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# Trigonometric Functions

## Definitions

The variable of a trigonometric function is a plane angle. The definitions of an angle is presented on Fig. Angles .

Since the angle is defined by the ratio of two lengths, it does not have physical dimensions. As shown, the units of measuring an angle used here are called radians. π  radians correspond to 180° .

The angles  α  between 0 and  2π  are divided in  4  quadrants. The graphical presentation of any other angle =  α±2 , where  n  is a natural number, shares the same position in the quadrant as the angle  α .

The basic trigonimetric functions  sin (sine), cos (cosine) and tan (tangent)  are defined graphically in Fig. Trigonometric functions and the following relations are given:

 ${\mathrm{sin}}^{2}\alpha +{\mathrm{cos}}^{2}\alpha =1$ (1.2.3.1)
 $\mathrm{tan}\alpha =\frac{\mathrm{sin}\alpha }{\mathrm{cos}\alpha }$ (1.2.3.2)

We also found that

 $\begin{array}{l}\mathrm{sin}0=\mathrm{cos}\frac{\pi }{2}=0\\ \mathrm{sin}\frac{\pi }{2}=\mathrm{cos}0=1\end{array}\right\}$ (1.2.3.3)

From the definition of the functions it is clear that they are periodic, meaning that they repeat themselves over and over again after an interval of the variable, called period. The period is  2π  for  sin  and  cos  and  π  for  tan . This can be seen again from the graphic presentations later on.

The following table summarizing the signs of the trigonometric functions by quadrants is very useful. It yields that the sign of any two of the functions sine, cosine and tangent defines the quadrant of the variable.

 Quadrants: Q1 Q2 Q3 Q4 sine + + − − cosine + − − + tangent + − + −

It can be shown that for any α and β :

 $\begin{array}{l}\mathrm{sin}\left(\alpha ±\beta \right)=\mathrm{sin}\text{ }\alpha \text{\hspace{0.17em}}\mathrm{cos}\beta ±\mathrm{cos}\text{ }\alpha \text{\hspace{0.17em}}\mathrm{sin}\text{ }\beta \\ \mathrm{cos}\left(\alpha ±\beta \right)=\mathrm{cos}\text{ }\alpha \text{\hspace{0.17em}}\mathrm{cos}\text{ }\beta \mp \mathrm{sin}\text{ }\alpha \text{\hspace{0.17em}}\mathrm{sin}\text{ }\beta \\ \mathrm{tan}\left(\alpha ±\beta \right)=\frac{\mathrm{tan}\text{ }\alpha ±\mathrm{tan}\text{ }\beta }{1\mp \mathrm{tan}\text{ }\alpha \text{\hspace{0.17em}}\mathrm{tan}\text{ }\beta }\end{array}\right\}$ (1.2.3.4)

These relations are assumed to be familiar from high-school and their proof is not repeated here. The last relation is easily obtained from the first two by use of (1.2.3.2).

By substituting  $\beta =\frac{\pi }{2}$  in (1.2.3.4) and in view of (1.2.3.3) one obtains easily that

 $\mathrm{cos}\alpha =\mathrm{sin}\left(\alpha +\frac{\pi }{2}\right)$ (1.2.3.5)

which together with (1.2.3.2) yields that the function  cos  and  tan  can be obtained from  sin .

The reciprocals of these trigonometric functions are seldom used. Their names are : sec (secant), csc (cosecant) and cot (cotangent). Here are their definitions:

 $\begin{array}{l}\mathrm{sec}\alpha =\frac{1}{\mathrm{cos}\alpha }\\ \mathrm{csc}\alpha =\frac{1}{\mathrm{sin}\alpha }\\ \mathrm{cot}\alpha =\frac{1}{\mathrm{tan}\alpha }\end{array}\right\}$ (1.2.3.6)

From (1.2.3.2), (1.2.3.5) and (1.2.3.6) one obtains easily that

 $\mathrm{cot}\text{\hspace{0.17em}}\alpha =-\mathrm{tan}\left(\alpha +\frac{\pi }{2}\right)$ (1.2.3.7)

As in the case of angles, the trigonometric functions are also physically dimensionless. As an example one can see that in the expression of harmonic motion

 $y=l\mathrm{cos}\left(\frac{2\pi t}{T}\right)$ (1.2.3.8)
where the variable t is the time, T is the time period of the motion and l the amplitude of the oscilator, both the cosine and the argument of cosine are dimensionless.

