A$, $n\in \mathbb N$. Let $f_A(n+x) = 0$ for $x\in [0,1]\setminus A$.
This was super helpful as well. Thanks! – Charlie Cunningham Dec 7 '12 at 19:38
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Vector Equation of a Plane
June 2nd 2013, 03:31 AM #1
Apr 2013
Vector Equation of a Plane
Find a vector perpendicular to both i+2j-k and 3i-j+k. Hence find the Cartesian equation of the plane parallel to both i+2j-k and 3i-j+k which passes through the point (2,0,-3).
I managed to solve the first one but not the parallel part.
Explanation with the aid of diagram is much appreciated. Thanks in advance
Re: Vector Equation of a Plane
Find a vector perpendicular to both i+2j-k and 3i-j+k. Hence find the Cartesian equation of the plane parallel to both i+2j-k and 3i-j+k which passes through the point (2,0,-3).
I managed to solve the first one but not the parallel part.
Explanation with the aid of diagram is much appreciated. Thanks in advance
If $u~\&~v$ are non-parallel vectors and $P: (a,b,c)$ then $(u\times v)\cdot<x-a,y-b,z-c>=0$ is a plane which contains $P$ and is parallel to $u~\&~v$.
Re: Vector Equation of a Plane
Different way of saying the same thing: if <A, B, C> is a vector perpendicular to both vectors u and v, then $A(x- x_0)+ B(y- y_0)+ C(z- z_0)= 0$ is a plane parallel to both u and v and passing
through $(x_0, y_0, z_0)$.
June 2nd 2013, 03:50 AM #2
June 2nd 2013, 04:55 AM #3
MHF Contributor
Apr 2005 | {"url":"http://mathhelpforum.com/math-topics/219520-vector-equation-plane.html","timestamp":"2014-04-17T07:01:15Z","content_type":null,"content_length":"38593","record_id":"<urn:uuid:52eaa06a-81b6-450f-bb2c-25905aaef129>","cc-path":"CC-MAIN-2014-15/segments/1397609537804.4/warc/CC-MAIN-20140416005217-00597-ip-10-147-4-33.ec2.internal.warc.gz"} |
Faculty Research
• TU Home
Department of Mathematics
• Jess & Mildred Fisher
College of Science & Faculty Research and Scholarship
• Department of
Mathematics Dr. Geoffrey Goodson's current research interests are in the fields of ergodic
theory, dynamical systems, matrix theory and operator theory.
• Faculty & Student
Research Some of Dr. Goodson's publications include:
Algebra, Geometry & • Geoffrey Goodson, Spectral properties of normal operators having symmetries
Logic arising from conjugation operators, Operators and Matrices (to appear).
• Geoffrey Goodson, Matrices that commute with their conjugate and transpose,
Analysis Advances in Linear Algebra & Matrix Theory (to appear).
• Geoffrey Goodson, Spectral doubling of normal operators and connections
Applied Mathematics with antiunitary operators, Integral Equations and Operator Theory 72
(2011), 115–160.
Mathematics Education • Geoffrey Goodson, Groups having elements conjugate to their squares and
applications to dynamical systems, Applied Mathematics 1 (2010), 416–424.
Probability & • Geoffrey Goodson, On commutators in matrix theory, Operators and Matrices 4
Statistics (2010), 283–292.
• G. R. Goodson and R. Horn, Canonical forms for normal matrices that commute
Applied Mathematics with their complex conjugate, Linear Algebra and Applications 430 (2009),
Laboratory 1025–1038.
• Geoffrey Goodson, On the multiplicity function of real normal operators,
Operators and Matrices 2 (2008) 1–13.
Dr. Angel Kumchev's current research interests are in the fields of additive
number theory, sieves, the distribution of primes, and analytic number theory
in number fields.
Some of Dr. Kumchev's publications include:
• Angel Kumchev, On Weyl sums over primes in short intervals, Proceedings of
the Sixth China-Japan Conference on Number Theory, Shanghai, August 2011
(to appear).
• A. Kumchev, J. Cilleruelo, F. Luca, J. Rue, and I. Shparlinski, On the
fractional parts of a^n/n, Bulletin of the London Mathematical Society 45
(2013), 249-256.
• Angel Kumchev, On the convergence of some alternating series, Ramanujan
Journal 30 (2013), 101-116.
• A. Kumchev and T.Y. Li, Sums of almost equal squares of primes, Journal of
Number Theory 132 (2012), 608-636.
• A. Kumchev and T.H. Chan, On sums of Ramanujan sums, Acta Arithmetica 152
(2012), 1-10.
• A. Kumchev, T.H. Chan and M. Wierdl, Additive bases arising from functions
in a Hardy field, Acta Mathematica Hungarica 129 (2010), 263-276.
• A. Kumchev and G. Harman, On sums of squares of primes II, Journal of
Number Theory 130 (2010), 1969-2002.
Dr. Houshang Sohrab's current research interests are in the fields of
functional analysis, normal solvability of partial (and pseudo) differential
Some of Dr. Sohrab's publications include:
• Houshang Sohrab, Basic Real Analysis, Birkhauser, Boston, 2003.
• Houshang Sohrab, Cordes algebras and superexponential growth, Proceedings
of the 30th Iranian International Conference on Mathematics, (1999),
• Houshang Sohrab, Cordes algebras based on Schrodinger operators with
rapidly increasing potentials, Journal of Mathematical Analysis &
Applications 207 (1999), 549–561.
• Houshang Sohrab, Spatially nontemperate pseudodifferential operators,
sphere extensions and Fredholm theory, Rocky Mountain Journal of
Mathematics 25 (1999), 501–514.
Department of Mathematics
7800 York Road, Room 316 (campus map)
Hours: Monday - Friday, 8:30 a.m. – 5 p.m.
Phone: 410-704-3091
Fax: 410-704-4149
E-mail: math@towson.edu
Department of Mathematics
Faculty Research and Scholarship
Dr. Geoffrey Goodson's current research interests are in the fields of ergodic
theory, dynamical systems, matrix theory and operator theory.
Some of Dr. Goodson's publications include:
• Geoffrey Goodson, Spectral properties of normal operators having symmetries
arising from conjugation operators, Operators and Matrices (to appear).
• Geoffrey Goodson, Matrices that commute with their conjugate and transpose,
Advances in Linear Algebra & Matrix Theory (to appear).
• Geoffrey Goodson, Spectral doubling of normal operators and connections
with antiunitary operators, Integral Equations and Operator Theory 72
(2011), 115–160.
• Geoffrey Goodson, Groups having elements conjugate to their squares and
applications to dynamical systems, Applied Mathematics 1 (2010), 416–424.
• Geoffrey Goodson, On commutators in matrix theory, Operators and Matrices 4
(2010), 283–292.
• G. R. Goodson and R. Horn, Canonical forms for normal matrices that commute
with their complex conjugate, Linear Algebra and Applications 430 (2009),
• Geoffrey Goodson, On the multiplicity function of real normal operators,
Operators and Matrices 2 (2008) 1–13.
Dr. Angel Kumchev's current research interests are in the fields of additive
number theory, sieves, the distribution of primes, and analytic number theory
in number fields.
Some of Dr. Kumchev's publications include:
• Angel Kumchev, On Weyl sums over primes in short intervals, Proceedings of
the Sixth China-Japan Conference on Number Theory, Shanghai, August 2011
(to appear).
• A. Kumchev, J. Cilleruelo, F. Luca, J. Rue, and I. Shparlinski, On the
fractional parts of a^n/n, Bulletin of the London Mathematical Society 45
(2013), 249-256.
• Angel Kumchev, On the convergence of some alternating series, Ramanujan
Journal 30 (2013), 101-116.
• A. Kumchev and T.Y. Li, Sums of almost equal squares of primes, Journal of
Number Theory 132 (2012), 608-636.
• A. Kumchev and T.H. Chan, On sums of Ramanujan sums, Acta Arithmetica 152
(2012), 1-10.
• A. Kumchev, T.H. Chan and M. Wierdl, Additive bases arising from functions
in a Hardy field, Acta Mathematica Hungarica 129 (2010), 263-276.
• A. Kumchev and G. Harman, On sums of squares of primes II, Journal of
Number Theory 130 (2010), 1969-2002.
Dr. Houshang Sohrab's current research interests are in the fields of
functional analysis, normal solvability of partial (and pseudo) differential
Some of Dr. Sohrab's publications include:
• Houshang Sohrab, Basic Real Analysis, Birkhauser, Boston, 2003.
• Houshang Sohrab, Cordes algebras and superexponential growth, Proceedings
of the 30th Iranian International Conference on Mathematics, (1999),
• Houshang Sohrab, Cordes algebras based on Schrodinger operators with
rapidly increasing potentials, Journal of Mathematical Analysis &
Applications 207 (1999), 549–561.
• Houshang Sohrab, Spatially nontemperate pseudodifferential operators,
sphere extensions and Fredholm theory, Rocky Mountain Journal of
Mathematics 25 (1999), 501–514.
Department of Mathematics
7800 York Road, Room 316 (campus map)
Hours: Monday - Friday, 8:30 a.m. – 5 p.m.
Phone: 410-704-3091
Fax: 410-704-4149
E-mail: math@towson.edu
Dr. Geoffrey Goodson's current research interests are in the fields of ergodic theory, dynamical systems, matrix theory and operator theory.
Department of Mathematics 7800 York Road, Room 316 (campus map) Hours: Monday - Friday, 8:30 a.m. – 5 p.m. Phone: 410-704-3091 Fax: 410-704-4149 E-mail: math@towson.edu | {"url":"http://www.towson.edu/math/research/analysis.asp","timestamp":"2014-04-18T03:33:25Z","content_type":null,"content_length":"16423","record_id":"<urn:uuid:de6ae4fb-1e5e-40f0-97dd-18819eb00468>","cc-path":"CC-MAIN-2014-15/segments/1397609532480.36/warc/CC-MAIN-20140416005212-00642-ip-10-147-4-33.ec2.internal.warc.gz"} |
The Conic Sections
The two curves we have just considered -- the circle and the parabola -- are special cases of the conic sections.
A conic section is a curve obtained by the intersection of a plane with the surface of a (double-napped) cone, as shown in Figure 4. When the plane is parallel to the edge of one cone , the
intersection is a parabola. When the plane and cone intersect in a closed curve, the result is an ellipse; in the special case where the plane is perpendicular to the axis of symmetry of the cone,
the ellipse is actually a circle. When the plane and cone intersect in two curves, the result is a hyperbola.
Figure 4: The Conic Sections: (1) Parabola, (2) Ellipse and Circle, (3) Hyperbola.
If the plane intersects the vertex of the cone, then the result will be one of three "degenerate" cases: a pair of intersecting lines, a single line, or a single point.
The conic sections were first studied by mathematicians of ancient Greece. Pappus credits Euclid with writing four volumes on conic sections (which have been lost to history), and Apollonius with
completing these volumes and writing an additional four. In Conics, Apollonius proved that two (distinct, nondegenerate) conic sections intersect in at most four points [Katz 2008, p.122]. Therefore,
four points do not uniquely determine a conic section, as illustrated in the applet in Figure 5. | {"url":"http://www.maa.org/sites/default/files/images/upload_library/46/stemkoski/cramer/article05.html","timestamp":"2014-04-20T10:22:31Z","content_type":null,"content_length":"3035","record_id":"<urn:uuid:82496d34-d7d0-4fbe-b6a7-1424c066f2c1>","cc-path":"CC-MAIN-2014-15/segments/1397609538110.1/warc/CC-MAIN-20140416005218-00613-ip-10-147-4-33.ec2.internal.warc.gz"} |
Comonads in everyday life.
This post is a literate haskell file. As is usual with such things, you can go ahead and paste it into a .lhs file and load it right up in ghci. As such, first some boring preliminaries.
> module CoMenu where
> import Control.Applicative; import Data.List; import Data.Tree; import Data.Maybe
> import Network.Frameworks.HVAC; import Network.Frameworks.HVAC.AltController
Anyway, say you’re serving up a website. And say this website has hierarchical menus. Maybe you want to display them with javascript, maybe statically, whatever. But of course, you also need some
component of these menus to be generated server-side regardless, so that the structure matches up with the location of the current page in the hierarchy of content. One common way to do this, which
isn’t all that bad actually, is to hardcode the whole menu, or to generate it from some list of sections/subsections, etc. that associates them with individual actions, be they .php or .jsp pages, or
be they in servlet mappings or, you know, whatever. And in this menu, be it hardcoded or partly generated, there’s some additional code that conditionally displays subhierarchies (i.e. if you’re in
the “About” section it will display the subsection selection as well, or etc.) or at a minimum disables the link to the current page, to give some indication of where the user currently is.
There are ways to fancy this up and reduce the overhead, but that’s the general notion — you have a dispatch system, and you have a menu system, and the information in the dispatch and menu systems
don’t necessarily correspond. Or if you’re using “configuration by convention” maybe you’ve built something that relies on a correspondence, so that, e.g., a page can “autodiscover” its location in
the hierarchy by introspection on its url.
So anyway, you’re describing the structure of your website twice, in perhaps very different formats, and the extent that you can improve on things is really improving the brevity of the formats and
throwing some logic to the wind on the assumption that the descriptions really are rendundant.
The natural big little idea that I’m going to discuss then, is how we might use the same structure to describe both the dispatch and the menu, and so cut out all that nonsense. Now doing this is at
least slightly trickier than it seems. That’s because in the typical imperative approach we’re thinking in passes from the top down — so even assuming we have a dispatch tree that we walk based on a
url parse — and with, e.g., the hvac framework I’m working on, that’s a only a few lines — well, even assuming we have the dispatch down pat, consider the standard imperative approach to generating
the menu itself.
For the purposes of this discussion the menu will be a horizontal one, with the top row containing the top level, the next row containing the paths of the selected sublevel, etc. The approach itself
will generalize to any sort of menu, however. In any case, we start by rendering the top row. But wait! We have to distinguish the selected section. So, uh, on rendering the link to each section we
check if it matches the first portion of the path. And then we react accordingly. And then, using the section we’ve found that matches that first portion we recurse into the next row and soforth.
But wait, that’s not how menus work! Generally, though not always, if you pick the a top section then it won’t show a special “top section” page but will show the page that you get if you “drill
down” through first options until you eventually find a leaf. So we’re not talking about a rose tree here, but about a tree with content only at the leaf nodes, which is an irregular structure. On
the other hand, maybe there’s content only at the leaf nodes 90% of the time, and 10% of the time there’s content elsewhere. So we’ll actually use a rose tree structure (from Data.Tree) with each
node containing the following:
> data MItem a = MItem {
> miName :: String,
> miPath :: String,
> miAction :: Maybe a } deriving Show
Of course with this there’s no guarantee that a leaf node contains a page action either, but it’ll do for the purposes of this discussion.
In any case, you see the issue — the dispatcher “drills down” to an action page, but the rendering of that page depends on the whole context of the menu, not just the initial path we were given.
There are various ways to hack around this, but I won’t enumerate them all here. And then of course there’s the insight that even though the menu is “dynamic” on the page that it is rendered in, its
static with regards to any given page and as such, a proper pre-traversal of the entire tree can (lazily, mind you) render the menus for every page at once.
So what we want is to traverse the tree in such a way that we preserve the context of the location we’re in with regards to everywhere else in the tree, so that we can render the menu properly,
relative urls and all. And we want to do this entire traversal only once, rather than piecewise and repeatedly with each request. So what recursion scheme works? Well, fmap doesn’t, nor are any
varients of traverse or mapAccum strong enough — they only tell us where we’ve been, not where we’re going. If only we could associate the context with each node beforehand — then a simple map would
do the trick!
Well, the contents of a node plus all the associated context of a node have a common representation — the zipper, which is precisely the “one-hole context” of a datastructure. By definition, the
zipper, or derivative, of any data structure, will contain enough information to reconstruct both the structure as well as an indext to a node within it. So here’s a simple and partial implementation
of a zipper over a Tree. The included functions actually take it to and from a Forest, which is really just an alias for [Tree a].
> data Zipper a = Zipper {
> prevLevels :: [Zipper a],
> leftForest :: Forest a,
> rightForest :: Forest a } deriving Show
> instance Functor Zipper where
> fmap f (Zipper p l r) = Zipper (fmap f <$> p) (fmap f <$> l) (fmap f <$> r)
> forest2zip :: Forest a -> Zipper a
> forest2zip ts = Zipper [] [] ts
> zip2forest :: Zipper a -> Forest a
> zip2forest z = case prevLevels z of
> [] -> rightForest z
> pl -> rightForest . head . reverse $ pl
Now actually we’d typically include a host of functions for “walking around” within the zipper, including generalized depth-first traversal, etc. But it turns out that for our purposes we only ever
need to step down, so the rest is ommited for brevity.
> zDown :: Zipper a -> Maybe (Zipper a)
> zDown x = case rightForest x of
> [] -> Nothing
> (t:_) -> case subForest t of
> (t':ts) -> Just $ Zipper {
> prevLevels = x : prevLevels x,
> leftForest = [],
> rightForest = t':ts }
> _ -> Nothing
This gives us a representation of each node and its context. But still, how to associate them with each node, so as to preserve the overall structure? We could concievably turn our entire menu into a
zipper, and then step through it, at each location replacing the action by one which includes the proper menu. But this would mean introducing another irregularity — holding
Either (Zipper a, a) a
or some equivalent (c.f. Conor McBride’s Clowns and Jokers). Yeesh! We want to do it all at once. I.e. we want a signature of
Zipper a -> Zipper (Zipper a)
. Wait! Hang on! That’s a specialization of “duplicate” which is to a comonad as “join” is to a monad! And in fact, as any fule kno, The dual of (monadic) substitution is (comonadic) redecoration.
Oh, and it gets better! Behind every zipper is a comonad. And better yet! We also know that a rose tree is a cofree comonad over the list functor, which should somewhat help to tie this all together
in terms of why and how a zipper is a comonad.
In any case, we provide the basic comonad class, and an instance for our zipper:
> class Functor w => Comonad w where
> extract :: w a -> a
> duplicate :: w a -> w (w a)
> extend :: (w a -> b) -> w a -> w b
> extend f = fmap f . duplicate
> duplicate = extend id
> instance Comonad Zipper where
> extract = rootLabel . head . rightForest
> duplicate z@(Zipper p l r) = Zipper p' l' r'
> where p' = fmap duplicate p
> (l',r') = splitAt (length l) fs
> lrs = l ++ r
> fs = map (go p) $ zip3 (inits lrs) (tails lrs) lrs
> go :: [Zipper a] -> ([Tree a],[Tree a],Tree a) -> Tree (Zipper a)
> go ls (i,t,tr) =
> let z = Zipper {prevLevels = ls,
> leftForest = i,
> rightForest = t}
> sf' = subForest tr
> in tr {rootLabel = z,
> subForest = map (go (z:ls)) $
> zip3 (inits sf') (tails sf') sf'}
Ok, so that duplicate code isn’t the easiest to write. Nonetheless, the beautiful thing is you only have to write it once. And it’s not that painful once you realize that we’ve really just got a
generalization of the list comonad. But with all that traversal out of the way, a function to render menus almost writes itself. This one is parameterized over the function that draws the actual
menu, and includes a little tweak to deal with irregular trees — if there’s no action at the given node, it traverses down until it finds a node that does provide an action.
> renderMenu ::
> (Zipper (MItem (String -> a)) -> String)
> -> Forest (MItem (String -> a)) -> Forest (MItem a)
> renderMenu renderRows = zip2forest . extend go . forest2zip
> where
> drillToAct :: Zipper (MItem a) -> Zipper (MItem a)
> drillToAct z = case rightForest z of
> [] -> z
> (t:ts) -> case miAction . rootLabel $ t of
> Nothing -> case zDown z of
> Nothing -> z
> Just z' -> drillToAct z'
> _ -> z
> go z = (extract z) {miAction = ($ renderRows z) <$>
> miAction (extract . drillToAct $ z)}
And here’s one example of a function that turns that zipper into a rendered menu. It makes lots of controversial design choices, has some quirks, and is really just hacked up for the purposes of this
discussion, but the main point is that the legwork we did before lets us abstract the rendering from the traversal, so we can swap in whatever we want with no hassle.
> menuRows :: Zipper (MItem a) -> String
> menuRows z = concatMap (renderRow curDepth "") (reverse $ prevLevels z) ++
> renderRow curDepth "" z ++
> (case zDown z of
> Just z' -> renderRow curDepth ((miPath . extract) z ++ "/") z'
> Nothing -> "")
> where curDepth = (length . prevLevels) z
> renderRow curDepth prevPath z =
> "<div>" ++
> intercalate " | " (map mkLink (leftForest z)) ++
> (if null prevPath
> then " | " ++ (miName . extract) z ++ " | " ++
> intercalate " | "
> (map mkLink (tail . rightForest $ z))
> else " | " ++ intercalate " | "
> (map mkLink (rightForest z))) ++
> "</div>"
> where mkLink x = "<a href='" ++ concat (replicate depth "../") ++
> prevPath ++
> (miPath . rootLabel) x ++ "/'>" ++
> (miName . rootLabel) x ++ "</a>"
> depth = curDepth - (length . prevLevels $ z)
And to round out the picture, the promised simple dispatch function, and below it, some sample data.
> menu2dispatch :: Forest (MItem (String -> HCGI q s CGIResult))
> -> HCGI q s CGIResult
> menu2dispatch mis' = endPath *>
> (fromJust . miAction . rootLabel . head) renderedMenu
> <|> go renderedMenu
> where renderedMenu = renderMenu menuRows mis'
> hasPath t pth = (miPath . rootLabel) t == pth
> go mis = do
> p <- takePath
> case find (`hasPath` p) mis of
> Nothing -> continue
> Just mi -> endPath *> (fromJust . miAction . rootLabel) mi
> <|> go (subForest mi)
> foo = [
> Node 2 [Node 10 [Node 11 []], Node 23 []],
> Node 4 [Node 5 [], Node 7 []],
> Node 8 [Node 9 [], Node 13 []],
> Node 25 []
> ]
> bar = map (fmap (\x -> MItem {
> miName = show x,
> miPath = show x,
> miAction = if even x
> then Nothing
> else Just $
> \y -> y ++ "\n" ++ "body: " ++ show x}))
> foo
> {-
> *CoMenu> fromJust . miAction . rootLabel . head $ renderMenu menuRows bar
> "<div> | 2 | <a href='4/'>4</a> | <a href='8/'>8</a> | <a href='25/'>25</a></div><div> | <a href='2/10/'>10</a> | <a href='2/23/'>23</a></div>\nbody: 11"
> -}
So yeah, comonads in practical day-to-day programming. Not as rare as you would suspect, and occasionally just what the doctor ordered.
July 13, 2008 @ 7:38 pm
The problem being solved is very interesting to me. However, I didn’t understand much else aside from the problem statement. I’m sure the solution is quite elegant, and someday I hope to understand
your post! Learning haskell has been great fun, but it can really make one feel quite illiterate. My head actually hurts from reading this post (no kidding)!
Could you recommend any remedial materials for a non-academic Haskell newbie that might prove useful in understanding the subject matter?
July 14, 2008 @ 6:43 pm
By the way, is your blog hooked into planet haskell? I don’t recall seeing some of these posts before.
January 26, 2009 @ 5:05 am
Thanks for the great post.
After a week of lurking I finally understand how to use this stuff.
The function “menuRows” turned out to be the most difficult to understand, BTW. :) | {"url":"http://fmapfixreturn.wordpress.com/2008/07/09/comonads-in-everyday-life/?like=1&source=post_flair&_wpnonce=671450efa2","timestamp":"2014-04-20T21:48:11Z","content_type":null,"content_length":"64863","record_id":"<urn:uuid:720c2cc8-776d-468d-bade-47ba0d3ff086>","cc-path":"CC-MAIN-2014-15/segments/1398223205375.6/warc/CC-MAIN-20140423032005-00510-ip-10-147-4-33.ec2.internal.warc.gz"} |
My 7-year-old daughter keeps beating me at Spot it!
I have an excuse. While playing, I start thinking about the mathematics behind the game rather than the cards in front of me.
The goal of Spot it! is to be the fastest player to spot and call out the matching symbol between two cards. There are 55 cards, each with 8 symbols. Between any two cards there is one, and only one,
matching symbol. How did the designers accomplish this? Sue VanHattum explores this question on her blog, Math Mama Writes.
In addition to thinking “How did they do that?” I started thinking about creating a smaller math version of Spot it! What if, rather than symbols, students matched equivalent expressions? A game
might consist of 21 cards, each with 5 expressions (e.g., $\sqrt {64}$, $2^{3}$, $\dfrac {4} {3}\div \dfrac {1} {6}$, $\left( -2\right)\left( -4\right)$, and $8$).
I began by creating 7 cards, each with 3 letters. While I was trying to create 13 cards, each with 4 letters, I finally asked “Why am I doing this?” Okay, so the game might be fun for some students,
but would it increase their conceptual understanding? Of course not. We’re talkin’ about practice.
I have decided to walk away from creating these types of activities. It won’t be easy. The card stock! The laminator! The paper cutter! I love these things more than a grown man should. I’m quitting.
Cold turkey.
But first, check out my latest Tarsia jigsaws…
factoring trinomials tarsia (normal)
factoring trinomials tarsia (larger)
factoring trinomials tarsia (solution)
rational exponents tarsia (normal)
rational exponents tarsia (larger)
rational exponents tarsia (solution)
12 thoughts on “My 7-year-old daughter keeps beating me at Spot it!”
1. When those laminator fumes come, and they will, be strong! You can do this!
2. I think a really useful activity would be to make this game…so maybe if you REALLY want to play the game, you can give your students some time to make it for themselves.
3. David, you’re right. There would be value in having students make the game. When I played the game, my first question was “How many possible cards can be created?” – kind of a cool application of
combinatorics. If there are 55 different symbols, then there are 55 choose 8, or 1 217 566 350, possible cards. My second question – “How do I design choose/create 55 of 1 217 566 350 possible
cards so that there is exactly one match?” – is more interesting. I was hoping the solution to this would also involve combinatorics. If it does, it goes beyond Math 12 (i.e., my understanding).
Students’ systematic solutions might lead to something interesting and completely different. (I haven’t returned to Sue’s post – I might revisit this problem at a later date.)
4. Thanks for your great feedback! You might be interested to know that Blue Orange Games is coming out with a set of limited edition games that will help kids learn the alphabet, numbers and
shapes, as well as basic, English, French and Spanish. They’ll be offered at a discounted price for teachers. To stay tuned, find us on Facebook at http://www.facebook.com/spotitgame.
5. Lorraine,
Thanks for letting me know about this. Any chance you or someone else at Blue Orange can answer why there are 55 not 57 cards? Or maybe the big question – How did they do that?
6. Pingback: 61* | Reflections in the Why
7. I am an elementary school teacher and my students love the Spot It. I would like to recreate it for my vocabulary words using data merge in Indesign. However, I need an excel or csv file to work
off of.
Would someone please post an excel file of the answer? That would be very helpful. Thank you, Eliezer
8. Pingback: My Diigo Bookmarks (weekly)
9. This is amazing! I am a high school math teacher and this will be fantastic for engaging my kids on a wednesday morning! Thank you so much for creating this lesson/activity. | {"url":"http://reflectionsinthewhy.wordpress.com/2012/05/01/my-7-year-old-daughter-keeps-beating-me-at-spot-it/","timestamp":"2014-04-19T09:24:06Z","content_type":null,"content_length":"66432","record_id":"<urn:uuid:f96d3cec-27a5-4c56-8990-754c57066c81>","cc-path":"CC-MAIN-2014-15/segments/1397609537097.26/warc/CC-MAIN-20140416005217-00470-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.
This is the testimonial you wrote.
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The contest problem book
The contest problem book: American high school mathematics examinations and American invitational mathematics examinations 1983-1988, Volume 5
George Berzsenyi, Stephen B. Maurer
Mathematical Association of America
, 1997 -
286 pages
Over the years perhaps the most popular of the MAA problem books have been the high school contest books, covering the yearly American High School Mathematics Examinations (AHSME) that began in 1950,
co-sponsored from the start by the MAA. Book V also includes the first six years of the American Invitational Mathematics Examination (AIME) which was developed as an intermediate step between the
AHSME & the USA Mathematical Olympiad (USAMO). The AIME has a unique answer format - all answers are integers between 0 & 999. New material, not included in premenu contents books is: * a
comprehensive guide to other problem materials world wide, * additional solutions, * dropped problems, * statistical information, * information on test development & history. This volume is a must
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Bibliographic information | {"url":"http://books.google.co.uk/books?id=wXfwAAAAMAAJ&q=%22Mathematics+and+Informatics+Quarterly%22&dq=%22Mathematics+and+Informatics+Quarterly%22&hl=en&sa=X&ei=AO8ST53MDo-1hAeB7u2JAg&redir_esc=y","timestamp":"2014-04-19T01:56:46Z","content_type":null,"content_length":"122690","record_id":"<urn:uuid:593c0c27-e318-4bae-a6ed-ae1824bda83b>","cc-path":"CC-MAIN-2014-15/segments/1397609535745.0/warc/CC-MAIN-20140416005215-00398-ip-10-147-4-33.ec2.internal.warc.gz"} |
Big-O notation
Steven Taschuk staschuk at telusplanet.net
Wed Apr 16 23:53:16 CEST 2003
Quoth Duncan Booth:
> I believe you are making some unreasonable assumptions here. Remember that
> if I have an algorithm that is O(N^2) that is really just a shorthand for
> saying that it will have a running time a*N^2 + b*N * c where a, b, and c
> are constants but for sufficiently large N only the N^2 term matters.
Not to dispute your general point, but a nit: O(n^2) doesn't imply
a polynomial expansion such as you have above; the actual runtime
could be, for example, 3n^2 + log n + 18/n. That is, the
additional terms might be anything which (asymptotically) grows
slower than n^2.
Steven Taschuk Aral: "Confusion to the enemy, boy."
staschuk at telusplanet.net Mark: "Turn-about is fair play, sir."
-- _Mirror Dance_, Lois McMaster Bujold
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Re: st: How to set calibrated weights
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Re: st: How to set calibrated weights
From Steve Samuels <sjsamuels@gmail.com>
To statalist@hsphsun2.harvard.edu
Subject Re: st: How to set calibrated weights
Date Sat, 20 Oct 2012 17:41:40 -0400
> On Oct 20, 2012, at 5:08 AM, Veronica Galassi wrote:
> Dear Steve,
> Thank you very much for your kind reply and the useful references!
> Your answer actually clarified many other doubts I had.
> Your intuition that my post-stratified weights are calibrated is
> correct. Unfortunately, I checked again the documents explaining the
> sampling methodology and there the PSU is simply defined as a
> geographic area containing more than 74 dwellings. Therefore I expect
> the number of PSU to be high (around 3,000) whereas I only have 9
> provinces and 4 geographical types in my survey. This implies that
> none of my cluster variables can be the PSU.
You still haven't persuaded me. I'd have to see the quote from the study
documents. Or, better, post a link to them if they are online. You'd
better figure out what role, if any, the cluster variables have in the
design. Why did you name them "cluster 1" and "cluster 2"?
> However, if I got your point, it does not really matter which PSU I
> indicate when conducting descriptive statistics. Is it correct?
No, it is not. It is scientifically irresponsible to publish estimates
of descriptive statistics without indications of uncertainty (SEs, CIs).
> For
> this reason, I also tried not to indicate any PSU but Stata gave me
> back the error: "invalid use of _n; observations can only be sampled
> in the final stage".
See FAQ Section 3.3 First stence
> To cut it short, do you still believe I can use the statement "svyset
> w2_gc_prov [pw = w2_wgt], strata(w2_gc_dc) || w2_hhgeo" you previously
> indicated to set my calibrated weigths? ( In my case I cannot use the
> fpc option).
I don't know, because you have not yet correctly described the sampling
design. As an aside, ave you even tried the statement, which assumed
that w2_gc_prov is the OSY? When you do, follow it by -svydes-.
2012/10/20 Steve Samuels <sjsamuels@gmail.com>:
> Veronica,
> The PSU variable is not missing. It is the sampling unit at the first
> stage of sampling and it's one of your cluster variables, probably
> "cluster 1" (check). Your statement that one must know the PSU variable
> to use probability weights is also incorrect. One can get proper
> weighted estimates, though not standard errors, without knowing the PSU.
> I'm not sure what wrong with your -concat- statement. I would have
> used "egen combination = group()". For it to have worked, the value of
> the "post-stratification weight" would have to be the population count
> for each combination of the three variables.
> If the "post-stratification" weights are not integers, they are probably
> "calibration" weights that have already adjusted the probability
> weights. In that case, further post-stratification are likely to be
> superfluous. You would then use the "post-stratification weight" in place of
> the probability weights. All weights should be
> described in the study documents (though usually not the"codebook"). If
> they are not, then contact the organization that did the study for
> details.
> If sampling was without replacement at one or more stages,
> you could use the fpc() option for those stages. In practice,
> it makes a difference only for the first stage.
> In any case, one guess at a -svyset- statement (assuming the
> "post-stratification weight" is a "calibration" weight) is:
> *************************************************************
> svyset w2_gc_prov [pw = w2_wgt], strata(w2_gc_dc) || w2_hhgeo
> **************************************************************
> But I could be wrong, depending on how w2_wgt was calculated.
> Before proceeding, I suggest that you learn more about sampling or take
> a survey course. I gave some references in:
> http://www.stata.com/statalist/archive/2012-09/msg01058.html.
> The Stata survey manual is also a very good resource, though the section on
> post-stratification is skimpy.
> Steve
> On Oct 19, 2012, at 1:57 PM, Veronica Galassi wrote:
> Dear Statalisters,
> I am writing you concerning the application of calibrated weights to
> my dataset for the computation of descriptive statistics only.
> The dataset I am working on collects information at household and
> individual level and comes from a stratified, two-stage clustered
> sample. The followings are the variables I have got:
> - probability weights: w2_dwgt
> - strata: w2_gc_dc
> - cluster 1: w2_gc_prov
> - cluster 2: w2_hhgeo
> - post-stratified weights: w2_wgt
> - age intervals: w2_age_intervals
> - gender: w2_best_gen
> - population group: w2_best_race
> In order to set the probability weights using the command svyset, I
> need the psu variable. As you may have noticed, this variable is
> missing and this makes me impossible to set pweights.
> In addition, from a couple of previous statalist conversations ( see
> in particular: http://www.ats.ucla.edu/stat/stata/faq/svy_stata_post.htm
> and http://www.stata.com/statalist/archive/2012-02/msg00584.html), I
> understood that:
> - when using calibrated weights I still have to set pweights and
> specify the original strata and clusters
> - In order to apply calibrated data I need to know the characteristics
> on the base of which the sample have been post-stratified ( in my case
> age intervals, gender and population groups).
> Therefore, I tried to set my post-stratified weights using the
> following command:
> "svyset [pw=w2_dwgt], strata (w2_gc_dc) poststrata (w2_age_intervals
> w2_best_gen w2_best_race) postweight(w2_wgt)"
> which did not work because in Stata the poststrata must be mutually
> exclusive and thus only one variable can be specified.
> In order to overcome this problem, I tried to generate a variable
> which is a combination of the three characteristics by using the
> command
> "egen combination=concat( w2_age_intervals w2_best_race w2_best_gen),
> format (float)".
> However, this command generated a variable containing only missing
> values and for this reason Stata gave me back the error:
> "option postweight() requires option poststrata()".
> The only way to make Stata set the post-calibrated weight was by using
> the command
> "svyset, poststrata (combination) postweight(w2_wgt)" with combination
> being a string variable. However I am scared that this command is not
> complete.
> At this point, I would really appreciate any hint on what I am doing
> wrong and how to proceed to set my post-stratified weights.
> Many thanks for your help!
> Kind regards,
> Veronica Galassi
> *
> * For searches and help try:
> * http://www.stata.com/help.cgi?search
> * http://www.stata.com/support/faqs/resources/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/faqs/resources/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/faqs/resources/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/faqs/resources/statalist-faq/
* http://www.ats.ucla.edu/stat/stata/ | {"url":"http://www.stata.com/statalist/archive/2012-10/msg00942.html","timestamp":"2014-04-19T04:48:19Z","content_type":null,"content_length":"16696","record_id":"<urn:uuid:069d8756-9d2f-4af7-b9ac-59be667c127a>","cc-path":"CC-MAIN-2014-15/segments/1398223207046.13/warc/CC-MAIN-20140423032007-00009-ip-10-147-4-33.ec2.internal.warc.gz"} |
Recognizing and Understanding Congruent Polygons
6.10: Recognizing and Understanding Congruent Polygons
Created by: CK-12
Practice Properties of Congruence
Have you ever built a geodesic dome? Take a look at this dilemma.
After doing all of his research and drawing a design, Dylan began working on the construction of his geodesic dome. He decided to use a combination of rolled newspaper tubes and duct tape. He rolled
tubes of newspaper, created triangles with duct tape and then worked on connecting them together.
“It doesn’t look right,” Sarah, Dylan’s sister commented as he was putting the structure together in the living room.
“What do you mean?” Dylan asked tearing off another piece of duct tape.
“It is crooked and I think it will collapse.”
“You don’t know anything,” Dylan snapped turning his back on his sister.
However, when Dylan actually went to connect the triangles together, the structure began to collapse. His sister came back into the room.
“Can I help?” she asked.
“Are the triangles congruent?” Sarah asked.
Congruent? Dylan had to think about that one. What would it mean if the triangles weren’t congruent? What does it mean “congruent?” How can one tell if a figure is congruent or not?
In this Concept, you will learn all about the importance of congruence and how to determine congruence.
What do we mean when we say that two figures are congruent? To complete all of the work in this Concept, you must first understand what the word “congruent” means.
Congruent means exactly the same.
Yes, but it also includes sides of figures and angles too.
When we have two figures of any kind that have the same size, shape and measure, we can say that these two figures are congruent.
Take a look at this situation.
Are these two angles congruent?
If you look at these two angles, you will see that they are both $55^\circ$$A$$B$
$\angle A \cong \angle B$
This is the way we would write a statement about congruence using mathematical notation. Notice the symbol that we used for congruent.
Now write the definition for congruent and its symbol in your notebook.
Just as we said that any two figures can be congruent, we can use this when we look at different types of polygons too. Two polygons can be considered congruent or not congruent. Let’s look and see
if we can determine congruence.
Are these two octagons congruent?
Look at these two octagons. They are exactly the same in every way. You can see that if we put one octagon on top of the other octagon that they would match up perfectly. The side lengths are also
congruent and the angle measures are congruent. If two polygons are congruent, then it is a given that the side lengths and the angle measures are also congruent.
Now that you know how to identify whether or not two figures are congruent, we can look at figuring out congruent parts and angles. First, let’s think again about the four characteristics of
congruent polygons.
Congruent Polygons have:
1. Same size
2. Same shape
3. Common angle measures
4. Common side lengths
Be sure that you have these notes written down in your notebook.
The last two characteristics can be a bit tricky. Sometimes, you will have two congruent figures, but all of the angles measures won’t be exactly the same. For example, if you had two irregular
congruent hexagons, that means that there are different angle measures in the two hexagons - however, they are congruent so there are “matching” angles between the two figures.
Take a look at these figures.
Here we have two hexagons. They are irregular - which means that all of the side lengths and angles are not the same. However, they are congruent. You can see that one matches the other. Because of
this, we have corresponding angles that connect with each angle from the first hexagon to the second hexagon.
We can identify corresponding parts of congruent figures. Corresponding parts can include side lengths and angle measures. When two figures are congruent, then there are corresponding parts.
Name each pair of corresponding side lengths for these congruent figures.
Now let’s look at these two congruent pentagons. To name the corresponding sides, we name the sides that match from one pentagon to another pentagon. Here are the corresponding sides and how we can
write them using mathematical notation.
$BA & \cong GF\\BC & \cong GH\\CD & \cong HK\\AE & \cong FL\\ED & \cong LK$
We can also look at the corresponding angles for two congruent figures. When two figures are congruent, then the matching angles will also be congruent.
Use the following figures to answer each question.
Example A
What is the angle measure of angle F?
Solution: $100^\circ$
Example B
Angle D is congruent to which other two angles?
Solution: F and H
Example C
What is the measure of angle G?
Solution: $80^\circ$
Now let's go back to the dilemma from the beginning of the Concept.
First, let’s think about what the word congruent means. Congruent means exactly the same. For an object to be congruent, the side lengths have to be the same. The triangles in the geodesic dome have
to be congruent for it to stand up because the triangle is a structure that is well balanced to help with structure and security. Triangles are used in all kinds of construction like roofs and
Dylan can test the congruence of his triangles because he can see that the side lengths are the same.
exactly the same, having the same size, shape and measurement.
Corresponding parts
When two figures are congruent, there are matching parts for each of the two figures.
Guided Practice
Here is one for you to try on your own.
Are these two hexagons congruent?
These two figures are both hexagons, but they are different hexagons. One is a regular hexagon where all of the sides are congruent, and one is irregular. The irregular hexagon has six sides, but
they are different lengths, etc. These two hexagons are not congruent.
Video Review
Directions: Answer each question true or false.
1. Congruent means that a figure has the same side lengths but not the same angle measures.
2. Congruent means exactly the same in every measure.
3. Similar means having the same shape, but not the same size.
4. Two congruent figures would have the same size and shape.
5. Corresponding parts are parts that are in the same figure.
6. You need to understand corresponding parts before you can determine if two figures are congruent.
7. You can determine if two figures are congruent without knowing any of their measurements.
8. Similar figures are also congruent.
9. If two triangles are equilateral triangles, then they are automatically congruent.
10. If two quadrilaterals have measures of $360^\circ$
Directions:The two figures shown are congruent. Use the illustration to answer each question.
11. If angle B has a measure of $75^\circ$
12. If angle F is $120^\circ$
13. True or false. Angle E and angle K have the same measure.
14. True or false. Angle C and angle H have the same measure.
15. Name this figure.
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Mixed systems of conservation laws in industrial mathematical modelling motion
Fitt, A.D. (1996) Mixed systems of conservation laws in industrial mathematical modelling motion. Surveys on Mathematics for Industry, 6, 21-25.
Full text not available from this repository.
Many mathematical models of evolutionary industrial processes may be written as N x N systems of conservation laws in terms of N independent variables comprising time and N-1 space variables. If such
systems posses real and distinct eigenvalues, they are said to be strictly hyperbolic. For "mixed systems", however, the eigenvalues may be equal at points in phase space or even fail to be real, so
that the problem has both hyperbolic and elliptic characteristics. In this case the system is ill-posed and requires the specification of boundary conditions that can violate causality. Mathematical
models of physical processes that lead to mixed equations are discussed and reviewed, and some of the properties of mixed systems are compared to those of hyperbolic systems. The significance of
prototype systems that have been proposed specifically to analyse such properties is considered, and attention is then turned to the archetypal mixed system; the two-phase flow equations. Possible
resolutions of the two-phase flow dilema are compared, and a manner in which the modelling may be approached via a more general rational asymptotic scheme is indicated.
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Grothendieck-Teichmuller conjecture
up vote 41 down vote favorite
(1) In "Esquisse d'un programme", Grothendieck conjectures
Grothendieck-Teichmuller conjecture: the morphism $$ G_{\mathbb{Q}} \longrightarrow Aut(\widehat{T}) $$ is an isomorphism.
Here $G_{\mathbb{Q}}$ is the absolute Galois group and $\widehat{T}$ is the category whose
• objects are the profinite fundamental groupoids $T_{g,n}$ of the moduli stacks $\mathfrak{M}_{g,n}$ (restricted to certain basepoints, tangential basepoints or special automorphisms curves)
• morphisms are certain natural morphisms induced by natural operations like forgetting marked points and (auto-)gluing curves along such points.
(2) A parallel conjecture is that the morphism $$ G_{\mathbb{Z}}^{\mathrm{mot}} \longrightarrow Aut(T_0^{\mathbb{mot}}) $$ is an isomorphism. Here $G_{\mathbb{Z}}^{\mathrm{mot}}$ is the motivic
Galois group of mixed Tate motives over $\mathbb{Z}$ and $T_0^{\mathbb{mot}}$ is the category of the mixed Tate fundamental groupoids of the $\mathfrak{M}_{0,n}$'s. The ($Gr^W$ of the) right hand
side is known to be the Grothendieck-Teichmuller group $GRT$ so the conjecture translates as
Deligne-Drinfeld conjecture: the graded Lie algebra $\mathfrak{grt}_1$ is free with one generator for each odd degree $2k+1 > 1$.
Note that in both cases, the morphism $G \to Aut(T_{0,4})$ is known to be injective (by Belyi's theorem in the profinite case and by a recent theorem of F. Brown in the motivic case).
Another point in Grothendieck's Esquisse is his "Lego-Teichmuller conjecture" that the tower of Teichmuller groupoids $T$ is generated in dimension $dim(\mathfrak{M}_{g,n}) = 3g-3+n = 1$ with
relations in dimension 2 (so we only get a finite number of these). This is also known. This is a purely topological fact that goes back to Moore-Seiberg and Drinfeld in the late 80's I think. In
genus 0, it is obvious once stated in terms of a presentation of the little disc operad.
So the conjectures boils down to the fact that these natural geometric relations (forgetting marked points and gluing curved along them) between (motivic) fundamental groupoids of moduli spaces of
curves characterize elements of the (motivic) Galois group. These are really amazing statements because they give a purely topological characterization of the Galois group a purely arithmetical
Question: Is there any evidence or intuition for these conjectures?
I think Grothendieck's intuition was that the study of varieties reduces to curves. This was his first approach to the Weil conjectures and his first test for the theory of the etale fundamental
group. But this seems very vague to me. I'd be very happy to have more convincing evidence.
ag.algebraic-geometry nt.number-theory galois-theory at.algebraic-topology
Shouldn't one consider $\mathbb{Q}$-prounipotent fundamental groupoids ? – DamienC May 6 '11 at 19:35
1 That's the same thing. The Betti realization of the mixed Tate groupoid of $M_{0,n}$ is the $\mathbb{Q}$-prounipotent completion of its topological fundamental groupoid (mainly because it is mixed
Tate and a rational $K(\pi,1)$). – YBL May 6 '11 at 19:46
"a purely topological characterization of the Galois group, a purely arithmetical object." - my guess is that Grothendieck's intuition was that to a large extent arithmetic "is" topology, which
made him looking for e.g. a new foundation of homotopy theory, anabelian conjectures. – Thomas Riepe May 8 '11 at 6:44
In some of his more discursive writing he talked about the first conjecture informally and if I remember well, did say why he thought it true. You may already have looked at his Esquisse and may
not have found what I mention but if not do look at him in `chatty' mode as he was often interested in the process of clarifying why a result 'should' be true. – Tim Porter Nov 24 '11 at 16:27
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Chapter 1.3
As the basis for the classification of the symmetry groups G three elements were taken into consideration: the types of symmetries (isometries, similarity symmetries, conformal symmetries) that occur
in G, the space on which the group G acts, and the sequence of maximal included proper subspaces, invariant with respect to the group G. According to this, the Bohm symbols (J. Bohm, K.
Dornberger-Schiff, 1966) are used for the categorization of the groups of isometries. Symbols of the same type are applied to the similarity symmetry and conformal symmetry groups. For example, the
symmetry group of square D4 acts in plane and possesses only one invariant point, so it belongs to the category G20 - the symmetry groups of rosettes.
A transformation S is an isometry of certain space En (Sn) if for every two points P, Q of that space |P,Q| = |S(P),S(Q)| holds, where |P,Q| denotes the length of the line segment defined by points
P, Q. All isometries of some space form a group. A transformation S of n-dimensional space is called indirect (or reflective, sense reversing, opposite, odd) if it transforms any oriented (n+1)-point
system onto an oppositely oriented (n+1)-point system (line segment AB onto BA, triangle ABC onto ACB, tetrahedron ABCD onto ACBD in cases of n = 1,2,3 respectively). Otherwise, it is called direct
(sense preserving, even) (Figure 1.6). Figure 1.6 (a) Direct and (b) indirect plane isometry. As an elementary isometric transformation we can take the reflection, non-identical isometry of space En
(Sn) for which, every point of its subspace En-1 (Sn-1) is an invariant point. In particular, for n = 1 we have point reflection, for n = 2 line reflection (or simply - reflection), for n = 3 plane
reflection, involutional indirect isometries. According to the fundamental theorem on minimal or canonic representation of an isometric transformation of space En (Sn), which states that every
isometry of this space can be presented as a composition of maximum n+1 (plane) reflections, it is possible to classify the isometries of different spaces. The classification of isometric
transformations and corresponding symmetry groups is common for spaces En, Sn, Ln for n < 2, while for n ³ 2 different possibilities of relations of disjoint lines, which are defined by the axiom of
parallelism, condition specific differences. This work exclusively discusses Euclidean spaces. In the space E2 (plane) we distinguish the following isometric transformations (Figure 1.7): Figure 1.7
(a) Identity transformation; (b) reflection; (c) rotation; (d) translation; (e) glide reflection. 1) identity transformation E, with the minimal reflectional representation of the length 2 (R2 = E);
2) reflection R; 3) rotation S = R1R2, the product of two reflections in the reflection lines crossing in the invariant point (center of rotation). The oriented angle of rotation is equal to twice
the angle between the reflection lines R1, R2; 4) translation X = R1R2, the product of two reflections with parallel reflection lines, such that the translation vector is perpendicular to them and
equal to twice the oriented distance between the reflection lines R1, R2; 5) glide reflection P = R3X = XR3 = R1R2R3, the commutative product of a translation X and a reflection R3 with the
reflection line parallel to the translation axis. With respect to the invariant figures, all the points of the plane E2 are invariant points of the identity transformation E, reflection R maintains
the invariance of all the points of the reflection line, rotation S possesses a single invariant point - the center of rotation, while translation and glide reflection have no invariant points. A
glide reflection possesses a single invariant line - the axis, and a translation keeps invariant all the lines parallel to the translation axis. In the case of rotation, if the relation S = R1R2 =
R2R1 holds, i.e. if the reflection lines R1, R2 are perpendicular, as a result we get the special involutional rotation - central reflection Z (two-fold rotation, half-turn, point-reflection) (Figure
1.8). Figure 1.8 Central reflection Z. When orientation is considered we distinguish direct transformations (or sense preserving transformations): identity transformation E, translation, rotation,
and indirect transformations (sense reversing transformations): reflection and glide reflection. Since direct transformations are the product of an even, and the indirect ones of an odd number of
reflections, we can call them respectively even and odd transformations. If a symmetry transformation S can be represented as a composition S = S1¼Sn such that SiSj = SjSi, i,j = 1,¼,n, we can call
it a complex or composite transformation while the transformations S1,¼,Sn we call the dependent transformations or dependent elements of symmetry. We will use such approach whenever we are
interested to learn to what degree the dependent elements of symmetry influence the characteristics of the composite transformation, and whether they have lost or preserved their geometric and visual
characteristics during it. For example, a glide reflection is such commutative composition of translation and reflection, with reflection line parallel to the translation axis. An analogous procedure
makes possible the classification of isometries of the space E3, where each isometry can be represented as the composition of four plane reflections at the most. Besides the transformations of the
space E2 afore mentioned with the line reflections substituted by plane reflections, as the new transformations of the space E3 we have two more transformations. They are a direct isometry - twist
(screw), the commutative composition of a rotation and a translation, the canonic representation of which consists of four plane reflections and indirect isometry - rotatory reflection, the
commutative composition of a rotation and a plane reflection in the plane perpendicular to the rotation axis, the canonic representation of which consists of three plane reflections. In particular,
the involutional rotatory reflection, which is the composition of three plane reflections of which every two commute, is called point inversion Z (or rotatory inversion). For every element S1 of a
transformation group G we can define the conjugate of the element S1 by an element S as the product S-1S1S, which we denote by S1S. If S1S Î G, then the mapping S1 onto S1S represents an automorphism
of the group G. If the element S by means of which this automorphism is being realized belongs to the group G, such an automorphism is called an internal automorphism. Any other automorphism of a
group G is called an external automorphism. An important characteristic of a conjugate is that the order of the conjugate S1S is equal to the order of the element S1. If a figure f is invariant under
a transformation S1, then S(f) is the figure invariant under transformation S1S. The conjugate of a reflection R with invariant reflection line p, derived by isometry S, is the reflection RS with the
invariant reflection line S(p). Hence we can conclude that the isometry S1 and all its conjugates S1S derived by different isometries S constitute one class of equivalence, i.e. the class of
isometries having the same name, which means that (internal) automorphism of a group of isometries G transforms reflections onto reflections, rotations onto rotations, etc. The properties of the
(internal) automorphisms are frequently used when proving theorems on isometric transformations and the other symmetry transformations. For example, by gR, g Î G, is defined an internal automorphism
of the symmetry group of square G, given by presentation {S,R} S4 = R2 = (RS)2 = E: ER = E, RR = R, (RS)R = SR, (SR)R = RS, SR = S3, (S2)R = S2, (S3)R = S. In the same way, it is defined an external
automorphism of the rotational group of square H, given by presentation {S} S4 = E, SR = S3, (S2)R = S2, (S3)R = S, where the reflection line of reflection R contains the center of four-fold rotation
S. Hence, external automorphisms are very efficient tool for extending symmetry groups. Since the product of direct transformations is a direct transformation, and the inverse of a direct
transformation is a direct transformation, each group of transformations G, which contains at least one indirect transformation has a subgroup of the index 2, denoted by G+, which consists of direct
transformations of the group G. For example, the rotational subgroup of sqare H satisfies this condition regarding the symmetry group of square, so H = G+, [G:H] = 2. All direct isometries of the
space En can be identified as movements of a material object in the space En, as opposed to indirect isometries which do not have such a physical interpretation (e.g., a plane reflection does not
represent motion in E3). For a figure f with the symmetry group Gf, which consists only of direct symmetries, it is possible to have the enantiomorphism - enantiomorphic modifications of a figure f,
i.e. to have the "left" and "right" form of the figure f (Figure 1.9). The existence of indirect symmetries of a figure f implies the absence of enantiomorphism. Since reflections have a role of
elementary isometric transformations, while all other isometries are their finite compositions, of special interest will be symmetry groups generated by reflections - groups, a set of generators of
which consists exclusively of reflections. Since every reflection keeps invariant each point of the reflection line, the fundamental region of these groups will possess a fixed shape, will not allow
variations and will have rectilinear edges. All symmetry groups will be subgroups of groups generated by reflections. In the case of conformal symmetry groups, along with reflections, circle
inversions have the analogous function. For example, the symmetry group of square is the group generated by reflections, with the fundamental region of the fixed shape (Figure 1.3a). Figure 1.9
"Left" and "right" rosette with the symmetry group C4, consisting of direct symmetries. The results of composition of plane isometries are different categories of groups of isometries of the space
E2, represented by Bohm symbols as: G210 - symmetry groups of finite friezes, G20 - symmetry groups of rosettes, G21 - symmetry groups of friezes and G2 - symmetry groups of ornaments. Because of the
relation G210 Ì G20, in this work we will discuss only the categories G20, G21, G2, while the category G210 will be discussed within the category G20. The definitions of symmetry groups will be
derived directly from Bohm symbols: symmetry groups of rosettes are groups of isometries of the space E2 (plane) with an invariant 0-dimensional subspace (point), symmetry groups of friezes are
groups of isometries of the space E2 with an invariant 1-dimensional subspace (line) and without invariant points, while symmetry groups of ornaments are groups of isometries of the space E2 without
invariant subspaces (points, lines). The groups of the category Gn are called the space groups, the groups of the category Gn1 the line groups, and the groups of the category Gn0 the point groups of
the space En. With symmetry groups of friezes G21 and symmetry groups of ornaments G2, a group contains one or two generating translations respectively, so that each of these groups has a
translational subgroup. A lattice is the orbit of a point with respect to a discrete group of translations. For the friezes it is a linear series of equidistant points while for ornaments we get a
plane lattice or simply a lattice. Five different symmetry types of plane lattices bear the name of Bravais lattices; the points of these lattices are defined by five different isohedral
tessellations, which consist of parallelograms, rhombuses, rectangles, squares or regular hexagons. To Bravais lattices correspond the crystal systems of the same names (Figure 1.10). Figure 1.10
Five plane Bravais lattices. Because the symmetry groups of friezes G21 are groups of isometries of the plane E2 with an invariant line, they cannot have rotations of an order greater than 2. For the
symmetry groups of ornaments G2 so-called crystallographic restriction holds, according to which symmetry groups of ornaments can have only rotations of the order n=1,2,3,4,6. The term
"crystallographic groups" is used for all groups which satisfy this condition, despite the category they belong to. In isometry groups all distances between points under the effect of symmetries
remain unchanged and the congruence of homologous figures is preserved. Consequently, the same holds for all other geometric properties of such figures, so that the equiangularity (the congruence of
the angles of homologous figures) and their equiformity (the same form of homologous figures) are the direct consequences of isometrism. The next class of symmetry groups we shall consider are the
similarity symmetry groups. A similarity transformation of the space En is a transformation which to each line segment of length AB assigns a line segment of the length kAB whereby k is a real
positive number, the coefficient of similarity. In particular, if k = 1 then a similarity transformation is an isometry. According to the theorem on the existence of an invariant point of every
similarity transformation which is not an isometry, there are, besides isometries, three types of similarity symmetry transformations of the space E2: (i) central dilatation K (or simply dilatation),
a transformation which to each vector (A,B) assigns the vector (A',B'), such that A' = K(A), B' = K(B) and (A',B') = k(A,B), where the coefficient of the dilatation is k Î Â\{-1,0,1} ; (ii) dilative
rotation L, the commutative composition of a central dilatation K and a rotation, with a common invariant point; (iii) dilative reflection M, the commutative composition of a dilatation K and a
reflection in the reflection line containing the invariant point (center) of the dilatation K (Figure 1.11). Figure 1.11 (a) Dilatation; (b) dilative rotation; (c) dilative reflection. Those
transformations are, in the given order, isomorphic with the isometries of the space E3: translation, twist and glide reflection. They make possible the extension of the symmetry groups of rosettes
G20 by the external automorphism, having as the result similarity symmetry groups S20 that we will, thanks to the existence of the invariant point, call the similarity symmetry groups of rosettes.
Dilatations K and dilative rotations L are direct, while dilative reflections M are indirect transformations. They all possess the properties of equiangularity and equiformity. All other aspects of
similarity symmetry groups (the problems of enantiomorphism, fundamental regions, tessellations,¼) will be discussed analogously to the case of isometry groups. Further generalization leads to
conformal transformations or circle preserving transformations of the plane E2\{O} ; for them the property of equiangularity has been preserved, but not that of equiformity. We have, as the
elementary transformation of conformal symmetry in E2\{O} , the circle inversion RI (or simply inversion) - an involutional transformation isomorphic with a reflection, that gives to each point A in
the plane E2\{O} a point A1 so that (O,A)°(O,A1) = r2, where r is the radius of the inversion circle c(O,r) and O is the singular point of the plane E2\{O} (Figure 1.12). Just like a reflection, for
which each point of the reflection line is invariant, an inversion maintains invariant each point of the inversion circle. By discussing a line as a circle with an infinite radius (and treating as
circles, at the same time and under the same term, circles and lines) it is possible to identify reflections with circle inversions. In such a case, all circle inversions (including line reflections)
and their compositions, can be discussed as circle preserving transformations, i.e. transformations mapping circles (including lines) onto circles. Figure 1.12 Circle inversion. Besides the circle
inversion RI, by composing it with isometries maintaining invariant the circle line c of the inversion circle c(O,r) - with a reflection with reflection line containing the circle center O or with a
rotation with the rotation center O, we have two more conformal transformations: (i) inversional reflection ZI = RIR = RRI, the involutional transformation, the commutative composition of a
reflection and a circle inversion; (ii) inversional rotation SI = SRI = RIS, the commutative composition of a rotation and a circle inversion (Figure 1.13). Figure 1.13 (a) Inversional reflection;
(b) inversional rotation. Those three conformal symmetry transformations, besides isometries and similarity symmetry transformations, constitute the finite and infinite conformal symmetry groups C21,
C2 - conformal symmetry groups of rosettes in E2\{O} . As an extension of the symmetry groups of rosettes G20 we have the finite conformal symmetry groups C21 isomorphic with the symmetry groups of
tablets G320. As a further extension of finite conformal symmetry groups C21 by the similarity symmetry transformations K, L, M, we get the infinite conformal symmetry groups C2. The similarity
symmetry groups S20 and the infinite conformal symmetry groups C2 are isomorphic with the line symmetry groups of the space E3 - the symmetry groups of rods G31. In line with the isomorphism
mentioned, all similarity symmetry and conformal symmetry transformations offer a reflectional (canonic) representation by, at most, four reflections (reflections and circle inversions). By applying
this isomorphism, ornamental motifs which correspond to the similarity symmetry and conformal symmetry groups, satisfy one more scope of painting: adequate interpretation of space objects in the
plane. The plane structures obtained are called generalized projections of the symmetry groups of tablets G320 and rods G31. NEXT CONTENTS
A transformation S of n-dimensional space is called indirect (or reflective, sense reversing, opposite, odd) if it transforms any oriented (n+1)-point system onto an oppositely oriented (n+1)-point
system (line segment AB onto BA, triangle ABC onto ACB, tetrahedron ABCD onto ACBD in cases of n = 1,2,3 respectively). Otherwise, it is called direct (sense preserving, even) (Figure 1.6). Figure
1.6 (a) Direct and (b) indirect plane isometry. As an elementary isometric transformation we can take the reflection, non-identical isometry of space En (Sn) for which, every point of its subspace
En-1 (Sn-1) is an invariant point. In particular, for n = 1 we have point reflection, for n = 2 line reflection (or simply - reflection), for n = 3 plane reflection, involutional indirect isometries.
According to the fundamental theorem on minimal or canonic representation of an isometric transformation of space En (Sn), which states that every isometry of this space can be presented as a
composition of maximum n+1 (plane) reflections, it is possible to classify the isometries of different spaces. The classification of isometric transformations and corresponding symmetry groups is
common for spaces En, Sn, Ln for n < 2, while for n ³ 2 different possibilities of relations of disjoint lines, which are defined by the axiom of parallelism, condition specific differences. This
work exclusively discusses Euclidean spaces. In the space E2 (plane) we distinguish the following isometric transformations (Figure 1.7): Figure 1.7 (a) Identity transformation; (b) reflection; (c)
rotation; (d) translation; (e) glide reflection. 1) identity transformation E, with the minimal reflectional representation of the length 2 (R2 = E); 2) reflection R; 3) rotation S = R1R2, the
product of two reflections in the reflection lines crossing in the invariant point (center of rotation). The oriented angle of rotation is equal to twice the angle between the reflection lines R1,
R2; 4) translation X = R1R2, the product of two reflections with parallel reflection lines, such that the translation vector is perpendicular to them and equal to twice the oriented distance between
the reflection lines R1, R2; 5) glide reflection P = R3X = XR3 = R1R2R3, the commutative product of a translation X and a reflection R3 with the reflection line parallel to the translation axis. With
respect to the invariant figures, all the points of the plane E2 are invariant points of the identity transformation E, reflection R maintains the invariance of all the points of the reflection line,
rotation S possesses a single invariant point - the center of rotation, while translation and glide reflection have no invariant points. A glide reflection possesses a single invariant line - the
axis, and a translation keeps invariant all the lines parallel to the translation axis. In the case of rotation, if the relation S = R1R2 = R2R1 holds, i.e. if the reflection lines R1, R2 are
perpendicular, as a result we get the special involutional rotation - central reflection Z (two-fold rotation, half-turn, point-reflection) (Figure 1.8). Figure 1.8 Central reflection Z. When
orientation is considered we distinguish direct transformations (or sense preserving transformations): identity transformation E, translation, rotation, and indirect transformations (sense reversing
transformations): reflection and glide reflection. Since direct transformations are the product of an even, and the indirect ones of an odd number of reflections, we can call them respectively even
and odd transformations. If a symmetry transformation S can be represented as a composition S = S1¼Sn such that SiSj = SjSi, i,j = 1,¼,n, we can call it a complex or composite transformation while
the transformations S1,¼,Sn we call the dependent transformations or dependent elements of symmetry. We will use such approach whenever we are interested to learn to what degree the dependent
elements of symmetry influence the characteristics of the composite transformation, and whether they have lost or preserved their geometric and visual characteristics during it. For example, a glide
reflection is such commutative composition of translation and reflection, with reflection line parallel to the translation axis. An analogous procedure makes possible the classification of isometries
of the space E3, where each isometry can be represented as the composition of four plane reflections at the most. Besides the transformations of the space E2 afore mentioned with the line reflections
substituted by plane reflections, as the new transformations of the space E3 we have two more transformations. They are a direct isometry - twist (screw), the commutative composition of a rotation
and a translation, the canonic representation of which consists of four plane reflections and indirect isometry - rotatory reflection, the commutative composition of a rotation and a plane reflection
in the plane perpendicular to the rotation axis, the canonic representation of which consists of three plane reflections. In particular, the involutional rotatory reflection, which is the composition
of three plane reflections of which every two commute, is called point inversion Z (or rotatory inversion). For every element S1 of a transformation group G we can define the conjugate of the element
S1 by an element S as the product S-1S1S, which we denote by S1S. If S1S Î G, then the mapping S1 onto S1S represents an automorphism of the group G. If the element S by means of which this
automorphism is being realized belongs to the group G, such an automorphism is called an internal automorphism. Any other automorphism of a group G is called an external automorphism. An important
characteristic of a conjugate is that the order of the conjugate S1S is equal to the order of the element S1. If a figure f is invariant under a transformation S1, then S(f) is the figure invariant
under transformation S1S. The conjugate of a reflection R with invariant reflection line p, derived by isometry S, is the reflection RS with the invariant reflection line S(p). Hence we can conclude
that the isometry S1 and all its conjugates S1S derived by different isometries S constitute one class of equivalence, i.e. the class of isometries having the same name, which means that (internal)
automorphism of a group of isometries G transforms reflections onto reflections, rotations onto rotations, etc. The properties of the (internal) automorphisms are frequently used when proving
theorems on isometric transformations and the other symmetry transformations. For example, by gR, g Î G, is defined an internal automorphism of the symmetry group of square G, given by presentation
{S,R} S4 = R2 = (RS)2 = E: ER = E, RR = R, (RS)R = SR, (SR)R = RS, SR = S3, (S2)R = S2, (S3)R = S. In the same way, it is defined an external automorphism of the rotational group of square H, given
by presentation {S} S4 = E, SR = S3, (S2)R = S2, (S3)R = S, where the reflection line of reflection R contains the center of four-fold rotation S. Hence, external automorphisms are very efficient
tool for extending symmetry groups. Since the product of direct transformations is a direct transformation, and the inverse of a direct transformation is a direct transformation, each group of
transformations G, which contains at least one indirect transformation has a subgroup of the index 2, denoted by G+, which consists of direct transformations of the group G. For example, the
rotational subgroup of sqare H satisfies this condition regarding the symmetry group of square, so H = G+, [G:H] = 2. All direct isometries of the space En can be identified as movements of a
material object in the space En, as opposed to indirect isometries which do not have such a physical interpretation (e.g., a plane reflection does not represent motion in E3). For a figure f with the
symmetry group Gf, which consists only of direct symmetries, it is possible to have the enantiomorphism - enantiomorphic modifications of a figure f, i.e. to have the "left" and "right" form of the
figure f (Figure 1.9). The existence of indirect symmetries of a figure f implies the absence of enantiomorphism. Since reflections have a role of elementary isometric transformations, while all
other isometries are their finite compositions, of special interest will be symmetry groups generated by reflections - groups, a set of generators of which consists exclusively of reflections. Since
every reflection keeps invariant each point of the reflection line, the fundamental region of these groups will possess a fixed shape, will not allow variations and will have rectilinear edges. All
symmetry groups will be subgroups of groups generated by reflections. In the case of conformal symmetry groups, along with reflections, circle inversions have the analogous function. For example, the
symmetry group of square is the group generated by reflections, with the fundamental region of the fixed shape (Figure 1.3a). Figure 1.9 "Left" and "right" rosette with the symmetry group C4,
consisting of direct symmetries. The results of composition of plane isometries are different categories of groups of isometries of the space E2, represented by Bohm symbols as: G210 - symmetry
groups of finite friezes, G20 - symmetry groups of rosettes, G21 - symmetry groups of friezes and G2 - symmetry groups of ornaments. Because of the relation G210 Ì G20, in this work we will discuss
only the categories G20, G21, G2, while the category G210 will be discussed within the category G20. The definitions of symmetry groups will be derived directly from Bohm symbols: symmetry groups of
rosettes are groups of isometries of the space E2 (plane) with an invariant 0-dimensional subspace (point), symmetry groups of friezes are groups of isometries of the space E2 with an invariant
1-dimensional subspace (line) and without invariant points, while symmetry groups of ornaments are groups of isometries of the space E2 without invariant subspaces (points, lines). The groups of the
category Gn are called the space groups, the groups of the category Gn1 the line groups, and the groups of the category Gn0 the point groups of the space En. With symmetry groups of friezes G21 and
symmetry groups of ornaments G2, a group contains one or two generating translations respectively, so that each of these groups has a translational subgroup. A lattice is the orbit of a point with
respect to a discrete group of translations. For the friezes it is a linear series of equidistant points while for ornaments we get a plane lattice or simply a lattice. Five different symmetry types
of plane lattices bear the name of Bravais lattices; the points of these lattices are defined by five different isohedral tessellations, which consist of parallelograms, rhombuses, rectangles,
squares or regular hexagons. To Bravais lattices correspond the crystal systems of the same names (Figure 1.10). Figure 1.10 Five plane Bravais lattices. Because the symmetry groups of friezes G21
are groups of isometries of the plane E2 with an invariant line, they cannot have rotations of an order greater than 2. For the symmetry groups of ornaments G2 so-called crystallographic restriction
holds, according to which symmetry groups of ornaments can have only rotations of the order n=1,2,3,4,6. The term "crystallographic groups" is used for all groups which satisfy this condition,
despite the category they belong to. In isometry groups all distances between points under the effect of symmetries remain unchanged and the congruence of homologous figures is preserved.
Consequently, the same holds for all other geometric properties of such figures, so that the equiangularity (the congruence of the angles of homologous figures) and their equiformity (the same form
of homologous figures) are the direct consequences of isometrism. The next class of symmetry groups we shall consider are the similarity symmetry groups. A similarity transformation of the space En
is a transformation which to each line segment of length AB assigns a line segment of the length kAB whereby k is a real positive number, the coefficient of similarity. In particular, if k = 1 then a
similarity transformation is an isometry. According to the theorem on the existence of an invariant point of every similarity transformation which is not an isometry, there are, besides isometries,
three types of similarity symmetry transformations of the space E2: (i) central dilatation K (or simply dilatation), a transformation which to each vector (A,B) assigns the vector (A',B'), such that
A' = K(A), B' = K(B) and (A',B') = k(A,B), where the coefficient of the dilatation is k Î Â\{-1,0,1} ; (ii) dilative rotation L, the commutative composition of a central dilatation K and a rotation,
with a common invariant point; (iii) dilative reflection M, the commutative composition of a dilatation K and a reflection in the reflection line containing the invariant point (center) of the
dilatation K (Figure 1.11). Figure 1.11 (a) Dilatation; (b) dilative rotation; (c) dilative reflection. Those transformations are, in the given order, isomorphic with the isometries of the space E3:
translation, twist and glide reflection. They make possible the extension of the symmetry groups of rosettes G20 by the external automorphism, having as the result similarity symmetry groups S20 that
we will, thanks to the existence of the invariant point, call the similarity symmetry groups of rosettes. Dilatations K and dilative rotations L are direct, while dilative reflections M are indirect
transformations. They all possess the properties of equiangularity and equiformity. All other aspects of similarity symmetry groups (the problems of enantiomorphism, fundamental regions,
tessellations,¼) will be discussed analogously to the case of isometry groups. Further generalization leads to conformal transformations or circle preserving transformations of the plane E2\{O} ; for
them the property of equiangularity has been preserved, but not that of equiformity. We have, as the elementary transformation of conformal symmetry in E2\{O} , the circle inversion RI (or simply
inversion) - an involutional transformation isomorphic with a reflection, that gives to each point A in the plane E2\{O} a point A1 so that (O,A)°(O,A1) = r2, where r is the radius of the inversion
circle c(O,r) and O is the singular point of the plane E2\{O} (Figure 1.12). Just like a reflection, for which each point of the reflection line is invariant, an inversion maintains invariant each
point of the inversion circle. By discussing a line as a circle with an infinite radius (and treating as circles, at the same time and under the same term, circles and lines) it is possible to
identify reflections with circle inversions. In such a case, all circle inversions (including line reflections) and their compositions, can be discussed as circle preserving transformations, i.e.
transformations mapping circles (including lines) onto circles. Figure 1.12 Circle inversion. Besides the circle inversion RI, by composing it with isometries maintaining invariant the circle line c
of the inversion circle c(O,r) - with a reflection with reflection line containing the circle center O or with a rotation with the rotation center O, we have two more conformal transformations: (i)
inversional reflection ZI = RIR = RRI, the involutional transformation, the commutative composition of a reflection and a circle inversion; (ii) inversional rotation SI = SRI = RIS, the commutative
composition of a rotation and a circle inversion (Figure 1.13). Figure 1.13 (a) Inversional reflection; (b) inversional rotation. Those three conformal symmetry transformations, besides isometries
and similarity symmetry transformations, constitute the finite and infinite conformal symmetry groups C21, C2 - conformal symmetry groups of rosettes in E2\{O} . As an extension of the symmetry
groups of rosettes G20 we have the finite conformal symmetry groups C21 isomorphic with the symmetry groups of tablets G320. As a further extension of finite conformal symmetry groups C21 by the
similarity symmetry transformations K, L, M, we get the infinite conformal symmetry groups C2. The similarity symmetry groups S20 and the infinite conformal symmetry groups C2 are isomorphic with the
line symmetry groups of the space E3 - the symmetry groups of rods G31. In line with the isomorphism mentioned, all similarity symmetry and conformal symmetry transformations offer a reflectional
(canonic) representation by, at most, four reflections (reflections and circle inversions). By applying this isomorphism, ornamental motifs which correspond to the similarity symmetry and conformal
symmetry groups, satisfy one more scope of painting: adequate interpretation of space objects in the plane. The plane structures obtained are called generalized projections of the symmetry groups of
tablets G320 and rods G31. NEXT CONTENTS
As an elementary isometric transformation we can take the reflection, non-identical isometry of space En (Sn) for which, every point of its subspace En-1 (Sn-1) is an invariant point. In particular,
for n = 1 we have point reflection, for n = 2 line reflection (or simply - reflection), for n = 3 plane reflection, involutional indirect isometries. According to the fundamental theorem on minimal
or canonic representation of an isometric transformation of space En (Sn), which states that every isometry of this space can be presented as a composition of maximum n+1 (plane) reflections, it is
possible to classify the isometries of different spaces. The classification of isometric transformations and corresponding symmetry groups is common for spaces En, Sn, Ln for n < 2, while for n ³ 2
different possibilities of relations of disjoint lines, which are defined by the axiom of parallelism, condition specific differences. This work exclusively discusses Euclidean spaces. In the space
E2 (plane) we distinguish the following isometric transformations (Figure 1.7): Figure 1.7 (a) Identity transformation; (b) reflection; (c) rotation; (d) translation; (e) glide reflection. 1)
identity transformation E, with the minimal reflectional representation of the length 2 (R2 = E); 2) reflection R; 3) rotation S = R1R2, the product of two reflections in the reflection lines
crossing in the invariant point (center of rotation). The oriented angle of rotation is equal to twice the angle between the reflection lines R1, R2; 4) translation X = R1R2, the product of two
reflections with parallel reflection lines, such that the translation vector is perpendicular to them and equal to twice the oriented distance between the reflection lines R1, R2; 5) glide reflection
P = R3X = XR3 = R1R2R3, the commutative product of a translation X and a reflection R3 with the reflection line parallel to the translation axis. With respect to the invariant figures, all the points
of the plane E2 are invariant points of the identity transformation E, reflection R maintains the invariance of all the points of the reflection line, rotation S possesses a single invariant point -
the center of rotation, while translation and glide reflection have no invariant points. A glide reflection possesses a single invariant line - the axis, and a translation keeps invariant all the
lines parallel to the translation axis. In the case of rotation, if the relation S = R1R2 = R2R1 holds, i.e. if the reflection lines R1, R2 are perpendicular, as a result we get the special
involutional rotation - central reflection Z (two-fold rotation, half-turn, point-reflection) (Figure 1.8). Figure 1.8 Central reflection Z. When orientation is considered we distinguish direct
transformations (or sense preserving transformations): identity transformation E, translation, rotation, and indirect transformations (sense reversing transformations): reflection and glide
reflection. Since direct transformations are the product of an even, and the indirect ones of an odd number of reflections, we can call them respectively even and odd transformations. If a symmetry
transformation S can be represented as a composition S = S1¼Sn such that SiSj = SjSi, i,j = 1,¼,n, we can call it a complex or composite transformation while the transformations S1,¼,Sn we call the
dependent transformations or dependent elements of symmetry. We will use such approach whenever we are interested to learn to what degree the dependent elements of symmetry influence the
characteristics of the composite transformation, and whether they have lost or preserved their geometric and visual characteristics during it. For example, a glide reflection is such commutative
composition of translation and reflection, with reflection line parallel to the translation axis. An analogous procedure makes possible the classification of isometries of the space E3, where each
isometry can be represented as the composition of four plane reflections at the most. Besides the transformations of the space E2 afore mentioned with the line reflections substituted by plane
reflections, as the new transformations of the space E3 we have two more transformations. They are a direct isometry - twist (screw), the commutative composition of a rotation and a translation, the
canonic representation of which consists of four plane reflections and indirect isometry - rotatory reflection, the commutative composition of a rotation and a plane reflection in the plane
perpendicular to the rotation axis, the canonic representation of which consists of three plane reflections. In particular, the involutional rotatory reflection, which is the composition of three
plane reflections of which every two commute, is called point inversion Z (or rotatory inversion). For every element S1 of a transformation group G we can define the conjugate of the element S1 by an
element S as the product S-1S1S, which we denote by S1S. If S1S Î G, then the mapping S1 onto S1S represents an automorphism of the group G. If the element S by means of which this automorphism is
being realized belongs to the group G, such an automorphism is called an internal automorphism. Any other automorphism of a group G is called an external automorphism. An important characteristic of
a conjugate is that the order of the conjugate S1S is equal to the order of the element S1. If a figure f is invariant under a transformation S1, then S(f) is the figure invariant under
transformation S1S. The conjugate of a reflection R with invariant reflection line p, derived by isometry S, is the reflection RS with the invariant reflection line S(p). Hence we can conclude that
the isometry S1 and all its conjugates S1S derived by different isometries S constitute one class of equivalence, i.e. the class of isometries having the same name, which means that (internal)
automorphism of a group of isometries G transforms reflections onto reflections, rotations onto rotations, etc. The properties of the (internal) automorphisms are frequently used when proving
theorems on isometric transformations and the other symmetry transformations. For example, by gR, g Î G, is defined an internal automorphism of the symmetry group of square G, given by presentation
{S,R} S4 = R2 = (RS)2 = E: ER = E, RR = R, (RS)R = SR, (SR)R = RS, SR = S3, (S2)R = S2, (S3)R = S. In the same way, it is defined an external automorphism of the rotational group of square H, given
by presentation {S} S4 = E, SR = S3, (S2)R = S2, (S3)R = S, where the reflection line of reflection R contains the center of four-fold rotation S. Hence, external automorphisms are very efficient
tool for extending symmetry groups. Since the product of direct transformations is a direct transformation, and the inverse of a direct transformation is a direct transformation, each group of
transformations G, which contains at least one indirect transformation has a subgroup of the index 2, denoted by G+, which consists of direct transformations of the group G. For example, the
rotational subgroup of sqare H satisfies this condition regarding the symmetry group of square, so H = G+, [G:H] = 2. All direct isometries of the space En can be identified as movements of a
material object in the space En, as opposed to indirect isometries which do not have such a physical interpretation (e.g., a plane reflection does not represent motion in E3). For a figure f with the
symmetry group Gf, which consists only of direct symmetries, it is possible to have the enantiomorphism - enantiomorphic modifications of a figure f, i.e. to have the "left" and "right" form of the
figure f (Figure 1.9). The existence of indirect symmetries of a figure f implies the absence of enantiomorphism. Since reflections have a role of elementary isometric transformations, while all
other isometries are their finite compositions, of special interest will be symmetry groups generated by reflections - groups, a set of generators of which consists exclusively of reflections. Since
every reflection keeps invariant each point of the reflection line, the fundamental region of these groups will possess a fixed shape, will not allow variations and will have rectilinear edges. All
symmetry groups will be subgroups of groups generated by reflections. In the case of conformal symmetry groups, along with reflections, circle inversions have the analogous function. For example, the
symmetry group of square is the group generated by reflections, with the fundamental region of the fixed shape (Figure 1.3a). Figure 1.9 "Left" and "right" rosette with the symmetry group C4,
consisting of direct symmetries. The results of composition of plane isometries are different categories of groups of isometries of the space E2, represented by Bohm symbols as: G210 - symmetry
groups of finite friezes, G20 - symmetry groups of rosettes, G21 - symmetry groups of friezes and G2 - symmetry groups of ornaments. Because of the relation G210 Ì G20, in this work we will discuss
only the categories G20, G21, G2, while the category G210 will be discussed within the category G20. The definitions of symmetry groups will be derived directly from Bohm symbols: symmetry groups of
rosettes are groups of isometries of the space E2 (plane) with an invariant 0-dimensional subspace (point), symmetry groups of friezes are groups of isometries of the space E2 with an invariant
1-dimensional subspace (line) and without invariant points, while symmetry groups of ornaments are groups of isometries of the space E2 without invariant subspaces (points, lines). The groups of the
category Gn are called the space groups, the groups of the category Gn1 the line groups, and the groups of the category Gn0 the point groups of the space En. With symmetry groups of friezes G21 and
symmetry groups of ornaments G2, a group contains one or two generating translations respectively, so that each of these groups has a translational subgroup. A lattice is the orbit of a point with
respect to a discrete group of translations. For the friezes it is a linear series of equidistant points while for ornaments we get a plane lattice or simply a lattice. Five different symmetry types
of plane lattices bear the name of Bravais lattices; the points of these lattices are defined by five different isohedral tessellations, which consist of parallelograms, rhombuses, rectangles,
squares or regular hexagons. To Bravais lattices correspond the crystal systems of the same names (Figure 1.10). Figure 1.10 Five plane Bravais lattices. Because the symmetry groups of friezes G21
are groups of isometries of the plane E2 with an invariant line, they cannot have rotations of an order greater than 2. For the symmetry groups of ornaments G2 so-called crystallographic restriction
holds, according to which symmetry groups of ornaments can have only rotations of the order n=1,2,3,4,6. The term "crystallographic groups" is used for all groups which satisfy this condition,
despite the category they belong to. In isometry groups all distances between points under the effect of symmetries remain unchanged and the congruence of homologous figures is preserved.
Consequently, the same holds for all other geometric properties of such figures, so that the equiangularity (the congruence of the angles of homologous figures) and their equiformity (the same form
of homologous figures) are the direct consequences of isometrism. The next class of symmetry groups we shall consider are the similarity symmetry groups. A similarity transformation of the space En
is a transformation which to each line segment of length AB assigns a line segment of the length kAB whereby k is a real positive number, the coefficient of similarity. In particular, if k = 1 then a
similarity transformation is an isometry. According to the theorem on the existence of an invariant point of every similarity transformation which is not an isometry, there are, besides isometries,
three types of similarity symmetry transformations of the space E2: (i) central dilatation K (or simply dilatation), a transformation which to each vector (A,B) assigns the vector (A',B'), such that
A' = K(A), B' = K(B) and (A',B') = k(A,B), where the coefficient of the dilatation is k Î Â\{-1,0,1} ; (ii) dilative rotation L, the commutative composition of a central dilatation K and a rotation,
with a common invariant point; (iii) dilative reflection M, the commutative composition of a dilatation K and a reflection in the reflection line containing the invariant point (center) of the
dilatation K (Figure 1.11). Figure 1.11 (a) Dilatation; (b) dilative rotation; (c) dilative reflection. Those transformations are, in the given order, isomorphic with the isometries of the space E3:
translation, twist and glide reflection. They make possible the extension of the symmetry groups of rosettes G20 by the external automorphism, having as the result similarity symmetry groups S20 that
we will, thanks to the existence of the invariant point, call the similarity symmetry groups of rosettes. Dilatations K and dilative rotations L are direct, while dilative reflections M are indirect
transformations. They all possess the properties of equiangularity and equiformity. All other aspects of similarity symmetry groups (the problems of enantiomorphism, fundamental regions,
tessellations,¼) will be discussed analogously to the case of isometry groups. Further generalization leads to conformal transformations or circle preserving transformations of the plane E2\{O} ; for
them the property of equiangularity has been preserved, but not that of equiformity. We have, as the elementary transformation of conformal symmetry in E2\{O} , the circle inversion RI (or simply
inversion) - an involutional transformation isomorphic with a reflection, that gives to each point A in the plane E2\{O} a point A1 so that (O,A)°(O,A1) = r2, where r is the radius of the inversion
circle c(O,r) and O is the singular point of the plane E2\{O} (Figure 1.12). Just like a reflection, for which each point of the reflection line is invariant, an inversion maintains invariant each
point of the inversion circle. By discussing a line as a circle with an infinite radius (and treating as circles, at the same time and under the same term, circles and lines) it is possible to
identify reflections with circle inversions. In such a case, all circle inversions (including line reflections) and their compositions, can be discussed as circle preserving transformations, i.e.
transformations mapping circles (including lines) onto circles. Figure 1.12 Circle inversion. Besides the circle inversion RI, by composing it with isometries maintaining invariant the circle line c
of the inversion circle c(O,r) - with a reflection with reflection line containing the circle center O or with a rotation with the rotation center O, we have two more conformal transformations: (i)
inversional reflection ZI = RIR = RRI, the involutional transformation, the commutative composition of a reflection and a circle inversion; (ii) inversional rotation SI = SRI = RIS, the commutative
composition of a rotation and a circle inversion (Figure 1.13). Figure 1.13 (a) Inversional reflection; (b) inversional rotation. Those three conformal symmetry transformations, besides isometries
and similarity symmetry transformations, constitute the finite and infinite conformal symmetry groups C21, C2 - conformal symmetry groups of rosettes in E2\{O} . As an extension of the symmetry
groups of rosettes G20 we have the finite conformal symmetry groups C21 isomorphic with the symmetry groups of tablets G320. As a further extension of finite conformal symmetry groups C21 by the
similarity symmetry transformations K, L, M, we get the infinite conformal symmetry groups C2. The similarity symmetry groups S20 and the infinite conformal symmetry groups C2 are isomorphic with the
line symmetry groups of the space E3 - the symmetry groups of rods G31. In line with the isomorphism mentioned, all similarity symmetry and conformal symmetry transformations offer a reflectional
(canonic) representation by, at most, four reflections (reflections and circle inversions). By applying this isomorphism, ornamental motifs which correspond to the similarity symmetry and conformal
symmetry groups, satisfy one more scope of painting: adequate interpretation of space objects in the plane. The plane structures obtained are called generalized projections of the symmetry groups of
tablets G320 and rods G31. NEXT CONTENTS
The classification of isometric transformations and corresponding symmetry groups is common for spaces En, Sn, Ln for n < 2, while for n ³ 2 different possibilities of relations of disjoint lines,
which are defined by the axiom of parallelism, condition specific differences. This work exclusively discusses Euclidean spaces.
In the space E2 (plane) we distinguish the following isometric transformations (Figure 1.7):
(a) Identity
(b) reflection;
(c) rotation;
(e) glide
1) identity transformation E, with the minimal reflectional representation of the length 2 (R2 = E);
3) rotation S = R1R2, the product of two reflections in the reflection lines crossing in the invariant point (center of rotation). The oriented angle of rotation is equal to twice the angle between
the reflection lines R1, R2; 4) translation X = R1R2, the product of two reflections with parallel reflection lines, such that the translation vector is perpendicular to them and equal to twice the
oriented distance between the reflection lines R1, R2; 5) glide reflection P = R3X = XR3 = R1R2R3, the commutative product of a translation X and a reflection R3 with the reflection line parallel to
the translation axis. With respect to the invariant figures, all the points of the plane E2 are invariant points of the identity transformation E, reflection R maintains the invariance of all the
points of the reflection line, rotation S possesses a single invariant point - the center of rotation, while translation and glide reflection have no invariant points. A glide reflection possesses a
single invariant line - the axis, and a translation keeps invariant all the lines parallel to the translation axis. In the case of rotation, if the relation S = R1R2 = R2R1 holds, i.e. if the
reflection lines R1, R2 are perpendicular, as a result we get the special involutional rotation - central reflection Z (two-fold rotation, half-turn, point-reflection) (Figure 1.8). Figure 1.8
Central reflection Z. When orientation is considered we distinguish direct transformations (or sense preserving transformations): identity transformation E, translation, rotation, and indirect
transformations (sense reversing transformations): reflection and glide reflection. Since direct transformations are the product of an even, and the indirect ones of an odd number of reflections, we
can call them respectively even and odd transformations. If a symmetry transformation S can be represented as a composition S = S1¼Sn such that SiSj = SjSi, i,j = 1,¼,n, we can call it a complex or
composite transformation while the transformations S1,¼,Sn we call the dependent transformations or dependent elements of symmetry. We will use such approach whenever we are interested to learn to
what degree the dependent elements of symmetry influence the characteristics of the composite transformation, and whether they have lost or preserved their geometric and visual characteristics during
it. For example, a glide reflection is such commutative composition of translation and reflection, with reflection line parallel to the translation axis. An analogous procedure makes possible the
classification of isometries of the space E3, where each isometry can be represented as the composition of four plane reflections at the most. Besides the transformations of the space E2 afore
mentioned with the line reflections substituted by plane reflections, as the new transformations of the space E3 we have two more transformations. They are a direct isometry - twist (screw), the
commutative composition of a rotation and a translation, the canonic representation of which consists of four plane reflections and indirect isometry - rotatory reflection, the commutative
composition of a rotation and a plane reflection in the plane perpendicular to the rotation axis, the canonic representation of which consists of three plane reflections. In particular, the
involutional rotatory reflection, which is the composition of three plane reflections of which every two commute, is called point inversion Z (or rotatory inversion). For every element S1 of a
transformation group G we can define the conjugate of the element S1 by an element S as the product S-1S1S, which we denote by S1S. If S1S Î G, then the mapping S1 onto S1S represents an automorphism
of the group G. If the element S by means of which this automorphism is being realized belongs to the group G, such an automorphism is called an internal automorphism. Any other automorphism of a
group G is called an external automorphism. An important characteristic of a conjugate is that the order of the conjugate S1S is equal to the order of the element S1. If a figure f is invariant under
a transformation S1, then S(f) is the figure invariant under transformation S1S. The conjugate of a reflection R with invariant reflection line p, derived by isometry S, is the reflection RS with the
invariant reflection line S(p). Hence we can conclude that the isometry S1 and all its conjugates S1S derived by different isometries S constitute one class of equivalence, i.e. the class of
isometries having the same name, which means that (internal) automorphism of a group of isometries G transforms reflections onto reflections, rotations onto rotations, etc. The properties of the
(internal) automorphisms are frequently used when proving theorems on isometric transformations and the other symmetry transformations. For example, by gR, g Î G, is defined an internal automorphism
of the symmetry group of square G, given by presentation {S,R} S4 = R2 = (RS)2 = E: ER = E, RR = R, (RS)R = SR, (SR)R = RS, SR = S3, (S2)R = S2, (S3)R = S. In the same way, it is defined an external
automorphism of the rotational group of square H, given by presentation {S} S4 = E, SR = S3, (S2)R = S2, (S3)R = S, where the reflection line of reflection R contains the center of four-fold rotation
S. Hence, external automorphisms are very efficient tool for extending symmetry groups. Since the product of direct transformations is a direct transformation, and the inverse of a direct
transformation is a direct transformation, each group of transformations G, which contains at least one indirect transformation has a subgroup of the index 2, denoted by G+, which consists of direct
transformations of the group G. For example, the rotational subgroup of sqare H satisfies this condition regarding the symmetry group of square, so H = G+, [G:H] = 2. All direct isometries of the
space En can be identified as movements of a material object in the space En, as opposed to indirect isometries which do not have such a physical interpretation (e.g., a plane reflection does not
represent motion in E3). For a figure f with the symmetry group Gf, which consists only of direct symmetries, it is possible to have the enantiomorphism - enantiomorphic modifications of a figure f,
i.e. to have the "left" and "right" form of the figure f (Figure 1.9). The existence of indirect symmetries of a figure f implies the absence of enantiomorphism. Since reflections have a role of
elementary isometric transformations, while all other isometries are their finite compositions, of special interest will be symmetry groups generated by reflections - groups, a set of generators of
which consists exclusively of reflections. Since every reflection keeps invariant each point of the reflection line, the fundamental region of these groups will possess a fixed shape, will not allow
variations and will have rectilinear edges. All symmetry groups will be subgroups of groups generated by reflections. In the case of conformal symmetry groups, along with reflections, circle
inversions have the analogous function. For example, the symmetry group of square is the group generated by reflections, with the fundamental region of the fixed shape (Figure 1.3a). Figure 1.9
"Left" and "right" rosette with the symmetry group C4, consisting of direct symmetries. The results of composition of plane isometries are different categories of groups of isometries of the space
E2, represented by Bohm symbols as: G210 - symmetry groups of finite friezes, G20 - symmetry groups of rosettes, G21 - symmetry groups of friezes and G2 - symmetry groups of ornaments. Because of the
relation G210 Ì G20, in this work we will discuss only the categories G20, G21, G2, while the category G210 will be discussed within the category G20. The definitions of symmetry groups will be
derived directly from Bohm symbols: symmetry groups of rosettes are groups of isometries of the space E2 (plane) with an invariant 0-dimensional subspace (point), symmetry groups of friezes are
groups of isometries of the space E2 with an invariant 1-dimensional subspace (line) and without invariant points, while symmetry groups of ornaments are groups of isometries of the space E2 without
invariant subspaces (points, lines). The groups of the category Gn are called the space groups, the groups of the category Gn1 the line groups, and the groups of the category Gn0 the point groups of
the space En. With symmetry groups of friezes G21 and symmetry groups of ornaments G2, a group contains one or two generating translations respectively, so that each of these groups has a
translational subgroup. A lattice is the orbit of a point with respect to a discrete group of translations. For the friezes it is a linear series of equidistant points while for ornaments we get a
plane lattice or simply a lattice. Five different symmetry types of plane lattices bear the name of Bravais lattices; the points of these lattices are defined by five different isohedral
tessellations, which consist of parallelograms, rhombuses, rectangles, squares or regular hexagons. To Bravais lattices correspond the crystal systems of the same names (Figure 1.10). Figure 1.10
Five plane Bravais lattices. Because the symmetry groups of friezes G21 are groups of isometries of the plane E2 with an invariant line, they cannot have rotations of an order greater than 2. For the
symmetry groups of ornaments G2 so-called crystallographic restriction holds, according to which symmetry groups of ornaments can have only rotations of the order n=1,2,3,4,6. The term
"crystallographic groups" is used for all groups which satisfy this condition, despite the category they belong to. In isometry groups all distances between points under the effect of symmetries
remain unchanged and the congruence of homologous figures is preserved. Consequently, the same holds for all other geometric properties of such figures, so that the equiangularity (the congruence of
the angles of homologous figures) and their equiformity (the same form of homologous figures) are the direct consequences of isometrism. The next class of symmetry groups we shall consider are the
similarity symmetry groups. A similarity transformation of the space En is a transformation which to each line segment of length AB assigns a line segment of the length kAB whereby k is a real
positive number, the coefficient of similarity. In particular, if k = 1 then a similarity transformation is an isometry. According to the theorem on the existence of an invariant point of every
similarity transformation which is not an isometry, there are, besides isometries, three types of similarity symmetry transformations of the space E2: (i) central dilatation K (or simply dilatation),
a transformation which to each vector (A,B) assigns the vector (A',B'), such that A' = K(A), B' = K(B) and (A',B') = k(A,B), where the coefficient of the dilatation is k Î Â\{-1,0,1} ; (ii) dilative
rotation L, the commutative composition of a central dilatation K and a rotation, with a common invariant point; (iii) dilative reflection M, the commutative composition of a dilatation K and a
reflection in the reflection line containing the invariant point (center) of the dilatation K (Figure 1.11). Figure 1.11 (a) Dilatation; (b) dilative rotation; (c) dilative reflection. Those
transformations are, in the given order, isomorphic with the isometries of the space E3: translation, twist and glide reflection. They make possible the extension of the symmetry groups of rosettes
G20 by the external automorphism, having as the result similarity symmetry groups S20 that we will, thanks to the existence of the invariant point, call the similarity symmetry groups of rosettes.