The function y=sinx is constructed geometrically and plotted in Fig. Sine.

From the plot of the function y=sinx and by use of (1.2.3.5) one obtains the graphic presentation of y=cosx as given in Fig. Cosine.

The functions y=tanx and y=cotx are plotted and discussed in Fig. Tangent & co.

From the graphic presentation of the trigonometric functions it was shown that all of them are odd with the exception of the cosine (and of course also the secant) which are even.

The following relations are widely used in order to transform trigonometric expressions in algebraic form. By substituting  α  and  β  by  $\frac{\theta }{2}$  in (1.2.3.4) one obtains

 $\mathrm{tan}\text{ }\theta =\frac{2t}{1-{t}^{2}}$ (1.2.3.9)
where
 $t=\mathrm{tan}\frac{\theta }{2}$ (1.2.3.10)

From the other relations of (1.2.3.4) and doing the same substitution, one obtains:

 $\begin{array}{l}\mathrm{sin}\text{ }\theta =\frac{2t}{1+{t}^{2}}\\ \mathrm{cos}\text{ }\theta =\frac{1-{t}^{2}}{1+{t}^{2}}\end{array}\right\}$ (1.2.3.11)
where t is from (1.2.3.10). The proof of (1.2.3.11) is given as exercise #2 for the user.

## Inverse trigonometric functions

Each inverse triginometric function is called arc-function, e.g. the inverse of sine is called: arc-sine. The name arc emphasizes that the inverse of a trigonometric function expresses an angle, which is actually an arc of a circle with a radius of unity. We are going to use the letter "a" in front of the trigonometric function for denoting its inverse. E.g.  asinx  means  arc-sine of x . One can find elsewhere also the use of "arc" as a prefix instead of "a" .

Since the trigonometric functons are periodic, the inververse functions are multiple-valued. However if one limits the angles in an interval of  2π  (four quadrants) the arc-sine and arc-cosine become double valued. As we'll see since the tangent and cotangent have a period of  π  by a choice of an interval of  π  their inverse functions become single valued. It is common to choose one branch of the function as reference (called also the main branch) in order to limit the arc-functions to single-valued.

The arc-sine function is obtained and discussed in Fig. Arc-sine. We found two branches of arc-sine that give single valued angle, each one in different pair of quadrants:
• Q4 and Q1, e.g.   $-\frac{\pi }{2}\le \text{a}\text{}\mathrm{sin}x\le \frac{\pi }{2}$   increasing function - reference
• Q2 and Q3, e.g.   $\frac{\pi }{2}\le \text{a}\text{}\mathrm{sin}x\le \frac{3\pi }{2}$   decreasing function

The deduction of the arc-cosine function is similar to that of arc-sine as seen from Fig. Arc-cosine. However there is difference in grouping the quadrants.

The single valued branches of arc-cosine are in different quadrants:
• Q1 and Q2, e.g.   $0\le \text{a}\text{}\mathrm{cos}x\le \pi$     decreasing function - reference
• Q3 and Q4, e.g.   $\pi \le \text{a}\text{}\mathrm{cos}x\le 2\pi$   increasing function

The arc-tangent and arc-cotangent functions are obtained and discussed at Fig. Arc-tangent & co

Each branch of the arc-tangent function is increasing and  $-\frac{\pi }{2}\le \text{atan}\text{ }x\le \frac{\pi }{2}$  is used as reference.
Each branch of the arc-cotangent function is decreasing and  $0\le \text{acot}\text{ }x\le \pi$  is used as reference.