Dilatations K and dilative rotations L are direct, while dilative reflections M are indirect transformations. They all possess the properties of equiangularity and equiformity. All other aspects of
similarity symmetry groups (the problems of enantiomorphism, fundamental regions, tessellations,¼) will be discussed analogously to the case of isometry groups. Further generalization leads to
conformal transformations or circle preserving transformations of the plane E2\{O} ; for them the property of equiangularity has been preserved, but not that of equiformity. We have, as the
elementary transformation of conformal symmetry in E2\{O} , the circle inversion RI (or simply inversion) - an involutional transformation isomorphic with a reflection, that gives to each point A in
the plane E2\{O} a point A1 so that (O,A)°(O,A1) = r2, where r is the radius of the inversion circle c(O,r) and O is the singular point of the plane E2\{O} (Figure 1.12). Just like a reflection, for
which each point of the reflection line is invariant, an inversion maintains invariant each point of the inversion circle. By discussing a line as a circle with an infinite radius (and treating as
circles, at the same time and under the same term, circles and lines) it is possible to identify reflections with circle inversions. In such a case, all circle inversions (including line reflections)
and their compositions, can be discussed as circle preserving transformations, i.e. transformations mapping circles (including lines) onto circles. Figure 1.12 Circle inversion. Besides the circle
inversion RI, by composing it with isometries maintaining invariant the circle line c of the inversion circle c(O,r) - with a reflection with reflection line containing the circle center O or with a
rotation with the rotation center O, we have two more conformal transformations: (i) inversional reflection ZI = RIR = RRI, the involutional transformation, the commutative composition of a
reflection and a circle inversion; (ii) inversional rotation SI = SRI = RIS, the commutative composition of a rotation and a circle inversion (Figure 1.13). Figure 1.13 (a) Inversional reflection;
(b) inversional rotation. Those three conformal symmetry transformations, besides isometries and similarity symmetry transformations, constitute the finite and infinite conformal symmetry groups C21,
C2 - conformal symmetry groups of rosettes in E2\{O} . As an extension of the symmetry groups of rosettes G20 we have the finite conformal symmetry groups C21 isomorphic with the symmetry groups of
tablets G320. As a further extension of finite conformal symmetry groups C21 by the similarity symmetry transformations K, L, M, we get the infinite conformal symmetry groups C2. The similarity
symmetry groups S20 and the infinite conformal symmetry groups C2 are isomorphic with the line symmetry groups of the space E3 - the symmetry groups of rods G31. In line with the isomorphism
mentioned, all similarity symmetry and conformal symmetry transformations offer a reflectional (canonic) representation by, at most, four reflections (reflections and circle inversions). By applying
this isomorphism, ornamental motifs which correspond to the similarity symmetry and conformal symmetry groups, satisfy one more scope of painting: adequate interpretation of space objects in the
plane. The plane structures obtained are called generalized projections of the symmetry groups of tablets G320 and rods G31. NEXT CONTENTS
4) translation X = R1R2, the product of two reflections with parallel reflection lines, such that the translation vector is perpendicular to them and equal to twice the oriented distance between the
reflection lines R1, R2; 5) glide reflection P = R3X = XR3 = R1R2R3, the commutative product of a translation X and a reflection R3 with the reflection line parallel to the translation axis. With
respect to the invariant figures, all the points of the plane E2 are invariant points of the identity transformation E, reflection R maintains the invariance of all the points of the reflection line,
rotation S possesses a single invariant point - the center of rotation, while translation and glide reflection have no invariant points. A glide reflection possesses a single invariant line - the
axis, and a translation keeps invariant all the lines parallel to the translation axis. In the case of rotation, if the relation S = R1R2 = R2R1 holds, i.e. if the reflection lines R1, R2 are
perpendicular, as a result we get the special involutional rotation - central reflection Z (two-fold rotation, half-turn, point-reflection) (Figure 1.8). Figure 1.8 Central reflection Z. When
orientation is considered we distinguish direct transformations (or sense preserving transformations): identity transformation E, translation, rotation, and indirect transformations (sense reversing
transformations): reflection and glide reflection. Since direct transformations are the product of an even, and the indirect ones of an odd number of reflections, we can call them respectively even
and odd transformations. If a symmetry transformation S can be represented as a composition S = S1¼Sn such that SiSj = SjSi, i,j = 1,¼,n, we can call it a complex or composite transformation while
the transformations S1,¼,Sn we call the dependent transformations or dependent elements of symmetry. We will use such approach whenever we are interested to learn to what degree the dependent
elements of symmetry influence the characteristics of the composite transformation, and whether they have lost or preserved their geometric and visual characteristics during it. For example, a glide
reflection is such commutative composition of translation and reflection, with reflection line parallel to the translation axis. An analogous procedure makes possible the classification of isometries
of the space E3, where each isometry can be represented as the composition of four plane reflections at the most. Besides the transformations of the space E2 afore mentioned with the line reflections
substituted by plane reflections, as the new transformations of the space E3 we have two more transformations. They are a direct isometry - twist (screw), the commutative composition of a rotation
and a translation, the canonic representation of which consists of four plane reflections and indirect isometry - rotatory reflection, the commutative composition of a rotation and a plane reflection
in the plane perpendicular to the rotation axis, the canonic representation of which consists of three plane reflections. In particular, the involutional rotatory reflection, which is the composition
of three plane reflections of which every two commute, is called point inversion Z (or rotatory inversion). For every element S1 of a transformation group G we can define the conjugate of the element
S1 by an element S as the product S-1S1S, which we denote by S1S. If S1S Î G, then the mapping S1 onto S1S represents an automorphism of the group G. If the element S by means of which this
automorphism is being realized belongs to the group G, such an automorphism is called an internal automorphism. Any other automorphism of a group G is called an external automorphism. An important
characteristic of a conjugate is that the order of the conjugate S1S is equal to the order of the element S1. If a figure f is invariant under a transformation S1, then S(f) is the figure invariant
under transformation S1S. The conjugate of a reflection R with invariant reflection line p, derived by isometry S, is the reflection RS with the invariant reflection line S(p). Hence we can conclude
that the isometry S1 and all its conjugates S1S derived by different isometries S constitute one class of equivalence, i.e. the class of isometries having the same name, which means that (internal)
automorphism of a group of isometries G transforms reflections onto reflections, rotations onto rotations, etc. The properties of the (internal) automorphisms are frequently used when proving
theorems on isometric transformations and the other symmetry transformations. For example, by gR, g Î G, is defined an internal automorphism of the symmetry group of square G, given by presentation
{S,R} S4 = R2 = (RS)2 = E: ER = E, RR = R, (RS)R = SR, (SR)R = RS, SR = S3, (S2)R = S2, (S3)R = S. In the same way, it is defined an external automorphism of the rotational group of square H, given
by presentation {S} S4 = E, SR = S3, (S2)R = S2, (S3)R = S, where the reflection line of reflection R contains the center of four-fold rotation S. Hence, external automorphisms are very efficient
tool for extending symmetry groups. Since the product of direct transformations is a direct transformation, and the inverse of a direct transformation is a direct transformation, each group of
transformations G, which contains at least one indirect transformation has a subgroup of the index 2, denoted by G+, which consists of direct transformations of the group G. For example, the
rotational subgroup of sqare H satisfies this condition regarding the symmetry group of square, so H = G+, [G:H] = 2. All direct isometries of the space En can be identified as movements of a
material object in the space En, as opposed to indirect isometries which do not have such a physical interpretation (e.g., a plane reflection does not represent motion in E3). For a figure f with the
symmetry group Gf, which consists only of direct symmetries, it is possible to have the enantiomorphism - enantiomorphic modifications of a figure f, i.e. to have the "left" and "right" form of the
figure f (Figure 1.9). The existence of indirect symmetries of a figure f implies the absence of enantiomorphism. Since reflections have a role of elementary isometric transformations, while all
other isometries are their finite compositions, of special interest will be symmetry groups generated by reflections - groups, a set of generators of which consists exclusively of reflections. Since
every reflection keeps invariant each point of the reflection line, the fundamental region of these groups will possess a fixed shape, will not allow variations and will have rectilinear edges. All
symmetry groups will be subgroups of groups generated by reflections. In the case of conformal symmetry groups, along with reflections, circle inversions have the analogous function. For example, the
symmetry group of square is the group generated by reflections, with the fundamental region of the fixed shape (Figure 1.3a). Figure 1.9 "Left" and "right" rosette with the symmetry group C4,
consisting of direct symmetries. The results of composition of plane isometries are different categories of groups of isometries of the space E2, represented by Bohm symbols as: G210 - symmetry
groups of finite friezes, G20 - symmetry groups of rosettes, G21 - symmetry groups of friezes and G2 - symmetry groups of ornaments. Because of the relation G210 Ì G20, in this work we will discuss
only the categories G20, G21, G2, while the category G210 will be discussed within the category G20. The definitions of symmetry groups will be derived directly from Bohm symbols: symmetry groups of
rosettes are groups of isometries of the space E2 (plane) with an invariant 0-dimensional subspace (point), symmetry groups of friezes are groups of isometries of the space E2 with an invariant
1-dimensional subspace (line) and without invariant points, while symmetry groups of ornaments are groups of isometries of the space E2 without invariant subspaces (points, lines). The groups of the
category Gn are called the space groups, the groups of the category Gn1 the line groups, and the groups of the category Gn0 the point groups of the space En. With symmetry groups of friezes G21 and
symmetry groups of ornaments G2, a group contains one or two generating translations respectively, so that each of these groups has a translational subgroup. A lattice is the orbit of a point with
respect to a discrete group of translations. For the friezes it is a linear series of equidistant points while for ornaments we get a plane lattice or simply a lattice. Five different symmetry types
of plane lattices bear the name of Bravais lattices; the points of these lattices are defined by five different isohedral tessellations, which consist of parallelograms, rhombuses, rectangles,
squares or regular hexagons. To Bravais lattices correspond the crystal systems of the same names (Figure 1.10). Figure 1.10 Five plane Bravais lattices. Because the symmetry groups of friezes G21
are groups of isometries of the plane E2 with an invariant line, they cannot have rotations of an order greater than 2. For the symmetry groups of ornaments G2 so-called crystallographic restriction
holds, according to which symmetry groups of ornaments can have only rotations of the order n=1,2,3,4,6. The term "crystallographic groups" is used for all groups which satisfy this condition,
despite the category they belong to. In isometry groups all distances between points under the effect of symmetries remain unchanged and the congruence of homologous figures is preserved.
Consequently, the same holds for all other geometric properties of such figures, so that the equiangularity (the congruence of the angles of homologous figures) and their equiformity (the same form
of homologous figures) are the direct consequences of isometrism. The next class of symmetry groups we shall consider are the similarity symmetry groups. A similarity transformation of the space En
is a transformation which to each line segment of length AB assigns a line segment of the length kAB whereby k is a real positive number, the coefficient of similarity. In particular, if k = 1 then a
similarity transformation is an isometry. According to the theorem on the existence of an invariant point of every similarity transformation which is not an isometry, there are, besides isometries,
three types of similarity symmetry transformations of the space E2: (i) central dilatation K (or simply dilatation), a transformation which to each vector (A,B) assigns the vector (A',B'), such that
A' = K(A), B' = K(B) and (A',B') = k(A,B), where the coefficient of the dilatation is k Î Â\{-1,0,1} ; (ii) dilative rotation L, the commutative composition of a central dilatation K and a rotation,
with a common invariant point; (iii) dilative reflection M, the commutative composition of a dilatation K and a reflection in the reflection line containing the invariant point (center) of the
dilatation K (Figure 1.11). Figure 1.11 (a) Dilatation; (b) dilative rotation; (c) dilative reflection. Those transformations are, in the given order, isomorphic with the isometries of the space E3:
translation, twist and glide reflection. They make possible the extension of the symmetry groups of rosettes G20 by the external automorphism, having as the result similarity symmetry groups S20 that
we will, thanks to the existence of the invariant point, call the similarity symmetry groups of rosettes. Dilatations K and dilative rotations L are direct, while dilative reflections M are indirect
transformations. They all possess the properties of equiangularity and equiformity. All other aspects of similarity symmetry groups (the problems of enantiomorphism, fundamental regions,
tessellations,¼) will be discussed analogously to the case of isometry groups. Further generalization leads to conformal transformations or circle preserving transformations of the plane E2\{O} ; for
them the property of equiangularity has been preserved, but not that of equiformity. We have, as the elementary transformation of conformal symmetry in E2\{O} , the circle inversion RI (or simply
inversion) - an involutional transformation isomorphic with a reflection, that gives to each point A in the plane E2\{O} a point A1 so that (O,A)°(O,A1) = r2, where r is the radius of the inversion
circle c(O,r) and O is the singular point of the plane E2\{O} (Figure 1.12). Just like a reflection, for which each point of the reflection line is invariant, an inversion maintains invariant each
point of the inversion circle. By discussing a line as a circle with an infinite radius (and treating as circles, at the same time and under the same term, circles and lines) it is possible to
identify reflections with circle inversions. In such a case, all circle inversions (including line reflections) and their compositions, can be discussed as circle preserving transformations, i.e.
transformations mapping circles (including lines) onto circles. Figure 1.12 Circle inversion. Besides the circle inversion RI, by composing it with isometries maintaining invariant the circle line c
of the inversion circle c(O,r) - with a reflection with reflection line containing the circle center O or with a rotation with the rotation center O, we have two more conformal transformations: (i)
inversional reflection ZI = RIR = RRI, the involutional transformation, the commutative composition of a reflection and a circle inversion; (ii) inversional rotation SI = SRI = RIS, the commutative
composition of a rotation and a circle inversion (Figure 1.13). Figure 1.13 (a) Inversional reflection; (b) inversional rotation. Those three conformal symmetry transformations, besides isometries
and similarity symmetry transformations, constitute the finite and infinite conformal symmetry groups C21, C2 - conformal symmetry groups of rosettes in E2\{O} . As an extension of the symmetry
groups of rosettes G20 we have the finite conformal symmetry groups C21 isomorphic with the symmetry groups of tablets G320. As a further extension of finite conformal symmetry groups C21 by the
similarity symmetry transformations K, L, M, we get the infinite conformal symmetry groups C2. The similarity symmetry groups S20 and the infinite conformal symmetry groups C2 are isomorphic with the
line symmetry groups of the space E3 - the symmetry groups of rods G31. In line with the isomorphism mentioned, all similarity symmetry and conformal symmetry transformations offer a reflectional
(canonic) representation by, at most, four reflections (reflections and circle inversions). By applying this isomorphism, ornamental motifs which correspond to the similarity symmetry and conformal
symmetry groups, satisfy one more scope of painting: adequate interpretation of space objects in the plane. The plane structures obtained are called generalized projections of the symmetry groups of
tablets G320 and rods G31. NEXT CONTENTS
5) glide reflection P = R3X = XR3 = R1R2R3, the commutative product of a translation X and a reflection R3 with the reflection line parallel to the translation axis. With respect to the invariant
figures, all the points of the plane E2 are invariant points of the identity transformation E, reflection R maintains the invariance of all the points of the reflection line, rotation S possesses a
single invariant point - the center of rotation, while translation and glide reflection have no invariant points. A glide reflection possesses a single invariant line - the axis, and a translation
keeps invariant all the lines parallel to the translation axis. In the case of rotation, if the relation S = R1R2 = R2R1 holds, i.e. if the reflection lines R1, R2 are perpendicular, as a result we
get the special involutional rotation - central reflection Z (two-fold rotation, half-turn, point-reflection) (Figure 1.8). Figure 1.8 Central reflection Z. When orientation is considered we
distinguish direct transformations (or sense preserving transformations): identity transformation E, translation, rotation, and indirect transformations (sense reversing transformations): reflection
and glide reflection. Since direct transformations are the product of an even, and the indirect ones of an odd number of reflections, we can call them respectively even and odd transformations. If a
symmetry transformation S can be represented as a composition S = S1¼Sn such that SiSj = SjSi, i,j = 1,¼,n, we can call it a complex or composite transformation while the transformations S1,¼,Sn we
call the dependent transformations or dependent elements of symmetry. We will use such approach whenever we are interested to learn to what degree the dependent elements of symmetry influence the
characteristics of the composite transformation, and whether they have lost or preserved their geometric and visual characteristics during it. For example, a glide reflection is such commutative
composition of translation and reflection, with reflection line parallel to the translation axis. An analogous procedure makes possible the classification of isometries of the space E3, where each
isometry can be represented as the composition of four plane reflections at the most. Besides the transformations of the space E2 afore mentioned with the line reflections substituted by plane
reflections, as the new transformations of the space E3 we have two more transformations. They are a direct isometry - twist (screw), the commutative composition of a rotation and a translation, the
canonic representation of which consists of four plane reflections and indirect isometry - rotatory reflection, the commutative composition of a rotation and a plane reflection in the plane
perpendicular to the rotation axis, the canonic representation of which consists of three plane reflections. In particular, the involutional rotatory reflection, which is the composition of three
plane reflections of which every two commute, is called point inversion Z (or rotatory inversion). For every element S1 of a transformation group G we can define the conjugate of the element S1 by an
element S as the product S-1S1S, which we denote by S1S. If S1S Î G, then the mapping S1 onto S1S represents an automorphism of the group G. If the element S by means of which this automorphism is
being realized belongs to the group G, such an automorphism is called an internal automorphism. Any other automorphism of a group G is called an external automorphism. An important characteristic of
a conjugate is that the order of the conjugate S1S is equal to the order of the element S1. If a figure f is invariant under a transformation S1, then S(f) is the figure invariant under
transformation S1S. The conjugate of a reflection R with invariant reflection line p, derived by isometry S, is the reflection RS with the invariant reflection line S(p). Hence we can conclude that
the isometry S1 and all its conjugates S1S derived by different isometries S constitute one class of equivalence, i.e. the class of isometries having the same name, which means that (internal)
automorphism of a group of isometries G transforms reflections onto reflections, rotations onto rotations, etc. The properties of the (internal) automorphisms are frequently used when proving
theorems on isometric transformations and the other symmetry transformations. For example, by gR, g Î G, is defined an internal automorphism of the symmetry group of square G, given by presentation
{S,R} S4 = R2 = (RS)2 = E: ER = E, RR = R, (RS)R = SR, (SR)R = RS, SR = S3, (S2)R = S2, (S3)R = S. In the same way, it is defined an external automorphism of the rotational group of square H, given
by presentation {S} S4 = E, SR = S3, (S2)R = S2, (S3)R = S, where the reflection line of reflection R contains the center of four-fold rotation S. Hence, external automorphisms are very efficient
tool for extending symmetry groups. Since the product of direct transformations is a direct transformation, and the inverse of a direct transformation is a direct transformation, each group of
transformations G, which contains at least one indirect transformation has a subgroup of the index 2, denoted by G+, which consists of direct transformations of the group G. For example, the
rotational subgroup of sqare H satisfies this condition regarding the symmetry group of square, so H = G+, [G:H] = 2. All direct isometries of the space En can be identified as movements of a
material object in the space En, as opposed to indirect isometries which do not have such a physical interpretation (e.g., a plane reflection does not represent motion in E3). For a figure f with the
symmetry group Gf, which consists only of direct symmetries, it is possible to have the enantiomorphism - enantiomorphic modifications of a figure f, i.e. to have the "left" and "right" form of the
figure f (Figure 1.9). The existence of indirect symmetries of a figure f implies the absence of enantiomorphism. Since reflections have a role of elementary isometric transformations, while all
other isometries are their finite compositions, of special interest will be symmetry groups generated by reflections - groups, a set of generators of which consists exclusively of reflections. Since
every reflection keeps invariant each point of the reflection line, the fundamental region of these groups will possess a fixed shape, will not allow variations and will have rectilinear edges. All
symmetry groups will be subgroups of groups generated by reflections. In the case of conformal symmetry groups, along with reflections, circle inversions have the analogous function. For example, the
symmetry group of square is the group generated by reflections, with the fundamental region of the fixed shape (Figure 1.3a). Figure 1.9 "Left" and "right" rosette with the symmetry group C4,
consisting of direct symmetries. The results of composition of plane isometries are different categories of groups of isometries of the space E2, represented by Bohm symbols as: G210 - symmetry
groups of finite friezes, G20 - symmetry groups of rosettes, G21 - symmetry groups of friezes and G2 - symmetry groups of ornaments. Because of the relation G210 Ì G20, in this work we will discuss
only the categories G20, G21, G2, while the category G210 will be discussed within the category G20. The definitions of symmetry groups will be derived directly from Bohm symbols: symmetry groups of
rosettes are groups of isometries of the space E2 (plane) with an invariant 0-dimensional subspace (point), symmetry groups of friezes are groups of isometries of the space E2 with an invariant
1-dimensional subspace (line) and without invariant points, while symmetry groups of ornaments are groups of isometries of the space E2 without invariant subspaces (points, lines). The groups of the
category Gn are called the space groups, the groups of the category Gn1 the line groups, and the groups of the category Gn0 the point groups of the space En. With symmetry groups of friezes G21 and
symmetry groups of ornaments G2, a group contains one or two generating translations respectively, so that each of these groups has a translational subgroup. A lattice is the orbit of a point with
respect to a discrete group of translations. For the friezes it is a linear series of equidistant points while for ornaments we get a plane lattice or simply a lattice. Five different symmetry types
of plane lattices bear the name of Bravais lattices; the points of these lattices are defined by five different isohedral tessellations, which consist of parallelograms, rhombuses, rectangles,
squares or regular hexagons. To Bravais lattices correspond the crystal systems of the same names (Figure 1.10). Figure 1.10 Five plane Bravais lattices. Because the symmetry groups of friezes G21
are groups of isometries of the plane E2 with an invariant line, they cannot have rotations of an order greater than 2. For the symmetry groups of ornaments G2 so-called crystallographic restriction
holds, according to which symmetry groups of ornaments can have only rotations of the order n=1,2,3,4,6. The term "crystallographic groups" is used for all groups which satisfy this condition,
despite the category they belong to. In isometry groups all distances between points under the effect of symmetries remain unchanged and the congruence of homologous figures is preserved.
Consequently, the same holds for all other geometric properties of such figures, so that the equiangularity (the congruence of the angles of homologous figures) and their equiformity (the same form
of homologous figures) are the direct consequences of isometrism. The next class of symmetry groups we shall consider are the similarity symmetry groups. A similarity transformation of the space En
is a transformation which to each line segment of length AB assigns a line segment of the length kAB whereby k is a real positive number, the coefficient of similarity. In particular, if k = 1 then a
similarity transformation is an isometry. According to the theorem on the existence of an invariant point of every similarity transformation which is not an isometry, there are, besides isometries,
three types of similarity symmetry transformations of the space E2: (i) central dilatation K (or simply dilatation), a transformation which to each vector (A,B) assigns the vector (A',B'), such that
A' = K(A), B' = K(B) and (A',B') = k(A,B), where the coefficient of the dilatation is k Î Â\{-1,0,1} ; (ii) dilative rotation L, the commutative composition of a central dilatation K and a rotation,
with a common invariant point; (iii) dilative reflection M, the commutative composition of a dilatation K and a reflection in the reflection line containing the invariant point (center) of the
dilatation K (Figure 1.11). Figure 1.11 (a) Dilatation; (b) dilative rotation; (c) dilative reflection. Those transformations are, in the given order, isomorphic with the isometries of the space E3:
translation, twist and glide reflection. They make possible the extension of the symmetry groups of rosettes G20 by the external automorphism, having as the result similarity symmetry groups S20 that
we will, thanks to the existence of the invariant point, call the similarity symmetry groups of rosettes. Dilatations K and dilative rotations L are direct, while dilative reflections M are indirect
transformations. They all possess the properties of equiangularity and equiformity. All other aspects of similarity symmetry groups (the problems of enantiomorphism, fundamental regions,
tessellations,¼) will be discussed analogously to the case of isometry groups. Further generalization leads to conformal transformations or circle preserving transformations of the plane E2\{O} ; for
them the property of equiangularity has been preserved, but not that of equiformity. We have, as the elementary transformation of conformal symmetry in E2\{O} , the circle inversion RI (or simply
inversion) - an involutional transformation isomorphic with a reflection, that gives to each point A in the plane E2\{O} a point A1 so that (O,A)°(O,A1) = r2, where r is the radius of the inversion
circle c(O,r) and O is the singular point of the plane E2\{O} (Figure 1.12). Just like a reflection, for which each point of the reflection line is invariant, an inversion maintains invariant each
point of the inversion circle. By discussing a line as a circle with an infinite radius (and treating as circles, at the same time and under the same term, circles and lines) it is possible to
identify reflections with circle inversions. In such a case, all circle inversions (including line reflections) and their compositions, can be discussed as circle preserving transformations, i.e.
transformations mapping circles (including lines) onto circles. Figure 1.12 Circle inversion. Besides the circle inversion RI, by composing it with isometries maintaining invariant the circle line c
of the inversion circle c(O,r) - with a reflection with reflection line containing the circle center O or with a rotation with the rotation center O, we have two more conformal transformations: (i)
inversional reflection ZI = RIR = RRI, the involutional transformation, the commutative composition of a reflection and a circle inversion; (ii) inversional rotation SI = SRI = RIS, the commutative
composition of a rotation and a circle inversion (Figure 1.13). Figure 1.13 (a) Inversional reflection; (b) inversional rotation. Those three conformal symmetry transformations, besides isometries
and similarity symmetry transformations, constitute the finite and infinite conformal symmetry groups C21, C2 - conformal symmetry groups of rosettes in E2\{O} . As an extension of the symmetry
groups of rosettes G20 we have the finite conformal symmetry groups C21 isomorphic with the symmetry groups of tablets G320. As a further extension of finite conformal symmetry groups C21 by the
similarity symmetry transformations K, L, M, we get the infinite conformal symmetry groups C2. The similarity symmetry groups S20 and the infinite conformal symmetry groups C2 are isomorphic with the
line symmetry groups of the space E3 - the symmetry groups of rods G31. In line with the isomorphism mentioned, all similarity symmetry and conformal symmetry transformations offer a reflectional
(canonic) representation by, at most, four reflections (reflections and circle inversions). By applying this isomorphism, ornamental motifs which correspond to the similarity symmetry and conformal
symmetry groups, satisfy one more scope of painting: adequate interpretation of space objects in the plane. The plane structures obtained are called generalized projections of the symmetry groups of
tablets G320 and rods G31. NEXT CONTENTS
With respect to the invariant figures, all the points of the plane E2 are invariant points of the identity transformation E, reflection R maintains the invariance of all the points of the reflection
line, rotation S possesses a single invariant point - the center of rotation, while translation and glide reflection have no invariant points. A glide reflection possesses a single invariant line -
the axis, and a translation keeps invariant all the lines parallel to the translation axis.
In the case of rotation, if the relation S = R1R2 = R2R1 holds, i.e. if the reflection lines R1, R2 are perpendicular, as a result we get the special involutional rotation - central reflection Z
(two-fold rotation, half-turn, point-reflection) (Figure 1.8). Figure 1.8 Central reflection Z. When orientation is considered we distinguish direct transformations (or sense preserving
transformations): identity transformation E, translation, rotation, and indirect transformations (sense reversing transformations): reflection and glide reflection. Since direct transformations are
the product of an even, and the indirect ones of an odd number of reflections, we can call them respectively even and odd transformations. If a symmetry transformation S can be represented as a
composition S = S1¼Sn such that SiSj = SjSi, i,j = 1,¼,n, we can call it a complex or composite transformation while the transformations S1,¼,Sn we call the dependent transformations or dependent
elements of symmetry. We will use such approach whenever we are interested to learn to what degree the dependent elements of symmetry influence the characteristics of the composite transformation,
and whether they have lost or preserved their geometric and visual characteristics during it. For example, a glide reflection is such commutative composition of translation and reflection, with
reflection line parallel to the translation axis. An analogous procedure makes possible the classification of isometries of the space E3, where each isometry can be represented as the composition of
four plane reflections at the most. Besides the transformations of the space E2 afore mentioned with the line reflections substituted by plane reflections, as the new transformations of the space E3
we have two more transformations. They are a direct isometry - twist (screw), the commutative composition of a rotation and a translation, the canonic representation of which consists of four plane
reflections and indirect isometry - rotatory reflection, the commutative composition of a rotation and a plane reflection in the plane perpendicular to the rotation axis, the canonic representation
of which consists of three plane reflections. In particular, the involutional rotatory reflection, which is the composition of three plane reflections of which every two commute, is called point
inversion Z (or rotatory inversion). For every element S1 of a transformation group G we can define the conjugate of the element S1 by an element S as the product S-1S1S, which we denote by S1S. If
S1S Î G, then the mapping S1 onto S1S represents an automorphism of the group G. If the element S by means of which this automorphism is being realized belongs to the group G, such an automorphism is
called an internal automorphism. Any other automorphism of a group G is called an external automorphism. An important characteristic of a conjugate is that the order of the conjugate S1S is equal to
the order of the element S1. If a figure f is invariant under a transformation S1, then S(f) is the figure invariant under transformation S1S. The conjugate of a reflection R with invariant
reflection line p, derived by isometry S, is the reflection RS with the invariant reflection line S(p). Hence we can conclude that the isometry S1 and all its conjugates S1S derived by different
isometries S constitute one class of equivalence, i.e. the class of isometries having the same name, which means that (internal) automorphism of a group of isometries G transforms reflections onto
reflections, rotations onto rotations, etc. The properties of the (internal) automorphisms are frequently used when proving theorems on isometric transformations and the other symmetry
transformations. For example, by gR, g Î G, is defined an internal automorphism of the symmetry group of square G, given by presentation {S,R} S4 = R2 = (RS)2 = E: ER = E, RR = R, (RS)R = SR, (SR)R =
RS, SR = S3, (S2)R = S2, (S3)R = S. In the same way, it is defined an external automorphism of the rotational group of square H, given by presentation {S} S4 = E, SR = S3, (S2)R = S2, (S3)R = S,
where the reflection line of reflection R contains the center of four-fold rotation S. Hence, external automorphisms are very efficient tool for extending symmetry groups. Since the product of direct
transformations is a direct transformation, and the inverse of a direct transformation is a direct transformation, each group of transformations G, which contains at least one indirect transformation
has a subgroup of the index 2, denoted by G+, which consists of direct transformations of the group G. For example, the rotational subgroup of sqare H satisfies this condition regarding the symmetry
group of square, so H = G+, [G:H] = 2. All direct isometries of the space En can be identified as movements of a material object in the space En, as opposed to indirect isometries which do not have
such a physical interpretation (e.g., a plane reflection does not represent motion in E3). For a figure f with the symmetry group Gf, which consists only of direct symmetries, it is possible to have
the enantiomorphism - enantiomorphic modifications of a figure f, i.e. to have the "left" and "right" form of the figure f (Figure 1.9). The existence of indirect symmetries of a figure f implies the
absence of enantiomorphism. Since reflections have a role of elementary isometric transformations, while all other isometries are their finite compositions, of special interest will be symmetry
groups generated by reflections - groups, a set of generators of which consists exclusively of reflections. Since every reflection keeps invariant each point of the reflection line, the fundamental
region of these groups will possess a fixed shape, will not allow variations and will have rectilinear edges. All symmetry groups will be subgroups of groups generated by reflections. In the case of
conformal symmetry groups, along with reflections, circle inversions have the analogous function. For example, the symmetry group of square is the group generated by reflections, with the fundamental
region of the fixed shape (Figure 1.3a). Figure 1.9 "Left" and "right" rosette with the symmetry group C4, consisting of direct symmetries. The results of composition of plane isometries are
different categories of groups of isometries of the space E2, represented by Bohm symbols as: G210 - symmetry groups of finite friezes, G20 - symmetry groups of rosettes, G21 - symmetry groups of
friezes and G2 - symmetry groups of ornaments. Because of the relation G210 Ì G20, in this work we will discuss only the categories G20, G21, G2, while the category G210 will be discussed within the
category G20. The definitions of symmetry groups will be derived directly from Bohm symbols: symmetry groups of rosettes are groups of isometries of the space E2 (plane) with an invariant
0-dimensional subspace (point), symmetry groups of friezes are groups of isometries of the space E2 with an invariant 1-dimensional subspace (line) and without invariant points, while symmetry groups
of ornaments are groups of isometries of the space E2 without invariant subspaces (points, lines). The groups of the category Gn are called the space groups, the groups of the category Gn1 the line
groups, and the groups of the category Gn0 the point groups of the space En. With symmetry groups of friezes G21 and symmetry groups of ornaments G2, a group contains one or two generating
translations respectively, so that each of these groups has a translational subgroup. A lattice is the orbit of a point with respect to a discrete group of translations. For the friezes it is a
linear series of equidistant points while for ornaments we get a plane lattice or simply a lattice. Five different symmetry types of plane lattices bear the name of Bravais lattices; the points of
these lattices are defined by five different isohedral tessellations, which consist of parallelograms, rhombuses, rectangles, squares or regular hexagons. To Bravais lattices correspond the crystal
systems of the same names (Figure 1.10). Figure 1.10 Five plane Bravais lattices. Because the symmetry groups of friezes G21 are groups of isometries of the plane E2 with an invariant line, they
cannot have rotations of an order greater than 2. For the symmetry groups of ornaments G2 so-called crystallographic restriction holds, according to which symmetry groups of ornaments can have only
rotations of the order n=1,2,3,4,6. The term "crystallographic groups" is used for all groups which satisfy this condition, despite the category they belong to. In isometry groups all distances
between points under the effect of symmetries remain unchanged and the congruence of homologous figures is preserved. Consequently, the same holds for all other geometric properties of such figures,
so that the equiangularity (the congruence of the angles of homologous figures) and their equiformity (the same form of homologous figures) are the direct consequences of isometrism. The next class
of symmetry groups we shall consider are the similarity symmetry groups. A similarity transformation of the space En is a transformation which to each line segment of length AB assigns a line segment
of the length kAB whereby k is a real positive number, the coefficient of similarity. In particular, if k = 1 then a similarity transformation is an isometry. According to the theorem on the
existence of an invariant point of every similarity transformation which is not an isometry, there are, besides isometries, three types of similarity symmetry transformations of the space E2: (i)
central dilatation K (or simply dilatation), a transformation which to each vector (A,B) assigns the vector (A',B'), such that A' = K(A), B' = K(B) and (A',B') = k(A,B), where the coefficient of the
dilatation is k Î Â\{-1,0,1} ; (ii) dilative rotation L, the commutative composition of a central dilatation K and a rotation, with a common invariant point; (iii) dilative reflection M, the
commutative composition of a dilatation K and a reflection in the reflection line containing the invariant point (center) of the dilatation K (Figure 1.11). Figure 1.11 (a) Dilatation; (b) dilative
rotation; (c) dilative reflection. Those transformations are, in the given order, isomorphic with the isometries of the space E3: translation, twist and glide reflection. They make possible the
extension of the symmetry groups of rosettes G20 by the external automorphism, having as the result similarity symmetry groups S20 that we will, thanks to the existence of the invariant point, call
the similarity symmetry groups of rosettes. Dilatations K and dilative rotations L are direct, while dilative reflections M are indirect transformations. They all possess the properties of
equiangularity and equiformity. All other aspects of similarity symmetry groups (the problems of enantiomorphism, fundamental regions, tessellations,¼) will be discussed analogously to the case of
isometry groups. Further generalization leads to conformal transformations or circle preserving transformations of the plane E2\{O} ; for them the property of equiangularity has been preserved, but
not that of equiformity. We have, as the elementary transformation of conformal symmetry in E2\{O} , the circle inversion RI (or simply inversion) - an involutional transformation isomorphic with a
reflection, that gives to each point A in the plane E2\{O} a point A1 so that (O,A)°(O,A1) = r2, where r is the radius of the inversion circle c(O,r) and O is the singular point of the plane E2\{O}
(Figure 1.12). Just like a reflection, for which each point of the reflection line is invariant, an inversion maintains invariant each point of the inversion circle. By discussing a line as a circle
with an infinite radius (and treating as circles, at the same time and under the same term, circles and lines) it is possible to identify reflections with circle inversions. In such a case, all
circle inversions (including line reflections) and their compositions, can be discussed as circle preserving transformations, i.e. transformations mapping circles (including lines) onto circles.
Figure 1.12 Circle inversion. Besides the circle inversion RI, by composing it with isometries maintaining invariant the circle line c of the inversion circle c(O,r) - with a reflection with
reflection line containing the circle center O or with a rotation with the rotation center O, we have two more conformal transformations: (i) inversional reflection ZI = RIR = RRI, the involutional
transformation, the commutative composition of a reflection and a circle inversion; (ii) inversional rotation SI = SRI = RIS, the commutative composition of a rotation and a circle inversion (Figure
1.13). Figure 1.13 (a) Inversional reflection; (b) inversional rotation. Those three conformal symmetry transformations, besides isometries and similarity symmetry transformations, constitute the
finite and infinite conformal symmetry groups C21, C2 - conformal symmetry groups of rosettes in E2\{O} . As an extension of the symmetry groups of rosettes G20 we have the finite conformal symmetry
groups C21 isomorphic with the symmetry groups of tablets G320. As a further extension of finite conformal symmetry groups C21 by the similarity symmetry transformations K, L, M, we get the infinite
conformal symmetry groups C2. The similarity symmetry groups S20 and the infinite conformal symmetry groups C2 are isomorphic with the line symmetry groups of the space E3 - the symmetry groups of
rods G31. In line with the isomorphism mentioned, all similarity symmetry and conformal symmetry transformations offer a reflectional (canonic) representation by, at most, four reflections
(reflections and circle inversions). By applying this isomorphism, ornamental motifs which correspond to the similarity symmetry and conformal symmetry groups, satisfy one more scope of painting:
adequate interpretation of space objects in the plane. The plane structures obtained are called generalized projections of the symmetry groups of tablets G320 and rods G31. NEXT CONTENTS
When orientation is considered we distinguish direct transformations (or sense preserving transformations): identity transformation E, translation, rotation, and indirect transformations (sense
reversing transformations): reflection and glide reflection. Since direct transformations are the product of an even, and the indirect ones of an odd number of reflections, we can call them
respectively even and odd transformations.
If a symmetry transformation S can be represented as a composition S = S1¼Sn such that SiSj = SjSi, i,j = 1,¼,n, we can call it a complex or composite transformation while the transformations S1,¼,Sn
we call the dependent transformations or dependent elements of symmetry. We will use such approach whenever we are interested to learn to what degree the dependent elements of symmetry influence the
characteristics of the composite transformation, and whether they have lost or preserved their geometric and visual characteristics during it. For example, a glide reflection is such commutative
composition of translation and reflection, with reflection line parallel to the translation axis. An analogous procedure makes possible the classification of isometries of the space E3, where each
isometry can be represented as the composition of four plane reflections at the most. Besides the transformations of the space E2 afore mentioned with the line reflections substituted by plane
reflections, as the new transformations of the space E3 we have two more transformations. They are a direct isometry - twist (screw), the commutative composition of a rotation and a translation, the
canonic representation of which consists of four plane reflections and indirect isometry - rotatory reflection, the commutative composition of a rotation and a plane reflection in the plane
perpendicular to the rotation axis, the canonic representation of which consists of three plane reflections. In particular, the involutional rotatory reflection, which is the composition of three
plane reflections of which every two commute, is called point inversion Z (or rotatory inversion). For every element S1 of a transformation group G we can define the conjugate of the element S1 by an
element S as the product S-1S1S, which we denote by S1S. If S1S Î G, then the mapping S1 onto S1S represents an automorphism of the group G. If the element S by means of which this automorphism is
being realized belongs to the group G, such an automorphism is called an internal automorphism. Any other automorphism of a group G is called an external automorphism. An important characteristic of
a conjugate is that the order of the conjugate S1S is equal to the order of the element S1. If a figure f is invariant under a transformation S1, then S(f) is the figure invariant under
transformation S1S. The conjugate of a reflection R with invariant reflection line p, derived by isometry S, is the reflection RS with the invariant reflection line S(p). Hence we can conclude that
the isometry S1 and all its conjugates S1S derived by different isometries S constitute one class of equivalence, i.e. the class of isometries having the same name, which means that (internal)
automorphism of a group of isometries G transforms reflections onto reflections, rotations onto rotations, etc. The properties of the (internal) automorphisms are frequently used when proving
theorems on isometric transformations and the other symmetry transformations. For example, by gR, g Î G, is defined an internal automorphism of the symmetry group of square G, given by presentation
{S,R} S4 = R2 = (RS)2 = E: ER = E, RR = R, (RS)R = SR, (SR)R = RS, SR = S3, (S2)R = S2, (S3)R = S. In the same way, it is defined an external automorphism of the rotational group of square H, given
by presentation {S} S4 = E, SR = S3, (S2)R = S2, (S3)R = S, where the reflection line of reflection R contains the center of four-fold rotation S. Hence, external automorphisms are very efficient
tool for extending symmetry groups. Since the product of direct transformations is a direct transformation, and the inverse of a direct transformation is a direct transformation, each group of
transformations G, which contains at least one indirect transformation has a subgroup of the index 2, denoted by G+, which consists of direct transformations of the group G. For example, the
rotational subgroup of sqare H satisfies this condition regarding the symmetry group of square, so H = G+, [G:H] = 2. All direct isometries of the space En can be identified as movements of a
material object in the space En, as opposed to indirect isometries which do not have such a physical interpretation (e.g., a plane reflection does not represent motion in E3). For a figure f with the
symmetry group Gf, which consists only of direct symmetries, it is possible to have the enantiomorphism - enantiomorphic modifications of a figure f, i.e. to have the "left" and "right" form of the
figure f (Figure 1.9). The existence of indirect symmetries of a figure f implies the absence of enantiomorphism. Since reflections have a role of elementary isometric transformations, while all
other isometries are their finite compositions, of special interest will be symmetry groups generated by reflections - groups, a set of generators of which consists exclusively of reflections. Since
every reflection keeps invariant each point of the reflection line, the fundamental region of these groups will possess a fixed shape, will not allow variations and will have rectilinear edges. All
symmetry groups will be subgroups of groups generated by reflections. In the case of conformal symmetry groups, along with reflections, circle inversions have the analogous function. For example, the
symmetry group of square is the group generated by reflections, with the fundamental region of the fixed shape (Figure 1.3a). Figure 1.9 "Left" and "right" rosette with the symmetry group C4,
consisting of direct symmetries. The results of composition of plane isometries are different categories of groups of isometries of the space E2, represented by Bohm symbols as: G210 - symmetry
groups of finite friezes, G20 - symmetry groups of rosettes, G21 - symmetry groups of friezes and G2 - symmetry groups of ornaments. Because of the relation G210 Ì G20, in this work we will discuss
only the categories G20, G21, G2, while the category G210 will be discussed within the category G20. The definitions of symmetry groups will be derived directly from Bohm symbols: symmetry groups of
rosettes are groups of isometries of the space E2 (plane) with an invariant 0-dimensional subspace (point), symmetry groups of friezes are groups of isometries of the space E2 with an invariant
1-dimensional subspace (line) and without invariant points, while symmetry groups of ornaments are groups of isometries of the space E2 without invariant subspaces (points, lines). The groups of the
category Gn are called the space groups, the groups of the category Gn1 the line groups, and the groups of the category Gn0 the point groups of the space En. With symmetry groups of friezes G21 and
symmetry groups of ornaments G2, a group contains one or two generating translations respectively, so that each of these groups has a translational subgroup. A lattice is the orbit of a point with
respect to a discrete group of translations. For the friezes it is a linear series of equidistant points while for ornaments we get a plane lattice or simply a lattice. Five different symmetry types
of plane lattices bear the name of Bravais lattices; the points of these lattices are defined by five different isohedral tessellations, which consist of parallelograms, rhombuses, rectangles,
squares or regular hexagons. To Bravais lattices correspond the crystal systems of the same names (Figure 1.10). Figure 1.10 Five plane Bravais lattices. Because the symmetry groups of friezes G21
are groups of isometries of the plane E2 with an invariant line, they cannot have rotations of an order greater than 2. For the symmetry groups of ornaments G2 so-called crystallographic restriction
holds, according to which symmetry groups of ornaments can have only rotations of the order n=1,2,3,4,6. The term "crystallographic groups" is used for all groups which satisfy this condition,
despite the category they belong to. In isometry groups all distances between points under the effect of symmetries remain unchanged and the congruence of homologous figures is preserved.
Consequently, the same holds for all other geometric properties of such figures, so that the equiangularity (the congruence of the angles of homologous figures) and their equiformity (the same form
of homologous figures) are the direct consequences of isometrism. The next class of symmetry groups we shall consider are the similarity symmetry groups. A similarity transformation of the space En
is a transformation which to each line segment of length AB assigns a line segment of the length kAB whereby k is a real positive number, the coefficient of similarity. In particular, if k = 1 then a
similarity transformation is an isometry. According to the theorem on the existence of an invariant point of every similarity transformation which is not an isometry, there are, besides isometries,
three types of similarity symmetry transformations of the space E2: (i) central dilatation K (or simply dilatation), a transformation which to each vector (A,B) assigns the vector (A',B'), such that
A' = K(A), B' = K(B) and (A',B') = k(A,B), where the coefficient of the dilatation is k Î Â\{-1,0,1} ; (ii) dilative rotation L, the commutative composition of a central dilatation K and a rotation,
with a common invariant point; (iii) dilative reflection M, the commutative composition of a dilatation K and a reflection in the reflection line containing the invariant point (center) of the
dilatation K (Figure 1.11). Figure 1.11 (a) Dilatation; (b) dilative rotation; (c) dilative reflection. Those transformations are, in the given order, isomorphic with the isometries of the space E3:
translation, twist and glide reflection. They make possible the extension of the symmetry groups of rosettes G20 by the external automorphism, having as the result similarity symmetry groups S20 that
we will, thanks to the existence of the invariant point, call the similarity symmetry groups of rosettes. Dilatations K and dilative rotations L are direct, while dilative reflections M are indirect
transformations. They all possess the properties of equiangularity and equiformity. All other aspects of similarity symmetry groups (the problems of enantiomorphism, fundamental regions,
tessellations,¼) will be discussed analogously to the case of isometry groups. Further generalization leads to conformal transformations or circle preserving transformations of the plane E2\{O} ; for
them the property of equiangularity has been preserved, but not that of equiformity. We have, as the elementary transformation of conformal symmetry in E2\{O} , the circle inversion RI (or simply
inversion) - an involutional transformation isomorphic with a reflection, that gives to each point A in the plane E2\{O} a point A1 so that (O,A)°(O,A1) = r2, where r is the radius of the inversion
circle c(O,r) and O is the singular point of the plane E2\{O} (Figure 1.12). Just like a reflection, for which each point of the reflection line is invariant, an inversion maintains invariant each
point of the inversion circle. By discussing a line as a circle with an infinite radius (and treating as circles, at the same time and under the same term, circles and lines) it is possible to
identify reflections with circle inversions. In such a case, all circle inversions (including line reflections) and their compositions, can be discussed as circle preserving transformations, i.e.
transformations mapping circles (including lines) onto circles. Figure 1.12 Circle inversion. Besides the circle inversion RI, by composing it with isometries maintaining invariant the circle line c
of the inversion circle c(O,r) - with a reflection with reflection line containing the circle center O or with a rotation with the rotation center O, we have two more conformal transformations: (i)
inversional reflection ZI = RIR = RRI, the involutional transformation, the commutative composition of a reflection and a circle inversion; (ii) inversional rotation SI = SRI = RIS, the commutative
composition of a rotation and a circle inversion (Figure 1.13). Figure 1.13 (a) Inversional reflection; (b) inversional rotation. Those three conformal symmetry transformations, besides isometries
and similarity symmetry transformations, constitute the finite and infinite conformal symmetry groups C21, C2 - conformal symmetry groups of rosettes in E2\{O} . As an extension of the symmetry
groups of rosettes G20 we have the finite conformal symmetry groups C21 isomorphic with the symmetry groups of tablets G320. As a further extension of finite conformal symmetry groups C21 by the
similarity symmetry transformations K, L, M, we get the infinite conformal symmetry groups C2. The similarity symmetry groups S20 and the infinite conformal symmetry groups C2 are isomorphic with the
line symmetry groups of the space E3 - the symmetry groups of rods G31. In line with the isomorphism mentioned, all similarity symmetry and conformal symmetry transformations offer a reflectional
(canonic) representation by, at most, four reflections (reflections and circle inversions). By applying this isomorphism, ornamental motifs which correspond to the similarity symmetry and conformal
symmetry groups, satisfy one more scope of painting: adequate interpretation of space objects in the plane. The plane structures obtained are called generalized projections of the symmetry groups of
tablets G320 and rods G31. NEXT CONTENTS
An analogous procedure makes possible the classification of isometries of the space E3, where each isometry can be represented as the composition of four plane reflections at the most. Besides the
transformations of the space E2 afore mentioned with the line reflections substituted by plane reflections, as the new transformations of the space E3 we have two more transformations. They are a
direct isometry - twist (screw), the commutative composition of a rotation and a translation, the canonic representation of which consists of four plane reflections and indirect isometry - rotatory
reflection, the commutative composition of a rotation and a plane reflection in the plane perpendicular to the rotation axis, the canonic representation of which consists of three plane reflections.
In particular, the involutional rotatory reflection, which is the composition of three plane reflections of which every two commute, is called point inversion Z (or rotatory inversion). For every
element S1 of a transformation group G we can define the conjugate of the element S1 by an element S as the product S-1S1S, which we denote by S1S. If S1S Î G, then the mapping S1 onto S1S represents
an automorphism of the group G. If the element S by means of which this automorphism is being realized belongs to the group G, such an automorphism is called an internal automorphism. Any other
automorphism of a group G is called an external automorphism. An important characteristic of a conjugate is that the order of the conjugate S1S is equal to the order of the element S1. If a figure f
is invariant under a transformation S1, then S(f) is the figure invariant under transformation S1S. The conjugate of a reflection R with invariant reflection line p, derived by isometry S, is the
reflection RS with the invariant reflection line S(p). Hence we can conclude that the isometry S1 and all its conjugates S1S derived by different isometries S constitute one class of equivalence,
i.e. the class of isometries having the same name, which means that (internal) automorphism of a group of isometries G transforms reflections onto reflections, rotations onto rotations, etc. The
properties of the (internal) automorphisms are frequently used when proving theorems on isometric transformations and the other symmetry transformations. For example, by gR, g Î G, is defined an
internal automorphism of the symmetry group of square G, given by presentation {S,R} S4 = R2 = (RS)2 = E: ER = E, RR = R, (RS)R = SR, (SR)R = RS, SR = S3, (S2)R = S2, (S3)R = S. In the same way, it
is defined an external automorphism of the rotational group of square H, given by presentation {S} S4 = E, SR = S3, (S2)R = S2, (S3)R = S, where the reflection line of reflection R contains the
center of four-fold rotation S. Hence, external automorphisms are very efficient tool for extending symmetry groups. Since the product of direct transformations is a direct transformation, and the
inverse of a direct transformation is a direct transformation, each group of transformations G, which contains at least one indirect transformation has a subgroup of the index 2, denoted by G+, which
consists of direct transformations of the group G. For example, the rotational subgroup of sqare H satisfies this condition regarding the symmetry group of square, so H = G+, [G:H] = 2. All direct
isometries of the space En can be identified as movements of a material object in the space En, as opposed to indirect isometries which do not have such a physical interpretation (e.g., a plane
reflection does not represent motion in E3). For a figure f with the symmetry group Gf, which consists only of direct symmetries, it is possible to have the enantiomorphism - enantiomorphic
modifications of a figure f, i.e. to have the "left" and "right" form of the figure f (Figure 1.9). The existence of indirect symmetries of a figure f implies the absence of enantiomorphism. Since
reflections have a role of elementary isometric transformations, while all other isometries are their finite compositions, of special interest will be symmetry groups generated by reflections -
groups, a set of generators of which consists exclusively of reflections. Since every reflection keeps invariant each point of the reflection line, the fundamental region of these groups will possess
a fixed shape, will not allow variations and will have rectilinear edges. All symmetry groups will be subgroups of groups generated by reflections. In the case of conformal symmetry groups, along
with reflections, circle inversions have the analogous function. For example, the symmetry group of square is the group generated by reflections, with the fundamental region of the fixed shape
(Figure 1.3a). Figure 1.9 "Left" and "right" rosette with the symmetry group C4, consisting of direct symmetries. The results of composition of plane isometries are different categories of groups of
isometries of the space E2, represented by Bohm symbols as: G210 - symmetry groups of finite friezes, G20 - symmetry groups of rosettes, G21 - symmetry groups of friezes and G2 - symmetry groups of
ornaments. Because of the relation G210 Ì G20, in this work we will discuss only the categories G20, G21, G2, while the category G210 will be discussed within the category G20. The definitions of
symmetry groups will be derived directly from Bohm symbols: symmetry groups of rosettes are groups of isometries of the space E2 (plane) with an invariant 0-dimensional subspace (point), symmetry
groups of friezes are groups of isometries of the space E2 with an invariant 1-dimensional subspace (line) and without invariant points, while symmetry groups of ornaments are groups of isometries of
the space E2 without invariant subspaces (points, lines). The groups of the category Gn are called the space groups, the groups of the category Gn1 the line groups, and the groups of the category Gn0
the point groups of the space En. With symmetry groups of friezes G21 and symmetry groups of ornaments G2, a group contains one or two generating translations respectively, so that each of these
groups has a translational subgroup. A lattice is the orbit of a point with respect to a discrete group of translations. For the friezes it is a linear series of equidistant points while for
ornaments we get a plane lattice or simply a lattice. Five different symmetry types of plane lattices bear the name of Bravais lattices; the points of these lattices are defined by five different
isohedral tessellations, which consist of parallelograms, rhombuses, rectangles, squares or regular hexagons. To Bravais lattices correspond the crystal systems of the same names (Figure 1.10).
Figure 1.10 Five plane Bravais lattices. Because the symmetry groups of friezes G21 are groups of isometries of the plane E2 with an invariant line, they cannot have rotations of an order greater
than 2. For the symmetry groups of ornaments G2 so-called crystallographic restriction holds, according to which symmetry groups of ornaments can have only rotations of the order n=1,2,3,4,6. The
term "crystallographic groups" is used for all groups which satisfy this condition, despite the category they belong to. In isometry groups all distances between points under the effect of symmetries
remain unchanged and the congruence of homologous figures is preserved. Consequently, the same holds for all other geometric properties of such figures, so that the equiangularity (the congruence of
the angles of homologous figures) and their equiformity (the same form of homologous figures) are the direct consequences of isometrism. The next class of symmetry groups we shall consider are the
similarity symmetry groups. A similarity transformation of the space En is a transformation which to each line segment of length AB assigns a line segment of the length kAB whereby k is a real
positive number, the coefficient of similarity. In particular, if k = 1 then a similarity transformation is an isometry. According to the theorem on the existence of an invariant point of every
similarity transformation which is not an isometry, there are, besides isometries, three types of similarity symmetry transformations of the space E2: (i) central dilatation K (or simply dilatation),
a transformation which to each vector (A,B) assigns the vector (A',B'), such that A' = K(A), B' = K(B) and (A',B') = k(A,B), where the coefficient of the dilatation is k Î Â\{-1,0,1} ; (ii) dilative
rotation L, the commutative composition of a central dilatation K and a rotation, with a common invariant point; (iii) dilative reflection M, the commutative composition of a dilatation K and a
reflection in the reflection line containing the invariant point (center) of the dilatation K (Figure 1.11). Figure 1.11 (a) Dilatation; (b) dilative rotation; (c) dilative reflection. Those
transformations are, in the given order, isomorphic with the isometries of the space E3: translation, twist and glide reflection. They make possible the extension of the symmetry groups of rosettes
G20 by the external automorphism, having as the result similarity symmetry groups S20 that we will, thanks to the existence of the invariant point, call the similarity symmetry groups of rosettes.
Dilatations K and dilative rotations L are direct, while dilative reflections M are indirect transformations. They all possess the properties of equiangularity and equiformity. All other aspects of
similarity symmetry groups (the problems of enantiomorphism, fundamental regions, tessellations,¼) will be discussed analogously to the case of isometry groups. Further generalization leads to
conformal transformations or circle preserving transformations of the plane E2\{O} ; for them the property of equiangularity has been preserved, but not that of equiformity. We have, as the
elementary transformation of conformal symmetry in E2\{O} , the circle inversion RI (or simply inversion) - an involutional transformation isomorphic with a reflection, that gives to each point A in
the plane E2\{O} a point A1 so that (O,A)°(O,A1) = r2, where r is the radius of the inversion circle c(O,r) and O is the singular point of the plane E2\{O} (Figure 1.12). Just like a reflection, for
which each point of the reflection line is invariant, an inversion maintains invariant each point of the inversion circle. By discussing a line as a circle with an infinite radius (and treating as
circles, at the same time and under the same term, circles and lines) it is possible to identify reflections with circle inversions. In such a case, all circle inversions (including line reflections)
and their compositions, can be discussed as circle preserving transformations, i.e. transformations mapping circles (including lines) onto circles. Figure 1.12 Circle inversion. Besides the circle
inversion RI, by composing it with isometries maintaining invariant the circle line c of the inversion circle c(O,r) - with a reflection with reflection line containing the circle center O or with a
rotation with the rotation center O, we have two more conformal transformations: (i) inversional reflection ZI = RIR = RRI, the involutional transformation, the commutative composition of a
reflection and a circle inversion; (ii) inversional rotation SI = SRI = RIS, the commutative composition of a rotation and a circle inversion (Figure 1.13). Figure 1.13 (a) Inversional reflection;
(b) inversional rotation. Those three conformal symmetry transformations, besides isometries and similarity symmetry transformations, constitute the finite and infinite conformal symmetry groups C21,
C2 - conformal symmetry groups of rosettes in E2\{O} . As an extension of the symmetry groups of rosettes G20 we have the finite conformal symmetry groups C21 isomorphic with the symmetry groups of
tablets G320. As a further extension of finite conformal symmetry groups C21 by the similarity symmetry transformations K, L, M, we get the infinite conformal symmetry groups C2. The similarity
symmetry groups S20 and the infinite conformal symmetry groups C2 are isomorphic with the line symmetry groups of the space E3 - the symmetry groups of rods G31. In line with the isomorphism
mentioned, all similarity symmetry and conformal symmetry transformations offer a reflectional (canonic) representation by, at most, four reflections (reflections and circle inversions). By applying
this isomorphism, ornamental motifs which correspond to the similarity symmetry and conformal symmetry groups, satisfy one more scope of painting: adequate interpretation of space objects in the
plane. The plane structures obtained are called generalized projections of the symmetry groups of tablets G320 and rods G31. NEXT CONTENTS
For every element S1 of a transformation group G we can define the conjugate of the element S1 by an element S as the product S-1S1S, which we denote by S1S. If S1S Î G, then the mapping S1 onto S1S
represents an automorphism of the group G. If the element S by means of which this automorphism is being realized belongs to the group G, such an automorphism is called an internal automorphism. Any
other automorphism of a group G is called an external automorphism. An important characteristic of a conjugate is that the order of the conjugate S1S is equal to the order of the element S1. If a
figure f is invariant under a transformation S1, then S(f) is the figure invariant under transformation S1S. The conjugate of a reflection R with invariant reflection line p, derived by isometry S,
is the reflection RS with the invariant reflection line S(p). Hence we can conclude that the isometry S1 and all its conjugates S1S derived by different isometries S constitute one class of
equivalence, i.e. the class of isometries having the same name, which means that (internal) automorphism of a group of isometries G transforms reflections onto reflections, rotations onto rotations,
etc. The properties of the (internal) automorphisms are frequently used when proving theorems on isometric transformations and the other symmetry transformations. For example, by gR, g Î G, is
defined an internal automorphism of the symmetry group of square G, given by presentation {S,R} S4 = R2 = (RS)2 = E: ER = E, RR = R, (RS)R = SR, (SR)R = RS, SR = S3, (S2)R = S2, (S3)R = S. In the
same way, it is defined an external automorphism of the rotational group of square H, given by presentation {S} S4 = E, SR = S3, (S2)R = S2, (S3)R = S, where the reflection line of reflection R
contains the center of four-fold rotation S. Hence, external automorphisms are very efficient tool for extending symmetry groups. Since the product of direct transformations is a direct
transformation, and the inverse of a direct transformation is a direct transformation, each group of transformations G, which contains at least one indirect transformation has a subgroup of the index
2, denoted by G+, which consists of direct transformations of the group G. For example, the rotational subgroup of sqare H satisfies this condition regarding the symmetry group of square, so H = G+,
[G:H] = 2. All direct isometries of the space En can be identified as movements of a material object in the space En, as opposed to indirect isometries which do not have such a physical
interpretation (e.g., a plane reflection does not represent motion in E3). For a figure f with the symmetry group Gf, which consists only of direct symmetries, it is possible to have the
enantiomorphism - enantiomorphic modifications of a figure f, i.e. to have the "left" and "right" form of the figure f (Figure 1.9). The existence of indirect symmetries of a figure f implies the
absence of enantiomorphism. Since reflections have a role of elementary isometric transformations, while all other isometries are their finite compositions, of special interest will be symmetry
groups generated by reflections - groups, a set of generators of which consists exclusively of reflections. Since every reflection keeps invariant each point of the reflection line, the fundamental
region of these groups will possess a fixed shape, will not allow variations and will have rectilinear edges. All symmetry groups will be subgroups of groups generated by reflections. In the case of
conformal symmetry groups, along with reflections, circle inversions have the analogous function. For example, the symmetry group of square is the group generated by reflections, with the fundamental
region of the fixed shape (Figure 1.3a). Figure 1.9 "Left" and "right" rosette with the symmetry group C4, consisting of direct symmetries. The results of composition of plane isometries are
different categories of groups of isometries of the space E2, represented by Bohm symbols as: G210 - symmetry groups of finite friezes, G20 - symmetry groups of rosettes, G21 - symmetry groups of
friezes and G2 - symmetry groups of ornaments. Because of the relation G210 Ì G20, in this work we will discuss only the categories G20, G21, G2, while the category G210 will be discussed within the
category G20. The definitions of symmetry groups will be derived directly from Bohm symbols: symmetry groups of rosettes are groups of isometries of the space E2 (plane) with an invariant
0-dimensional subspace (point), symmetry groups of friezes are groups of isometries of the space E2 with an invariant 1-dimensional subspace (line) and without invariant points, while symmetry groups
of ornaments are groups of isometries of the space E2 without invariant subspaces (points, lines). The groups of the category Gn are called the space groups, the groups of the category Gn1 the line
groups, and the groups of the category Gn0 the point groups of the space En. With symmetry groups of friezes G21 and symmetry groups of ornaments G2, a group contains one or two generating
translations respectively, so that each of these groups has a translational subgroup. A lattice is the orbit of a point with respect to a discrete group of translations. For the friezes it is a
linear series of equidistant points while for ornaments we get a plane lattice or simply a lattice. Five different symmetry types of plane lattices bear the name of Bravais lattices; the points of
these lattices are defined by five different isohedral tessellations, which consist of parallelograms, rhombuses, rectangles, squares or regular hexagons. To Bravais lattices correspond the crystal
systems of the same names (Figure 1.10). Figure 1.10 Five plane Bravais lattices. Because the symmetry groups of friezes G21 are groups of isometries of the plane E2 with an invariant line, they
cannot have rotations of an order greater than 2. For the symmetry groups of ornaments G2 so-called crystallographic restriction holds, according to which symmetry groups of ornaments can have only
rotations of the order n=1,2,3,4,6. The term "crystallographic groups" is used for all groups which satisfy this condition, despite the category they belong to. In isometry groups all distances
between points under the effect of symmetries remain unchanged and the congruence of homologous figures is preserved. Consequently, the same holds for all other geometric properties of such figures,
so that the equiangularity (the congruence of the angles of homologous figures) and their equiformity (the same form of homologous figures) are the direct consequences of isometrism. The next class
of symmetry groups we shall consider are the similarity symmetry groups. A similarity transformation of the space En is a transformation which to each line segment of length AB assigns a line segment
of the length kAB whereby k is a real positive number, the coefficient of similarity. In particular, if k = 1 then a similarity transformation is an isometry. According to the theorem on the
existence of an invariant point of every similarity transformation which is not an isometry, there are, besides isometries, three types of similarity symmetry transformations of the space E2: (i)
central dilatation K (or simply dilatation), a transformation which to each vector (A,B) assigns the vector (A',B'), such that A' = K(A), B' = K(B) and (A',B') = k(A,B), where the coefficient of the
dilatation is k Î Â\{-1,0,1} ; (ii) dilative rotation L, the commutative composition of a central dilatation K and a rotation, with a common invariant point; (iii) dilative reflection M, the
commutative composition of a dilatation K and a reflection in the reflection line containing the invariant point (center) of the dilatation K (Figure 1.11). Figure 1.11 (a) Dilatation; (b) dilative
rotation; (c) dilative reflection. Those transformations are, in the given order, isomorphic with the isometries of the space E3: translation, twist and glide reflection. They make possible the
extension of the symmetry groups of rosettes G20 by the external automorphism, having as the result similarity symmetry groups S20 that we will, thanks to the existence of the invariant point, call
the similarity symmetry groups of rosettes. Dilatations K and dilative rotations L are direct, while dilative reflections M are indirect transformations. They all possess the properties of
equiangularity and equiformity. All other aspects of similarity symmetry groups (the problems of enantiomorphism, fundamental regions, tessellations,¼) will be discussed analogously to the case of
isometry groups. Further generalization leads to conformal transformations or circle preserving transformations of the plane E2\{O} ; for them the property of equiangularity has been preserved, but
not that of equiformity. We have, as the elementary transformation of conformal symmetry in E2\{O} , the circle inversion RI (or simply inversion) - an involutional transformation isomorphic with a
reflection, that gives to each point A in the plane E2\{O} a point A1 so that (O,A)°(O,A1) = r2, where r is the radius of the inversion circle c(O,r) and O is the singular point of the plane E2\{O}
(Figure 1.12). Just like a reflection, for which each point of the reflection line is invariant, an inversion maintains invariant each point of the inversion circle. By discussing a line as a circle
with an infinite radius (and treating as circles, at the same time and under the same term, circles and lines) it is possible to identify reflections with circle inversions. In such a case, all
circle inversions (including line reflections) and their compositions, can be discussed as circle preserving transformations, i.e. transformations mapping circles (including lines) onto circles.
Figure 1.12 Circle inversion. Besides the circle inversion RI, by composing it with isometries maintaining invariant the circle line c of the inversion circle c(O,r) - with a reflection with
reflection line containing the circle center O or with a rotation with the rotation center O, we have two more conformal transformations: (i) inversional reflection ZI = RIR = RRI, the involutional
transformation, the commutative composition of a reflection and a circle inversion; (ii) inversional rotation SI = SRI = RIS, the commutative composition of a rotation and a circle inversion (Figure
1.13). Figure 1.13 (a) Inversional reflection; (b) inversional rotation. Those three conformal symmetry transformations, besides isometries and similarity symmetry transformations, constitute the
finite and infinite conformal symmetry groups C21, C2 - conformal symmetry groups of rosettes in E2\{O} . As an extension of the symmetry groups of rosettes G20 we have the finite conformal symmetry
groups C21 isomorphic with the symmetry groups of tablets G320. As a further extension of finite conformal symmetry groups C21 by the similarity symmetry transformations K, L, M, we get the infinite
conformal symmetry groups C2. The similarity symmetry groups S20 and the infinite conformal symmetry groups C2 are isomorphic with the line symmetry groups of the space E3 - the symmetry groups of
rods G31. In line with the isomorphism mentioned, all similarity symmetry and conformal symmetry transformations offer a reflectional (canonic) representation by, at most, four reflections
(reflections and circle inversions). By applying this isomorphism, ornamental motifs which correspond to the similarity symmetry and conformal symmetry groups, satisfy one more scope of painting:
adequate interpretation of space objects in the plane. The plane structures obtained are called generalized projections of the symmetry groups of tablets G320 and rods G31. NEXT CONTENTS
{S,R} S4 = R2 = (RS)2 = E: ER = E, RR = R, (RS)R = SR, (SR)R = RS, SR = S3, (S2)R = S2, (S3)R = S.
In the same way, it is defined an external automorphism of the rotational group of square H, given by presentation
{S} S4 = E, SR = S3, (S2)R = S2, (S3)R = S,
where the reflection line of reflection R contains the center of four-fold rotation S. Hence, external automorphisms are very efficient tool for extending symmetry groups.
Since the product of direct transformations is a direct transformation, and the inverse of a direct transformation is a direct transformation, each group of transformations G, which contains at least
one indirect transformation has a subgroup of the index 2, denoted by G+, which consists of direct transformations of the group G. For example, the rotational subgroup of sqare H satisfies this
condition regarding the symmetry group of square, so H = G+, [G:H] = 2. All direct isometries of the space En can be identified as movements of a material object in the space En, as opposed to
indirect isometries which do not have such a physical interpretation (e.g., a plane reflection does not represent motion in E3).
For a figure f with the symmetry group Gf, which consists only of direct symmetries, it is possible to have the enantiomorphism - enantiomorphic modifications of a figure f, i.e. to have the "left"
and "right" form of the figure f (Figure 1.9). The existence of indirect symmetries of a figure f implies the absence of enantiomorphism. Since reflections have a role of elementary isometric
transformations, while all other isometries are their finite compositions, of special interest will be symmetry groups generated by reflections - groups, a set of generators of which consists
exclusively of reflections. Since every reflection keeps invariant each point of the reflection line, the fundamental region of these groups will possess a fixed shape, will not allow variations and
will have rectilinear edges. All symmetry groups will be subgroups of groups generated by reflections. In the case of conformal symmetry groups, along with reflections, circle inversions have the
analogous function. For example, the symmetry group of square is the group generated by reflections, with the fundamental region of the fixed shape (Figure 1.3a). Figure 1.9 "Left" and "right"
rosette with the symmetry group C4, consisting of direct symmetries. The results of composition of plane isometries are different categories of groups of isometries of the space E2, represented by
Bohm symbols as: G210 - symmetry groups of finite friezes, G20 - symmetry groups of rosettes, G21 - symmetry groups of friezes and G2 - symmetry groups of ornaments. Because of the relation G210 Ì
G20, in this work we will discuss only the categories G20, G21, G2, while the category G210 will be discussed within the category G20. The definitions of symmetry groups will be derived directly from
Bohm symbols: symmetry groups of rosettes are groups of isometries of the space E2 (plane) with an invariant 0-dimensional subspace (point), symmetry groups of friezes are groups of isometries of the
space E2 with an invariant 1-dimensional subspace (line) and without invariant points, while symmetry groups of ornaments are groups of isometries of the space E2 without invariant subspaces (points,
lines). The groups of the category Gn are called the space groups, the groups of the category Gn1 the line groups, and the groups of the category Gn0 the point groups of the space En. With symmetry
groups of friezes G21 and symmetry groups of ornaments G2, a group contains one or two generating translations respectively, so that each of these groups has a translational subgroup. A lattice is
the orbit of a point with respect to a discrete group of translations. For the friezes it is a linear series of equidistant points while for ornaments we get a plane lattice or simply a lattice. Five
different symmetry types of plane lattices bear the name of Bravais lattices; the points of these lattices are defined by five different isohedral tessellations, which consist of parallelograms,
rhombuses, rectangles, squares or regular hexagons. To Bravais lattices correspond the crystal systems of the same names (Figure 1.10). Figure 1.10 Five plane Bravais lattices. Because the symmetry
groups of friezes G21 are groups of isometries of the plane E2 with an invariant line, they cannot have rotations of an order greater than 2. For the symmetry groups of ornaments G2 so-called
crystallographic restriction holds, according to which symmetry groups of ornaments can have only rotations of the order n=1,2,3,4,6. The term "crystallographic groups" is used for all groups which
satisfy this condition, despite the category they belong to. In isometry groups all distances between points under the effect of symmetries remain unchanged and the congruence of homologous figures
is preserved. Consequently, the same holds for all other geometric properties of such figures, so that the equiangularity (the congruence of the angles of homologous figures) and their equiformity
(the same form of homologous figures) are the direct consequences of isometrism. The next class of symmetry groups we shall consider are the similarity symmetry groups. A similarity transformation of
the space En is a transformation which to each line segment of length AB assigns a line segment of the length kAB whereby k is a real positive number, the coefficient of similarity. In particular, if
k = 1 then a similarity transformation is an isometry. According to the theorem on the existence of an invariant point of every similarity transformation which is not an isometry, there are, besides
isometries, three types of similarity symmetry transformations of the space E2: (i) central dilatation K (or simply dilatation), a transformation which to each vector (A,B) assigns the vector
(A',B'), such that A' = K(A), B' = K(B) and (A',B') = k(A,B), where the coefficient of the dilatation is k Î Â\{-1,0,1} ; (ii) dilative rotation L, the commutative composition of a central dilatation
K and a rotation, with a common invariant point; (iii) dilative reflection M, the commutative composition of a dilatation K and a reflection in the reflection line containing the invariant point
(center) of the dilatation K (Figure 1.11). Figure 1.11 (a) Dilatation; (b) dilative rotation; (c) dilative reflection. Those transformations are, in the given order, isomorphic with the isometries
of the space E3: translation, twist and glide reflection. They make possible the extension of the symmetry groups of rosettes G20 by the external automorphism, having as the result similarity
symmetry groups S20 that we will, thanks to the existence of the invariant point, call the similarity symmetry groups of rosettes. Dilatations K and dilative rotations L are direct, while dilative
reflections M are indirect transformations. They all possess the properties of equiangularity and equiformity. All other aspects of similarity symmetry groups (the problems of enantiomorphism,
fundamental regions, tessellations,¼) will be discussed analogously to the case of isometry groups. Further generalization leads to conformal transformations or circle preserving transformations of
the plane E2\{O} ; for them the property of equiangularity has been preserved, but not that of equiformity. We have, as the elementary transformation of conformal symmetry in E2\{O} , the circle
inversion RI (or simply inversion) - an involutional transformation isomorphic with a reflection, that gives to each point A in the plane E2\{O} a point A1 so that (O,A)°(O,A1) = r2, where r is the
radius of the inversion circle c(O,r) and O is the singular point of the plane E2\{O} (Figure 1.12). Just like a reflection, for which each point of the reflection line is invariant, an inversion
maintains invariant each point of the inversion circle. By discussing a line as a circle with an infinite radius (and treating as circles, at the same time and under the same term, circles and lines)
it is possible to identify reflections with circle inversions. In such a case, all circle inversions (including line reflections) and their compositions, can be discussed as circle preserving
transformations, i.e. transformations mapping circles (including lines) onto circles. Figure 1.12 Circle inversion. Besides the circle inversion RI, by composing it with isometries maintaining
invariant the circle line c of the inversion circle c(O,r) - with a reflection with reflection line containing the circle center O or with a rotation with the rotation center O, we have two more
conformal transformations: (i) inversional reflection ZI = RIR = RRI, the involutional transformation, the commutative composition of a reflection and a circle inversion; (ii) inversional rotation SI
= SRI = RIS, the commutative composition of a rotation and a circle inversion (Figure 1.13). Figure 1.13 (a) Inversional reflection; (b) inversional rotation. Those three conformal symmetry
transformations, besides isometries and similarity symmetry transformations, constitute the finite and infinite conformal symmetry groups C21, C2 - conformal symmetry groups of rosettes in E2\{O} .
As an extension of the symmetry groups of rosettes G20 we have the finite conformal symmetry groups C21 isomorphic with the symmetry groups of tablets G320. As a further extension of finite conformal
symmetry groups C21 by the similarity symmetry transformations K, L, M, we get the infinite conformal symmetry groups C2. The similarity symmetry groups S20 and the infinite conformal symmetry groups
C2 are isomorphic with the line symmetry groups of the space E3 - the symmetry groups of rods G31. In line with the isomorphism mentioned, all similarity symmetry and conformal symmetry
transformations offer a reflectional (canonic) representation by, at most, four reflections (reflections and circle inversions). By applying this isomorphism, ornamental motifs which correspond to
the similarity symmetry and conformal symmetry groups, satisfy one more scope of painting: adequate interpretation of space objects in the plane. The plane structures obtained are called generalized
projections of the symmetry groups of tablets G320 and rods G31. NEXT CONTENTS
Since reflections have a role of elementary isometric transformations, while all other isometries are their finite compositions, of special interest will be symmetry groups generated by reflections -
groups, a set of generators of which consists exclusively of reflections. Since every reflection keeps invariant each point of the reflection line, the fundamental region of these groups will possess
a fixed shape, will not allow variations and will have rectilinear edges. All symmetry groups will be subgroups of groups generated by reflections. In the case of conformal symmetry groups, along
with reflections, circle inversions have the analogous function. For example, the symmetry group of square is the group generated by reflections, with the fundamental region of the fixed shape
(Figure 1.3a).
"Left" and
with the
group C[4],
of direct
The results of composition of plane isometries are different categories of groups of isometries of the space E2, represented by Bohm symbols as: G210 - symmetry groups of finite friezes, G20 -
symmetry groups of rosettes, G21 - symmetry groups of friezes and G2 - symmetry groups of ornaments. Because of the relation G210 Ì G20, in this work we will discuss only the categories G20, G21, G2,
while the category G210 will be discussed within the category G20. The definitions of symmetry groups will be derived directly from Bohm symbols: symmetry groups of rosettes are groups of isometries
of the space E2 (plane) with an invariant 0-dimensional subspace (point), symmetry groups of friezes are groups of isometries of the space E2 with an invariant 1-dimensional subspace (line) and
without invariant points, while symmetry groups of ornaments are groups of isometries of the space E2 without invariant subspaces (points, lines). The groups of the category Gn are called the space
groups, the groups of the category Gn1 the line groups, and the groups of the category Gn0 the point groups of the space En. With symmetry groups of friezes G21 and symmetry groups of ornaments G2, a
group contains one or two generating translations respectively, so that each of these groups has a translational subgroup. A lattice is the orbit of a point with respect to a discrete group of
translations. For the friezes it is a linear series of equidistant points while for ornaments we get a plane lattice or simply a lattice. Five different symmetry types of plane lattices bear the name
of Bravais lattices; the points of these lattices are defined by five different isohedral tessellations, which consist of parallelograms, rhombuses, rectangles, squares or regular hexagons. To
Bravais lattices correspond the crystal systems of the same names (Figure 1.10). Figure 1.10 Five plane Bravais lattices. Because the symmetry groups of friezes G21 are groups of isometries of the
plane E2 with an invariant line, they cannot have rotations of an order greater than 2. For the symmetry groups of ornaments G2 so-called crystallographic restriction holds, according to which
symmetry groups of ornaments can have only rotations of the order n=1,2,3,4,6. The term "crystallographic groups" is used for all groups which satisfy this condition, despite the category they belong
to. In isometry groups all distances between points under the effect of symmetries remain unchanged and the congruence of homologous figures is preserved. Consequently, the same holds for all other
geometric properties of such figures, so that the equiangularity (the congruence of the angles of homologous figures) and their equiformity (the same form of homologous figures) are the direct
consequences of isometrism. The next class of symmetry groups we shall consider are the similarity symmetry groups. A similarity transformation of the space En is a transformation which to each line
segment of length AB assigns a line segment of the length kAB whereby k is a real positive number, the coefficient of similarity. In particular, if k = 1 then a similarity transformation is an
isometry. According to the theorem on the existence of an invariant point of every similarity transformation which is not an isometry, there are, besides isometries, three types of similarity
symmetry transformations of the space E2: (i) central dilatation K (or simply dilatation), a transformation which to each vector (A,B) assigns the vector (A',B'), such that A' = K(A), B' = K(B) and
(A',B') = k(A,B), where the coefficient of the dilatation is k Î Â\{-1,0,1} ; (ii) dilative rotation L, the commutative composition of a central dilatation K and a rotation, with a common invariant
point; (iii) dilative reflection M, the commutative composition of a dilatation K and a reflection in the reflection line containing the invariant point (center) of the dilatation K (Figure 1.11).
Figure 1.11 (a) Dilatation; (b) dilative rotation; (c) dilative reflection. Those transformations are, in the given order, isomorphic with the isometries of the space E3: translation, twist and glide
reflection. They make possible the extension of the symmetry groups of rosettes G20 by the external automorphism, having as the result similarity symmetry groups S20 that we will, thanks to the
existence of the invariant point, call the similarity symmetry groups of rosettes. Dilatations K and dilative rotations L are direct, while dilative reflections M are indirect transformations. They
all possess the properties of equiangularity and equiformity. All other aspects of similarity symmetry groups (the problems of enantiomorphism, fundamental regions, tessellations,¼) will be discussed
analogously to the case of isometry groups. Further generalization leads to conformal transformations or circle preserving transformations of the plane E2\{O} ; for them the property of
equiangularity has been preserved, but not that of equiformity. We have, as the elementary transformation of conformal symmetry in E2\{O} , the circle inversion RI (or simply inversion) - an
involutional transformation isomorphic with a reflection, that gives to each point A in the plane E2\{O} a point A1 so that (O,A)°(O,A1) = r2, where r is the radius of the inversion circle c(O,r) and
O is the singular point of the plane E2\{O} (Figure 1.12). Just like a reflection, for which each point of the reflection line is invariant, an inversion maintains invariant each point of the
inversion circle. By discussing a line as a circle with an infinite radius (and treating as circles, at the same time and under the same term, circles and lines) it is possible to identify
reflections with circle inversions. In such a case, all circle inversions (including line reflections) and their compositions, can be discussed as circle preserving transformations, i.e.
transformations mapping circles (including lines) onto circles. Figure 1.12 Circle inversion. Besides the circle inversion RI, by composing it with isometries maintaining invariant the circle line c
of the inversion circle c(O,r) - with a reflection with reflection line containing the circle center O or with a rotation with the rotation center O, we have two more conformal transformations: (i)
inversional reflection ZI = RIR = RRI, the involutional transformation, the commutative composition of a reflection and a circle inversion; (ii) inversional rotation SI = SRI = RIS, the commutative
composition of a rotation and a circle inversion (Figure 1.13). Figure 1.13 (a) Inversional reflection; (b) inversional rotation. Those three conformal symmetry transformations, besides isometries
and similarity symmetry transformations, constitute the finite and infinite conformal symmetry groups C21, C2 - conformal symmetry groups of rosettes in E2\{O} . As an extension of the symmetry
groups of rosettes G20 we have the finite conformal symmetry groups C21 isomorphic with the symmetry groups of tablets G320. As a further extension of finite conformal symmetry groups C21 by the
similarity symmetry transformations K, L, M, we get the infinite conformal symmetry groups C2. The similarity symmetry groups S20 and the infinite conformal symmetry groups C2 are isomorphic with the
line symmetry groups of the space E3 - the symmetry groups of rods G31. In line with the isomorphism mentioned, all similarity symmetry and conformal symmetry transformations offer a reflectional
(canonic) representation by, at most, four reflections (reflections and circle inversions). By applying this isomorphism, ornamental motifs which correspond to the similarity symmetry and conformal
symmetry groups, satisfy one more scope of painting: adequate interpretation of space objects in the plane. The plane structures obtained are called generalized projections of the symmetry groups of
tablets G320 and rods G31. NEXT CONTENTS
Because the symmetry groups of friezes G21 are groups of isometries of the plane E2 with an invariant line, they cannot have rotations of an order greater than 2.
For the symmetry groups of ornaments G2 so-called crystallographic restriction holds, according to which symmetry groups of ornaments can have only rotations of the order n=1,2,3,4,6. The term
"crystallographic groups" is used for all groups which satisfy this condition, despite the category they belong to. In isometry groups all distances between points under the effect of symmetries
remain unchanged and the congruence of homologous figures is preserved. Consequently, the same holds for all other geometric properties of such figures, so that the equiangularity (the congruence of
the angles of homologous figures) and their equiformity (the same form of homologous figures) are the direct consequences of isometrism. The next class of symmetry groups we shall consider are the
similarity symmetry groups. A similarity transformation of the space En is a transformation which to each line segment of length AB assigns a line segment of the length kAB whereby k is a real
positive number, the coefficient of similarity. In particular, if k = 1 then a similarity transformation is an isometry. According to the theorem on the existence of an invariant point of every
similarity transformation which is not an isometry, there are, besides isometries, three types of similarity symmetry transformations of the space E2: (i) central dilatation K (or simply dilatation),
a transformation which to each vector (A,B) assigns the vector (A',B'), such that A' = K(A), B' = K(B) and (A',B') = k(A,B), where the coefficient of the dilatation is k Î Â\{-1,0,1} ; (ii) dilative
rotation L, the commutative composition of a central dilatation K and a rotation, with a common invariant point; (iii) dilative reflection M, the commutative composition of a dilatation K and a
reflection in the reflection line containing the invariant point (center) of the dilatation K (Figure 1.11). Figure 1.11 (a) Dilatation; (b) dilative rotation; (c) dilative reflection. Those
transformations are, in the given order, isomorphic with the isometries of the space E3: translation, twist and glide reflection. They make possible the extension of the symmetry groups of rosettes
G20 by the external automorphism, having as the result similarity symmetry groups S20 that we will, thanks to the existence of the invariant point, call the similarity symmetry groups of rosettes.
Dilatations K and dilative rotations L are direct, while dilative reflections M are indirect transformations. They all possess the properties of equiangularity and equiformity. All other aspects of
similarity symmetry groups (the problems of enantiomorphism, fundamental regions, tessellations,¼) will be discussed analogously to the case of isometry groups. Further generalization leads to
conformal transformations or circle preserving transformations of the plane E2\{O} ; for them the property of equiangularity has been preserved, but not that of equiformity. We have, as the
elementary transformation of conformal symmetry in E2\{O} , the circle inversion RI (or simply inversion) - an involutional transformation isomorphic with a reflection, that gives to each point A in
the plane E2\{O} a point A1 so that (O,A)°(O,A1) = r2, where r is the radius of the inversion circle c(O,r) and O is the singular point of the plane E2\{O} (Figure 1.12). Just like a reflection, for
which each point of the reflection line is invariant, an inversion maintains invariant each point of the inversion circle. By discussing a line as a circle with an infinite radius (and treating as
circles, at the same time and under the same term, circles and lines) it is possible to identify reflections with circle inversions. In such a case, all circle inversions (including line reflections)
and their compositions, can be discussed as circle preserving transformations, i.e. transformations mapping circles (including lines) onto circles. Figure 1.12 Circle inversion. Besides the circle
inversion RI, by composing it with isometries maintaining invariant the circle line c of the inversion circle c(O,r) - with a reflection with reflection line containing the circle center O or with a
rotation with the rotation center O, we have two more conformal transformations: (i) inversional reflection ZI = RIR = RRI, the involutional transformation, the commutative composition of a
reflection and a circle inversion; (ii) inversional rotation SI = SRI = RIS, the commutative composition of a rotation and a circle inversion (Figure 1.13). Figure 1.13 (a) Inversional reflection;
(b) inversional rotation. Those three conformal symmetry transformations, besides isometries and similarity symmetry transformations, constitute the finite and infinite conformal symmetry groups C21,
C2 - conformal symmetry groups of rosettes in E2\{O} . As an extension of the symmetry groups of rosettes G20 we have the finite conformal symmetry groups C21 isomorphic with the symmetry groups of
tablets G320. As a further extension of finite conformal symmetry groups C21 by the similarity symmetry transformations K, L, M, we get the infinite conformal symmetry groups C2. The similarity
symmetry groups S20 and the infinite conformal symmetry groups C2 are isomorphic with the line symmetry groups of the space E3 - the symmetry groups of rods G31. In line with the isomorphism
mentioned, all similarity symmetry and conformal symmetry transformations offer a reflectional (canonic) representation by, at most, four reflections (reflections and circle inversions). By applying
this isomorphism, ornamental motifs which correspond to the similarity symmetry and conformal symmetry groups, satisfy one more scope of painting: adequate interpretation of space objects in the
plane. The plane structures obtained are called generalized projections of the symmetry groups of tablets G320 and rods G31. NEXT CONTENTS
In isometry groups all distances between points under the effect of symmetries remain unchanged and the congruence of homologous figures is preserved. Consequently, the same holds for all other
geometric properties of such figures, so that the equiangularity (the congruence of the angles of homologous figures) and their equiformity (the same form of homologous figures) are the direct
consequences of isometrism. The next class of symmetry groups we shall consider are the similarity symmetry groups. A similarity transformation of the space En is a transformation which to each line
segment of length AB assigns a line segment of the length kAB whereby k is a real positive number, the coefficient of similarity. In particular, if k = 1 then a similarity transformation is an
isometry. According to the theorem on the existence of an invariant point of every similarity transformation which is not an isometry, there are, besides isometries, three types of similarity
symmetry transformations of the space E2: (i) central dilatation K (or simply dilatation), a transformation which to each vector (A,B) assigns the vector (A',B'), such that A' = K(A), B' = K(B) and
(A',B') = k(A,B), where the coefficient of the dilatation is k Î Â\{-1,0,1} ; (ii) dilative rotation L, the commutative composition of a central dilatation K and a rotation, with a common invariant
point; (iii) dilative reflection M, the commutative composition of a dilatation K and a reflection in the reflection line containing the invariant point (center) of the dilatation K (Figure 1.11).