## Exercises

Exercise 1. Prove the equations (1.2.3.5) and (1.2.3.7) with the aid of (1.2.3.4)!

Exercise 2. Prove the relations (1.2.3.11) with t defined at (1.2.3.10) !

Exercise 3. Given cotβ=0.75 and cosβ<0 .

1. In what quadrant is the angle β ?
2. Calculate the values of all the triginometric functions for this β !
3. What is β in radians for the interval (−π,π] ? Use a math table or calculator.

Exercise 4. Given: $a\mathrm{sec}\theta +b\mathrm{tan}\theta =c$ ,   where a,b and c are constants≠0.

Use the variable t from (1.2.3.10) together with (1.2.3.9) and (1.2.3.11) for solving θ.
1. What is the condition for not having a solution?
2. Find the condition for a unique solution! What is t in this case?
3. Calculate numerically θ in the interval [0,2π) for the case a=b=c !

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1. ## yet another limit...

Hi,

I have this limit i would like some help with

$\displaystyle\lim_{x\to 0}~arctan(1/|x|)~\frac{\sqrt{1+\pi x}-1}{sin(2x)}$

So i tried replacing the sin(2x) term by it's equivalent taylor polynomials for (sin x and cos x)

like so:
$\displaystyle\lim_{x\to 0}~arctan(1/|x|)~\frac{\sqrt{1+\pi x}-1}{2\bigg(\frac{x^3}{3}+\frac{x^5}{5}\bigg)\bigg(1-\frac{x^2}{2}+\frac{x^4}{4}\bigg)}$

But that didn't really seem to have helped =/
any ideas?

2. $\lim_{x\to 0}tan^{-1}(\frac{1}{|x|})\frac{\sqrt{1+{\pi}x}-1}{sin(2x)}$

L'Hopital:

$\lim_{x\to 0}\frac{1}{4}\frac{\pi}{\sqrt{1+{\pi}x}\cdot cos(2x)}\cdot tan^{-1}(\frac{1}{|x|})$

The radical term can be eliminated because it evaluates to 1:

$\frac{\pi}{4}\frac{\lim_{x\to 0}tan^{-1}(\frac{1}{|x|})}{\lim_{x\to 0}cos(2x)}$

The cos term in the denominator evaluates to 1 and we have:

$\frac{\pi}{4}\lim_{x\to 0}tan^{-1}(\frac{1}{|x|})$

$\lim_{x\to 0}tan^{-1}(\frac{1}{|x|})=\frac{\pi}{2}$

So, we have $\frac{\pi}{4}\cdot \frac{\pi}{2}=\boxed{\frac{{\pi}^{2}}{8}}$

3. Originally Posted by Jones
Hi,

I have this limit i would like some help with

$\displaystyle\lim_{x\to 0}~arctan(1/|x|)~\frac{\sqrt{1+\pi x}-1}{sin(2x)}$

So i tried replacing the sin(2x) term by it's equivalent taylor polynomials for (sin x and cos x)

like so:
$\displaystyle\lim_{x\to 0}~arctan(1/|x|)~\frac{\sqrt{1+\pi x}-1}{2\bigg(\frac{x^3}{3}+\frac{x^5}{5}\bigg)\bigg(1-\frac{x^2}{2}+\frac{x^4}{4}\bigg)}$

But that didn't really seem to have helped =/
any ideas?

Perhaps there's a mistake since the limit, as given, doesn't exist: the left and right limits are different, but if we look for the limit when $x\rightarrow 0^+$ then:
your idea looks just fine to me, but instead of writing the MacClaurin series for sine and cosine let us use the series for $\sin 2x=2x+\frac{(2x)^3}{3!}+\ldots$ , and now we can try L'Hospitals rule on that fraction, since $\arctan\frac{1}{|x|}\xrightarrow [x\to 0]{}\frac{\pi}{2}$ , so derivating numerator and denominator we get:

$\lim_{x\to 0}\frac{\displaystyle{\frac{\pi}{2\sqrt{1+\pi x}}}}{2+4x^2+\ldots}=\frac{\pi}{4}$ , and thus the limit is $\frac{\pi^2}{8}$

Tonio

4. Hello, Jones!

$\displaystyle\lim_{x\to 0}\,\arctan\!\left(\frac{1}{|x|}\right)\,\frac{\sq rt{1+\pi x}-1}{\sin2x}$
Note that: . $\arctan\left(\frac{1}{|x|}\right) \:=\:\text{arccot}|x|$

Multiply by $\frac{\sqrt{1+\pi x} + 1}{\sqrt{1+\pi x} + 1}$

. . $\text{arccot}|x|\cdot\frac{\sqrt{1+\pi x} - 1}{\sin2x}\cdot\frac{\sqrt{1+\pi x} + 1}{\sqrt{1+\pi x} + 1} \;=\;\text{arccot}|x|\cdot\frac{(1+\pi x) - 1}{\sin2x(\sqrt{1+\pi x} + 1}$

. . $=\; \text{arccot}|x|\cdot\frac{\pi x}{\sin2x(\sqrt{1+\pi x} + 1)} \;=\;\frac{\pi}{2}\cdot\text{arccot}|x|\cdot\frac{ 1}{\sqrt{1+\pi x} + 1}\cdot \frac{2x}{\sin2x}$

$\lim_{x\to0}\left[\frac{\pi}{2}\cdot\text{arccot}|x| \cdot\frac{1}{\sqrt{1+\pi x} + 1} \cdot\frac{2x}{\sin2x}\right]$

. . $=\; \frac{\pi}{2}\cdot \underbrace{\left(\lim_{x\to0}\text{arccot}|x|\rig ht)}_{\frac{\pi}{2}} \cdot \underbrace{\left(\lim_{x\to0}\frac{1}{\sqrt{1+\pi x} + 1}\right)}_{\frac{1}{2}} \cdot \underbrace{\left(\lim_{x\to0}\frac{2x}{\sin2x}\ri ght)}_{1}$

. . $=\;\frac{\pi^2}{8}$

5. Hm, difficult limit. I don't think i would have pulled this one off

Thanks for the help though

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Where can causal inference be used with machine learning in a data science project lifecycle?
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Causal inference can be used after the modeling phase in order to confirm some correlations between variables and the target/outcome.

For example, if the model has good accuracy and gives you a high correlation/association between input $A$ and the target $B$ you may want to perform a causal inference to validate that $A$ has an effect on $B$. But then I ponder whether that’s all Causality can help me achieve or is there some other application of Causality in Machine learning.

Source: https://towardsdatascience.com/causality-in-machine-learning-101-for-dummies-like-me-f7f161e7383e
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Filtered colimits commute with forgetful functors

In many cases, filtered colimits commute with forgetful functors to $\mathbf{Set}$, for example with $\mathbf{CRing} \to \mathbf{Set}$ or $R-\mathbf{Mod} \to \mathbf{Set}$. Is there a slick way of showing this?

I am mainly interested in this because you use this fact for the computation of stalks of a sheaf, and usually this is proven by saying: "check it if you don't believe it"

• I don't know about "slick", but there is certainly a general way of showing this for categories of finitary algebraic structures. – Zhen Lin Nov 11 '14 at 11:02
• I would be interested in this way. Do you have any resources? The nLab wasn't very helpful (or I overlooked something) – Andre Knispel Nov 11 '14 at 18:05
• Well, if you know how to do it for a few examples it is easy to extrapolate (provided you understand what finitary algebraic structures are). – Zhen Lin Nov 11 '14 at 18:35

1) Let $T$ be a monad on a cocomplete category $C$. It is a general and trivial fact that the forgetful functor $\mathsf{Mod}(T) \to C$ preserves all colimits which $T$ presveres. In particular, if $T$ preserves filtered colimits (one then says that $T$ is finitary), then the forgetful functor does so. If $T$ is given by a theory of finite operations, $C$ has products which preserve filtered colimits in each variable, then $T$ is finitary.
2) Stalks of sheaves on $X$ commute with colimits because the stalk functor is left adjoint; in fact it is given by $i^*$ where $i : \{x\} \to X$, with right adjoint $i_*$.