Figure 1.11 (a) Dilatation; (b) dilative rotation; (c) dilative reflection. Those transformations are, in the given order, isomorphic with the isometries of the space E3: translation, twist and glide
reflection. They make possible the extension of the symmetry groups of rosettes G20 by the external automorphism, having as the result similarity symmetry groups S20 that we will, thanks to the
existence of the invariant point, call the similarity symmetry groups of rosettes. Dilatations K and dilative rotations L are direct, while dilative reflections M are indirect transformations. They
all possess the properties of equiangularity and equiformity. All other aspects of similarity symmetry groups (the problems of enantiomorphism, fundamental regions, tessellations,¼) will be discussed
analogously to the case of isometry groups. Further generalization leads to conformal transformations or circle preserving transformations of the plane E2\{O} ; for them the property of
equiangularity has been preserved, but not that of equiformity. We have, as the elementary transformation of conformal symmetry in E2\{O} , the circle inversion RI (or simply inversion) - an
involutional transformation isomorphic with a reflection, that gives to each point A in the plane E2\{O} a point A1 so that (O,A)°(O,A1) = r2, where r is the radius of the inversion circle c(O,r) and
O is the singular point of the plane E2\{O} (Figure 1.12). Just like a reflection, for which each point of the reflection line is invariant, an inversion maintains invariant each point of the
inversion circle. By discussing a line as a circle with an infinite radius (and treating as circles, at the same time and under the same term, circles and lines) it is possible to identify
reflections with circle inversions. In such a case, all circle inversions (including line reflections) and their compositions, can be discussed as circle preserving transformations, i.e.
transformations mapping circles (including lines) onto circles. Figure 1.12 Circle inversion. Besides the circle inversion RI, by composing it with isometries maintaining invariant the circle line c
of the inversion circle c(O,r) - with a reflection with reflection line containing the circle center O or with a rotation with the rotation center O, we have two more conformal transformations: (i)
inversional reflection ZI = RIR = RRI, the involutional transformation, the commutative composition of a reflection and a circle inversion; (ii) inversional rotation SI = SRI = RIS, the commutative
composition of a rotation and a circle inversion (Figure 1.13). Figure 1.13 (a) Inversional reflection; (b) inversional rotation. Those three conformal symmetry transformations, besides isometries
and similarity symmetry transformations, constitute the finite and infinite conformal symmetry groups C21, C2 - conformal symmetry groups of rosettes in E2\{O} . As an extension of the symmetry
groups of rosettes G20 we have the finite conformal symmetry groups C21 isomorphic with the symmetry groups of tablets G320. As a further extension of finite conformal symmetry groups C21 by the
similarity symmetry transformations K, L, M, we get the infinite conformal symmetry groups C2. The similarity symmetry groups S20 and the infinite conformal symmetry groups C2 are isomorphic with the
line symmetry groups of the space E3 - the symmetry groups of rods G31. In line with the isomorphism mentioned, all similarity symmetry and conformal symmetry transformations offer a reflectional
(canonic) representation by, at most, four reflections (reflections and circle inversions). By applying this isomorphism, ornamental motifs which correspond to the similarity symmetry and conformal
symmetry groups, satisfy one more scope of painting: adequate interpretation of space objects in the plane. The plane structures obtained are called generalized projections of the symmetry groups of
tablets G320 and rods G31. NEXT CONTENTS
The next class of symmetry groups we shall consider are the similarity symmetry groups. A similarity transformation of the space En is a transformation which to each line segment of length AB assigns
a line segment of the length kAB whereby k is a real positive number, the coefficient of similarity. In particular, if k = 1 then a similarity transformation is an isometry. According to the theorem
on the existence of an invariant point of every similarity transformation which is not an isometry, there are, besides isometries, three types of similarity symmetry transformations of the space E2:
(i) central dilatation K (or simply dilatation), a transformation which to each vector (A,B) assigns the vector (A',B'), such that A' = K(A), B' = K(B) and (A',B') = k(A,B), where the coefficient of
the dilatation is k Î Â\{-1,0,1} ; (ii) dilative rotation L, the commutative composition of a central dilatation K and a rotation, with a common invariant point; (iii) dilative reflection M, the
commutative composition of a dilatation K and a reflection in the reflection line containing the invariant point (center) of the dilatation K (Figure 1.11). Figure 1.11 (a) Dilatation; (b) dilative
rotation; (c) dilative reflection. Those transformations are, in the given order, isomorphic with the isometries of the space E3: translation, twist and glide reflection. They make possible the
extension of the symmetry groups of rosettes G20 by the external automorphism, having as the result similarity symmetry groups S20 that we will, thanks to the existence of the invariant point, call
the similarity symmetry groups of rosettes. Dilatations K and dilative rotations L are direct, while dilative reflections M are indirect transformations. They all possess the properties of
equiangularity and equiformity. All other aspects of similarity symmetry groups (the problems of enantiomorphism, fundamental regions, tessellations,¼) will be discussed analogously to the case of
isometry groups. Further generalization leads to conformal transformations or circle preserving transformations of the plane E2\{O} ; for them the property of equiangularity has been preserved, but
not that of equiformity. We have, as the elementary transformation of conformal symmetry in E2\{O} , the circle inversion RI (or simply inversion) - an involutional transformation isomorphic with a
reflection, that gives to each point A in the plane E2\{O} a point A1 so that (O,A)°(O,A1) = r2, where r is the radius of the inversion circle c(O,r) and O is the singular point of the plane E2\{O}
(Figure 1.12). Just like a reflection, for which each point of the reflection line is invariant, an inversion maintains invariant each point of the inversion circle. By discussing a line as a circle
with an infinite radius (and treating as circles, at the same time and under the same term, circles and lines) it is possible to identify reflections with circle inversions. In such a case, all
circle inversions (including line reflections) and their compositions, can be discussed as circle preserving transformations, i.e. transformations mapping circles (including lines) onto circles.
Figure 1.12 Circle inversion. Besides the circle inversion RI, by composing it with isometries maintaining invariant the circle line c of the inversion circle c(O,r) - with a reflection with
reflection line containing the circle center O or with a rotation with the rotation center O, we have two more conformal transformations: (i) inversional reflection ZI = RIR = RRI, the involutional
transformation, the commutative composition of a reflection and a circle inversion; (ii) inversional rotation SI = SRI = RIS, the commutative composition of a rotation and a circle inversion (Figure
1.13). Figure 1.13 (a) Inversional reflection; (b) inversional rotation. Those three conformal symmetry transformations, besides isometries and similarity symmetry transformations, constitute the
finite and infinite conformal symmetry groups C21, C2 - conformal symmetry groups of rosettes in E2\{O} . As an extension of the symmetry groups of rosettes G20 we have the finite conformal symmetry
groups C21 isomorphic with the symmetry groups of tablets G320. As a further extension of finite conformal symmetry groups C21 by the similarity symmetry transformations K, L, M, we get the infinite
conformal symmetry groups C2. The similarity symmetry groups S20 and the infinite conformal symmetry groups C2 are isomorphic with the line symmetry groups of the space E3 - the symmetry groups of
rods G31. In line with the isomorphism mentioned, all similarity symmetry and conformal symmetry transformations offer a reflectional (canonic) representation by, at most, four reflections
(reflections and circle inversions). By applying this isomorphism, ornamental motifs which correspond to the similarity symmetry and conformal symmetry groups, satisfy one more scope of painting:
adequate interpretation of space objects in the plane. The plane structures obtained are called generalized projections of the symmetry groups of tablets G320 and rods G31. NEXT CONTENTS
(i) central dilatation K (or simply dilatation), a transformation which to each vector (A,B) assigns the vector (A',B'), such that A' = K(A), B' = K(B) and (A',B') = k(A,B), where the coefficient of
the dilatation is k Î Â\{-1,0,1} ; (ii) dilative rotation L, the commutative composition of a central dilatation K and a rotation, with a common invariant point; (iii) dilative reflection M, the
commutative composition of a dilatation K and a reflection in the reflection line containing the invariant point (center) of the dilatation K (Figure 1.11). Figure 1.11 (a) Dilatation; (b) dilative
rotation; (c) dilative reflection. Those transformations are, in the given order, isomorphic with the isometries of the space E3: translation, twist and glide reflection. They make possible the
extension of the symmetry groups of rosettes G20 by the external automorphism, having as the result similarity symmetry groups S20 that we will, thanks to the existence of the invariant point, call
the similarity symmetry groups of rosettes. Dilatations K and dilative rotations L are direct, while dilative reflections M are indirect transformations. They all possess the properties of
equiangularity and equiformity. All other aspects of similarity symmetry groups (the problems of enantiomorphism, fundamental regions, tessellations,¼) will be discussed analogously to the case of
isometry groups. Further generalization leads to conformal transformations or circle preserving transformations of the plane E2\{O} ; for them the property of equiangularity has been preserved, but
not that of equiformity. We have, as the elementary transformation of conformal symmetry in E2\{O} , the circle inversion RI (or simply inversion) - an involutional transformation isomorphic with a
reflection, that gives to each point A in the plane E2\{O} a point A1 so that (O,A)°(O,A1) = r2, where r is the radius of the inversion circle c(O,r) and O is the singular point of the plane E2\{O}
(Figure 1.12). Just like a reflection, for which each point of the reflection line is invariant, an inversion maintains invariant each point of the inversion circle. By discussing a line as a circle
with an infinite radius (and treating as circles, at the same time and under the same term, circles and lines) it is possible to identify reflections with circle inversions. In such a case, all
circle inversions (including line reflections) and their compositions, can be discussed as circle preserving transformations, i.e. transformations mapping circles (including lines) onto circles.
Figure 1.12 Circle inversion. Besides the circle inversion RI, by composing it with isometries maintaining invariant the circle line c of the inversion circle c(O,r) - with a reflection with
reflection line containing the circle center O or with a rotation with the rotation center O, we have two more conformal transformations: (i) inversional reflection ZI = RIR = RRI, the involutional
transformation, the commutative composition of a reflection and a circle inversion; (ii) inversional rotation SI = SRI = RIS, the commutative composition of a rotation and a circle inversion (Figure
1.13). Figure 1.13 (a) Inversional reflection; (b) inversional rotation. Those three conformal symmetry transformations, besides isometries and similarity symmetry transformations, constitute the
finite and infinite conformal symmetry groups C21, C2 - conformal symmetry groups of rosettes in E2\{O} . As an extension of the symmetry groups of rosettes G20 we have the finite conformal symmetry
groups C21 isomorphic with the symmetry groups of tablets G320. As a further extension of finite conformal symmetry groups C21 by the similarity symmetry transformations K, L, M, we get the infinite
conformal symmetry groups C2. The similarity symmetry groups S20 and the infinite conformal symmetry groups C2 are isomorphic with the line symmetry groups of the space E3 - the symmetry groups of
rods G31. In line with the isomorphism mentioned, all similarity symmetry and conformal symmetry transformations offer a reflectional (canonic) representation by, at most, four reflections
(reflections and circle inversions). By applying this isomorphism, ornamental motifs which correspond to the similarity symmetry and conformal symmetry groups, satisfy one more scope of painting:
adequate interpretation of space objects in the plane. The plane structures obtained are called generalized projections of the symmetry groups of tablets G320 and rods G31. NEXT CONTENTS
(ii) dilative rotation L, the commutative composition of a central dilatation K and a rotation, with a common invariant point; (iii) dilative reflection M, the commutative composition of a dilatation
K and a reflection in the reflection line containing the invariant point (center) of the dilatation K (Figure 1.11). Figure 1.11 (a) Dilatation; (b) dilative rotation; (c) dilative reflection. Those
transformations are, in the given order, isomorphic with the isometries of the space E3: translation, twist and glide reflection. They make possible the extension of the symmetry groups of rosettes
G20 by the external automorphism, having as the result similarity symmetry groups S20 that we will, thanks to the existence of the invariant point, call the similarity symmetry groups of rosettes.
Dilatations K and dilative rotations L are direct, while dilative reflections M are indirect transformations. They all possess the properties of equiangularity and equiformity. All other aspects of
similarity symmetry groups (the problems of enantiomorphism, fundamental regions, tessellations,¼) will be discussed analogously to the case of isometry groups. Further generalization leads to
conformal transformations or circle preserving transformations of the plane E2\{O} ; for them the property of equiangularity has been preserved, but not that of equiformity. We have, as the
elementary transformation of conformal symmetry in E2\{O} , the circle inversion RI (or simply inversion) - an involutional transformation isomorphic with a reflection, that gives to each point A in
the plane E2\{O} a point A1 so that (O,A)°(O,A1) = r2, where r is the radius of the inversion circle c(O,r) and O is the singular point of the plane E2\{O} (Figure 1.12). Just like a reflection, for
which each point of the reflection line is invariant, an inversion maintains invariant each point of the inversion circle. By discussing a line as a circle with an infinite radius (and treating as
circles, at the same time and under the same term, circles and lines) it is possible to identify reflections with circle inversions. In such a case, all circle inversions (including line reflections)
and their compositions, can be discussed as circle preserving transformations, i.e. transformations mapping circles (including lines) onto circles. Figure 1.12 Circle inversion. Besides the circle
inversion RI, by composing it with isometries maintaining invariant the circle line c of the inversion circle c(O,r) - with a reflection with reflection line containing the circle center O or with a
rotation with the rotation center O, we have two more conformal transformations: (i) inversional reflection ZI = RIR = RRI, the involutional transformation, the commutative composition of a
reflection and a circle inversion; (ii) inversional rotation SI = SRI = RIS, the commutative composition of a rotation and a circle inversion (Figure 1.13). Figure 1.13 (a) Inversional reflection;
(b) inversional rotation. Those three conformal symmetry transformations, besides isometries and similarity symmetry transformations, constitute the finite and infinite conformal symmetry groups C21,
C2 - conformal symmetry groups of rosettes in E2\{O} . As an extension of the symmetry groups of rosettes G20 we have the finite conformal symmetry groups C21 isomorphic with the symmetry groups of
tablets G320. As a further extension of finite conformal symmetry groups C21 by the similarity symmetry transformations K, L, M, we get the infinite conformal symmetry groups C2. The similarity
symmetry groups S20 and the infinite conformal symmetry groups C2 are isomorphic with the line symmetry groups of the space E3 - the symmetry groups of rods G31. In line with the isomorphism
mentioned, all similarity symmetry and conformal symmetry transformations offer a reflectional (canonic) representation by, at most, four reflections (reflections and circle inversions). By applying
this isomorphism, ornamental motifs which correspond to the similarity symmetry and conformal symmetry groups, satisfy one more scope of painting: adequate interpretation of space objects in the
plane. The plane structures obtained are called generalized projections of the symmetry groups of tablets G320 and rods G31. NEXT CONTENTS
(iii) dilative reflection M, the commutative composition of a dilatation K and a reflection in the reflection line containing the invariant point (center) of the dilatation K (Figure 1.11). Figure
1.11 (a) Dilatation; (b) dilative rotation; (c) dilative reflection. Those transformations are, in the given order, isomorphic with the isometries of the space E3: translation, twist and glide
reflection. They make possible the extension of the symmetry groups of rosettes G20 by the external automorphism, having as the result similarity symmetry groups S20 that we will, thanks to the
existence of the invariant point, call the similarity symmetry groups of rosettes. Dilatations K and dilative rotations L are direct, while dilative reflections M are indirect transformations. They
all possess the properties of equiangularity and equiformity. All other aspects of similarity symmetry groups (the problems of enantiomorphism, fundamental regions, tessellations,¼) will be discussed
analogously to the case of isometry groups. Further generalization leads to conformal transformations or circle preserving transformations of the plane E2\{O} ; for them the property of
equiangularity has been preserved, but not that of equiformity. We have, as the elementary transformation of conformal symmetry in E2\{O} , the circle inversion RI (or simply inversion) - an
involutional transformation isomorphic with a reflection, that gives to each point A in the plane E2\{O} a point A1 so that (O,A)°(O,A1) = r2, where r is the radius of the inversion circle c(O,r) and
O is the singular point of the plane E2\{O} (Figure 1.12). Just like a reflection, for which each point of the reflection line is invariant, an inversion maintains invariant each point of the
inversion circle. By discussing a line as a circle with an infinite radius (and treating as circles, at the same time and under the same term, circles and lines) it is possible to identify
reflections with circle inversions. In such a case, all circle inversions (including line reflections) and their compositions, can be discussed as circle preserving transformations, i.e.
transformations mapping circles (including lines) onto circles. Figure 1.12 Circle inversion. Besides the circle inversion RI, by composing it with isometries maintaining invariant the circle line c
of the inversion circle c(O,r) - with a reflection with reflection line containing the circle center O or with a rotation with the rotation center O, we have two more conformal transformations: (i)
inversional reflection ZI = RIR = RRI, the involutional transformation, the commutative composition of a reflection and a circle inversion; (ii) inversional rotation SI = SRI = RIS, the commutative
composition of a rotation and a circle inversion (Figure 1.13). Figure 1.13 (a) Inversional reflection; (b) inversional rotation. Those three conformal symmetry transformations, besides isometries
and similarity symmetry transformations, constitute the finite and infinite conformal symmetry groups C21, C2 - conformal symmetry groups of rosettes in E2\{O} . As an extension of the symmetry
groups of rosettes G20 we have the finite conformal symmetry groups C21 isomorphic with the symmetry groups of tablets G320. As a further extension of finite conformal symmetry groups C21 by the
similarity symmetry transformations K, L, M, we get the infinite conformal symmetry groups C2. The similarity symmetry groups S20 and the infinite conformal symmetry groups C2 are isomorphic with the
line symmetry groups of the space E3 - the symmetry groups of rods G31. In line with the isomorphism mentioned, all similarity symmetry and conformal symmetry transformations offer a reflectional
(canonic) representation by, at most, four reflections (reflections and circle inversions). By applying this isomorphism, ornamental motifs which correspond to the similarity symmetry and conformal
symmetry groups, satisfy one more scope of painting: adequate interpretation of space objects in the plane. The plane structures obtained are called generalized projections of the symmetry groups of
tablets G320 and rods G31. NEXT CONTENTS
Those transformations are, in the given order, isomorphic with the isometries of the space E3: translation, twist and glide reflection. They make possible the extension of the symmetry groups of
rosettes G20 by the external automorphism, having as the result similarity symmetry groups S20 that we will, thanks to the existence of the invariant point, call the similarity symmetry groups of
Dilatations K and dilative rotations L are direct, while dilative reflections M are indirect transformations. They all possess the properties of equiangularity and equiformity. All other aspects of
similarity symmetry groups (the problems of enantiomorphism, fundamental regions, tessellations,¼) will be discussed analogously to the case of isometry groups.
Further generalization leads to conformal transformations or circle preserving transformations of the plane E2\{O} ; for them the property of equiangularity has been preserved, but not that of
equiformity. We have, as the elementary transformation of conformal symmetry in E2\{O} , the circle inversion RI (or simply inversion) - an involutional transformation isomorphic with a reflection,
that gives to each point A in the plane E2\{O} a point A1 so that (O,A)°(O,A1) = r2, where r is the radius of the inversion circle c(O,r) and O is the singular point of the plane E2\{O} (Figure
1.12). Just like a reflection, for which each point of the reflection line is invariant, an inversion maintains invariant each point of the inversion circle. By discussing a line as a circle with an
infinite radius (and treating as circles, at the same time and under the same term, circles and lines) it is possible to identify reflections with circle inversions. In such a case, all circle
inversions (including line reflections) and their compositions, can be discussed as circle preserving transformations, i.e. transformations mapping circles (including lines) onto circles. Figure 1.12
Circle inversion. Besides the circle inversion RI, by composing it with isometries maintaining invariant the circle line c of the inversion circle c(O,r) - with a reflection with reflection line
containing the circle center O or with a rotation with the rotation center O, we have two more conformal transformations: (i) inversional reflection ZI = RIR = RRI, the involutional transformation,
the commutative composition of a reflection and a circle inversion; (ii) inversional rotation SI = SRI = RIS, the commutative composition of a rotation and a circle inversion (Figure 1.13). Figure
1.13 (a) Inversional reflection; (b) inversional rotation. Those three conformal symmetry transformations, besides isometries and similarity symmetry transformations, constitute the finite and
infinite conformal symmetry groups C21, C2 - conformal symmetry groups of rosettes in E2\{O} . As an extension of the symmetry groups of rosettes G20 we have the finite conformal symmetry groups C21
isomorphic with the symmetry groups of tablets G320. As a further extension of finite conformal symmetry groups C21 by the similarity symmetry transformations K, L, M, we get the infinite conformal
symmetry groups C2. The similarity symmetry groups S20 and the infinite conformal symmetry groups C2 are isomorphic with the line symmetry groups of the space E3 - the symmetry groups of rods G31. In
line with the isomorphism mentioned, all similarity symmetry and conformal symmetry transformations offer a reflectional (canonic) representation by, at most, four reflections (reflections and circle
inversions). By applying this isomorphism, ornamental motifs which correspond to the similarity symmetry and conformal symmetry groups, satisfy one more scope of painting: adequate interpretation of
space objects in the plane. The plane structures obtained are called generalized projections of the symmetry groups of tablets G320 and rods G31. NEXT CONTENTS
Besides the circle inversion RI, by composing it with isometries maintaining invariant the circle line c of the inversion circle c(O,r) - with a reflection with reflection line containing the circle
center O or with a rotation with the rotation center O, we have two more conformal transformations:
(i) inversional reflection ZI = RIR = RRI, the involutional transformation, the commutative composition of a reflection and a circle inversion; (ii) inversional rotation SI = SRI = RIS, the
commutative composition of a rotation and a circle inversion (Figure 1.13). Figure 1.13 (a) Inversional reflection; (b) inversional rotation. Those three conformal symmetry transformations, besides
isometries and similarity symmetry transformations, constitute the finite and infinite conformal symmetry groups C21, C2 - conformal symmetry groups of rosettes in E2\{O} . As an extension of the
symmetry groups of rosettes G20 we have the finite conformal symmetry groups C21 isomorphic with the symmetry groups of tablets G320. As a further extension of finite conformal symmetry groups C21 by
the similarity symmetry transformations K, L, M, we get the infinite conformal symmetry groups C2. The similarity symmetry groups S20 and the infinite conformal symmetry groups C2 are isomorphic with
the line symmetry groups of the space E3 - the symmetry groups of rods G31. In line with the isomorphism mentioned, all similarity symmetry and conformal symmetry transformations offer a reflectional
(canonic) representation by, at most, four reflections (reflections and circle inversions). By applying this isomorphism, ornamental motifs which correspond to the similarity symmetry and conformal
symmetry groups, satisfy one more scope of painting: adequate interpretation of space objects in the plane. The plane structures obtained are called generalized projections of the symmetry groups of
tablets G320 and rods G31. NEXT CONTENTS
(ii) inversional rotation SI = SRI = RIS, the commutative composition of a rotation and a circle inversion (Figure 1.13). Figure 1.13 (a) Inversional reflection; (b) inversional rotation. Those three
conformal symmetry transformations, besides isometries and similarity symmetry transformations, constitute the finite and infinite conformal symmetry groups C21, C2 - conformal symmetry groups of
rosettes in E2\{O} . As an extension of the symmetry groups of rosettes G20 we have the finite conformal symmetry groups C21 isomorphic with the symmetry groups of tablets G320. As a further
extension of finite conformal symmetry groups C21 by the similarity symmetry transformations K, L, M, we get the infinite conformal symmetry groups C2. The similarity symmetry groups S20 and the
infinite conformal symmetry groups C2 are isomorphic with the line symmetry groups of the space E3 - the symmetry groups of rods G31. In line with the isomorphism mentioned, all similarity symmetry
and conformal symmetry transformations offer a reflectional (canonic) representation by, at most, four reflections (reflections and circle inversions). By applying this isomorphism, ornamental motifs
which correspond to the similarity symmetry and conformal symmetry groups, satisfy one more scope of painting: adequate interpretation of space objects in the plane. The plane structures obtained are
called generalized projections of the symmetry groups of tablets G320 and rods G31. NEXT CONTENTS
Those three conformal symmetry transformations, besides isometries and similarity symmetry transformations, constitute the finite and infinite conformal symmetry groups C21, C2 - conformal symmetry
groups of rosettes in E2\{O} . As an extension of the symmetry groups of rosettes G20 we have the finite conformal symmetry groups C21 isomorphic with the symmetry groups of tablets G320. As a
further extension of finite conformal symmetry groups C21 by the similarity symmetry transformations K, L, M, we get the infinite conformal symmetry groups C2. The similarity symmetry groups S20 and
the infinite conformal symmetry groups C2 are isomorphic with the line symmetry groups of the space E3 - the symmetry groups of rods G31. In line with the isomorphism mentioned, all similarity
symmetry and conformal symmetry transformations offer a reflectional (canonic) representation by, at most, four reflections (reflections and circle inversions). By applying this isomorphism,
ornamental motifs which correspond to the similarity symmetry and conformal symmetry groups, satisfy one more scope of painting: adequate interpretation of space objects in the plane. The plane
structures obtained are called generalized projections of the symmetry groups of tablets G320 and rods G31. NEXT CONTENTS
As an extension of the symmetry groups of rosettes G20 we have the finite conformal symmetry groups C21 isomorphic with the symmetry groups of tablets G320. As a further extension of finite conformal
symmetry groups C21 by the similarity symmetry transformations K, L, M, we get the infinite conformal symmetry groups C2. The similarity symmetry groups S20 and the infinite conformal symmetry groups
C2 are isomorphic with the line symmetry groups of the space E3 - the symmetry groups of rods G31. In line with the isomorphism mentioned, all similarity symmetry and conformal symmetry
transformations offer a reflectional (canonic) representation by, at most, four reflections (reflections and circle inversions). By applying this isomorphism, ornamental motifs which correspond to
the similarity symmetry and conformal symmetry groups, satisfy one more scope of painting: adequate interpretation of space objects in the plane. The plane structures obtained are called generalized
projections of the symmetry groups of tablets G320 and rods G31. NEXT CONTENTS | {"url":"http://www.emis.de/monographs/jablan/chap13.htm","timestamp":"2014-04-16T07:25:21Z","content_type":null,"content_length":"28330","record_id":"<urn:uuid:891c1608-c319-4ff6-b7b3-a4484d3cb127>","cc-path":"CC-MAIN-2014-15/segments/1397609521558.37/warc/CC-MAIN-20140416005201-00397-ip-10-147-4-33.ec2.internal.warc.gz"} |
What is Lagrange Inversion good for?
up vote 39 down vote favorite
I am planning an introductory combinatorics course (mixed grad-undergrad) and am trying to decide whether it is worth budgeting a day for Lagrange inversion. The reason I hesitate is that I know of
very few applications for it -- basically just enumeration of trees and some slight variants on this. I checked van Lint and Wilson, Enumerative Combinatorics II (but not the exercises) and Concrete
Mathematics, and they all only present this application.
So, besides counting trees, where can we use Lagrange inversion?
mathematics-education co.combinatorics generating-functions
1 Have you checked Bergeron, Labelle, and Leroux? Maybe Flajolet and Sedgewick? – Qiaochu Yuan Jul 16 '10 at 1:39
1 I can suggest looking at Christian Krattenthaler's papers (mat.univie.ac.at/~kratt/papers.html). It looks like $q$-Lagrange inversions have much more applications. – Wadim Zudilin Jul 16 '10 at
Have you looked at Krantz-Parks, "The Implicit Function Theorem"? I don't remember whether they give references to combinatorial applications. – Andres Caicedo Jul 16 '10 at 6:04
1 A good survey paper is Lagrange Inversion by Josef Hofbauer, Seminaire Lotharingien de Combinatoire B06a (1982) – Johann Cigler Jul 17 '10 at 16:17
1 I got some very good hint answers in a similar questions time ago: mathoverflow.net/questions/25491/… – Pietro Majer Oct 23 '11 at 10:29
show 3 more comments
7 Answers
active oldest votes
You can use Lagrange inversion to explicitly solve
$$x^5-x-a=0\qquad (*)$$
(yes, a fifth degree equation, gasp). More precisely, it yields an infinite series expansion
up vote
47 down $$x=-\sum_{k\geq 0}\binom{5k}{k}\frac{a^{4k+1}}{4k+1}$$
for the root of $(*)$ which is $0$ at $a=0.$ Although this isn't combinatorics, I'd gladly devote a class in any subject I teach to be able to derive it, because by Bring–Jerrard, any
quintic equation can be reduced to this form, and you get a solution of something that many people believe, albeit for differing reasons, to be unsolvable.
6 Just as the generating function identity c = 1 + xc^2 defines binary trees, the generating function identity c = 1 + xc^n defines n-ary trees, and there is a proof of the closed formula
for the Catalan numbers which generalizes to n-ary trees. The rest is just a change of variables, as far as I can tell. – Qiaochu Yuan Jul 17 '10 at 6:10
16 The application to the quintic has a very interesting history. It was first discovered by Lambert in 1757 (before Lagrange, and before the Bring-Gerrard reduction of the quintic was
known), rediscovered by Eisenstein when he was 14 (!) and later mentioned by him in a footnote to an 1844 paper. – John Stillwell Jul 17 '10 at 11:26
2 Thank you, John! I was hoping that you would come along and comment on the history. For those who don't know: John Stillwell wrote an article "Eisenstein's footnote" in the Math
Intelligencer explaining the history. Unfortunately, I couldn't view the whole thing, but here is the first page: resources.metapress.com/… – Victor Protsak Jul 17 '10 at 18:02
Hi Thierry, it was a figure of speech, but it reflects my frustration with what might be called "the mythology of mathematics". While impossibility proofs receive a good deal of
6 emphasis, very ingenious constructions, such as the conchoid of Nicomedes (trisection of the angle), the solution of Archytas of the doubling of the cube based on the curve of
intersection of a cylinder and a degenerate torus, and, of course, Hermite's and Eisenstein's formulas for the quintic have become obscure and many mathematicians don't even learn about
them any more. – Victor Protsak Feb 24 '11 at 4:58
Another combinatorial interpretation: it's the generalization of counting triangulations of a polygon where instead of triangles you tile with quadrilaterals, pentagons, etc. The
3 quintic case counts hexagons.$$ $$ While I'm at it: see the "one-page papers" catalan and catalan2 at math.harvard.edu/~elkies/Misc/index.html#papers for elementary derivations of these
power series without Lagrange or residue calculus. – Noam D. Elkies Nov 29 '11 at 0:06
show 5 more comments
1. The Lagrange inversion theorem is the essential tool needed to prove results like the following: Let $F(x)$ be the unique power series with rational coefficients such that for all $n\geq
0$, the coefficient of $x^n$ in $F(x)^{n+1}$ is 1. Then $F(x)=x/(1-e^{-x})$. For an application to algebraic geometry, see Lemma 1.7.1 of F. Hirzebruch, Topological Methods in Algebraic
up vote 2. An alternating tree is a tree on the vertex set $\{1,\dots,n\}$ such that every vertex is either greater than all its neighbors or less than all its neighbors. Alternating trees arise in
27 down such contexts as the general hypergeometric systems of Gelfand and his collaborators, and in the combinatorics of the Linial hyperplane arrangement. Let $f(n-1)$ be the number of
vote alternating trees on the vertex set $\{1,\dots,n\}$. Then $$ f(n) =\frac{1}{2^n}\sum_{k=0}^n {n\choose k}(k+1)^{n-1}. $$ So far as I know, the only known proof uses Lagrange inversion.
(While this is a tree enumeration result, it is of a different nature than the standard applications of Lagrange inversion to tree enumeration.)
2 I just tried skimming through that part of Hirzebruch. I wasn't able to tell whether that lemma is important for the proof of H-R-R? – Kevin H. Lin Jul 16 '10 at 18:39
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In combinatorics, applications are more general than just counting trees. In general context, Lagrange inversion is used to obtain a generating function $\sum c_n t^n$ for the numbers
$c_n$ of the form $$c_n = [x^n] f(x) g(x)^n.$$
Perhaps, the simplest example is a generating function for $c_n = \binom{2n}{n}$ treated as the coefficient of $x^n$ in $(1+x)^{2n}$ (i.e., $f(x)=1$ and $g(x)=(1+x)^2$). However, in
this case it is not hard to get the anticipated generating function $(1-4t)^{-1/2}$ by other means.
up vote 7 More sophisticated examples:
down vote
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Looking for a reference to this question I have realized that there are important applications of Lagrange's inversion formula in asymptotical analysis (although its role in the implicit
function theorems is already noted by Andres Caicedo). N.G. de Bruijn's Asymptotic Methods in Analysis, Section 2.3, gives some explicit examples: for instance, one can use the Lagrange
up vote 6 inversion for computing the asymptotics of the positive root of the equation $xe^x=1/t$ as $t\to\infty$.
down vote
add comment
The reps of the Lagrange inversion formula (LIF) in different “coordinate systems” are intrinsically interesting.
Consider a compositional inverse pair of functions, $h$ and $h^{-1}$, analytic at the origin with $h(0)=0=h^{-1}(0)$.
Then with $\omega=h(z)$ and $g(z)=1/[dh(z)/dz]$,
$$\exp \left[ {t \cdot g(z)\frac{d}{{dz}}} \right]f(z) = \exp \left[ {t\frac{d}{{d\omega }}} \right]f[{h^{ - 1}}(\omega )] = f[{h^{ - 1}}[t + \omega]] = f[{h^{ - 1}}[t + h(z)]],$$ so $$\exp
\left[ {t \cdot g(z)\frac{d}{{dz}}} \right]z |_{z=0}=h^{-1}(t)$$
(see OEIS A145271 and A139605 for more relations).
With the power series rep $h(z)= c_1z + c_2z^2 + c_3z^3 + ... ,$
$$\frac{1}{5!}[g(z)\frac{d}{{dz}}]^{5}z|_{z=0} = \frac{1}{c_1^{9}} [14 c_2^{4} - 21 c_1 c_2^2 c_3 + c_1^2[6 c_2 c_4+ 3 c_3^2] - 1 c_1^3 c_5],$$
which is the coefficient of the fifth order term of the power series for $h^{-1}(t)$. This is related to a refined f-vector (face-vector) for the 3-D Stasheff polytope, or 3-D
associahedron, with 14 vertices (0-D faces), 21 edges (1-D faces), 6 pentagons (2-D faces), 3 rectangles (2-D faces), 1 3-D polytope (3-D faces).
This correspondence between the refined f-vectors of the n-Dim Stasheff polytope, or associahedron, and the coefficients of the (n+2) term of the compositional inverse holds in general,
(see A133437, inversion for power series, and compare with A033282, coarse f-vectors for associahedra, and with MO-6373). These refined partition polynomials are also a refined presentation
up vote 2 of the number of diagonal dissections of a convex n-gon (A033282) or, equivalently, the refined numbers for a set of Schroeder lattice paths (A126216), which sum to the little Schroeder
down vote numbers (A001003).
If $h(z)$ is presented as a Taylor series, the LIF A134685 is obtained, which is related to the tropical Grassmannian G(2,n) and phylogenetic trees (A134991) and to the partitioning of 2n
elements into n groups.
When the invertible function $h(t)$ is represented as a power series of its own reciprocal, $t/h(t)$, the refined Narayana numbers are obtained (A134264), which are the refined
h-polynomials of the simplicial complexes (A001263) dual to the Stasheff associahedra and also a refined presentation of a set of Dyck lattice paths A125181, which sum to the Catalan
numbers A000108.
Also, the "infinitesimal generators" A145271 for these reps have very interesting associations (e.g., to permutahedra, surjections, and multiplicative reciprocals A019538/A049019, for the
LIF A134685) and allow reps of the partition polynomials for A133437 as colored umbral binary trees related to refined Lah polynomials.
To illustrate an important application, you might look at OEIS-A074060 "Graded dimension of the cohomology ring of the moduli space of n-pointed curves of genus 0 satisfying the
associativity equations of physics (also known as the WDVV equations)," as well as links in the LIF entries, to relate Lagrange inversion (or, equivalently, the Legendre transform) of
series to the cohomology of moduli spaces.
For a less fancy application, the LIFs can sometimes be used, just as other transforms, such as the Fourier transform, to jump between "reciprocal" domains to simplify expressions to solve
a problem, e.g., in conjunction with the OEIS to suggest generating functions for integer arrays by looking at their compositional inverses numerically.
add comment
There are plenty of uses of the Lagrange inversion formula in the following paper in statistics 'Letac, G. and Mora, M. (1990) 'Natural exponential families with cubic variances.'
up vote 1 down Ann. Statist. 1-37.'
add comment
Look at I2.24 (an exercise!) in the book I.G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd Ed., Clarendon Press, Oxford, 1995.
Let $\lambda$ be a partition. Then for sufficiently large $n$ there is a corresponding conjugacy class $K_\lambda(n)$ of $S_n$ (got by ignoring $1$'s in $\lambda$). Use $+$ to denote the sum
of the elements in the group algebra ${\mathbb Q}S_n$, we may write $$ K_\lambda(n)^+K_\mu(n)^+=\sum_\nu c_{\lambda,\mu}^\nu(n)K_\nu(n)^+, $$ where the $c_{\lambda,\mu}^\nu(n)$ are
non-negative integers that depend on all $3$ partitions and on $n$. Say that each element of $K(\lambda)$ can be written as a product of $\ell(\lambda)$ transpositions, but no fewer. Throw
away the $\nu$ for which $\ell(\nu)<\ell(\lambda)+\ell(\mu)$. Then the resulting $c_{\lambda,\mu}^\nu(n)=c_{\lambda,\mu}^\nu$ are independent of $n$ (H. K. Farahat, G. Higman, The centres of
up vote symmetric group rings, Proc. Royal Soc. London A 250 (1959) 212-221.).
1 down
vote Now let $H(t)=\prod_{i=1}^\infty(1-tx_i)^{-1}$ be the generating function for the complete symmetric functions. Suppose that $H(t)$ has Lagrange inverse $H^*(t)=\sum_{n=0}^\infty h_n^*t^n$.
Then the corresponding symmetric functions $h_n^* $ are algebraically independent. Set $h_\lambda^*=\prod h_{\lambda_i}^*$.
Denote the dual (w.r.t. the usual symmetric bilinear form on symmetric functions) of $h_\lambda$ by $K_\lambda$, for each partition $\lambda$. Then Macdonald shows that the $K_\lambda$ form a
basis for symmetric functions whose multiplication constants are: $$ K_\lambda^+K_\mu^+=\sum_\nu c_{\lambda,\mu}^\nu K_\nu. $$
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Topic: Cable around the Equator
Replies: 10 Last Post: Jun 17, 2012 2:29 PM
Messages: [ Previous | Next ]
Re: Cable around the Equator
Posted: Nov 5, 1996 9:37 AM
How can adding 50' of slack to the cable spread evenly around the equator
add 8' at each point? Your math was impeccable but your answer does not
pass the sanity check! This is why we must not blindly accept mathematical
results. That is the lesson to be learned from this geometry problem.
lipp@educ.umass.edu wrote in article
> On Mon, 4 Nov 1996, mfr1 wrote:
> > Ma Bell wants to place a telephone cable around the equator. She adds
> > feet to the length of the cable beyond what is required. This slack in
> > cable allows the cable to be strung up above the ground. How high up
> > the surface of the earth will the cable stand? You can assume that the
> > earth is a perfect sphere.
> >
> > In your mind, run a sanity check on your answer to see if it makes
> >
> Surprisingly, you will be able to walk under the cble easilty. Since the
> circumference of a circle is C = 2piR, each increase in R obf 1 foot
> increases the circumference by 2pi feet. since the cable has been
> increased by 50 feet the radius of the cable-circle will be increased by
> 50/2pi or about 8 feet!
> Alan Lipp
Date Subject Author
11/4/96 mfried1@earthlink.net
11/4/96 Re: Cable around the Equator Alan Lipp
11/5/96 Re: Cable around the Equator Pat Ballew
11/5/96 Re: Cable around the Equator Art Mabbott
11/5/96 Re: Cable around the Equator mfried1@earthlink.net
6/17/12 Re: Cable around the Equator Taylor
11/5/96 Re: Cable around the Equator Bernard Domroy
11/5/96 Re: Cable around the Equator Juan Miguel Vilar
11/6/96 Re: Cable around the Equator Charles Biehl
11/6/96 Re: Cable around the Equator Pat Ballew
11/6/96 Re: Cable around the Equator DougKuhlmann-PhillipsAcademy-Math | {"url":"http://mathforum.org/kb/message.jspa?messageID=1083181","timestamp":"2014-04-20T06:38:57Z","content_type":null,"content_length":"29262","record_id":"<urn:uuid:c64bf2ba-0a5b-4977-9ca3-af59372a6fbb>","cc-path":"CC-MAIN-2014-15/segments/1397609538022.19/warc/CC-MAIN-20140416005218-00246-ip-10-147-4-33.ec2.internal.warc.gz"} |
Using TIPS to gauge deflation expectations
« What does "structural" mean? | Main | A good time for price-level targeting? »
October 07, 2010
Using TIPS to gauge deflation expectations
In the recent Survey of Professional Forecasters, economists were asked to give their subjective probability of deflation during the next year. Specifically, they were asked about the chances that
the quarterly consumer price index excluding food and energy (core CPI) will decline in 2011. According to the respondents, the probability of core CPI deflation in 2011 was only 2 percent.
This rather sanguine view of the probability of deflation is encouraging. But is it a view shared by noneconomists? While there are many sources used to measure inflation expectations, there aren't
many that gauge inflation uncertainty or the risk of deflation. However, one might estimate a probability of deflation as seen by investors by exploiting the different deflation safeguards of a pair
of Treasury Inflation Protected Securities (TIPS), which have about the same maturity date but different issue dates.
Here's the idea: A TIPS cannot pay less than its face value at maturity, so the principal repayment of a five-year TIPS issued today is not reduced if the five-year rate of inflation is negative over
the life of the security. But a 10-year TIPS issued five years ago will have its capital gain from accrued inflation reduced if there is a net decline in the CPI over the next five years. As a
result, part of the real yield spread between the 10-year and five-year TIPS issues should reflect the value of the better deflation safeguard of the latter security.
In a comment on a paper by Campbell, Shiller, and Viceira, Jonathan Wright derives a very simple formula for calculating a lower bound on the probability of deflation using this real yield spread.
(The lower-bound formula is rm/ln(CPI[5yr]/CPI[10yr]), where r is the yield spread between the 10-year and five-year TIPS real yields, m is the number of years until the midpoint of the maturity
dates of the two TIPS, and CPI[5yr]/CPI[10yr] are the levels of the NSA CPI on the issue dates of the five-year and 10-year year TIPS. These reference CPIs are available here. Deflation is defined as
the level of the CPI being lower than its value on the issue date of the five-year TIPS.) Wright's calculation makes a number of simplifying assumptions, some of which are counterfactual, but it is
easy to compute—almost literally a back-of-the-envelope calculation if you have two real TIPS yields in hand. The formula also has the advantage that it does not require any assumptions about the
probability distribution of inflation.
To get exact probabilities of deflation instead of a lower bound, I developed a simple model for TIPS pricing. The model is an extension of the TIPS pricing model developed by Brian Sack. One has to
make a lot of assumptions to derive these estimates—which you can read about in the appendix to this post (link provided in last paragraph)—but let's get to the main results. The figure below plots
the probability that the level of the reference CPI on April 15, 2015, is lower than its April 15, 2010, level. (The reference CPI is the nonseasonally adjusted consumer price index interpolated to a
daily frequency; it is calculated by taking a weighted average of the CPI two months ago and three months ago.) If the April 2015 reference CPI ended up below this threshold, then the deflation
safeguard for the five-year TIPS would kick in. Also included in the graph is the lower bound of this "deflation probability" calculated using Wright's formula.
An alternative way of generating deflation probabilities is to exploit the estimated "confidence interval" from a forecasting model of inflation. When I use a variant of the inflation model proposed
by Stock and Watson (for those interested in more detail, the model I am using is the Stock-Watson unobserved components with stochastic volatility, or UC-SV model), it says there is about a 10
percent chance that average CPI inflation over the next five years will be below zero.
Is this the last word on estimating deflation probability? Of course not; there are more than a few pitfalls in this method of calculating a deflation probability, some of which are described in the
aforementioned technical appendix posted on the Atlanta Fed's Inflation Project. But this approach does have the advantage of exploiting information from market prices on traded securities. As such,
it may prove a valuable addition to our toolkit of indicators. Consequently, we intend to update these estimates and post them on the Inflation Project web page every Thursday afternoon.
By Patrick Higgins, an economist in the Atlanta Fed's research department
October 7, 2010 in Forecasts | Permalink
TrackBack URL for this entry:
Listed below are links to blogs that reference Using TIPS to gauge deflation expectations:
Even though point is pretty neat, personally I wouldn't rely on this too much. Liquidity of 5year TIPS markets would be higher now relative to the pre-crisis levels exactly for those same reasons you
want to use it gauge deflationary expectations: it's a better hedge against significant medium-term inflation uncertainty.
Posted by: Dan | October 08, 2010 at 01:39 AM
I don't know who is included in the survey of economists, but I would put the probability that we will have deflation in 2011 at substantially above 2%.