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# Why does this mixed model produce discrepant output in SPSS and R?

I've been working with an example from the Heck et al (2013) textbook, which gives the following model:

$$Y_{ij} = \gamma_{00} + \gamma_{01}\text{ses_mean}_j + \gamma_{02}\text{pro4yrc}_j + \gamma_{03}\text{public}_j + \gamma_{10}\text{ses}_{ij} + u_{0j} + u_{1j}\text{ses}_{ij} + \epsilon_{ij}$$

I tried to run this in R using the following code

heckc3  <- read_sav(file ="https://github.com/user1205901/codeforposting/blob/main/ch3multilevel.sav?raw=true")

model1 <- lmer("math ~ 1 + ses + ses_mean + pro4yrc + public + (1 + ses|schcode)", data = heckc3)

summary(model1)

I found that the results did not match the SPSS output, though the output I got from SPSS did match the output in the book.

In my previous experience running uncomplicated models of this sort in SPSS and lme4 I had thought that parameter estimates tended to come out very similar or identical, but here there is a substantial difference.

Trying to investigate why this happened, I noticed something defective in the dataset such that within the Level 2 units 418 and 419 certain individuals have different values on the "public" (Level 2) predictor. However, when I run heckc3 <- subset(heckc3, schcode < 417) rerun the R analysis it still does not match and SPSS analysis with those Level 2 units deleted.

Heck, R. H., Thomas, S. L., & Tabata, L. N. (2012). Multilevel and longitudinal modeling with IBM SPSS. Routledge.

Okay, so I think that I have an answer here that is explaining the differences. The model in lme4 results in a singular fit. Looking at the model summary, this is happening because the correlation parameter between the random effects could not be estimated. When this parameter cannot be estimated, it will normally show up in the summary as 1.00 or -1.00. Here is -1.00 as shown in the printout of your model below.

As this parameter could not be estimated properly, I dropped the correlation parameter and fit the same model with an uncorrelated random slope below. The estimates from this model are very close to the estimates you posted from SPSS, and are definitely within the range of differences that I would expect from slightly different algorithms for maximum likelihood estimation.

What I believe is happening is that SPSS is automatically dropping this correlation parameter, or that SPSS defaults to fitting uncorrelated random slopes. This is somewhat supported by that not being reported in the table you posted, although I have not been able to confirm this in the SPSS documentation.

Here is the printout of the model fitted by your code on my machine.

summary(model1)
Linear mixed model fit by REML ['lmerMod']
Formula: math ~ 1 + ses + ses_mean + pro4yrc + public + (1 + ses | schcode)
Data: d

REML criterion at convergence: 48095.1

Scaled residuals:
Min      1Q  Median      3Q     Max
-3.8525 -0.5590  0.1383  0.6541  5.7017

Random effects:
Groups   Name        Variance Std.Dev. Corr
schcode  (Intercept)  2.045   1.430
ses          1.317   1.147    -1.00
Residual             62.204   7.887
Number of obs: 6871, groups:  schcode, 419

Fixed effects:
Estimate Std. Error t value
(Intercept)  56.6683     0.4662 121.561
ses           3.1874     0.1684  18.926
ses_mean      2.4601     0.2796   8.798
pro4yrc       1.3146     0.4700   2.797
public       -0.3210     0.2609  -1.230

Correlation of Fixed Effects:
(Intr) ses    ses_mn pr4yrc
ses      -0.044
ses_mean  0.209 -0.521
pro4yrc  -0.875 -0.011 -0.246
public   -0.412 -0.002 -0.053  0.005
optimizer (nloptwrap) convergence code: 0 (OK)
boundary (singular) fit: see help('isSingular')