Posted by: don | October 12, 2010 at 10:30 PM
I wonder if there's some reason that you don't simply calculate the value of the embedded floors in TIPS, and then use the delta of that option? It's not easy, as you can't use black-scholes and you
have to first strip out seasonalities and such...but it's actually a market price rather than a strange economist view of a market price.
For that matter, why not just compute the delta of 0% ZC floors, which are quoted and trade in the market?
Interesting shorthand approaches, but it seems to me that if economists really wanted the market's view on this, they ought to calculate an actual number.
Posted by: Michael Ashton | August 24, 2012 at 08:23 AM | {"url":"http://macroblog.typepad.com/macroblog/2010/10/using-tips-to-gauge-deflation-expectations.html","timestamp":"2014-04-19T01:47:22Z","content_type":null,"content_length":"58700","record_id":"<urn:uuid:7fbf1d9d-c03d-47fe-8dae-524737b6711d>","cc-path":"CC-MAIN-2014-15/segments/1397609535745.0/warc/CC-MAIN-20140416005215-00454-ip-10-147-4-33.ec2.internal.warc.gz"} |
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English major - need help with basic Geometry question
November 25th 2011, 03:38 PM #1
Nov 2011
Help! I have to teach a 'struggling student' how to solve a Geometry question as part of a job interview. The interview is for a general teacher's aide position. The employer understands that I
am certified to teach English and cannot be relied upon to teach Math to anyone's child. They just want to assess my basic teaching skills.
Here's the question:
A 15-ft ladder is leaning against a 30-ft wall. The bottom end of the ladder is 9 ft from the wall. How many feet above the ground does the ladder touch the wall?
I think the answer is 12. (I drew a picture, using a ruler). Is that the correct answer? And how can I use math language to explain how to solve the problem? A picture will not be enough. Would
anyone be willing to assist a struggling job seeker?
Re: English major - need help with basic Geometry question
The ladder forms the right triangle with the wall and the floor. You know the hypotenuse (the ladder length) and the horizontal side (the distance of the bottom end of the ladder from the wall).
Pythagorean theorem allows you to find the other side.
Re: English major - need help with basic Geometry question
Help! I have to teach a 'struggling student' how to solve a Geometry question as part of a job interview. The interview is for a general teacher's aide position. The employer understands that I
am certified to teach English and cannot be relied upon to teach Math to anyone's child. They just want to assess my basic teaching skills.
Here's the question:
A 15-ft ladder is leaning against a 30-ft wall. The bottom end of the ladder is 9 ft from the wall. How many feet above the ground does the ladder touch the wall?
I think the answer is 12. (I drew a picture, using a ruler). Is that the correct answer? And how can I use math language to explain how to solve the problem? A picture will not be enough. Would
anyone be willing to assist a struggling job seeker?
If your employer is not going to rely on you to teach mathematics, I do not understand why you would be asked to demonstrate your teaching ability by doing so. Surely you would be asked to teach
a 'struggling' child how to do something that was within the field of what you would be teaching.
Furthermore, you are meant to be applying for this position on the basis of your own abilities, not the abilities of others.
12 is correct.
Something does not look right here.
Thread closed.
November 25th 2011, 03:58 PM #2
MHF Contributor
Oct 2009
November 25th 2011, 08:37 PM #3 | {"url":"http://mathhelpforum.com/geometry/192681-english-major-need-help-basic-geometry-question.html","timestamp":"2014-04-19T09:02:25Z","content_type":null,"content_length":"39694","record_id":"<urn:uuid:5d141860-b250-4cfb-bda5-94e4f6f26218>","cc-path":"CC-MAIN-2014-15/segments/1397609536300.49/warc/CC-MAIN-20140416005216-00501-ip-10-147-4-33.ec2.internal.warc.gz"} |
principal infinity-bundle
Classes of bundles
Universal bundles
Examples and Applications
Special and general types
Special notions
The notion of principal $\infty$-bundle is a categorification of principal bundle from groups and groupoids to ∞-groupoids, or rather from parameterized groupoids (generalized spaces called stacks)
to parameterized $\infty$-groupoids (generalized spaces called ∞-stacks).
For motivation, background and further details see
A model for principal $\infty$-bundles is given by
See also
We define $G$-principal $\infty$-bundles in the general context of an ∞-stack (∞,1)-topos $\mathbf{H}$, with $G$ a group object in the (∞,1)-topos.
Recall that for $A \in \mathbf{H}$ an object equipped with a point $pt_A : {*} \to A$, its corresponding loop space object $\Omega A$ is the homotopy pullback
$\array{ \Omega A &\to& {*} \\ \downarrow && \downarrow \\ {*} &\to& A } \,.$
Conversely, for $G \in \mathbf{H}$ we say an object $\mathbf{B}G$ is a delooping of $G$ if it has an essentially unique point and if $G \simeq \Omega \mathbf{B}G$. We call $G$ an ∞-group. More in
detail, its structure as a group object in an (∞,1)-category is exhibited by the Čech nerve
$\left( \array{ &\cdots& {*} \times_{\mathbf{B}G} {*} \times_{\mathbf{B}G} {*} &\stackrel{\to}{\stackrel{\to}{\to}}& {*} \times_{\mathbf{B}G} {*} &\stackrel{\to}{\to}& {*} } \right) \simeq \left( \
array{ &\cdots& G \times G &\stackrel{\to}{\stackrel{\to}{\to}}& G &\stackrel{\to}{\to}& {*} } \right)$
of ${*} \to \mathbf{B}G$.
$G$-principal $\infty$-bundles
To every cocycle $g : X \to \mathbf{B}G$ is canonically associated its homotopy fiber $P \to X$, the (∞,1)-pullback
$\array{ P &\to& {*} \\ \downarrow && \downarrow \\ X &\stackrel{g}{\to}& \mathbf{B}G \,. } \,.$
We discuss now that $P$ canonically has the structure of a $G$-principal ∞-bundle and that $\mathbf{B}G$ is the fine moduli space for $G$-principal $\infty$-bundles.
(principal $G$-action)
Let $G$ be a group object in the (∞,1)-topos $\mathbf{H}$. A principal action of $G$ on a morphism $(P \to X) \in \mathbf{H}$ is a groupoid object $P//G$ that sits over $*//G$ in that we have a
morphism of simplicial diagrams $\Delta^{op} \to \mathbf{H}$
$\array{ \vdots && \vdots \\ P \times G \times G &\stackrel{(p_2, p_3)}{\to}& G \times G \\ \downarrow\downarrow\downarrow && \downarrow\downarrow\downarrow \\ P \times G &\stackrel{p_2}{\to}& G \\ \
downarrow\downarrow && \downarrow\downarrow \\ P &\stackrel{}{\to}& {*} }$
in $\mathbf{H}$;
and such that $P \to X$ exhibits the (∞,1)-colimit
$X \simeq \lim_\to (P//G : \Delta^{op} \to \mathbf{H})$
called the base space over which the action takes place.
We may think of $P//G$ as the action groupoid of the $G$-action on $P$. For us it defines this $G$-action.
The $G$-principal action as defined above satisfies the principality condition in that we have an equivalence of groupoid objects
$\array{ \vdots && \vdots \\ P \times_X P \times_X P &\stackrel{\simeq}{\to}& P \times G \times G \\ \downarrow\downarrow\downarrow && \downarrow\downarrow\downarrow \\ P \times_X P &\stackrel{\
simeq}{\to}& P \times G \\ \downarrow\downarrow && \downarrow\downarrow \\ P &\stackrel{\simeq}{\to}& P } \,.$
For $X \to \mathbf{B}G$ any morphism, its homotopy fiber $P \to X$ is canonically equipped with a principal $G$-action with base space $X$.
First we show that we have a morphism of simplicial diagrams
$\array{ \vdots && \vdots && \vdots \\ P \times_X P \times_X P &\stackrel{\simeq}{\to}& P \times G \times G &\to& G \times G \\ \downarrow\downarrow\downarrow && \downarrow\downarrow\downarrow && \
downarrow\downarrow\downarrow \\ P \times_X P &\stackrel{\simeq}{\to}& P \times G &\stackrel{p_2}{\to}& G \\ \downarrow\downarrow && \downarrow\downarrow && \downarrow\downarrow \\ P &\stackrel{=}{\
to}& P &\stackrel{}{\to}& {*} \\ \downarrow && \downarrow && \downarrow \\ X &\stackrel{=}{\to}& X &\stackrel{g}{\to}& \mathbf{B}G } \,,$
with the right square swhere the left horizontal morphisms are equivalences, as indicated. We proceed by induction through the height of this diagram.
The defining (∞,1)-pullback square for $P \times_X P$ is
$\array{ P \times_X P &\to& P \\ \downarrow && \downarrow \\ P &\to& X }$
To compute this, we may attach the defining $(\infty,1)$-pullback square of $P$ to obtain the pasting diagram
$\array{ P \times_X P &\to& P &\to& {*} \\ \downarrow && \downarrow && \downarrow \\ P &\to& X &\to& \mathbf{B}G \,. }$
and use the pasting law for pullbacks, to conclude that $P \times_X P$ is the pullback
$\array{ P \times_X P &\to& &\to& {*} \\ \downarrow && && \downarrow \\ P &\to& X &\to& \mathbf{B}G \,. }$
By definition of $P$ as the homotopy fiber of $X \to \mathbf{B}G$, the lower horizontal morphism is equivbalent to $P \to {*} \to \mathbf{B}G$, so that $P \times_X P$ is also equivalent to the
$\array{ P \times_X P &\to& &\to& {*} \\ \downarrow && && \downarrow \\ P &\to& {*} &\to& \mathbf{B}G \,. }$
This finally may be computed as the pasting of two pullbacks
$\array{ P \times_X P &\simeq& P \times G &\to& G &\to& {*} \\ &&\downarrow && \downarrow && \downarrow \\ &&P &\to& {*} &\to& \mathbf{B}G \,. }$
of which the one on the right is the defining one for $G$ and the remaining one on the left is just an (∞,1)-product.
Proceeding by induction from this case we find analogously that $P^{\times_X^{n+1}} \simeq P \times G^{\times_n}$: suppose this has been shown for $(n-1)$, then the defining pullback square for $P^{\
times_X^{n+1}}$ is
$\array{ P \times_X P^{\times_X^n} &\to& P \\ \downarrow && \downarrow \\ P^{\times_X^n}&\to& X } \,.$
We may again paste this to obtain
$\array{ P \times_X P^{\times_X^n} &\to& P &\to& * \\ \downarrow && \downarrow && \downarrow \\ P^{\times_X^n}&\to& X &\to& \mathbf{B}G }$
and use from the previous induction step that
$(P^{\times_X^n} \to X \to \mathbf{B}G) \simeq (P^{\times_X^n} \to * \to \mathbf{B}G)$
to conclude the induction step with the same arguments as before.
This shows that $P//G$ is the Cech nerve of $P \to X$. It remains to show that indeed $X = {\lim_\to}_n P \times G^{\times^n}$. For this notice that $* \to \mathbf{B}G$ is an effective epimorphism in
an (infinity,1)-category. Hence so is $P \to X$. This proves the claim, by definition of effective epimorphism.
using this we have
\begin{aligned} X & \simeq \mathbf{B}G \prod_{\mathbf{B}G} X \\ & \simeq \left({\lim_{\to}}_n G^{\times^n}\right) \prod_{\mathbf{B}G} X \\ & \simeq {\lim_{\to}}_n ( G^{\times^n} \prod_{\mathbf{B}G} X
) \\ & \simeq {\lim_\to}_n ( P\times G^{\times^n} ) \\ & \simeq {\lim_\to} P//G \end{aligned} \,.
We have established that every cocycle $X \to \mathbf{B}G$ canonically induced a $G$-principal action on the homotopy fiber $P \to X$. The following definition declares the $G$-principal $\infty$
-bundles to be those $G$-principal actions that do arise this way.
We say a $G$-principal action of $G$ on $P$ over $X$ is a $G$-principal ∞-bundle if the colimit over $P//G \to *//G$ produces a pullback square: the bottom square in
$\array{ \vdots && \vdots \\ P \times G \times G &\to& G \times G \\ \downarrow\downarrow\downarrow && \downarrow\downarrow\downarrow \\ P \times G &\stackrel{p_2}{\to}& G \\ \downarrow\downarrow &&
\downarrow\downarrow \\ P &\stackrel{}{\to}& {*} \\ \downarrow && \downarrow \\ X = \lim_\to (P \times G^\bullet) &\stackrel{g}{\to}& \mathbf{B}G = \lim_\to( G^\bullet) } \,.$
For $G$ an infinity-group in $\mathbf{H}$ and $X \in \mathbf{H}$ any object, write
$G Bund(X) \subset Grpd(\mathbf{H})/{*//G}$
for the sub-(infinity,1)-category on the over-(infinity,1)-category of the groupoid objects over $*//G$ on the $G$-principal $\infty$-bundles as above.
The arrow category $\mathbf{H}^I$ is still an (infinity,1)-topos and hence the Griraud-Lurie axioms still hold. This means that by the discussion at groupoid object in an (infinity,1)-category (using
the statement below HTT, cor. 6.2.3.5) we have an equivalence
$Grpd(\mathbf{H}^I) \simeq (\mathbf{H}^{I})^{I}_{eff}$
between groupoid objects in $\mathbf{H}^I$ and effective epimorphisms in the arrow category.
Notice that groupoid objects and effective epis in $\mathbf{H}^I$ are given objectwise over the two objects of the inerval $I = \Delta[1]$.
Restricting this equivalence along the inclusion
$\mathbf{H}(X, \mathbf{B}G) \hookrightarrow (\mathbf{H}^I)^I$
given by sending a cocycle to its homotopy fiber diagram
$(X \to \mathbf{B}G) \mapsto \left( \array{ P &\to& * \\ \downarrow && \downarrow \\ X &\to& \mathbf{B}G } \right)$
therefore yields precisely the equivalence in question
$\array{ G Bund(X) &\hookrightarrow& Grpd(\mathbf{H}^I) \\ \downarrow^\simeq && \downarrow^\simeq \\ \mathbf{H}(X, \mathbf{B}G) &\stackrel{hofib}{\hookrightarrow}& (\mathbf{H}^I)^I } \,.$
In words this says that the cohomology on $X$ with coefficients in $G$ classified $G$-principal $\infty$-bundles, and in fact does so on the level of cocycles.
Connections on $G$-principal $\infty$-bundles
For some comments on the generalization of the notion of connection on a bundle to principal $\infty$-bundles see differential cohomology in an (∞,1)-topos -- survey.
Concrete realizations
We discuss realizations of the general definition in various (∞,1)-toposes $\mathbf{H}$.
In topological spaces
The following general construction was originally due to Quillen and defines principal groupoid $\infty$-bundles in the (∞,1)-topos Top in its presentation by the model structure on simplicial sets.
Let $C$ be a small category and let
$\rho_P : C \to SSet$
be a functor with values in SSet such that it sends all morphisms in $C$ to weak equivalences in SSet (weak homotopy equivalences of simplicial sets).
Consider first the case that $C$ has a single object, so that it is the delooping $\mathbf{B}G$ of a monoid or group $G$. Then
$P := \rho_P(\bullet)$
be the simplicial set assigned to this single object and let
$X := P//G := hocolim \rho_P$
be the corresponding action groupoid (see there for the description as a weak colimit).
Notice that, as every action group, this comes with a canonical map $P//G \to \mathbf{B}G$.
Given the above, the diagram
$\array{ P &\to& {*} \\ \downarrow && \downarrow \\ X &\stackrel{g}{\to}& \mathbf{B}G }$
is a homotopy pullback (i.e. defines a fibration sequence).
This is originally due to
• D. Quillen, Higher algebraic K-theory I, Springer Lecture notes in Math. 341 (1973) 85–147.
The statement is reproduced in section IV of
• P. G. Goerss and J. F. Jardine, 1999, Simplicial Homotopy Theory, number 174 in Progress in Mathematics, Birkhauser. (ps)
Quillen’s original construction is more general than this, concerning in fact 1-groupoid-principal $\infty$-bundles:
Let now $C$ be a category and for
$\rho_P : C \to SSet$
a functor that sends all morphisms to weak equivalences of simplicial sets.
Let now for each object $c \in C$
$P_c := \rho_C(c)$
be the “bundle of $c$-fibers”.
Then for each $c$ the diagram
$\array{ P_c &\to& {*} \\ \downarrow && \downarrow^{{*} \mapsto c} \\ X &\stackrel{g}{\to}& C }$
is a homotopy pullback (i.e. defines a fibration sequence).
This classical construction is recalled in the introduction of
• Jardine, Diagrams and torsors (pdf)
In simplicial sets / Kan complexes
See simplicial principal bundle.
In a petit $(\infty,1)$-topos
For $X$ a topological space $C = Op(X)$ the category of open subsets of $X$, let $\mathbf{H} = Sh_{(\infty,1)}(X)$ be the (∞,1)-topos of ∞-stacks on $C$. This is the petit topos incarnation of $X$.
In its presentation by the model structure on simplicial presheaves this is the context in which princpal $\infty$-bundles are discussed in
• Jardine, Diagrams and torsors (pdf)
In a gros $(\infty,1)$-topos
For $C$ a site of test space, – for instance duals of algebras over a Lawvere theory as described at function algebras on infinity-stacks – let $\mathbf{H} = Sh_{(\infty,1)}(C)$ be the (∞,1)-topos of
∞-stacks on $C$. This is a gros topos.
Smooth principal $\infty$-bundles
Smooth principal $\infty$-bundles are realized in the $\infty$-Cahiers topos as described in some detail at ∞-Lie groupoid.
In this context there is a notion of connection on a principal ∞-bundle.
Ordinary principal bundles
For $G$ an ordinary Lie group, a $G$-principal bundle in the $(\infty,1)$-topos $\mathbf{H} =$∞LieGrpd is an ordinary $G$-principal bundle.
Circle $n$-bundles
For $G = \mathbf{B}^{n-1} U(1) \in$∞LieGrpd, the circle Lie n-group, a $G$-principal $\infty$-bundle is a circle $n$-bundle.
See circle n-bundle with connection.
Classes of examples include
Bundle gerbes
• A bundle gerbe is a concrete model for the total space groupoid of the total space of a $\mathbf{B}U(1)$-principal 2-bundle.
More generally, a nonabelian bundle gerbe is a concrete model for the groupoid of the total space of a general principal 2-bundle.
• A bundle 2-gerbe is a concrete model for the total space 2-groupoid of the total space of a $\mathbf{B}^2 U(1)$-principal 3-bundle.
More generally, a nonabelian bundle 2-gerbe is a concrete model for the 2-groupoid of the total space of a general principal 3-bundle.
Classes of examples include
Normal morphisms of $\infty$-groups
A principal $\infty$-bundle over a 0-connected object / delooping object $\mathf{B}K$ is a normal morphism of ∞-groups. See there for more details.
The notion of principal $\infty$-bundle (often addressed in the relevant literature in the language of torsors) appears in the context of the simplicial presheaf model for generalized spaces in
An earlier description in terms of simplicial objects is
• P. Glenn, Realization of cohomology classes in arbitrary exact categories, J. Pure Appl. Algebra 25, 1982, no. 1, 33–105.
In that article not the total space of the bundle $P \to X$ is axiomatized, but the $\infty$-action groupoid of the action of $G$ on it.
See the remarks at principal 2-bundle.
See also
on associated ∞-bundles.
The fully general abstract formalization in (∞,1)-topos theory as discussed here was first indicated in
A more comprehensive conceptual account is in
The classifying spaces for a large class of principal $\infty$-bundles are discussed in
A fairly comprehensive account of the literature is also in the introduction of NSS 12, “Presentations”.
For $\mathbf{H}= \infty Grpd$ the statement that homotopy types over $B G$ are equivalently $G$-infinity-actions is maybe due to
• E. Dror, William Dwyer, and Daniel Kan, Equivariant maps which are self homotopy equivalences, Proc. Amer. Math. Soc. 80 (1980), no. 4, 670–672 (JSTOR)
This is mentioned for instance as exercise 4.2in
• William Dwyer, Homotopy theory of classifying spaces, Lecture notes Copenhagen (June, 2008) pdf
Closely related discussion of homotopy fiber sequences and homotopy action but in terms of Segal spaces is in section 5 of
There, conditions are given for a morphism $A_\bullet \to B_\bullet$ to a reduced Segal space to have a fixed homotopy fiber, and hence encode an action of the loop group of $B$ on that fiber. | {"url":"http://nlab.mathforge.org/nlab/show/principal+infinity-bundle","timestamp":"2014-04-20T01:49:50Z","content_type":null,"content_length":"129009","record_id":"<urn:uuid:c11ab999-8afb-480e-a2d6-db4643b99fd3>","cc-path":"CC-MAIN-2014-15/segments/1397609537804.4/warc/CC-MAIN-20140416005217-00576-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Linear Programming : Solving by a Graphical Method
1. 58359
Linear Programming : Solving by a Graphical Method
Let x be the number short-answer questions and y be the number of essay quesitons.
Given : 9 points for short - answer
25 points for essay.
x <=10, obviously x >=9
y<=10, obviously y >=3
Maximum score
Subject to the constraints,
x>=9, y >= 3
That's it,
Thanks for using Brainmass service :-)
How do you set it up from here using the numbers in the equation to solve? How do I use those equations to come up with the answer?
A linear programming problem is solved using graphical methods. The solution is detailed and well presented. | {"url":"https://brainmass.com/math/linear-programming/58359","timestamp":"2014-04-19T04:49:14Z","content_type":null,"content_length":"28104","record_id":"<urn:uuid:a6da5eb3-967e-4625-8bca-07647bb5a074>","cc-path":"CC-MAIN-2014-15/segments/1398223206647.11/warc/CC-MAIN-20140423032006-00003-ip-10-147-4-33.ec2.internal.warc.gz"} |
Three step gas cycle
1. The problem statement, all variables and given/known data
A monatomic ideal gas has pressure p_1 and temperature T_1. It is contained in a cylinder of volume V_1 with a movable piston, so that it can do work on the outside world.
Consider the following three-step transformation of the gas:
1. The gas is heated at constant volume until the pressure reaches Ap_1 (where A >1).
2. The gas is then expanded at constant temperature until the pressure returns to p_1.
3. The gas is then cooled at constant pressure until the volume has returned to V_1.
It may be helpful to sketch this process on the pV plane.
Part 1-
How much heat DeltaQ_1 is added to the gas during step 1 of the process?
Express the heat added in terms of p_1, V_1, and A.
Part 2-
How much work W_2 is done by the gas during step 2?
Express the work done in terms of p_1, V_1, and A.
Part 3-
How much work W_3 is done by the gas during step 3?
If you've drawn a graph of the process, you won't need to calculate an integral to answer this question.
Express the work done in terms of p_1, V_1, and A.
2. Relevant equations
R = 8.31
3. The attempt at a solution
Part 1-
I tried Q = p_1*V_1*(C_V/R) = 1.5*Ap_1*V_1 but I was told this is the final internal energy, not the change in internal energy. so I worked out that
Q = [1.5*p_1*V_1*(AT_1-T_1)] / T_1 but the answer does not depend on AT_1 or T_1
Part 2-
all I've got so far is
W = nRT*ln(V_f/V_i) = pV*ln(V_f/V_i)
but thats about as far as I get.
Part 3-
I got Ap_1*V_1 but this is what the value would be if it were coming from V = 0. So I re-arranged pV=nRT to eventually get
W = p_1[(p_1V_1)/(Ap_1) - V_1]
but this is also wrong how do I take into account the initial state, wouldn't I just be able to write W = (Ap_1V_1) - V_1 ? | {"url":"http://www.physicsforums.com/showthread.php?t=382289","timestamp":"2014-04-18T10:51:52Z","content_type":null,"content_length":"27916","record_id":"<urn:uuid:5c197326-0fd7-415b-9fc4-c69fcdfcbac7>","cc-path":"CC-MAIN-2014-15/segments/1397609533308.11/warc/CC-MAIN-20140416005213-00242-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Example 9.36: Levene’s test for equal variances
June 25, 2012
By Ken Kleinman
The assumption of equal variances among the groups in analysis of variance is an expression of the assumption of homoscedasticity for linear models more generally. For ANOVA, this assumption can be
tested via
Levene's test
. The test is a function of the residuals and means within each group, though various modifications are used, including the Brown-Forsythe test. This uses the medians within group, rather than the
mean, and is recommended when normality may be suspect.
We illustrate using the HELP data set available
, modeling depressive symptoms (assessed via CESD) as a function of abused substance.
In SAS, the tests are available as an option to the
statement in
proc glm
data help;
set "C:\book\help.sas7bdat";
proc glm data = help;
class substance;
model cesd = substance;
means substance / hovtest=levene(type=abs) hovtest=bf;
The two requested tests are a version of Levene's test that is produced in R, below, and the aforementioned Brown-Forsythe test. The relevant results are shown below.
Levene's Test for Homogeneity of CESD Variance
ANOVA of Absolute Deviations from Group Means
Sum of Mean
Source DF Squares Square F Value Pr > F
SUBSTANCE 2 272.4 136.2 2.61 0.0747
Error 450 23480.7 52.1793
Brown and Forsythe's Test for Homogeneity of CESD Variance
ANOVA of Absolute Deviations from Group Medians
Sum of Mean
Source DF Squares Square F Value Pr > F
SUBSTANCE 2 266.0 133.0 2.46 0.0864
Error 450 24310.9 54.0243
There's some suggestion of a lack of homoscedasticity; it might be wise to consider methods robust to violations of this assumption.
In R, the test can be found in the
function in the lawstat package.
help = read.csv("http://www.math.smith.edu/r/data/help.csv")
with(help, levene.test(cesd, as.factor(substance), location="mean"))
classical Levene's test based on the absolute deviations from the mean
( none not applied because the location is not set to median )
data: cesd
Test Statistic = 2.6099, p-value = 0.07465
with(help, levene.test(cesd, as.factor(substance),location="median"))
modified robust Brown-Forsythe Levene-type test based on the absolute
deviations from the median
data: cesd
Test Statistic = 2.462, p-value = 0.08641
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categories: Reality check
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categories: Reality check
In an earlier posting I showed how to define co-inductively the closed
interval, in particular I showed that its elements are named by
sequences of 0s and 1s with the usual binary-expansion equivalence
relation. There is a well-known computational problem with this
approach, already in the definition of the midpoint operation: at what
point can you determine the first digit of the midpoint of .0000...
and .1111...?
At the Como meeting I learned from Andrej Bauer about a better
approach. Take the elements of [-1,1] to be named by infinite
sequences of _signed_ binary digits, that is -1, 0, +1.
[Just to confuse matters, Scedrov and I once used signed _ternary_
digits (n Cats and Allegators for the "Freyd curve"). The signed binary
expansions .+1 -1 and .0 +1 describe the same number, to wit, 1/4.]
Using signed binary expansions one can compute midpoints with a little
3-state machine that takes as input the sequence of pairs of signed
binary digits of the given numbers x and y, and produces as output
a sequence of signed binary digits for the midpoint x|y. (There may,
indeed, be momentary delays in the output, but there will not be an
indefinite delay -- indeed, the number of output digits will never be
more than one less than the number of pairs of input digits).
The challenge is to revise the co-induction so that it is this better
version that emerges.
In the previous version I worked in the category of posets with
_distinct_ top and bottom, that is, those posets for which
not[(B = x) and (x = T)].
In the revised version I'll strengthen the condition by working in the
category of posets with _separated_ top and bottom:
[(B < x) or (x < T)].
(The conditions are equivalent in the presence of De Morgan's law. In
a topos the top and bottom of omega are always distinct but they are
separated only when De Morgan's law is satisfied throughout.)
In the previous setting I defined what I'll now call the _thin_
version of the ordered-wedge of X and Y, to wit, the set of pairs,
<x,y> satisfying the condition:
(x = T) or (B = y).
The _thick_ version is the set of pairs, <x,y> satisfying the two
weaker conditions:
(x < T) => (B = y)
(B < y) => (x = T)
(each of which is classically equivalent to the single condition used
in the thin version).
Easy exercise: if top and bottom are separated in X and Y, then
they are separated in the thick version of the ordered-wedge XvY.
(Indeed, it's enough for top and bottom to be separated in just one of
X and Y . No, it is not enough to assume just that they are distinct
in each.)
A map X -> XvX is thus given by a pair d,u: X -> X such that for
all x:
(dx < T) => (B = ux)
(B < ux) => (dx = T)
The final coalgebra for XvX is still the closed interval, but now in
the better computational sense. Let me explain.
Given an arbitrary coalgebra d,u : X -> X, I need to describe a
coalgebra homomorphism f: X -> I. where I is the set of equivalence
types of infinite sequences of signed binary digits.
The first step is to work not with elements of X but elements of
XvX. Consider a machine that given <x,y>:XvX asks in parallel the
"B < y?"
"B < ux and dy < T?"
"x < T?"
Exercise: If the top and bottom of X are separated and d,u
describe a map to the thick ordered-wedge XvX, then at least one of
these questions has a positive answer.
Given z:X obtain a sequence of signed binary digits by starting with
the pair <x,y> = <dz,uz> and iterating the non-deterministic
If B < y
then emit +1 as output and replace <x,y> with <dy,uy>;
If B < ux and dy < T
then emit 0 as output and replace <x,y> with <ux,dy>;
If x < B
then emit -1 as output and replace <x,y> with <dx,ux>.
Not-so-easy exercises: regardless of the non-determinism, the element
fz:[-1,+1] named by the resulting sequence is determined. Moreover,
f(uz) = u(fz) and f(dz) = df(z).
PS. There is some geometry behind this stuff. Let me here mention just
this: given a pair <x,y> in XvX map it into the four-fold
ordered-wedge XvXvXvX. Think of each of the four copies of X as one
"quarter" of the whole. If B < y then the point lies inside the "top
half" (the 3rd and 4th quarters). If B < ux and dy < T then the
point lies inside the "middle half" (the 2nd and 3rd quarters). If
x < B then the point lies inside the bottom half" (the 1st and 2nd
quarters). Clearly any point is inside at least one of these three
halves. The output-digit registers which of the three halves is moved
to and the corresponding pair-replacement effects that move.
PPS. On 22 Dec I gave a Dedekind-cut proof that the interval [0,1]
constructed in the standard fashion from (unsigned) binary sequences
is the final coalgebra for the functor that sends X (with distinct
top and bottom) to the thin version of XvX. That proof should be
replaced. First note that the midpoint-algebra homomorphism from [0,1]
to [-1,1] can be effected simply by replacing each 0 with -1 and
keeping each 1 as +1. Given d,u:X -> X such that for all x it is
the case that either dx = T or ux = B , consider a machine that upon
given x:X asks in parallel the questions "dx = T?" and "ux = B?".
If dx = T then emit +1 as output and replace x with ux,
If ux = B then emit -1 as output and replace x with dx.
Given x:X one may iterate this (non-deterministic) procedure to
obtain a sequence of +1s and -1s. Pretty-easy exercises: regardless
of the non-determinism, the element fx:[-1,+1] named by the
resulting sequence is determined; f(ux) = u(fx); f(dx) = df(x). | {"url":"http://facultypages.ecc.edu/alsani/ct99-00(8-12)/msg00200.html","timestamp":"2014-04-19T15:31:48Z","content_type":null,"content_length":"9379","record_id":"<urn:uuid:cb9a67cb-1604-4afa-876d-7715c29b1004>","cc-path":"CC-MAIN-2014-15/segments/1397609537271.8/warc/CC-MAIN-20140416005217-00300-ip-10-147-4-33.ec2.internal.warc.gz"} |
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The cobordism group of homology cylinders
The Library
The cobordism group of homology cylinders
Cha, Jae Choon, 1971-, Friedl, Stefan and Kim, Taehee. (2011) The cobordism group of homology cylinders. Compositio Mathematica, Vol.147 (No.3). pp. 914-942. ISSN 0010-437X
WRAP_Friedl_S0010437X10004975a.pdf - Published Version
Download (1005Kb)
Garoufalidis and Levine introduced the homology cobordism group of homology cylinders over a surface. This group can be regarded as an enlargement of the mapping class group. Using torsion
invariants, we show that the abelianization of this group is infinitely generated provided that the first Betti number of the surface is positive. In particular, this shows that the group is not
perfect. This answers questions of Garoufalidis and Levine, and Goda and Sakasai. Furthermore, we show that the abelianization of the group has infinite rank for the case that the surface has more
than one boundary component. These results also hold for the homology cylinder analogue of the Torelli group.
Item Type: Journal Article
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Science > Mathematics
Library of Homology theory, Cobordism theory, Abelian groups
Journal or Compositio Mathematica
Publisher: Cambridge University Press
ISSN: 0010-437X
Date: 2011
Volume: Vol.147
Number: No.3
Page Range: pp. 914-942
Identification 10.1112/S0010437X10004975
Status: Peer Reviewed
Publication Published
Access rights Restricted or Subscription Access
to Published
Funder: National Research Foundation of Korea (NRF), Korea (South). Kyoyuk Kwahak Kisulbu [Ministry of Education, Science and Technology] (MEST)
Grant number: 2007-0054656 (NRF), 2009-0094069 (NRF), 2009-0068877 (NRF), 2009-0086441 (NRF)
BH71 J. S. Birman and H. M. Hilden, On the mapping class groups of closed surfaces as covering spaces, in Advances in the theory of Riemann surfaces, Proc. conf. (Stony Brook, NY,
1969), Annals of Mathematics Studies, vol. 66 (Princeton University Press, Princeton, NJ, 1971), 81{115. BP01 R. Benedetti and C. Petronio, Reidemeister{Turaev torsion of
3-dimensional Euler structure with simple boundary tangency and pseudo-Legendrian knots, Manuscripta Math. 106 (2001), 13{61. Cha07 J. C. Cha, The structure of the rational
concordance group of knots, Mem. Amer. Math. Soc. 189 (2007), x+95pp. CF10 J. C. Cha and S. Friedl, Twisted torsion invariants and link concordance, arXiv:1001.0926. EW99 G. Everest
and T.Ward, Heights of polynomials and entropy in algebraic dynamics, Universitext (Springer, New York, 1999). FM10 B. Farb and D. Margalit, A primer on mapping class groups, http://
www.math.utah.edu/margalit/primer/. FS90 R. Fintushel and R. Stern, Instanton homology of Seifert fibred homology three spheres, Proc. Lond. Math. Soc. 61 (1990), 109{137. FM66 R. H.
Fox and J. W. Milnor, Singularities of 2-spheres in 4-space and cobordism of knots, Osaka J. Math. 3 (1966), 257{267. Fre82 M. Freedman, The topology of four-dimensional manifolds, J.
Differential Geom. 17 (1982), 357{453. FQ90 M. H. Freedman and F. Quinn, Topology of 4-manifolds, Princeton Mathematical Series, vol. 39 (Princeton University Press, Princeton, NJ,
1990). FJR09 S. Friedl, A. Juhasz and J. Rasmussen, The decategorification of sutured Floer homology, arXiv:0903.5287. Fur90 M. Furuta, Homology cobordism group of homology 3-spheres,
Invent. Math. 100 (1990), 339{355. GL05 S. Garoufalidis and J. Levine, Tree-level invariants of three-manifolds, Massey products and the Johnson homomorphism, Graphs and patterns in
mathematics and theorical physics, Proc. Sympos. Pure Math. 73 (2005), 173{205. GS08 H. Goda and T. Sakasai, Homology cylinders in knot theory, arXiv:0807.4034. GS09 H. Goda and T.
Sakasai, Abelian quotients of monoids of homology cylinders, arXiv:0905.4775. Gou99 M. Goussarov, Finite type invariants and n-equivalence of 3-manifolds, C. R. Math. Acad. Sci. Paris
329 (1999), 517{522. Hab00 K. Habiro, Claspers and finite type invariants of links, Geom. Topol. 4 (2000), 1{83. Joh83a D. Johnson, The structure of the Torelli group. I. A finite set
of generators for I, Ann. of Math. (2) 118 (1983), 423{442. Joh83b D. Johnson, A survey of the Torelli group, Contemp. Math. 20 (1983), 165{179. Joh85 D. Johnson, The structure of the
Torelli group. III. The abelianization of I, Topology 24 (1985), 127{144. KL99 P. Kirk and C. Livingston, Twisted Alexander invariants, Reidemeister torsion and Casson{ Gordon
References: invariants, Topology 38 (1999), 635{661. KLW01 P. Kirk, C. Livingston and Z. Wang, The Gassner representation for string links, Commun. Contemp. Math. 3 (2001), 87{136. Lev69a J.
Levine, Knot cobordism groups in codimension two, Comment. Math. Helv. 44 (1969), 229{244. Lev69b J. Levine, Invariants of knot cobordism, Invent. Math. 8 (1969), 98{110. Lev01 J.
Levine, Homology cylinders: An enlargement of the mapping class group, Algebr. Geom. Topol. 1 (2001), 243{270. McC75 J. McCool, Some finitely presented subgroups of the automorphism
group of a free group, J. Algebra 35 (1975), 205{213. MM86 A. Miller and D. McCullough, The genus 2 Torelli group is not finitely generated, Topology Appl. 22 (1986), 43{49. Mes92 G.
Mess, The Torelli groups for genus 2 and 3 surfaces, Topology 31 (1992), 775{790. Mil62 J. Milnor, A duality theorem for Reidemeister torsion, Ann. of Math. (2) 76 (1962), 137{147.
Mil66 J. Milnor, Whitehead torsion, Bull. Amer. Math. Soc. 72 (1966), 358{426. Mor06 S. Morita, Cohomological structure of the mapping class group and beyond, in Problems on mapping
class groups and related topics, Proceedings of Symposia in Pure Mathematics, vol. 74 ed. B. Farb (American Mathematical Society, Providence, RI, 2006). Mor08 S. Morita, Symplectic
automorphism groups of nilpotent quotients of fundamental groups of surfaces, Adv. Stud. Pure Math. 52 (2008), 443{468. Ni07 Y. Ni, Knot Floer homology detects fibred knots, Invent.
Math. 170 (2007), 577{608. Nic03 L. Nicolaescu, The Reidemeister torsion of 3-manifolds, de Gruyter Studies in Mathematics, vol. 30 (Walter de Gruyter & Co., Berlin, 2003). Pow78 J.
Powell, Two theorems on the mapping class group of a surface, Proc. Amer. Math. Soc. 68 (1978), 347{350. Sak06 T. Sakasai, Mapping class groups, groups of homology cobordisms of
surfaces and invariants of 3-manifolds, Doctoral Dissertation, The University of Tokyo (2006). Sak08 T. Sakasai, The Magnus representation and higher-order Alexander invariants for
homology cobordisms of surfaces, Algebr. Geom. Topol. 8 (2008), 803{848. Sav02 N. Saveliev, Invariants of Homology 3-spheres, Encyclopaedia of Mathematical Sciences, vol. 140
(Springer, Berlin, 2002). SW02 D. Silver and S. Williams, Mahler measure, links and homology growth, Topology 41 (2002), 979{991. SW04 D. Silver and S. Williams, Mahler measure of
Alexander polynomials, J. London Math. Soc. (2) 69 (2004), 767{782. Sta65 J. Stallings, Homology and central series of groups, J. Algebra 2 (1965), 170{181. Tur86 V. Turaev,
Reidemeister torsion in knot theory, Russian Math. Surveys 41 (1986), 119{182. Tur01 V. Turaev, Introduction to combinatorial torsions, Lectures in Mathematics, ETH Zurich, 2001
(Birkhauser, Basel, 2001). Tur02 V. Turaev, Torsions of 3-manifolds, Progress in Mathematics, vol. 208 (Birkhauser, Basel, 2002).
URI: http://wrap.warwick.ac.uk/id/eprint/39993
Data sourced from Thomson Reuters' Web of Knowledge
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14 Bipolar Junction Transistor - conocimientos.com.ve
The frequency response of a BJT or MOSFET can be found using nearly the exact same process, with the only variations being caused by a single resistor and simple naming conventions that differ
between the two devices.
Before we start let's think a little bit about what we're doing:
Our goal is going to be to find the pole(s) of the circuit.
Okay? What is a pole and why do I care where it is?
A pole is a frequency at which the gain of the device rolls off. (remember that when it rolls off , it will be at the -3dB frequency with a slope of -20dB/decade)
We care because if the gain of a device rolls off at a certain frequency, then we won't be able to amplify a signal above that frequency very well because the gain will be decreasing by 20dB/decade.
The procedure is nearly identical whether we are using a BJT of a MOSFET, but we will work each of them side by side just in case there might be any confusion, and we'll follow these steps as we go
through. (we will also use some values that came from the output file when running a simulation of this circuit in Cadence (or PSPICE) )
1. Take a look at one of the circuits and see what you notice, how about the MOSFET. This step is just to help us with our knowledge understanding of the circuit.
- At a glance it just looks just like another MOSFET right? Sure is, but let's take a look at a few things just for kicks. Notice that it is using a bypass capacitor at the source so we don't have to
worry about $R_s$ (at when working with high frequency). Since the capacitor $C_s$ bypasses $R_s$ to ground, you should notice that this is a common-source amplifier. You could notice the Values for
$R_1$ and $R_2$ and start to think about what the Gate voltage is and how that may affect the circuit.
2. We are talking about frequency response so that means we are probably going to want to draw the small signal equivalent circuit.
Remember that the capacitors $C_1$ and $C_2$ will act like short circuits at high frequencies so we will ignore them, but we will have to account for some of the capacitance internal to the device.
Both devices have internal capacitances that are very similar. As you can see from the small signal models for a MOSFET (above) and BJT (below), the only significant difference is that the BJT has an
additional resistance Rpi between the Base and Emitter.
Most of the analysis we will do is based on the small signal model. Note that small signal models are not typically used in PSPICE so this picture may look a bit odd, especially the controlled source
but for our purpose it is good to have a visual reference. To start we will point out what everything is. Cgs is an internal capacitance betwe
en the gate and source. The
values for Cgs was similar to one the a PSPICE simulation may give. CM1 and CM2 are Miller capacitances which we will find values for later . ro is a Norton equivalent resistance that makes the model
more ideal. And just pretend that the G2 looks more like a voltage controlled current source and that their gains are gm*Vgs and gm*Vpi. For the BJT CM1 and CM2 are both Miller capacitances, Cpi is
similar to Cgs and Rpi the additional component used for BJTs but not MOSFETs. The other part should look familiar from the other figures.
ON TO THE ANALYSIS!!!
We will find the device gain, overall gain, equivalent input and output capacitances, and the input and output poles. The process for both is essentially the same.
Device Gain: This is the gain from the control source to the output so we are looking for Vout/Vgs (or Vout/Vpi for a BJT). We will ignore CM2 for this process. Notice the resistances ro, RD, and RL
are in parallel. Vout should be given by that equivalent resistance times the current though it which is gm*Vgs from the control source. So the equation for device gain is,
$V_{out} / V_{gs} = gm*(r_o//R_D//R_L)$ (MOSFET)
$V_{out} / V_{\pi} = gm*(r_o//R_C//R_L)$ (BJT)
Overall Gain: This will be the gain from the source (Vs) to the output (Vout). We already know what Vout/Vgs is so if we find Vgs/Vs, we can multiply them to get Vout/Vs = (Vout/Vgs) * (Vgs/Vs). Vgs/
Vs is a simple voltage divider. Hopefully you can see this from the small signal model (remember that we are ignoring the capacitors for now but they will play a part later). The equations we will
get for Vgs/Vs and the overall gain are.
$V_{gs} / V_s = \frac{ (R_1//R_2)}{(R_1//R_2) + R_s}$ (MOSFET)
Overall Gain: $V_{out} / V_s = \frac{ (R_1//R_2)}{(R_1//R_2) + R_s} * gm(r_o//R_D//R_L)$ (MOSFET)
$V_{gs} / V_s = \frac{ (R_1//R_2//r_\pi)}{(R_1//R_2//r_\pi) + R_s}$ (BJT)
Overall Gain: $V_{out} / V_s = \frac{ (R_1//R_2//r_\pi)}{(R_1//R_2//r_\pi) + R_s} * gm(r_o//R_C//R_L)$ (BJT)
Now we will find the input and output poles. For this we will need to look at the capacitances and use a formula to find the Miller capacitances, CM1 and CM2. Any explanation for the miller
capacitance will have to wait for another post or check out your Electronics Book, Wikipedia, Google, etc. but we will need to use a couple of special equations. Overall we will need to find the
input resistance and input capacitance for the input pole and the output resistance and output capacitance for the output pole.
Each pole will be at a frequency w=1/RC where the R and C are the equivalent R and C at that point, so to find the input pole, we will need to find the input resistance and the input capacitance.
These are found by looking into the input (the left side of the small signal model). The voltage source will act like a short so we see Rs in parallel with R1//R2 for the MOSFET (the BJT will have
Rpi in parallel also). The input capacitance will be Cgs in parallel with CM1 (the BJT will be the same). The output resistance and capacitance are found the same way only looking in from the output
(the right side of the small signal model).
$\omega_{IN} = \frac{1}{R_{IN}C_{IN}}$$\omega_{OUT} = \frac{1}{R_{OUT}C_{OUT}}$ (MOSFET or BJT)
So the input pole will be: (MOSFET)
$R_{IN} = R_S//R_1//R_2$ = 950 $R_{OUT} = r_o//R_D//R_L$ =
$C_{IN} = C_{gs} + C_{M1}$ = $C_{OUT} = C_{M2}$ =
$\omega_{IN}$ = $\omega_{OUT}$ =
and the output pole will be: (MOSFET)
$R_{IN} = R_S//R_1//R_2//r_\pi$ = $R_{OUT} = r_o//R_D//R_L$ =
$C_{IN} = C_{BE} + C_{M1}$ = $C_{OUT} = C_{M2}$ =
ELABORADO POR:
NERWIN ANTONIO MORA REINOSO
C.I: 17.557.095
SECCION 1
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Here's the question you clicked on:
Express the following in terms of Heavyside Function=> H(t-a) and find its laplace transform Where F(t) = t 0<t<1 0 1<t< 2 1 t>2
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Hi guys, im new to this topic i would appreciate it if someone can show me in how to proceed forward with this problem.
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Thanks that site looks useful
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The examples on the website are kinda difficult to understand, im still kinda confused....?
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do you know the \[\mathcal L\{u_c(t)f(t-c)\}=e^{-cs}F(s)\]transform?
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Yes i know the forumla but dont know how to apply it yet... Im i suppose to sub in the interval values in first.