Here is the code that fit the model with uncorrelated random slopes, and prints out the model summary. The numbers are almost exactly the same as SPSS:

model2 <- lmer(math ~ 1 + ses + ses_mean + pro4yrc + public + (ses||schcode),
data = d)

summary(model2)

Linear mixed model fit by REML ['lmerMod']
Formula: math ~ 1 + ses + ses_mean + pro4yrc + public + ((1 | schcode) +      (0 + ses | schcode))
Data: d

REML criterion at convergence: 48121.8

Scaled residuals:
Min      1Q  Median      3Q     Max
-3.7910 -0.5538  0.1342  0.6587  5.7021

Random effects:
Groups    Name        Variance Std.Dev.
schcode   (Intercept)  2.112   1.453
schcode.1 ses          1.314   1.146
Residual              62.115   7.881
Number of obs: 6871, groups:  schcode, 419

Fixed effects:
Estimate Std. Error t value
(Intercept)  56.4698     0.4716 119.749
ses           3.1639     0.1689  18.734
ses_mean      2.6596     0.3136   8.480
pro4yrc       1.3602     0.4679   2.907
public       -0.1200     0.2744  -0.437

Correlation of Fixed Effects:
(Intr) ses    ses_mn pr4yrc
ses      -0.001
ses_mean  0.233 -0.486
pro4yrc  -0.868 -0.001 -0.255
public   -0.444 -0.002 -0.045  0.023


text:
245 articles – 3647 Notices  [english version]
 HAL : in2p3-00000298, version 1
 DOI : 10.1063/1.368736
 Journal of Applied Physics 84 (1998) 4921-4928
 Characterization of three-dimensional grain boundary topography in a YBa$_2$Cu$_3$O$_{7-d}$ thin film bicrystal grown on a SrTiO$_3$ substrate
 (1998)
 The topography and crystallography of YBa2Cu3O7 – d (YBCO) bicrystal films grown epitaxially on oriented SrTiO3 (STO) bicrystals have been characterized by scanning and transmission electron microscopies (SEM and TEM) and atomic force microscopy (AFM). The YBCO films were formed by laser ablation on melt-grown Sigma 13 STO bicrystals with a misorientation of 24° around the <001> tilt axis. In agreement with previous reports, TEM analysis revealed that the grain boundary in the film did not always follow the planar substrate grain boundary faithfully, but undulated about the average boundary plane. High resolution electron microscopy observations of the apparently complex undulating boundary structures could be explained as a result of an overlap between different orientation variants of the orthorhombic YBCO film. Cross correlation between SEM, AFM, and TEM imaging gave a clear evidence that an island growth mechanism is responsible for the observed grain boundary structure and morphology for which a schematic model is presented. It is seen that meandering of the YBCO grain boundary (GB) is necessarily coupled to a wide range of inclination of the GB plane in the z direction. The implications of this interfacial structure for the behavior of GB based Josephson junctions are discussed and compared to models proposed in the literature. It is also seen that inclination of the GB may be responsible for the poor correlation usually found in the literature between calculations and experimental curves of current density Jc versus the GB angle since the most elaborate models proposed up to now take into account only pure tilt GB plane facets, that is to say facets in the zone of the tilt axis. Moreover, such a GB structure may affect the interpretation of recent phase sensitive experiments done on bicrystal or tricrystal high Tc superconductors to determine the symmetry of the order parameter.
 Thème(s) : Physique/Matière Condensée/Science des matériaux
 Mot(s)-clé(s) : HIGH-TC SUPERCONDUCTORS – YTTRIUM OXIDES – BARIUM OXIDES – COPPER OXIDES – BICRYSTALS – STRONTIUM OXIDES – TITANIUM OXIDES – GRAIN BOUNDARIES – TRANSMISSION ELECTRON MICROSCOPY – SUPERCONDUCTING FILMS – superconducting thin films – high-temperature superconductors – yttrium compounds – barium compounds – tilt boundaries – interface structure – atomic force microscopy – scanning electron microscopy – Josephson effect – superconducting junction devices – island structure
 in2p3-00000298, version 1 http://hal.in2p3.fr/in2p3-00000298 oai:hal.in2p3.fr:in2p3-00000298 Contributeur : Jocelyne Lorgeril <> Soumis le : Mercredi 6 Janvier 1999, 15:37:30 Dernière modification le : Mardi 16 Mai 2006, 14:52:04