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For 0<t<1 we have f(t)=t, so let's start with that Now, since for 1<t<2 we have f(t)=0 we need to subtract the t we had at the beginning, and we need to initiate that using the step function with
c=1. So, so far that gives us\[f(t)=t-tu_1(t)\]so far so good?
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why is f(t)= 0 ? is that from the limits 0<t<1
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your step function is\[f(t)=\left\{\begin{matrix}t&&0<t<1\\0&&1<t<2\\1&&t>2\end{matrix}\right.\]correct? if so, then for t between 1 and 2 your function should equal zero
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yes thats correct, But why?
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you asked why f(t)=0 for 1<t<2... that's why; it's stated in your step function (sorry for the slow replies, my connection is horrible right now) so do you see how\[f(t)=t-tu_1(t)\]satisfies the
first two parts of the step function?
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Oh right i see, so do i integrate now?
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Well first off, I am thinking you want the laplace transforms, not regular integration. We need to do some more manipulation before we can do that. Secondly, we have not yet added the part that
will make it such that f(t)=1 for t>2 any ideas how you might do that? (hint: it requires adding a single term, which is another step function)
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H(t-a) ?
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but the all-important question is"what is a ?".
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1 i guess?
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well is it?
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lol im so confused at the moment, im totally new to this!
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no, you need to understand the heavyside function for H(t-1) what is the value when t=2 ?
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i mean 1 ?
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Symbol Error rate for QAM (16, 64, 256,.., M-QAM)
In May 2008, we derived the theoretical symbol error rate for a general M-QAM modulation (in Embedded.com, DSPDesignLine.com and dsplog.com) under Additive White Gaussian Noise. While re-reading
that post, felt that the article is nice and warrants a re-run, using OFDM as the underlying physical layer. This post discuss the derivation of symbol error rate for a general M-QAM modulation. The
companion Matlab script compares the theoretical and the simulated symbol error rate for 16QAM, 64QAM and 256QAM over OFDM in AWGN channel.
Enjoy and HAPPY NEW YEAR 2012 !!!
Quadrature Amplitude Modulation (QAM) schemes like 16-QAM, 64-QAM are used in typical wireless digital communications specifications like IEEE802.11a, IEEE802.16d. In this post let us derive the
equation for probability of symbol being in error for a general M-QAM constellation, given that the signal (symbol) to noise ratio is $\frac{E_s}{N_0}$ .
The general M-QAM constellation
The number of points in the constellation is defined as, $M=2^b$ where $b$ is the number of bits in each constellation symbol. In this analysis, it is desirable to restrict $b$ to be an even number
for the following reasons (Refer Sec 5.2.2 in [DIG-COMM-BARRY-LEE-MESSERSCHMITT]
1. Half the bits are represented on the real axis and half the bits are represented on imaginary axis. The in-phase and quadrature signals are independent $b/2$ level Pulse Amplitude Modulation (PAM)
signals. This simplifies the design of mapper.
2. For decoding, symbol decisions may be applied independently on the real and imaginary axis, simplifying the receiver implementation.
Note that the above square constellation is not the most optimal scheme for a given signal to noise ratio.
Average energy of an M-QAM constellation
In a general M-QAM constellation where $M=2^b$ and $b$ the number of bits in each constellation is even, the alphabets used are:
$\alpha_{MQAM}=\left{\pm(2m-1)\pm(2m-1)j\right}$, where $m \in \left{1, \cdots,\frac{\sqrt{M}}{2}\right}$.
For example, considering a 64-QAM ($M=64$) constellation,
$m \in \left{1,\ 2,\ 3,\ 4\right}$ and the alphabets are
$\alpha_{64QAM}=\left{\pm7 \pm7j,\ \pm7 \pm5j,\ \pm7 \pm3j, \pm7 \pm1j,\\\pm5 \pm7j,\ \pm5 \pm5j,\ \pm5 \pm3j,\ \pm5 \pm1j,\ \\\pm3 \pm7j,\ \pm3 \pm5j,\ \pm3 \pm3j,\ \pm3 \pm1j, \\\pm1 \pm7j,\ \pm1 \
pm5j,\ \pm1 \pm3j,\ \pm1 \pm1j\right}$.
For computing the average energy of the M-QAM constellation, let us proceed as follows:
(a) Find the sum of energy of the individual alphabets
$\begin{eqnarray}E__\alpha & =&\sum_{m=1}^{\frac{\sqrt{M}}{2}}\left|(2m-1)+j(2m-1)\right|^2\\&=&\frac{\sqrt{M}}{3}(M-1)\end{eqnarray}$
(b) Each alphabet is used $2\sqrt{M}$ times in the M-QAM constellation.
(c) So, to find the average energy from $M$ constellation symbols, divide the product of (a) and (b) by$M$.The average energy is,
$\begin{eqnarray}E_{MQAM} & = & \frac{2\sqrt{M}}{M}E_\alpha\\&=&\frac{2\sqrt{M}}{M}\frac{\sqrt{M}}{3}(M-1)\\&=&\frac{2}{3}(M-1)\end{eqnarray}$.
Plugging in the number for 64-QAM,
Plugging in the number for 16-QAM,
From the above explanations, it is reasonably intuitive to guess that the scaling factor of $\frac{1}{\sqrt{10}}$, $\frac{1}{\sqrt{42}}$ which is seen along with 16-QAM, 64-QAM constellations
respectively is for normalizing the average transmit power to unity.
Finding the symbol error rate
For computing the symbol error rate for an M-QAM modulation, let us consider the 64-QAM constellation as shown in the figure below and extend it to the M-QAM case.
Figure: Constellation plot for 64-QAM modulation (without the scaling factor of $\frac{1}{\sqrt{42}}$)
As can be seen from the above figure, there are three types of constellation points in a general M-QAM constellation:
(a) Constellation points in the corner (red-square)
The number of constellation points in the corner in any M-QAM constellation is always 4, i.e
(b) Constellation points in the inside (magneta-diamond)
The number of constellation points in the inside is,
For example with M=64, there are 36 constellation points in the inside.
(c) Constellation points neither at the corner, nor at the center (blue-star) The number of constellation points of this category is,
$N_{\mbox{neither inside nor corner}}=4(\sqrt{M}-2)$.
For example with M=64, there are 24 constellation points in the inside.
Additive White Gaussian Noise (AWGN) channel
Let the received symbol is,
$y=k\sqrt{E_s}s+n$, where
$E_s$ is the energy,
$k=\sqrt{\frac{1}{\frac{2}{3}(M-1)}}$ is the normalizing factor,
$s$ is the transmit symbol and
$n$ is the noise.
Assume that the additive noise $n$ follows the Gaussian probability distribution function,
$p(x) = \frac{1}{\sqrt{2\pi\sigma^2}}e^{\frac{-(x-\mu)^2}{2\sigma^2}$ with
mean $\mu=0$ and
variance $\sigma^2 = \frac{N_0}{2}$.
Symbol in the inside $I=+1, Q=+1$ (magenta-diamond)
The conditional probability distribution function (PDF) of $y$given that the transmitted symbol is $\left{ +k\sqrt{E_s},+k\sqrt{E_s}\right}$ :
$p(y|inside)=\frac{1}{\sqrt{\pi N_0}}e^{\frac{-(y-k\sqrt{Es})^2}{N_0}}$.
As can be seen from the above figure, the symbol in the inside is decoded correctly only if real part of $y$$(\Re y)$ lies inbetween 0 to 2 and the imaginary part of $y$$(\Im y)$ lies inbetween 0 to
2. The probability of correct demodulation is,
$p(c|inside) = p\left(\Re y >0, \Re y\le 2k\sqrt{E_s} | +1\right)p\left(\Im y >0, \Im y\le 2k\sqrt{E_s} | +1\right)$.
The probability of the real component falling with in 0 to 2 can be found by integrating the probability distribution function of two parts:
(a) Find the probability that the real component lies from $2$ to $+\infty$.
(b) Find the probability that the real component lies from $-\infty$ to $0$.
(c) Given that the total probability is always 1, for finding the probability of the real component lies within $0$ to $2$, subtract the sum of (a) and (b) from 1.
$\begin{eqnarray}p\left(\Re y >0, \Re y\le 2k\sqrt{E_s} | +1\right)& =& 1-\left[\frac{1}{\sqrt{\pi N_0}}\int_{-\infty}^{0}e^{\frac{-(y-k\sqrt{Es})^2}{N_0}}dy + \frac{1}{\sqrt{\pi N_0}}\int_{2k\sqrt
{E_s}}^{+\infty}e^{\frac{-(y-k\sqrt{Es})^2}{N_0}}dy\right]\\& = & 1-erfc\left(k\sqrt{\frac{E_s}{N_0}} \right)\end{eqnarray}$
Note: The complementary error function, $erfc(x) = \frac{2}{\sqrt{\pi}}\int_x^\infty e^{-x^2}dx$.
$\begin{eqnarray}p\left(\Im y >0, \Im y\le 2k\sqrt{E_s} | +1\right)& =&1-erfc\left(k\sqrt{\frac{E_s}{N_0}} \right)\end{eqnarray}$.
From the above equations,
$p(c|inside) = \left[1-erfc\left(\sqrt{\frac{E_s}{10N_o}}\right)\right]\left[1-erfc\left(\sqrt{\frac{E_s}{10N_o}}\right)\right]$.
The probability of the symbol decoded being in error is,
$\begin{eqnarray}p(e|inside) &=&1-\left[1-erfc\left(k\sqrt{\frac{E_s}{N_0}} \right)\right]^2\\& =& 2erfc\left(k\sqrt{\frac{E_s}{N_0}} \right) - erfc^2\left(k\sqrt{\frac{E_s}{N_0}} \right)\end
Symbol in the corner $I=+7, Q=+7$ (red-square)
The conditional probability distribution function (PDF) of $y$given that the transmitted symbol is $\left{ +7k\sqrt{E_s},+7k\sqrt{E_s}\right}$ :
$p(y|inside)=\frac{1}{\sqrt{\pi N_0}}e^{\frac{-(y-7k\sqrt{Es})^2}{N_0}}$.
As can be seen from the above figure, the symbol in the inside is decoded correctly only if real part of $y$$(\Re y)$ lies from $6$ to $\infty$ and the imaginary part of $y$$(\Im y)$ lies from $6$
to $\infty$.
$p(c|corner) = p\left(\Re y >6, \Re y\le \infty| +7\right)p\left(\Im y >0, \Im y\le \infty | +7\right)$.
For finding the probability that the real component lies from $6$ to $\infty$, one can integrate the probability distribution function of the received symbol.
$\begin{eqnarray}p\left(\Re y >6, \Re y\le \infty| +7\right)& =& \frac{1}{\sqrt{\pi N_0}}\int_{6k\sqrt{E_s}}^{+\infty}e^{\frac{-(y-7k\sqrt{Es})^2}{N_0}}dy \\& = & 1-\frac{1}{2}erfc\left(k\sqrt{\frac
{E_s}{N_0}} \right)\end{eqnarray}$
$\begin{eqnarray}p\left(\Im y >6, \Im y\le \infty| +7\right)& =& \frac{1}{\sqrt{\pi N_0}}\int_{6k\sqrt{E_s}}^{+\infty}e^{\frac{-(y-7k\sqrt{Es})^2}{N_0}}dy \\& = & 1-\frac{1}{2}erfc\left(k\sqrt{\frac
{E_s}{N_0}} \right)\end{eqnarray}$.
So, probability that the decoded symbol is correct given $I=+7, Q=+7$is transmitted is,
$\begin{eqnarray}p(c|corner) &=& \left[1-\frac{1}{2}erfc\left(k\sqrt{\frac{E_s}{N_0}} \right) \right]\left[1-\frac{1}{2}erfc\left(k\sqrt{\frac{E_s}{N_0}} \right) \right]\\&=&1-erfc\left(k\sqrt{\frac
{E_s}{N_0}} \right) + \frac{1}{4}erfc^2\left(k\sqrt{\frac{E_s}{N_0}} \right)\end{eqnarray}$.
Now, the probability of the symbol decoded being in error is,
$\begin{eqnarray}p(e|corner) &=&1-p(c|corner)\\& =& erfc\left(k\sqrt{\frac{E_s}{N_0}} \right) - \frac{1}{4}erfc^2\left(k\sqrt{\frac{E_s}{N_0}} \right)\end{eqnarray}$.
Symbol neither at the corner nor inside $I=+7, Q=+1$ (blue-star)
As can be seen from the above figure, the symbol in the inside is decoded correctly only if real part of $y$$(\Re y)$ lies from $6$ to $\infty$ and the imaginary part of $y$$(\Im y)$ lies from 0 to
$p(c|corner) = p\left(\Re y >6, \Re y\le \infty| +7\right)p\left(\Im y >0, \Im y\le \infty | +7\right)$.
For finding the probability that the real component lies from $6$ to $\infty$, one can integrate the probability distribution function of the received symbol.
$\begin{eqnarray}p\left(\Re y >6, \Re y\le \infty| +7\right)& =& \frac{1}{\sqrt{\pi N_0}}\int_{6k\sqrt{E_s}}^{+\infty}e^{\frac{-(y-7k\sqrt{Es})^2}{N_0}}dy \\& = & 1-\frac{1}{2}erfc\left(k\sqrt{\frac
{E_s}{N_0}} \right)\end{eqnarray}$.
As described for the symbol in the inside scenario, the probability of the imaginary component falling with in 0 to 2 can be found by integrating the probability distribution function of two parts:
(a) Find the probability that the imaginary component lies from $2$ to $+\infty$.
(b) Find the probability that the imaginary component lies from $-\infty$ to $0$.
(c) As the total probability is 1, for finding the probability of the imaginary component lies within $0$ to $2$, subtract the sum of (a) and (b) from 1.
$\begin{eqnarray}p\left(\Im y >0, \Im y\le 2k\sqrt{E_s} | +1\right)& =& 1-\left[\frac{1}{\sqrt{\pi N_0}}\int_{-\infty}^{0}e^{\frac{-(y-k\sqrt{Es})^2}{N_0}}dy + \frac{1}{\sqrt{\pi N_0}}\int_{2k\sqrt
{E_s}}^{+\infty}e^{\frac{-(y-k\sqrt{Es})^2}{N_0}}dy\right]\\& = & 1-erfc\left(k\sqrt{\frac{E_s}{N_0}} \right)\end{eqnarray}$
So the probability that the symbol is decoded correctly is,
$\begin{eqnarray}p(c|\mbox{neither inside nor corner}) &=& \left[1-erfc\left(k\sqrt{\frac{E_s}{N_0}} \right) \right]\left[1-\frac{1}{2}erfc\left(k\sqrt{\frac{E_s}{N_0}} \right) \right]\\&=&1-\frac{3}
{2}erfc\left(k\sqrt{\frac{E_s}{N_0}} \right) + \frac{1}{2}erfc^2\left(k\sqrt{\frac{E_s}{N_0}} \right)\end{eqnarray}$
The probability of error is,
$\begin{eqnarray}p(e|\mbox{neither inside nor corner}) &=&1-p(c|\mbox{neither inside nor corner})\\& =& \frac{3}{2}erfc\left(k\sqrt{\frac{E_s}{N_0}} \right) - \frac{1}{2}erfc^2\left(k\sqrt{\frac{E_s}
{N_0}} \right)\end{eqnarray}$.
Total symbol error probability
Given that we have computed the individual symbol error probability for each of the three types of constellation points, to find the joint symbol error rate we compute the average error i.e.
$P(e|MQAM) = \frac{N_{inside}p(e|inside) + \\ N_{corner}p(e|corner) + \\ N_{\mbox{neither inside nor corner}}p(e|\mbox{neither inside nor corner})}{M}$
Pluging in the equations,
$\Large \begin{array}{rrr}P(e|MQAM)& = & 2\left(1-\frac{1}{\sqrt{M}}\right)erfc\left(k\sqrt{\frac{E_s}{N_0}}\right)-\\ & & \left(1-\frac{2}{\sqrt{M}}+ \frac{1}{M}\right)erfc^2\left(k\sqrt{\frac{E_s}
OFDM modulation
Let us use the OFDM system loosely based on IEEE 802.11a specifications.
│ Parameter │ Value │
│FFT size. nFFT │ 64 │
│Number of used subcarriers. nDSC │ 52 │
│FFT Sampling frequency │ 20MHz │
│Subcarrier spacing │ 312.5kHz │
│Used subcarrier index │{-26 to -1, +1 to +26}│
│Cylcic prefix duration, Tcp │ 0.8us │
│Data symbol duration, Td │ 3.2us │
│Total Symbol duration, Ts │ 4us │
Simulation model
Simple Matlab/Octave script for computing the symbol error rate for 16QAM/64QAM and 256QAM modulation scheme in additive white gaussian noise channel over OFDM.
Click here to download : Matlab code for computing symbol error rate for 16QAM/64QAM and 256QAM in AWGN over OFDM
[DIG-COMM-BARRY-LEE-MESSERSCHMITT]Digital Communication: Third Edition, by John R. Barry, Edward A. Lee, David G. Messerschmitt
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e-Book on BER of BPSK/QPSK/16QAM/16PSK in AWGN.
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Simons Foundation Launches Three Math Sciences Initiatives
October 19, 2010
Mathematical models, miscast as villains in some accounts of the current financial crisis, have been responsible for keeping the hedge funds of at least one carefully watched firm spectacularly
profitable. In a classic case of giving back, profits from those funds are now being used to benefit the discipline from which the models emerged.
Many readers will quickly identify the main players in this tale: Renaissance Technologies, the investment firm founded and until recently run by James Simons (former professor and chair of
mathematics at Stony Brook University, best known for the Chern–Simons invariants) and the Renaissance hedge funds, including Medallion, whose investments are guided by mathematical models.
Another key player, probably less familiar to readers, is the Simons Foundation, created about 15 years ago to organize James and Marilyn Simons's giving and expanded over the years to support
scientific research and organizations. Readers may be aware of two of the foundation's largest projects in mathematics and the physical sciences: the Simons Center for Geometry and Physics at Stony
Brook University, and Math for America, an organization devoted to the improvement of high school mathematics teaching. Other major interests of the foundation include the biological
sciences---mathematical biology and the study of autism in particular. Even readers who have followed one or another aspect of the story may not be aware of the extent of the Simons Foundation's
support for the mathematical and physical sciences.
In a late-August phone call, David Eisenbud, since January the Simons Foundation's director for mathematics and the physical sciences, introduced SIAM News to the foundation and some of its
activities. (Eisenbud is on two-thirds leave from UC Berkeley, where he is a professor of mathematics and, from 1997 to 2007, was director of the Mathematical Sciences Research Institute.) Parts of
the SIAM community, he said, will be interested in three new programs:
• Math + X. The Simons Foundation intends to endow university chairs, shared by mathematics and some other discipline, "X". The foundation will actually provide half the endowment, with the
university expected to match the investment; the foundation's share will be $1.5 million toward the endowment, along with, for six years, $325,000/year for postdocs and graduate students and
$100,000/year for conferences, visitors, and similar activities. The intention is to create one or two chairs each year for some time.
Details can be found at https://simonsfoundation.org. Letters of intent from interested pairs of departments must be received by October 27.
• A New Institute for the Theory of Computing. Given the many overlapping and computing-related areas in which members of the SIAM community work, some clarification is in order. The institute,
which is to be located at a U.S. institution, will be focused on theoretical computer science and the scientific applications of its ideas. The foundation has adopted a broad definition of the
field, including work in complexity theory, algorithms, machine learning, randomness and pseudorandomness, zero-knowledge proof, computer networks, computer vision, robotics, and streaming
The institute project, Eisenbud explains, emerged from a round table discussion in which theoretical physicists, mathematicians, and theoretical computer scientists put on the table lists of
"unfulfilled needs" in their disciplines. The computer science people on the panel were extremely enthusiastic about the chosen direction, he says, as is Simons himself. Once the foundation had
settled on the theory of computing as the institute's scientific area, a panel of experts was convened to help define its scope. As to the resources the new institute will have, the Simons
Foundation intends to provide $6 million/year for ten years, with a possibility of endowment support thereafter. Again, details can be found on the web and letters of intent are due by October
• Collaboration Grants for Mathematicians. A response to longtime wishes of parts of the mathematical community, this program will award many small grants to, in Eisenbud's words, "researchers who
don't have major public funding." Each grant will be $7000/year for five years, including $1000/year to the grantee's department to "enhance the research atmosphere" and $1000/year in indirect
costs. By supporting travel for visitors and graduate students, Eisenbud said, the program is an attempt to enhance the "fluidity and effectiveness of the community." About 140 awards are planned
each year, until a steady state of 700 is reached.
Eisenbud also points to a postdoc program, established a year ago as a response to the sparse job market. Nearly 70 two- or three-year postdoc positions (half in mathematics, about a quarter each in
theoretical physics and theoretical computer science) were promised; the fellowships will take effect from 2010 to 2012. The awards were made to university departments, on the recommendation of a
small committee in each field. The goal was to allow the strongest postdocs to take positions in institutions that could best support their development, but that might not have had the funds to hire
as many postdocs as they could foster. From the first, this program was seen as an interim response to the bad economic situation and is not slated for continuation, at least in its present form.
Demonstrating the reach of Simons's interests, Eisenbud also mentions his efforts in support of a restructuring of the State University of New York system that would include quality-based differences
in tuition, and consequently in revenues, at individual campuses.
Asked how his ten years at MSRI might have prepared him for the position he holds at the Simons Foundation, Eisenbud laughs. The two activities are inverses of one another, he explains. At MSRI,
finding the money to support mathematical programs was a constant challenge. At the Simons Foundation, the money is there; the question is identifying the areas and people who will put it to the best
possible use.---GRC | {"url":"https://www.siam.org/news/news.php?id=1829","timestamp":"2014-04-19T17:05:07Z","content_type":null,"content_length":"12915","record_id":"<urn:uuid:726b90ea-724f-4c0d-8cdc-64025f304cb2>","cc-path":"CC-MAIN-2014-15/segments/1398223211700.16/warc/CC-MAIN-20140423032011-00130-ip-10-147-4-33.ec2.internal.warc.gz"} |
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quantum number
is a positive integer, but is often represented as K, L, M, and so on, corresponding to 1, 2, 3, etc respectively. This represents the
principal quantum number
which is sometimes referred to as an energy level (such as it is later in this writeup).
l relates to the quantization of the orbital angular momentum. It is an integer as well, between 0 and n-1. When it is 0, the state is s, a spherical shell with no net angular momentum. When l is 1,
the state is p. In this case, there is directionality, a sort of dumbell shape elongated on the coordinate axes. If l is 2, the state is referred to as d. For more details on this see d orbital. The
letter f corresponds to 3, see f orbital.
m There are two different m variables. m[i] is greater than -l and less than +l. ( -l <= m[i] <= l) and it reprsesnts the magnetic quantum number. m[s] however is either 1/2 or -1/2 indicating an
"upward" or "downward" spin. It is called the "spin quantum number"
The Pauli Exclusion Principle says that no given electron in an atom can share exactly the same values of all four numbers with another. At least one must be different.
This information was adapted from my
Materials Science and Engineering
lecture notes. | {"url":"http://everything2.com/title/quantum+number","timestamp":"2014-04-19T20:49:39Z","content_type":null,"content_length":"20588","record_id":"<urn:uuid:66ed04f8-7a9e-4022-b458-f7af1cf889c4>","cc-path":"CC-MAIN-2014-15/segments/1397609537376.43/warc/CC-MAIN-20140416005217-00366-ip-10-147-4-33.ec2.internal.warc.gz"} |
ML_FiniteElements Namespace Reference
Default namespace for ML finite element example. More...
Detailed Description
Default namespace for ML finite element example.
The ML_FiniteElements namespace contains all the classes and the examples that show a possible structure for a finite element code using Epetra and ML.
Scalar second-order, symmetric and non-symmetric PDEs of type
can be discretized using this example. Neumann boundary conditions require minimal changes to the code.
Two discretizations are available:
• GalerkinVariational is a pure Galerkin approach. Since this approach is unstable for large Peclet numbers, the class defines b = 0.
• SUPGVariational is a SUPG discretization, only for triangles and tetrahedra, and coth formula for tau. It should be used for advective problems.
The code is based on the following pure virtual classes:
The solution can be visualized using MEDIT (see web page http://www.ann.jussieu.fr/~frey/logiciels/medit.html for details and download).
A nice feature of this small example is that some grids can be created on-the-fly, with no need of input files. The following domains can be triangulated:
These classes are valid for both serial and parallel runs.
An interface to triangle (see http://www-2.cs.cmu.edu/~quake/triangle.html for more details and download) is under development; the interested user can modify the file ml_TRIANGLEGrid.h. This
interface is currently working for serial runs only.
The interested user should look to the following examples, all located in ml/examples/FiniteElements:
• Laplacian3D.cpp describes how to solve the Laplace equation in a unitary cube, discretized using hexahedra. After solution, the norm of the numerical solution, exact solution and error is
computed, and the numerical solution is visualized using MEDIT.
• AdvDiff2D.cpp describes how to solve an advection-diffusion problem in a 2D domain, using SUPG.
• MLAPI shows how to interface with the MLAPI classes.
• TestAll is for testing purposed only.
Marzio Sala, SNL 9214.
Last updated on Apr-05.
Generated on Thu Sep 18 12:40:58 2008 for ML by 1.3.9.1 | {"url":"http://trilinos.sandia.gov/packages/docs/r6.0/packages/ml/doc/html/namespaceML__FiniteElements.html","timestamp":"2014-04-18T08:36:11Z","content_type":null,"content_length":"15971","record_id":"<urn:uuid:4c665735-38b7-4cc1-9cd7-c1a383808786>","cc-path":"CC-MAIN-2014-15/segments/1397609533121.28/warc/CC-MAIN-20140416005213-00265-ip-10-147-4-33.ec2.internal.warc.gz"} |
Arithmetic mean
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In mathematics and statistics, the arithmetic mean of a set of numbers is the sum of all the members of the set divided by the number of items in the set (cardinality). (The word set is used perhaps
somewhat loosely; for example, the number 3.8 could occur more than once in such a "set".) If one particular number occurs more times than others in the set, it is called a mode. The arithmetic mean
is what pupils are taught very early to call the "average." If the set is a statistical population, then we speak of the population mean. If the set is a statistical sample, we call the resulting
statistic a sample mean.
When the mean is not an accurate estimate of the median, the set of numbers, or frequency distribution, is said to be skewed.
The symbol μ (Greek: mu) is used to denote the arithmetic mean of a population.
If we denote a set of data by X = { x[1], x[2], ..., x[n]}, then the sample mean is typically denoted with a horizontal bar over the variable (x̅, generally enunciated "x bar").
In practice, the difference between μ and x̅ is that μ is typically unobservable because one observes only a sample rather than the whole population, and if the sample is drawn randomly, then one may
treat x̅, but not μ, as a random variable, attributing a probability distribution to it.
Both are computed in the same way:
$\mathrm{mean} = (x_1+\cdots+x_n)/n.$
The arithmetic mean is greatly influenced by outliers. For instance, reporting the "average" net worth in Redmond, Washington as the arithmetic mean of all annual net worths would yield a
surprisingly high number because of Bill Gates. These distortions occur when the mean is different from the median, and the median is a superior alternative when that happens.
In certain situations, the arithmetic mean is the wrong concept of "average" altogether. For example, if a stock rose 10% in the first year, 30% in the second year and fell 10% in the third year,
then it would be incorrect to report its "average" increase per year over this three year period as the arithmetic mean (10% + 30% + (−10%))/3 = 10%; the correct average in this case is the geometric
mean which yields an average increase per year of only 8.8%.
If X is a random variable, then the expected value of X can be seen as the long-term arithmetic mean that occurs on repeated measurements of X. This is the content of the law of large numbers. As a
result, the sample mean is used to estimate unknown expected values.
Note that several other "means" have been defined, including the generalized mean, the generalized f-mean, the harmonic mean, the arithmetic-geometric mean, and the weighted mean.
Alternate notations
The arithmetic mean may also be expressed using the sum notation:
$\bar{x} = \frac1n\sum_{i=1}^n x_i.$
See also
mean, average, summary statistics, variance, central tendency, standard deviation, inequality of arithmetic and geometric means, Muirhead's inequality
External links
cs:Aritmetický průměr de:Mittelwert es:Media aritmética fr:Moyenne arithmétique hr:Aritmetička_sredina nl:Rekenkundig gemiddeldeno:Gjennomsnittpt:Média aritmética fi:Aritmeettinen keskiarvo zh:算术平 | {"url":"http://psychology.wikia.com/wiki/Sample_mean","timestamp":"2014-04-19T07:33:56Z","content_type":null,"content_length":"63540","record_id":"<urn:uuid:11cdd47f-2f5e-4b9c-a2e2-728db869b1ee>","cc-path":"CC-MAIN-2014-15/segments/1397609536300.49/warc/CC-MAIN-20140416005216-00074-ip-10-147-4-33.ec2.internal.warc.gz"} |
Bisection Method And Matlab Code
Bisection method is the one of the bracketing numerical methods. This method allows us to find the roots of polynomial but this method cannot find noncomplex roots of the polynomial equation, it
allows us to find only reel roots of the equation. Due to this feature, bisection method considered primitive method by mathematicians. If you use this method you cannot find the real roots of the
equation but you can find the closest values of the roots with some inaccuracy
To find the roots of the equation, you must start with two initial estimates of the roots.One of these estimates should be higher value of the root and the other one should be remain under value of
the root. Otherwise bisection method will not work properly.
Description Of The Bisection Method
It is known that, the function values above and below of the root is opposite in sign. So we can define the root as the point where the equation changes its sign. This statement is actually not true.
For example, let’s look at the following very simple example
As everyone knows, the root is zero. However, this problem cannot solve by the bisection method. Because, this function is a dual function. (symmetrical by y axis) basically, there are no negative
root of this equation so when we multiply this roots, multiplication value must be positive.
But we have one simple rule to use this method;
F(xu) = under value F(xl)= higher value
F(xu) x F(xl) < 0
After providing this condition, we must find the location of the root. Estimated location of the root by bisection method can be found by this equation;
This formula makes sense indeed because it is certain that the root must be found between the under value and higher value. After the calculation phase we must make a decision. The root is between xr
and xu or it is between xr and xl ?
It is decided as follows.
If f(xr) x f(xu) < 0 it is between xr and xu.
Otherwise it is between xr and xl.
After deciding this, method will make new assignments and repeats itself in this way.
This methot called bisection method because of that. Bisection method approximates the root by dividing it.
Below, you can see the algorithm of Bisection method.
Matlab Code For Bisection Method
% wwww.engineershandbook.net
a=input('Enter Function:','s');
xl=input('Enter lower guess:') ;
xu=input('Enter upper guess:');
tol=input('Enter tolerance(recommended 0.001):');
if f(xu)*f(xl)<0
fprintf('Wrong Guess!Enter new guess\n');
xl=input('Enter lower guess:\n') ;
xu=input('Enter upper guess:\n');
for i=2:1000
if f(xu)*f(xr)<0
if f(xl)*f(xr)<0
if abs((xnew(i)-xnew(i-1))/xnew(i))<tol,break,end
str = ['Root: ', num2str(xr), '']
There are two examples for Matlab output. In the first one, the estimates have been selected entirely true. In the other one, knowingly false predictions have been selected. | {"url":"http://www.engineershandbook.net/bisection-method-and-matlab-code/","timestamp":"2014-04-19T05:07:58Z","content_type":null,"content_length":"19578","record_id":"<urn:uuid:b38a1891-37bb-45fd-bd6c-8973a9383a4b>","cc-path":"CC-MAIN-2014-15/segments/1398223206647.11/warc/CC-MAIN-20140423032006-00001-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Safe Haskell Safe-Inferred
SC3 pitch model implementation.
class Pitched p whereSource
Pitched values, minimal definition is midinote.
midinote (defaultPitch {degree = 5}) == 69
freq (defaultPitch {degree = 5,detune = 10}) == 440 + 10
data Pitch Source
The supercollider language pitch model is organised as a tree with three separate layers, and is designed to allow separate processes to manipulate aspects of the model independently.
The haskell variant implements Pitch as a labeled data type, with a default value such that scale degree 5 is the A above middle C.
freq (defaultPitch {degree = 5}) == 440
The note is given as a degree, with a modal transposition, indexing a scale interpreted relative to an equally tempered octave divided into the indicated number of steps.
The midinote is derived from the note by adding the indicated root, octave and gamut transpositions.
The frequency is derived by a chromatic transposition of the midinote, with a harmonic multiplier.
let {p = defaultPitch
;n = p {stepsPerOctave = 12
,scale = [0,2,4,5,7,9,11]
,degree = 0
,mtranspose = 5}
;m = n {root = 0
,octave = 5
,gtranspose = 0}
;f = m {ctranspose = 0
,harmonic = 1}}
in (note n,midinote m,freq f) == (9,69,440)
By editing the values of aspects of a pitch, processes can cooperate. Below one process controls the note by editing the modal transposition, a second edits the octave.
let {edit_mtranspose p d = p {mtranspose = mtranspose p + d}
;edit_octave p o = p {octave = octave p + o}
;p' = repeat defaultPitch
;q = zipWith edit_mtranspose p' [0,2,4,3,5]
;r = zipWith edit_octave q [0,-1,0,1,0]
;f = map midinote}
in (f q,f r) == ([60,64,67,65,69],[60,52,67,77,69])
Eq Pitch
Show Pitch
Pitched Pitch
defaultPitch :: PitchSource
A default Pitch value of middle C given as degree 0 of a C major scale.
let {p = defaultPitch
;r = ([0,2,4,5,7,9,11],12,0,5,0)}
in (scale p,stepsPerOctave p,root p,octave p,degree p) == r
type T616 a b c = (a, a, a, a, a, a, b, c, c, c, c, c, c)Source
Tuple in 6-1-6 arrangement. | {"url":"http://hackage.haskell.org/package/hsc3-lang-0.14/docs/Sound-SC3-Lang-Control-Pitch.html","timestamp":"2014-04-20T23:49:11Z","content_type":null,"content_length":"15760","record_id":"<urn:uuid:23e56877-3f9f-465d-b541-9419ff6db646>","cc-path":"CC-MAIN-2014-15/segments/1397609539337.22/warc/CC-MAIN-20140416005219-00422-ip-10-147-4-33.ec2.internal.warc.gz"} |
FOM: Hersh/Rota/Feferman/Lakatos
Reuben Hersh rhersh at math.unm.edu
Fri Jan 2 20:18:32 EST 1998
On Fri, 2 Jan 1998, Robert S Tragesser wrote:
> ABSTRACT: A sketch of the origins of mathematics that
> explains what is distinctive about mathematical knowledge
> but does not essentially depend on a reference to
> traditional mathematicals (such as numbers). This
> explanation explains why mathematical experience
> supports Platonism but also why Platonism can likely be
> never more than a regulative idea(l).
> Feferman's questions about the distinctiveness of
> mathematical thought are put to Hersh/Lakatos, and
> answered via Rota's phenomenological thinking, and
> answered in a way without undercutting the phenomena
> which fund Platonism, but which at the same time reveal
> the extreme difficulties in sustaining Platonism as anything
> more than a regulative idea(l). [End of Abstract.]
> I've been enjoying Hersh's _What is Mathematics,
> Really?_ I think that Reuben Hersh is importantly
> correcting the course of the philosophy of mathematics,
> but at the same time he _over-corrects_ it.
> How should the over-correction be itself corrected?
> Hersh and Lakatos stand in need of such correction,
> and in need of the very same correction -- they both overly
> collapse -- in a phenomenologically undiscerning way --
> uses of 'certainty' (and its "cognates").
> I've been writing a long review essay on the
> phenomenological studies of mathematical thought in
> Gian-Carlo Rota's _Indiscrete Thoughts_. It is worth
> observing that, although Rota and Hersh, through
> forewards and dust-jacket copy, strongly applaud one
> another's (philosophical) work, Rota's phenomenological
> appreciation of mathematical proof and mathematical
> understanding provides the needed correction to Hersh's
> course.
> I want to try to explain this here, and do so in a way
> that points to some answers to questions raised by Sol
> Feferman.
> In a recent FOM posting, Cook asked Hersh how
> mathematics is distinguished from other "academic
> subjects". Hersh answered that mathematics was one of
> the "humanities" (because it is entirely a human product)
> and is distinguished among the humanities by being about
> mathematicals.
NO, NOT AT ALL, NOT IN THE LEAST. I EXPLAINED ALL
ABOUT THIS IN ANSWERING PROF. TRAGESSER'S LAST POSTING.
> In his review of Lakatos, Sol Feferman asked a series
> of questions -- one close to that of Cook's -- which we,
> qua phenomenological Rota, put to Hersh (but don't wait
> around for his answer), --
> FEFERMAN'S QUESTIONS:
> [1] What is distinctive about mathematics?
> _i.e._,
> [2] What is distinctive about its verification structure?
> [3] What is distinctive about its conceptual content?
> The absolutely important _necessary condition_ for giving
> a good answer to these questions:
> Necessary Condition: They must allow us to
> characterize mathematics in a way that is independent of
> the kind of appeal Hersh makes, viz., that mathematics
> studies the mathematicals (studies what mathematicians
> study).
NO NO ONCE AGAIN NO!
> It is exactly by evading this issue that Hersh misses
> the distinctive features of the verification structure of
> mathematics (or: is able to play a bit too much of the old
> fast and loose with it).
WHERE? WHEN? HOW? DETAILS, PLEASE.
> ****
> I'll make this short, though details can be supplied (some
> will be supplied in a posting giving an account of
> Lebesgue's conception of arithmetic).
> First, I'll describe the two stages in the origin of
> mathematics, and then I'll give an example.
> [1] First stage of the origin of mathematics: Witty persons
> become aware of (practical) problems which have
> solutions which (a) can be framed or represented in
> thought and (b) can be seen by thought alone to be
> definitive solutions (seen by a peculiar sort of light;
> mathematical proofs will be architectures in that light).
> The recognition and cultivation of such problems is the
> first stage in the development of mathematical thought.
> Their cultivation may go beyond practical interests (e.g.,
> as in play or poetry. . .riddles. . .problems to be solved in
> contests, etc. . . .N.B., Ian Hacking's phantasy of
> language issuing more from play rather than work).
> [2] Second stage of the origin of mathematics (Berkeleyian
> abstraction): it is observed that the light by which one sees
> that such and such is THE solution to the problem "Q?"
> has an authority that is not bound to the concrete
> particulars of the problem. In successfully reframing such
> problems, their solutions, and the
> exhibition/demonstration of such solutions as _the_
> solutions, mathematics proper begins.
> EXAMPLE: [This example is meant to illustrate the ideas,
> so it is ideal; but it is exemplary, too, in that more
> realistic examples having to do with the "real" origins of
> arithmetic, algebra, geometry, combinatorics. . . will be
> patterned after it.]
> [E1] It becomes practically important for me to know the
> minimum number of fruitfly I have to capture in order to
> be certain that I have two which are the same sex.
> I proceed experimentally.
> First, I form collections each containing one fruitfly.
> I notce that none of them contains two fruitfly which have
> the same sex (but dull empiricist that I am, I do not notice
> that each collection fails to contain two fruitfly of the same
> sex _because_ each collection contains only one fruitfly).
> Second, I form collections each containing _three_
> fruitfly (because dull empiricist that I am I don't see that
> the next logical step would be to try collections of two). I
> find that each collection contains at least two fruitfly of the
> same sex. But now I worry whether this is the least
> number. And dull empiricst that I am, I first make a lot
> of collections containing _two_ fruitfly and a lot of
> collections containing _five_ fruitfly, comparing them to
> the collections containing three fruitfly to see whether the
> collections containing five fruitfly or the collections
> containing two fruitfly contain fewer fruitfly than the
> collections containing three fruitfly. I learn that the
> collections containing two fruitfly contain fewer fruitfly;
> so I will not look through those collections to see if two
> fruitfly of the same sex invariably occur. I find that they
> do not. So I conclude: _3_ is the answer to the initial
> question. Of course, it might have happened that male
> fruitfly were very rare, so that in fact all the pairs I
> collected were pairs of females, and I was led to answer:
> 2, instead of 3. But in any case, someone will cast
> doubt on my answer _3_ because I overlooked so many
> other possibilities, such as 4, 23, 197, 272. . .
> Observe that it is rather unlikely that there should be
> such very dull empiricists. . .who never let in any as it
> were _apriori_ thinking, for whom it could never be
> decisive that collections containing two things have fewer
> things than those containing three or four things.
> But here is the example of _apriori_ thinking I want
> to dwell on:
> A collection of three firefly must contain at least two
> which have the same sex, for here are the only possible
> combinations of three firefly identified up to sex (M, F):
> MMM, MMF, MFF, FFF.
> "Check that this lists all possible combinations."
> (There are of course a number of ways that the check
> can be made _apriori_ . . .)
> Here one has solved the problem, definitively, and
> _apriori_. Any skeptic must be either witless or fail to
> understand the terms of the problem (such as its being
> assumed that all firefly are either M or F -- see below ON
> HIDDEN ASSUMPTIONS).
> This problem illustrates the first stage. The second
> stage, mathematics proper, begins with the observation
> that in dmeonstrating to onesself that the solution to the
> problem is three, one actually has proved a more general
> proposition (which in the framing becomes more abstract):
> The demonstration of the correctness of the solution
> to the problem does not essentially depend on fruitflies or
> on sex.
> Ginning up concepts to state and prove, and then
> stating and proving, the more general/abstract proposition
> is the very essence of logico-mathematical activity. It is
> one thing to notice that the "proof" proves more than the
> particular proposition (about collections of sexed fruitfly)
> at issue. That's the first phase of the second stage on the
> way to mathematics. The second phase is to find the
> concepts in which to frame the more general/abstract
> proposition. For examples, a language containing
> 'individual', 'property', 'collection', 'has the property',
> 'does not have the property', and so on, and more or less
> explicit rules for using these terms.
> Some conjectures about the characteristic trait of
> mathematical concepts:
> What enable emergence of the definitive solution to
> the firefly problem?
> Being able to canvas all the possibilities in advance.
> This suggests then a tentative characterization of
> mathematical concepts: that they enable us to canvass
> (directly as above, or indirectly, as is more typical) in
> advance all the relevant possibilities of what they frame (of
> the problems which can be framed through them).
> This suggests why Platonism might seem supported
> but in actuality false:
> Concepts arising in mathematics (through the process
> of Berkleyian "abstraction/generalizatiuon" sketched
> above) may sustain _apriori_ solutions to a wide range of
> problems posed through them (the concepts), but the
> concepts may also be rough around the edges -- we in fact
> cannot canvass all the relevant possibilities relating to all
> problems framed in those concepts because the concepts
> are not decisive on all such ranges.
> REMARKS:
> Rota calls Evidenz (which he translates "insightful;
> understanding") the kind of demonstration/light by which
> we decisively (and _apriori_) find proposed solutions to
> mathematical problems to be solutions to problems. Yes,
> one can always be "skeptical" in the sense of demanding
> greater explicitness, etc. But there comes a point at
> which one either has understood the problem or one has
> not. . .whereafter we have to say that, as far as the
> problem at issue is concerned, the skeptic is not cooking
> on all four burners. (For those who can read Descartes'
> First meditation with great care, it can be seen that
> Descartes -- like Wittgenstein and Cavell -- makes
> essentially this point. . .that doubt uninhibited by authentic
> understanding is madness, skeptical terror only.)
> I had best break off here.
> rbrt tragesser
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Current status of Waring-Goldbach problem
up vote 8 down vote favorite
Is the following statement proved?
For any positive integer $k$ there exists positive integer $n$ such that all sufficiently large integers may be represented as $p_1^k+p_2^k+\dots+p_n^k$ for primes $p_1,\dots,p_n$.
This Wiki article claims that some progress is made only for $k$ up to 7, on the other hand, it refers to Hua Lo Keng's monograph, in which this statement is proved, if we may take zero terms instead
some prime powers. This zero does not seem to be so essential on the first glance... And on the third hand, this paper of Chubarikov contains announcement of the complete proof, though I did not
succeed in finding any reaction on it, even no MathSciNet-review.
nt.number-theory prime-numbers
I'm pretty sure this is known for all k but I'm afraid I don't know a reference. – gowers Jul 14 '11 at 20:03
Sorry, I take that back -- I was thinking of the result with zeros. But I think the zeros are needed for congruence reasons and it's not clear that one couldn't use some trick. E.g. if k=6 then
each kth power is 1 mod 7, except that we can throw in a few $p_i=7$s to deal with that. – gowers Jul 14 '11 at 20:22
wiki says n is an absolute constant for all k. Is this the case? – user16007 Aug 12 '11 at 8:45
@unknown(yahoo): definitely no! For fixed $n$ and large $k$ result is not true by simple counting argument (density of representable numbers is 0), even if you take all positive integers instead
primes. – Fedor Petrov Aug 14 '11 at 6:51
add comment
1 Answer
active oldest votes
This is a corrected version of my original response, incorporating a nice argument by Fedor Petrov.
Hua in his book (cf. review of MR0124306) proved that there are integers $s,K,N>0$ such that every $n>N$ with $n\equiv s\pmod{K}$ is a sum of $s$ $k$-th powers of primes. For any $t>0$ let
$M(t)$ denote the set of residues modulo $K$ which can be represented by a sum of $t$ $k$-th powers of primes. Clearly $M(t+1)$ contains $M(t)+p^k$ for any prime $p$, hence $|M(t+1)|> |M
(t)|$ unless $M(t+1)$ equals $M(t)+p^k$ for any prime $p$. In this case $M(t)$ is invariant under the shift of $p^k-q^k$ for any two distinct primes $p$ and $q$. The shifts are coprime
up vote 9 (e.g. $p^k-q^k$ is coprime with $q$), hence $M(t)=M(t)+1$, and $M(t)$ contains all residues modulo $K$. This argument shows that $|M(K)|=K$, i.e. modulo $K$ every residue class is a sum of
down vote $K$ $k$-th powers of primes. If $p$ denotes the largest of the $K^2$ primes used in the latter representation, and $M$ equals $N+Kp^k$, then we have the following. For every $m>M$ there is
accepted a sum of $K$ $k$-th powers of primes, denote it by $m'$, such that $m-m'\equiv s\pmod{K}$ and $m-m'>N$. By Hua's theorem, $m-m'$ is a sum of $s$ $k$-th powers of primes, hence in fact
every $m>M$ is a sum of $s+K$ $k$-th powers of primes.
To summarize: the statement in Fedor Petrov's original question follows from Hua's theorem.
3 Well, if Hua's theorem is that, then indeed the rest is easy. I do not take your argument literary, by the way (Dirichlet's theorem requires some coprimeness etc), but it looks that we
may avoid even Dirichlet's theorem for finishing the proof. Indeed, if $M(n)$ denote the set of residues of $p_1^k+\dots+p_n^k$ modulo $K$, then either $|M(n+1)|\geq \min(M(n)+1,K)$. –
Fedor Petrov Jul 14 '11 at 21:30
2 (continuation) Indeed, $M(n+1)$ contains $M(n)+p^k$ for each prime $p$, so if $|M(n+1)|=|M(n)|$, then $M(n+1)=M(n)+p^k$, so $M(n)$ is invariant under shift to $p_1^k-p_2^k$ for any two
primes $p_1$, $p_2$. But such shifts are coprime for obvious reasons (if they all are divisible by prime $q$, then choose $p_1=q$ and $p_2\neq q$). – Fedor Petrov Jul 14 '11 at 21:30
Dear Fedor, thanks for the correction. I realized the problem with coprimeness shortly after my response, but fortunately your argument saved the day. I updated my response accordingly.
– GH from MO Jul 14 '11 at 22:24
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Solving K[sp] Problems: Part One
There are five substances I will use as examples. I will do AgCl in this file and do two others in each of part two and part three.
The problem will always be the same in each example. It is:
Calculate the molar solubility (in mol/L) of a saturated solution of the substance.
Silver chloride, AgCl, has a K[sp] = 1.77 x 10¯^10. Calculate its solubility in moles per liter.
The dissociation equation is as before:
AgCl (s) <===> Ag^+ (aq) + Cl¯ (aq)
The K[sp] expression is as before:
K[sp] = [Ag^+] [Cl¯]
This is the equation we must solve. First we put in the K[sp] value:
1.77 x 10¯^10 = [Ag^+] [Cl¯]
Now, we have to reason out the values of the two guys on the right, First of all, I have no idea what the values are so I'll use unknowns. Like this:
[Ag^+] = x
[Cl¯] = y
However, I now have a BIG problem (No, not B.I.G., just BIG). The problem is that I have two unknowns, but only one equation. So it seems I'm stuck, untillllll . . .
I examine the chemical equation and see that there is a one-to-one ratio between Ag^+ and Cl¯. I know this from the coefficients (both one) of the balanced equation.
That means that the concentrations of the two ions are EQUAL. I can use the same unknown to represent both. Like this:
[Ag^+] = x = [Cl¯]
Substituting, we get:
1.77 x 10¯^10 = (x) (x)
Now, we take the square root of both sides. I hope I'm not too insulting when I emphasize both sides. I have had lots of people take the square root of the x^2 side, but not the other. After the
square root, we get:
x = 1.33 x 10¯^5 M
This is the answer because there is a one-to-one relationship between the Ag^+ dissolved and the AgCl it came from. So, the molar solubility of AgCl is 1.33 x 10¯^5 moles per liter.
One last thing. The K[sp] value does not have any units on it, but when you get to the value for x, be sure to put M (for molarity) on it. The reasons behind this are complex and beyond the scope of
the ChemTeam's chosen area.
Do you want to see more problems like this?
Go to Solving K[sp] Problems - Part Two
Go to Solving K[sp] Problems - Part Three
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Build your own disease
Some guidance on Build your own disease
In simulation 1 the infection will keep on spreading, as every infected person goes on to infect at least one other person. In simulation 2 the infection dies out as soon as you get heads. While it's
theoretically possible that you only get tails, so that the infection keeps on spreading, this is unlikely if you trace the infection over many generations. So in general the disease is likely to die
out of its own accord.
In simulation 1 an infected person infects 3/2 others on average, and it simulation 2 they infect 1/2 others on average. These numbers are the respective basic reproduction ratios.
Suppose there are n infected people to start with, then on average there are nR newly infected people in the next generation, nR^2 in the third generation, nR^3 in the fourth generation, and so on,
up to nR^k newly infected people in the k+1st generation. If R is greater than one, then R^k gets larger and larger as k gets larger, so the disease escalates. If on the other hand R is less than
one, then R^k gets smaller and smaller as k gets larger. In this case the disease peters out. So any intervention programme should aim to reduce the basic reproduction ratio to an effective
reproduction ratio which is less than 1.
Now suppose that R=3/2, as in simulation 1, and that you have vaccinated over 1-2/3=1/3 of the population. So over 1/3 of the 3/2 people an infected person goes on to infect on average are now
immune. Since a third of 3/2 is 1/2, this leaves an effective reproduction number of just under 3/2-1/2=1.
The simulations assume that every infected person falls into one of two classes in terms of how many people they go on to infect. But this is unrealistic, as people's contacts patterns vary widely: a
teacher may come into contact with hundreds of people a day, while an office worker may come into contact with only around ten, and someone working from home only with their immediate family. A more
realistic model should take this into account, for example by dividing the population into different classes reflecting their contact patterns.
The simulations also assume that a person is either infectious or healthy. In reality, people can be infected but not infectious, they can recover from the disease (or, god forbid, die), or they can
be immune. Standard epidemiological models divide the population up into classes (for example susceptible, infectious and recovered) and then use mathematical rules to describe the way in which
people pass from one class to the next. See the rest of this package on epidemiology for more information.
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Nonogram #4
Picture logic puzzle in which cells in a grid have to be colored or left blank according to numbers given at the side of the grid to reveal a hidden picture. The numbers measure how many unbroken
lines of filled-in squares there are in any given row or column. For example, a clue of “4 8 3″ would mean there are sets of four, eight, and three filled squares, in that order, with at least one
blank square between successive groups.
Play Nonogram #4 for free! Check out Nonogram #4 Game Cheats, Nonogram #4 Reviews and Nonogram #4 Walkthrough!
It would be great if you could login and share Nonogram #4 Cheats, Reviews and Walkthrough’s with us. That would help other players to finish this game and to have a lot of fun. Any Nonogram #4 hint
is welcome..
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Harringtons Compression Method Revistied *EXTREMELY LONG* -
"Einstein" <michaelhh{}> writes:
> I am going to convince you with real software.
You're going to convince of *of what*? You've mentioned a theorem.
Please state the theorem.
Keith Thompson (The_Other_Keith) kst-u{}mib.org <http://www.ghoti.net/~kst>
San Diego Supercomputer Center <*> <http://users.sdsc.edu/~kst>
We must do something. This is something. Therefore, we must do this.
Matt Mahoney wrote:
> John McCarten wrote:
> > maud wrote:
> > > Matt Mahoney wrote in part:
> > >
> > > > Here is an easier problem. I generate one 8 MB file and post it. You
> > > > compress it and post an archive containing the compressed file and a
> > > > decompression program, such that the archive is at least one byte
> > > > smaller than the file I post. Follow the rules of
> > > > http://mailcom.com/challenge/ for creating the archive and decompressor
> > > > (substituting the test file for the Calgary corpus).
> > > > ...
> > > > To make it easier still, I will tell you how I will create the test
> > > > file. I will use the program sharnd.exe to create it.
> > > > ...
> >
> > If you look at the rules for the challenge you will see an important
> > element missing for challenges such as the one proposed. There is no
> > time limit.
> I am patient.
Yes. Me too. One often has to wait patiently for a bite. I only got one
bite, from BR, but that's the way it sometimes goes when fishing
> > The second thing is that the challenge file is to be created by sharnd.
> > The source code of which has been made available.
> So the problem ought to be easy.
True. Lucky for me the source was available. I would have been in
serious trouble otherwise.
> After einstein gives up
You may be in for a long wait, but as you've said
> I'll post a
> decompressor under 30 KB just to prove it can be done.
This will be definitely be worth seeing. I won't be calling tho'. I
will be safely out of harm's way, sitting on the rail, when you play
this hand.
> > Now some readers will already have clicked where this is going, but I
> > will continue.
> >
> > loop through all possible values of secret key
> > compare the sharnd output to the challenge file
> > stop when match found else continue
> >
> > The decompressor is a modified version of sharnd itself with the
> > matching key and the 8MB length value embedded.
> >
> > The length that meets the challenge is length(modifiedsharnd).
> >
> > Now, it might take a while to find the key but hey! thats combinatorics
> > for you.
> >
> >
> > john.
> That's the obvious solution.
> Also, there are known weaknesses in SHA-1
> (2^69 resistance to collisions), but I'm not too concerned.
My only concern is that I probably wouldn't be around when the key was
> -- Matt Mahoney
Matt Mahoney wrote:
> Just curious, which did you prove, P = NP or P != NP?
Given his counterproofs of Shannon's theory, probably he proves that NP is a
strict subset of P.
> Now, the problem is: how on earth do you ever expect to be able to map
> SEVEN (or 2^(N+1) - 1) plain text strings to THREE (or 2^N - 1)
> compressed strings?
> E.g.:
> '11' -> (compression) -> '1'
> '10' -> (compression) -> '0'
> '01' -> (compression) -> ''
> '00' -> (compression) -> ??????? // no possibilities left...
> '0' -> (compression) -> ??????? // no possibilities left...
> '1' -> (compression) -> ??????? // no possibilities left...
> '' -> (compression) -> ??????? // no possibilities left...
> This goes for any N!!! Claiming that you can compress *any* and *all*
> strings of length N or less, is saying that you can map all possible
> 2^(N+1) - 1 plain text strings to 2^N - 1 compressed strings. I'm afraid
> you want to fill pigeon holes with on average about 2 pigeons per hole.
It's a very good and clear example.
And: if the program compress all strings of n bytes (that what is
needed to compress a random file using a rigorous definition of
random), then nothing prevent you to use it recursively (the output
file is an *any file*), so if any input file is compressed at least of
1 bit each time we have that any file of n bit must disappear (being 0
bit of size) after being recursively compressed by that program n
times, that is equal to proof that the compressor is not lossless since
all the n bits of information are now gone.
As said in other posts, anyone could use a PRN generator and so provide
a very efficient compressor that map each random looking output to his
seed and size, however it is indeed compressing a subgroup of files,
not at all compressing all files.
In other worlds a program that generate a random looking output, use
ent or diehard suite to say that "it's random" (really, the tests can
only say that is probable that the file doesn't contain any
recognizable structure, and many good PRNG will pass that kind of
tests) and compress it to an amazingly small file is not enough to say
that any arbirtrary input can be compressed.
Trying to recompress the output recursively is a way to disprove the
claim, if the compressor can compress an arbitrary input it must
compress any of the output of the recusion until the output size is 0,
proving itself to not be a lossless compressor.
Matt Mahoney wrote:
> Just curious, which did you prove, P = NP or P != NP?
> -- Matt Mahoney
Yes, do tell, I've been dying to see how this one turns out.
I'm not calling for a straw poll or anything, but we're all betting on
P != NP, right?
But if P = NP, what a world!
| Mark Nelson
markn{}ieee.org wrote:
> Matt Mahoney wrote:
> >
> > Just curious, which did you prove, P = NP or P != NP?
> >
> > -- Matt Mahoney
> Yes, do tell, I've been dying to see how this one turns out.
> I'm not calling for a straw poll or anything, but we're all betting on
> P != NP, right?
> But if P = NP, what a world!
> |
> | Mark Nelson
> |
Yep, and it's a simple proof too. All you need is a compressor that
will compress any string longer than m, and you can perform any
computation in O(n) time, which is in P:
1. Recursively compress the input from size n to size m in O(n) time.
(I have omitted some details, such as encoding the number of
recursions, dealing with compression by more than one bit, and showing
that each cycle does not depend on n so is O(1), but the basic result
is the same).
2. Compute the compressed output in O(1) time using a 2^m by m lookup
3. Recursively decompress the output in O(n) time. Total computation
is O(n).
As a corrolary, the halting problem is in P. This fact can be used to
solve the 6 remaining Clay millenium prize problems.
Einstein, if you are still interested in the Clay prizes, let me know.
Here is the deal.
1. You pay me a fee of $1000 by July 1, 2006.
2. You write and publish (with source code) a compressor/decompressor
pair that will compress every file or string larger than some size m by
at least one bit, where m is any positive integer you choose, and the
decompressor restores the original input to the compressor.
3. After step 2, I will write a paper with you as co-author solving the
7 millenium problems and get it published in an appropriate mathematics
journal within 6 months.
4. The Clay Institute requires a 2 year waiting period after
publication, acceptance within the mathematics community, and approval
by their scientific advisory board before awarding the prizes ($7
million). We split the prize money, regardless of the actual amount.
5. If after step 2 I fail to complete steps 3 and 4 (for the full $7
million) then I refund my fee.
6. There is no refund if you fail to complete step 2.
7. After July 1, 2006 my fee increases to $2000, then increases $1000
per week after that. The amount of the fee is determined by the day
that I receive payment.
8. There is no time limit for you to complete step 2.
9. You are free to file for patents worldwide on your algorithm for
step 2 with you as sole inventor.
If we have a deal, contact me by email to arrange payment.
Of course you can go it alone and claim the whole $7 million for
yourself (or at least $1 million for P = NP using either the proof I
posted or the one sitting on your other computer). I don't know if you
have tried to publish any papers before, but it can be a humbling
experience. It really helps to get outside assistance in this matter.
-- Matt Mahoney
So as long as guys like Harrington are out there, I need to make sure
that we keep the million random digit challenge on the table. In the
past it's been a little hard to find, you have to search through
comp.compression and there are various false hits.
Now, it has a permanent home on my site:
Einstein, the point of the million random digit challenge is basically
to encourage people with magical compressor claims to put up or shut
up. If you can beat the challenge within the spirit of the rules, you
It's been out there for four or five years, and nobody has taken a
serious crack at it. One or two people have noted that there is a tiny
bit of redundancy in the million random digit file, but not enough to
exploit with a conventional decompressor - nibbling away a dozen bytes
doesn't leave room for much of a decompression program.
Nope, to compress this file will take a magic compressor. So, why not
code up what you've been talking about and earn undying fame by meeting
the challenge? It's no Clay Prize, but then, I don't have a retail
empire or a family fortune to fund a prize of this nature.
But I'm still in for $100.
maud wrote:
> tchow{}lsa.umich.edu wrote:
> > In article <1150377194.973543.41280{}i40g2000cwc.googlegroups.com>,
> > maud <maudlin.poker{}hotmail.com> wrote:
> > >a) Create all sets of possible values that occupy 8MB space.
> > >b) Compress and decompress them all without error using your method.
> > >c) Count up how many sets when compressed result in a smaller space
> > >than 8MB.
> > >
> > >Now, it may take some time but hey! thats combinatorics for you.
> >
> > No kidding. 8MB is 64 million bits, so there are 2^(64 million) possible
> > values that occupy 8MB space. Even if you handle a googol cases per
> > femtosecond, you would handle a negligible fraction of all cases in a
> > googol universe lifetimes.
> Probably be a good idea if we went for lunch first then
> maud.
Hi all,
I found a way to create a recursive compressor software
right away if some one could transform any random file into this format
(for example) with minimal additional bytes.
each section in the binary string of the output must have runs of 1,
with the length atleast 2 and above, and run of the zero is atleast 1
and above...
x>=2 , y>=1 where x = run length of 1, and y = run
length of 0
so the minimal group will be like this 110
if anyone found any variable length code that have this characteristic
or output of
some codec generate something like this, please contact me
we will create the miracle and solve this problem once for all.
I am hinting Fibonanci for the 11 but still it is not suitable...
Einstein, this is a challenge for you.. I only need a data transform to
the quest. so before I complete it with the help of experts here, I do
hope you did first.
> I will have a functional code sequence as soon as I lick this issue I
> have with loops in PHP
Any guesses on what this "issue with loops" might be? I am curious since
the Halting Problem reduces to the problem Mr.Einstein claims to have
Back online again for a short while.
Seems like another one of the futile. MUST not map directly , must
modulate representation, must sleep.
nice quadrapak parabolic antenna me thinks he, he | {"url":"http://objectmix.com/theory/14419-harringtons-compression-method-revistied-*extremely-long*-6.html","timestamp":"2014-04-16T04:11:18Z","content_type":null,"content_length":"69419","record_id":"<urn:uuid:8aea983b-4b73-44be-9116-65c24d98236b>","cc-path":"CC-MAIN-2014-15/segments/1397609521512.15/warc/CC-MAIN-20140416005201-00025-ip-10-147-4-33.ec2.internal.warc.gz"} |
Hall F
Typical time required:
30 - 60 minutes
Mathematics expresses itself everywhere, in almost every facet of life - in nature all around us, and in the technologies in our hands. Mathematics is the language of science and engineering -
describing our understanding of all that we observe.
The Mathematics Everywhere & Everyday Exhibition explores the many wonders and uses of mathematics in our lives. This exhibition is divided into nine areas focusing on different aspects of
Gallery Pathways worksheets prepared for primary and secondary school classes are available on request and booking. Please view our Enrichment Programmes – Gallery Pathways for more information on
booking, etc.
Introduction & Concepts
Mathematics has been around since the beginnings of time and it most probably began with counting. Learn the history of mathematics, and get to know some of the greatest mathematical minds and
their contributions.
In this section, you are also introduced to some basic mathematical concepts and counting tools – from giant electronic calculators to the abaci of different cultures.
The Ames Room shows the effects of scale and illusion.
In Counting
Counting various quantities is one of the activities that people engage in from young. However, sometimes we wonder, just how big is one million? See how much space one million saga seeds occupy
and be awed by numbers that you can relate to in your daily lives like the number of heartbeats in a typical lifetime.
Sometimes, numbers can be used to fool people. Some exhibits in this section provide a few examples of well-known mathematical tricks.
In Nature
1, 1, 2, 3, 5, 8, 13... This is the Fibonacci Sequence, where each number is derived from adding the previous two numbers. This sequence of numbers can be found in many natural patterns like in
pineapples, sunflowers, nautilus and pine cones.
Our eyes are usually drawn to objects that are symmetrical. Leonardo Da Vinci’s Vitruvian Man is often used as a representation of symmetry in the human body. The Soothing Symmetries exhibit
anchors this section and explains what symmetry means to science and in mathematics, as well as in everyday life.
In Shapes, Curves & Patterns
Circles, squares and triangles are just a few of the shapes that are familiar to us through our daily lives. Discover the usefulness, beauty and 'cleanness' of the round shape, and learn more about
the other shapes that can be found around us.
This section also features the Möbius strip that has several curious properties.
In Games & Puzzles
Many, if not all puzzles and games require mathematical logic and deduction. This section uses the fun and excitement of various popular games and puzzles, and the exhilaration of solving them, to
attract and engage the visitors to realise the mathematics in fun and games.
The Odds Are Stacked Against You! explores the odds of various casino games like roulette, blackjack and slot machines. This will set many thinking about their chances of winning in gambling.
In Time & The Heavens
Mathematics was developed to understand the cycles of nature as observed in the seasons. Ancient people understood the need to define time in relation to celestial movements for agricultural,
astronomical, astrological and navigational reasons.
This section looks at the mathematics of astronomy, its relationship to the inventions of various cultural-historical calendars, and the division of time into units of hours, minutes and seconds.
Challenges & Controversies
When we talk about mathematics, equations and formulas will pop into our mind. There are hundreds of equations in mathematics, but which is the Most ‘Beautiful’ Equation of all? You are invited to
vote for your favourite.
What do modern mathematicians actually do? What problems are of interest today? These are some of the issues explored in this section. Some outstanding challenges that remain unresolved are still
intriguing many mathematicians. Discover what some of these challenges are.
In Real Applications
Mathematic is used in our everyday lives; from figuring out the amount needed to buy your lunch to calculating the bank’s interest. This section explores some of the real life applications of
mathematics. For example, internet banking is getting more and more common these days, and we depend on cryptology – the study of protecting information using codes – to keep our transactions safe.
Learn more about how it is done in this section.
Awesome, Fearsome Calculus
Calculus is the study of change and it is one of the most important fields in mathematics. Isaac Newton and Gottfried Leibniz are usually credited with the invention of calculus. Newton used
calculus in his laws of motion and gravitational attraction.
Explore an introductory calculus lesson by taking something to the limit in Know Your Limits, and learn more about slopes, tangents, integration and differentiation in this section's other | {"url":"http://www.science.edu.sg/exhibitions/Pages/Mathematics.aspx","timestamp":"2014-04-19T15:00:43Z","content_type":null,"content_length":"44843","record_id":"<urn:uuid:6a319078-bd6d-4a79-9c33-8f9b55d37667>","cc-path":"CC-MAIN-2014-15/segments/1397609537271.8/warc/CC-MAIN-20140416005217-00105-ip-10-147-4-33.ec2.internal.warc.gz"} |
Foundations of Mathematics
March 25th 2011, 06:49 AM #1
Mar 2011
Foundations of Mathematics
1)lndicate whether the following statements are true or false:
a)lt can not proved mathematically that 2+2=4
b)It can be proved that there is no paradox in mathematics.(give an example)
c)Hilbert proved that thereis a statement in matematics whicah is true but can not be proved
d) Gödel proved that there is a statement in mathematics which can neither be proved nor pisproved
3)What was the set that Russell used to show that there was a paradox in mathematics?
Thank you so much I am waiiting for your msg guys..
Last edited by Ackbeet; March 25th 2011 at 07:16 AM. Reason: Deleted probability question - it's for the other thread.
1.a) take a look at this link: 2p2e4 - Metamath Proof Explorer
b) the wording of this question is a little too vague, for my liking. if the statement is that: it can be proved mathematics is (logically) consistent, it is untrue. this isn't to say mathematics
is inconsistent, it's just that there are important statements in mathematics that cannot be proven. an example: the Axiom of Choice, is it true or false? it's been shown that including AC and
denying AC both lead to consistent versions of set theory.
c) my guess is you are studying Gödel's theorems. i decline to answer this question, and the following.
3) why, Russell's set, of course.
To add some things:
a)lt can not proved mathematically that 2+2=4
This is false. A particular proof depends on the axioms used.
b)It can be proved that there is no paradox in mathematics.(give an example)
I think the word "paradox" here is a synonym of "contradiction." A theory (including the whole of mathematics) is called consistent if one cannot prove a contradiction in it.
d) Gödel proved that there is a statement in mathematics which can neither be proved nor pisproved
Search for the First Gödel's Incompleteness Theorem.
3)What was the set that Russell used to show that there was a paradox in mathematics?
Search for Russell's paradox.
You are correct. From Wikipedia:
By work of Kurt Gödel and Paul Cohen, the axiom of choice is logically independent of ZF. This means that neither it nor its negation can be proven to be true in ZF, if ZF is consistent.
Consequently, if ZF is consistent, then ZFC is consistent and ZF¬C is also consistent.
It's a bit strange that they would give
"b)It can be proved that there is no paradox in mathematics"
"3)What was the set that Russell used to show that there was a paradox in mathematics?"
"Mathematics" is a fluid concept...
March 25th 2011, 10:16 AM #2
MHF Contributor
Mar 2011
March 25th 2011, 10:37 PM #3
March 26th 2011, 04:15 AM #4
MHF Contributor
Oct 2009
March 26th 2011, 04:18 AM #5
MHF Contributor
Oct 2009
March 26th 2011, 05:48 AM #6
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Apr 2005
March 26th 2011, 05:57 AM #7
MHF Contributor
Oct 2009
March 26th 2011, 11:29 AM #8
MHF Contributor
Mar 2011 | {"url":"http://mathhelpforum.com/discrete-math/175785-foundations-mathematics.html","timestamp":"2014-04-21T02:11:30Z","content_type":null,"content_length":"55599","record_id":"<urn:uuid:7153558e-f006-4f89-9d73-220a9fc3c7f1>","cc-path":"CC-MAIN-2014-15/segments/1397609539447.23/warc/CC-MAIN-20140416005219-00568-ip-10-147-4-33.ec2.internal.warc.gz"} |
Who invented the first calendar? Why do we have a leap year? - Homework Help - eNotes.com
Who invented the first calendar? Why do we have a leap year?
The first calender was invented by the Greeks. We have a leap year because,
LOOK --
one day insint 24 hours, one day = 23 hours, 56 minutes and 4 seconds. After 4 years, the remaining of 23 hours, 56 minutes and 4 seconds become complete and thus form a exxtra day which comes as
February 29. It is called a leap year.
Earth is slowing down and one day we will not have a leap year.
are you sad?
The question of who invented the first calendar is very hard to answer -- no one knows for sure. Many ancient societies had them. The Egyptians are, however, credited with the first solar calendar
(based on the sun as opposed to on the cycles of the moon).
The reason we have a leap year is much easier... As you know, a year is defined as the time it takes the earth to complete one orbit of the sun. Sadly, the earth doesn't take a whole number of days
to orbit but instead takes 365.25 days to orbit the sun.
So, every four years we add an extra day to make up for the fact that our year is actually one-quarter day shorter than it should be.
Calendar is a system of measuring and recording times in terms of days and years. Many different types of calendar were developed and used by people in different parts of the world. Exact knowledge
about many of these calendars is lost. The oldest calender about which information exists is perhaps the Hebrew calendar.
The Hebrew calendar started 3760 years and 3 months before christian era. This calendar is based on lunar movement. One year in this calendar consists of 12 months which are alternatively 29 and 30
days long. Seven times during every 19 year period and extra month of 29 days is added to keep the calendar year and the real solar year in line.
Egyptians started using a calender much before the Start of Hebrew calendar. Their calendar was based on the solar system. According to experts the earliest known date according to Egyptian calendar
is 4236 B.C.
I believe that the Mayans made the first calender and they all of a sudden, stopped at 2012....Woah....
As for leap year...I don't know sorry...
but check this website
Join to answer this question
Join a community of thousands of dedicated teachers and students.
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Posts by
Posts by Elle
Total # Posts: 119
hey, i would really appreciate some help solving for x when: sin2x=cosx Use the identity sin 2A = 2sinAcosA so: sin 2x = cos x 2sinxcosx - cosx = 0 cosx(2sinx - 1)=0 cosx = 0 or 2sinx=1, yielding
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hi, i'm having trouble with this question: The cost of running a train at a constant speed of v km/h is C=50+v^2/1000 dollars/hour find the time taken for an 800km journey in terms of v.
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circular (trigonometric) functions
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exponential and logarithmic graphs question
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Econ T/F
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Econ T/F
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According to a survey of U.S. firms, the advertising elasticity of demand is only about 0.003. Does this indicate that firms spend too much on advertising? It does not indicate too much is being
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The Meaning of Horsepower and Torque - ROUSH Performance Blog
The Meaning of Horsepower and Torque
Perhaps the most used term in describing a vehicle’s performance is horsepower. In this article, we’ll dive into what exactly this term means, how it compares to its companion term “torque,” and
how we can best look at these to glean insight into real usable performance.
The Definition of Horsepower
The term “horsepower” has been around for quite a while. In 1702, Thomas Savery made reference to the potential work that a horse can do as a measurement of power. In his book The Miner’s Friend he
writes, “So that an engine which will raise as much water as two horses, working together at one time in such a work, can do, and for which there must be constantly kept ten or twelve horses for
doing the same. Then I say, such an engine may be made large enough to do the work required in employing eight, ten, fifteen, or twenty horses to be constantly maintained and kept for doing such a
James Watt later made reference to the potential work that a horse can do (horsepower) as a way to market a new and improved steam engine. He determined that a horse can do 33,000 foot-pounds of work
in one minute (the equivalent of pulling one pound of weight 33,000 feet in distance).
The Definition of Torque
Torque is the tendency of a force to move around a point. In other words, torque refers to twisting force. The unit of measure for torque that we all know, the “ft-lb,” “lb-ft,” or “foot-pound” is
the amount of turning force applied to move one pound a distance of one foot around an axis at a radius of one foot.
So, one full rotation around an axis at a radius of one foot with one foot of resistance yields the following amount of work:
work = (2 * pi) * 1 lb-ft = 6.2832 lb-ft
Where Torque and Horsepower Converge
As you can see above, horsepower is a measure of work/time, where torque is a measure of work. So, if we use the 6.2832 lb-ft per revolution that we came up with above, we can now determine the RPM
(revolutions per minute => work / time) to find out at what RPM we would have one horsepower with 1 lb-ft of torque. Let’s see what we come up with:
33,000 lb-ft/min / 6.2832 lb-ft/revolution = 5252 RPM
Even though we used the case of one lb-ft of torque and one horsepower to find where these two numbers converge, the horsepower and torque numbers will always be the same at this RPM. Test this
statement. Look at a number of different horsepower and torque graphs. You’ll find that on all of them, the torque and horsepower lines will all cross at this point. See an example below for the
ROUSH Performance M90 supercharger.
What This Means
As we proved above, horsepower is simply an extrapolation of torque applied over time. When an engine is measured for its power potential on a dynamometer, horsepower and torque are not measured as
separate entities. Rather, torque is measured, and horsepower is then calculated given the torque at the specific RPM level.
Car owners often use “horsepower” as the end-all be-all rating for engine performance. This perspective is flawed. First of all, when you hear of a car having X horsepower, it only refers to the peak
horsepower on the dyno graph. Secondly, it doesn’t indicate what the shape of the torque curve is. You can feel the torque that an engine generates as you’re pushed back into your seat.
The Case of the 2011 Ford F-350 6.7L
One example of how misleading the measure of horsepower alone can be is the case of a diesel truck engine. Below is a dyno chart for a 2011 Ford F-350 6.7L V8 engine:
As you can see, this engine is a monster with over 640 lb-ft of torque at low RPMs. However, because the torque curve falls off before the torque and horsepower convergence point that we found above
(at 5252 RPM), the horsepower remains well below the torque level for the entirety of the power band. Does the relatively low peak horsepower of 307 indicate that this engine has little power?
Absolutely not. The truck engine above isn’t intended for running at high-RPMs. It’s running RPMs are right in its appropriate power band, and it has a high level of pulling power off the line, which
is ideal for towing large amounts of weight.
When you hear someone refer to a high torque engine, what they really mean is that the torque band starts high at low RPM (and probably drops off early in the RPM continuum relative to engines not
considered high in torque).
The Case of the Horsepower Bastard
On the other end of the spectrum, we see horsepower numbers that are unreal, but because the torque curve only favors the high RPM band, the usable power generated by the given engine is much less
than the horsepower number would lead you to believe. You would see this where there isn’t a steady torque curve across the usable RPM but rather an upwardly slanted torque curve the strongly favors
higher RPMs. You can see this with forced induction systems that take an inordinately long time to spool up.
A Better Perspective on Power
A good rule of thumb to use when trying to get a sense for real usable power is to look at the shape of the torque curve. For performance applications, like a modified Ford Mustang, the shape of the
curve should be fairly constant over the RPM band that is intended for use. You want to maximize the area underneath this curve across the RPM range. There really needs to be a better unit of measure
for determining usable engine power. One possibility would be to come up with an average torque rating across this band. Another would be the area under the torque curve. Of course there are many
other possibilities, but you get the idea.
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World Series
How Many Games Does it Take to Win the World Series?
In order to determine the champion of Major League Baseball in the U.S. and Canada, the winners from the National League and the American League play the "World Series." The Series consists of seven
games. If a team wins four out of the seven games, they are the champions.
how many games do you expect to be played before a team wins four games and becomes the champion?
Predicting how long a series will take is not easy. There are many factors that are involved: Where are they playing? Is anyone injured? What is the weather like?
For the purpose of this problem, however, we will simplify these into one question... What is the probability that the National League (NL) will win an individual game?
You are welcome to make that probability whatever you like (depending upon whether or not you prefer NL teams or AL teams), but we suggest that you begin by making the chances 50-50 (probability = 1/
2 = .5) and then play with different probabilities later.
1. What is the least number of games that could be played and still have a champion?
2. What is the greatest number of games that would be played?
3. What sort of model could you use for this problem?
4. How would you solve this problem analytically?
5. How would changing the probability of the NL winning an individual game affect your model?
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Curricula, Scheme of Examinations & Syllabi for Semesters V to VIII of B. Tech.
ME2K 603 : METAL CASTING & JOINING
(common with PM2K 603)
3 hours lecture and 1 hour tutorial per week
Module I (10 hours)
Introduction - solidification of metals - mechanism of solidification - solidification with predominant interface resistance - solidification with constant surface temperature - solidification with
predominant resistance in mould and solidified metal - flow of molten metal in moulds - furnaces and melting practices - patterns - pattern allowance - design considerations - shrinkage and machining
allowance - foundries
Module II (14 hours)
Casting processes - comparison - sand casting - shell moulding - silicate bonded sand process (CO[2] process) - expended polystyrene process - plaster mould casting - ceramic mould casting -
investment casting - permanent mould casting - slush casting - pressure casting - die casting - centrifugal casting - squeeze casting - semisolid casting (rheocasting, thixoforming) - casting
techniques for single crystal components - rapid solidification - residual stress - defects - inspection of castings - casting design - gating system design - risering - casting alloys - economics of
casting - design rules for castings - case studies with specific examples of sand cast and permanent mould cast parts
Module III (16 hours)
Classification - filler materials - consumable electrodes - liquid state - chemical - arc - resistance - electrical characteristics of the arc - analysis of metal transfer - free flight and
short-circuiting metal transfer - equations for heat flow in welding - equations for temperature distribution in the Heat Affected Zone-Gas-Metal reactions - sensitivity to hydrogen porosity - weld
pool solidification - contraction and residual stress crack sensitivity - dilution and uniformity of the weld deposit - solid state - liquid-solid state - process: OFW - SMAW - SAW- GMAW - FCAW -
GTAW - PAW - ESW - EGW - RW - RSEW - HFRW - RPW - FW - SW - PEW - FOW - CW - USW - FRW - EXW - TW - EBW - LBW - DFW
Module IV (12 hours)
The metallurgy of welding - metallurgy of weld metal and HAZ for carbon steels, ferritic and high alloy steels, austenitic and high alloy steels non-ferrous metals (Aluminium and its alloys, Copper
and its alloys, Magnesium and its alloys) - weld quality - weldability - testing welded joints - welding design and process selection - brazing, soldering, adhesive bonding and mechanical joining
processes - joining plastics - surface energy and contact angle - capillary action in brazing and soldering - residual stress and stress concentration factors in adhesive bonding
Reference books
Flemings M.C., “Solidification Processing”, McGraw Hill
Serope Kalpakjian, Manufacturing Engineering & Technology, Addison Wesley
Heine R.W., Loper C.R. Jr. & Rosenthal P.C., Principles of Metal Casting, Tata McGraw Hill
American Welding Society, Welding Hand Book
Doyle L.E., Manufacturing Processes and Materials for Engineers, Prentice Hall of India
Metals HandBook- Vol.5., Welding Institute of Metals
Lancaster J.F., “The Metallaurgy of Welding, Brazing and Soldering”, George Allen & Unwin Ltd.
Sessional work assessment
2 tests 2x15 = 30
2 assignments 2x10 = 20
Total marks = 50
University examination pattern
Q I - 8 short type questions of 5 marks each, 2 from each module
Q II - 2 questions A and B of 15marks each from module I with choice to answer any one
Q III - 2 questions A and B of 15marks each from module II with choice to answer any one
Q IV - 2 questions A and B of 15marks each from module III with choice to answer any one
Q V - 2 questions A and B of 15marks each from module IV with choice to answer any one
ME2K 604 : THERMAL ENGINEERING I
3 hours lecture and 1 hour tutorial per week
Module I (14 hours)
Internal combustion engines - classification - four stroke and two stroke engines - spark ignition and compression ignition engines - value timing diagram - thermodynamic analysis of air standard
cycles - Otto, diesel and duel combustion cycles - engine testing - performance and characteristics of constant speed and variable speed engines - heat balance test - Morse test - retardation test -
actual engine cycles - effect of dissociation - variable specific heats and heat losses - scavenging - objectives - effects and methods
Module II (13 hours)
Systems and components of IC engines - fuel systems - ignition systems - cooling - starting - lubrication - governing of IC engines - supercharging of SI and CI engines - turbocharging - exhaust
emissions of IC engines - alternate potential engines - free piston engine - Wankel engine and stratified charged engine - automotive transmission system and its components
Module III (12 hours)
Combustion in IC engines - flame propagation - normal and abnormal combustion - detonation - pre ignition - after burning - HUCR - fuel rating - additives in petrol - combustion chambers of SI
engines - combustion in CI engines - phase of normal combustion - diesel knock - effect of engine variables on diesel knock - cetane number - additives in diesel - combustion chambers of CI engines
Module IV (13 hours)
Gas turbine plants - open and closed cycles - thermodynamic cycles - regeneration - reheating - intercooling - efficiency and performance of gas turbines - rotary compressors - analysis - centrifugal
and axial flow compressors - combustion chambers of gas turbines - cylindrical - annular and industrial type combustion chamber design - combustion intensity - combustion efficiency - pressure loss
combustion process and stability loop - axial flow turbines - elementary and vortex theories - design of nozzles and blades for turbines - limiting factors in turbine design
Reference books
Rogowsky, “Elements of Internal Combustion Engines”, Tata McGraw Hill
Gill, Smith & Ziurys, “Fundamentals of Internal Combustion Engines”, Oxford and IBH
Maleev, “Internal Combustion Engine Theory and Design” McGraw Hill
Judge, “Modern Petrol Engines,” Chapman & Hall
Benson & Whitehouse, “Internal Combustion Engines” Vol. I & II, Pergamon press
Mathur & Mehta, “Thermodynamics and Heat Power Engineering”, Vol. I & II
Cohen & Rogers, “Gas Turbine Theory,” Longmans
Sessional work assessment
2 tests 2x15 = 30
2 assignments 2x10 = 20
Total marks = 50
University examination pattern
Q I - 8 short type questions of 5 marks each, 2 from each module
Q II - 2 questions A and B of 15marks each from module I with choice to answer any one
Q III - 2 questions A and B of 15marks each from module II with choice to answer any one
Q IV - 2 questions A and B of 15marks each from module III with choice to answer any one
Q V - 2 questions A and B of 15marks each from module IV with choice to answer any one
ME2K 605 : MACHINE DESIGN I
3 hours lecture and 1 hour tutorial per week
Module I (13 hours)
Introduction to design - steps in design process - design factors - tolerances and fits - principles of standardisation - selection of materials - strength of mechanical elements - stress
concentration - theories of failure - impact load - fatigue loading - consideration of creep and thermal stresses in design
Module II (13 hours)
Threaded fasteners - thread standards - stresses in screw threads - preloading of bolts - bolted joints - eccentric loading - gasketed joints - analysis of power screws - keys: types of keys and pins
- stresses in keys and pins - design of keys - design of cotter and pin joints - riveted joints - stresses in riveted joints - strength analysis - boiler and tank joints - structural joints
Module III (13 hours)
Welded joints - types of welded joints - stresses in butt and fillet welds - torsion and bending in welded joints - welds subjected to fluctuating loads - design of welded machine parts and
structural joints - springs: stresses in helical springs - deflection of helical springs - extension, compression and torsion springs - design of helical springs for static and fatigue loading -
critical frequency of helical springs - stress analysis and design of leaf springs
Module IV (13 hours)
Power shafting - stresses in shafts - design for static loads - reversed bending and steady torsion - design for strength and deflection - design for fatigue loading - critical speed of shafts -
stresses in couplings - design of couplings
Text book
Shigley J.E., Mechanical Engineering Design, McGraw Hill Book Company
Reference books
Siegel, Maleev & Hartman, Mechanical Design of Machines, International Book Company
Phelan R.M., Fundamentals of Mechanical Design, Tata McGraw Hill Publishing Co. Ltd.
Doughtie V.L., & Vallance A.V., Design of Machine Elements, McGraw Hill Book Company
Juvinall R.C. & Marshek K.M., Fundamentals of Machine Component Design, John Wiley
Data hand books (allowed for reference during examinations)
Prof. Narayana Iyengar B. R. & Dr Lingaiah K., Machine Design Data Handbook, Vol. I &II
P.S.G., Tech., Machine Design Data Handbook
Sessional work assessment
2 tests (best 2 out of 3 tests conducted) 2x15 = 30
2 assignments 2x10 = 20
Total marks = 50
University examination pattern
Q I - 8 short type questions of 5 marks each, 2 from each module
Q II - 2 questions A and B of 15marks each from module I with choice to answer any one
Q III - 2 questions A and B of 15marks each from module II with choice to answer any one
Q IV - 2 questions A and B of 15marks each from module III with choice to answer any one
Q V - 2 questions A and B of 15marks each from module IV with choice to answer any one
ME2K 606A : OPTIMISATION TECHNIQUES
(common with AI2K/CE2K/EC2K/EE2K/IC2K/PM2K 606A)
3 hours lecture and 1 hour tutorial per week
Module I: Linear programming I (13 hours)
Systems of linear equations and inequalities - convex sets - convex functions - formulation of linear programming problems - theory of simplex method - simplex algorithm - Charne’s M method - two
phase method - duality in linear programming - dual simplex method
Module II: Linear programming II (13 hours)
Sensitivity analysis - parametric programming - bounded variable problems - transportation problem - development of the method - integrality property - degeneracy - unbalanced problems - assignment
problem - development of the Hungarian method - routing problems
Module III: Nonlinear programming (13 hours)
Mathematical preliminaries of non-linear programming - gradient and Hessian - unimodal functions - convex and concave functions - role of convexity - unconstrained optimization - Fibonacci search -
golden section search - optimal gradient method - classical optimisation - Lagrange multiplier method - Kuhn-Tucker conditions - quadratic programming - separable convex programming - Frank and Wolfe
Module IV: Dynamic programming & game theory (13 hours)
Nature of dynamic programming problem - Bellman’s optimality principle - cargo loading problem - replacement problems - multistage production planning and allocation problems - rectangular games -
two person zero sum games - pure and mixed strategies - 2m and m2 games - relation between theory of games and linear programming
Reference books
Bazarra M.S., Jarvis J.J. & Sherali H.D., ‘Linear Programming and Network Problems', John Wiley
Bazarra M.S., Sherali H.D. & Shetty C.M., ‘Nonlinear Programming, Theory and Algorithms', John Wiley
Hadley G., ‘Linear Programming', Addison Wesley
Hillier F.S. & Lieberman G.J. ‘Introduction to Operations Research', McGraw Hill
Ravindran A., Phillips D.T. & Solberg J.J., Operations Research Principles and Practice, John Wiley
Taha H.A., Operations Research, An introduction, P.H.I.
Wagner H.M., ‘Principles of Operations Research with Application to Managerial Decisions', P.H.I.
Sessional work assessment
Assignments 2x10=20
2 tests 2x15=30
Total marks =50
University examination pattern
Q I - 8 short type questions of 5 marks each, 2 from each module
Q II - 2 questions A and B of 15marks each from module I with choice to answer any one
Q III - 2 questions A and B of 15marks each from module II with choice to answer any one
Q IV - 2 questions A and B of 15marks each from module III with choice to answer any one
Q V - 2 questions A and B of 15marks each from module IV with choice to answer any one | {"url":"http://lib.znate.ru/docs/index-35360.html?page=4","timestamp":"2014-04-19T01:47:55Z","content_type":null,"content_length":"29084","record_id":"<urn:uuid:28080a4f-7f0b-4350-9010-b69cdbf76354>","cc-path":"CC-MAIN-2014-15/segments/1398223205375.6/warc/CC-MAIN-20140423032005-00122-ip-10-147-4-33.ec2.internal.warc.gz"} |
Different uses of the word "ergodic"
up vote 10 down vote favorite
There appear to be two definitions of the word ergodic.
The dynamical systems definition says that a measure space $(X,\mathit B, \mu)$ and measure preserving transformation $T: X \mapsto X$ is ergodic if
the only $T$-invariant sets have measure 0 or 1.
However, a Markov chain is ergodic if
there exists $t$ such that for all $x,y \in \Omega, P^t(x,y) >0$
I've used the Markov chain notation and definition found here
I would like to know if these definitions are equivalent.
Of course, I am asking here because it seems to me that they are not. For example, if $X=\{0,1\}$, $\mathit B = \{\emptyset, \{0\},\{1\},X\}$, $\mu(\{0\})=0,\mu(\{1\})=1$ and $T(x) = 1$ for all $x\in
X$, then $(X,\mathit B, \mu, T)$ is ergodic as a dynamical system, but the equivalent Markov chain is not ergodic, since the probability of traveling from $0$ to $1$ is zero.
ds.dynamical-systems graph-theory markov-chains ergodic-theory
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1 Answer
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Unfortunately, the way the term "ergodic" is used in the theory of (finite) Markov chains is completely misleading from the point of view of general ergodic theory. To be consistent,
one should have called "ergodic" the chains whose state space does not admit a decomposition into non-trivial non-communicating subsets. The notion of ergodicity you are referring to
would rather correspond to what is called "mixing" in ergodic theory.
More precisely, an initial distribution $m$ of a Markov chain on a state space $X$ determines the associated measure $\mathbf P_m$ on the space of sample paths $X^{\mathbb Z_+}$. The
up vote 12 measure $\mathbf P_m$ is shift invariant iff the measure $m$ is stationary. Now, if $m$ is finite (this condition is important; otherwise the following claim is false), then ergodicity
down vote of the time shift is equivalent to absence of non-trivial partitions of $X$ into non-communicating subsets.
By the way, your example is really too degenerate: the standard example for difference between ergodicity and mixing for Markov chains is presence of so-called periodic classes $A_1\to
A_2\to\dots\to A_k\to A_1$ (the only allowed transitions are from $A_i$ to $A_{i+1}$ mod k). For finite chains this is actually the only reason for difference between ergodicity and
mixing, but for general state spaces the situation is more complicated.
3 As you say, the more consistent definition (and I think the way I learned it) of ergodicity for Markov chains is to say that ergodic is a synonym for irreducible. – Anthony Quas Sep
4 '11 at 22:28
R W: I have the same question. Thanks for the nice clarification. (1) What is the definition of "mixing" in ergodic theory used in your reply? (2) Is a Markov chain generated by a
measure-preserving mapping? I know a measure-preserving mapping can generate a stationary process, but I don't know if it can also generate a (homogeneous?) Markov chain ? Thanks. –
Tim Jul 21 '13 at 1:42
A more common terminology for Markov processes is to call a process ergodic if it has a unique invariant measure that attracts all the other measure. This again is not consistent
with the terminology of ergodic theory but consistent with Boltzmann's usage, don't you think? – Algernon Mar 8 at 13:10
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