text:
# Prove (a³/c) + (b³/d) ≥ 1

#### anemone

##### MHB POTW Director
Staff member
If $a, b, c, d >0$ and $c^2+d^2=(a^2+b^2)^3$, then show that $$\displaystyle \frac{a^3}{c}+\frac{b^3}{d} \ge 1$$.

#### Jester

##### Well-known member
MHB Math Helper
Re: Prove (a³/c)+(b³/d) ≥1

My solution

This probably can be done with some clever manipulation of inequalities but the problem does suggest that the method of Lagrange multiplier would be effective.

If we define

$F = \dfrac{a^3}{c} + \dfrac{b^3}{d} + \lambda\left(c^2+d^2-(a^2+b^2)^3 \right)$,

so
\begin{eqnarray}
F_a &=& 3\,\frac {a^2}{c}-6a\,\lambda \left( a^2+b^{2} \right) ^{2}\\
F_b &=&3\,\frac {b^2}{d}-6b\,\lambda \left( a^2+b^{2} \right) ^{2}\\
F_c &=&-\,\frac {a^3}{c^2}+2\,\lambda c\;\;\;\;\;(*)\\
F_d &=&-\,\frac {b^3}{d^2}+2\,\lambda d\\
F_{\lambda} &=&c^2+d^2-(a^2+b^2)^3
\end{eqnarray}

Setting $F_c=0$ and solving for $\lambda$ gives $\lambda= \dfrac {a^3}{2c^3}$. From the $F_a=0$ equation we have

$-3\,{\dfrac {{a}^{2} \left( -c+{a}^{3}+a{b}^{2} \right) \left( c+{a}^{ 3}+a{b}^{2} \right) }{{c}^{3}}} =0$

from which we obtain $c = a^3+ab^2$. Factoring the $F_b=0$ equation gives

$3\,{\dfrac { \left( {a}^{2}b+{b}^{3}-d \right) b}{ \left( {a}^{2}+{b}^{ 2} \right) d}}=0$

leading to $d = a^2b+b^3$ noting that the remaining equation in (*) are satisfied. Thus, we have

$\dfrac{a^3}{c} + \dfrac{b^3}{d} = \dfrac{a^3}{a^3+ab^2} + \dfrac{b^3}{a^2b+b^3} = 1$

the minimum as required.

#### anemone

##### MHB POTW Director
Staff member
Re: Prove (a³/c)+(b³/d) ≥1

Thanks for participating, Jester and well done! And now it seems to me Lagrange Multiplier is a magic tool and can be applied to all kind of optimization problems to find the desired extrema points.

I want to share with you and others the solution proposed by other as well. It suggests the use of Hölder's Inequality to solve this problem.

If we let $$\displaystyle x=\left(\frac{a^3}{c} \right)^{\frac{2}{3}}$$ and $$\displaystyle y=\left(\frac{b^3}{d} \right)^{\frac{2}{3}}$$, we see that

$$\displaystyle a^2+b^2=c^{\frac{2}{3}}\left(\frac{a^3}{c} \right)^{\frac{2}{3}}+d^{\frac{2}{3}}\left(\frac{b^3}{d} \right)^{\frac{2}{3}}$$

$$\displaystyle \;\;\;\;\;\;\;\;\;\;\;\;=c^{\frac{2}{3}}(x)+d^{ \frac{2}{3}}(y)$$

$$\displaystyle \;\;\;\;\;\;\;\;\;\;\;\;\le (c^2+d^2)^{\frac{1}{3}}(x^{\frac{3}{2}}+y^{\frac{3}{2}})^{\frac{2}{3}}$$

$$\displaystyle \;\;\;\;\;\;\;\;\;\;\;\;\le (a^2+b^2)(x^{\frac{3}{2}}+y^{\frac{3}{2}})^{\frac{2}{3}}$$

Hence

$$\displaystyle a^2+b^2\le (a^2+b^2)(x^{\frac{3}{2}}+y^{\frac{3}{2}})^{\frac{2}{3}}$$

$$\displaystyle x^{\frac{3}{2}}+y^{\frac{3}{2}} \ge 1$$

$$\displaystyle \therefore \frac{a^3}{c}+\frac{b^3}{d} \ge 1$$ (Q.E.D.)


text:
# Letizia Lusito

Ontology type: schema:Person

NAME

Letizia

SURNAME

Lusito

### Publications in SciGraph latest 50 shown

• 2018-12 A direct scaling analysis for the sea level rise in STOCHASTIC ENVIRONMENTAL RESEARCH AND RISK ASSESSMENT
• 2015-06 Search for a standard model Higgs boson produced in association with a top-quark pair and decaying to bottom quarks using a matrix element method in THE EUROPEAN PHYSICAL JOURNAL C
• 2015-05 Erratum to: Measurement of jet multiplicity distributions in tt¯ production in pp collisions at s=7TeV in THE EUROPEAN PHYSICAL JOURNAL C
• 2015-04 Measurements of differential and double-differential Drell–Yan cross sections in proton–proton collisions at s=8TeV in THE EUROPEAN PHYSICAL JOURNAL C
• 2015-02 Measurement of electroweak production of two jets in association with a Z boson in proton–proton collisions at s=8TeV in THE EUROPEAN PHYSICAL JOURNAL C
• 2015-01 Erratum to: Search for new physics in events with same-sign dileptons and jets in pp collisions at s = 8 TeV in JOURNAL OF HIGH ENERGY PHYSICS
• 2014-11 Search for heavy neutrinos and W bosons with right-handed couplings in proton–proton collisions at s=8TeV in THE EUROPEAN PHYSICAL JOURNAL C
• 2014-11 Measurement of differential cross sections for the production of a pair of isolated photons in pp collisions at s=7TeV in THE EUROPEAN PHYSICAL JOURNAL C
• 2014-10 Observation of the diphoton decay of the Higgs boson and measurement of its properties in THE EUROPEAN PHYSICAL JOURNAL C
• 2014-10 Measurement of pseudorapidity distributions of charged particles in proton–proton collisions at s=8 TeV by the CMS and TOTEM experiments in THE EUROPEAN PHYSICAL JOURNAL C
• 2014-09 Searches for electroweak production of charginos, neutralinos, and sleptons decaying to leptons and W, Z, and Higgs bosons in pp collisions at 8 TeV in THE EUROPEAN PHYSICAL JOURNAL C
• 2014-09 Measurement of top quark–antiquark pair production in association with a W or Z boson in pp collisions at s=8TeV in THE EUROPEAN PHYSICAL JOURNAL C
• 2014-08 Measurement of jet multiplicity distributions in tt¯ production in pp collisions at s=7TeV in THE EUROPEAN PHYSICAL JOURNAL C
• 2014-08 Search for invisible decays of Higgs bosons in the vector boson fusion and associated ZH production modes in THE EUROPEAN PHYSICAL JOURNAL C
• 2014-08 Measurement of WZ and ZZ production in pp collisions at s=8TeV in final states with b-tagged jets in THE EUROPEAN PHYSICAL JOURNAL C
• 2014-07 Studies of dijet transverse momentum balance and pseudorapidity distributions in pPb collisions at sNN=5.02TeV in THE EUROPEAN PHYSICAL JOURNAL C
• 2014-06 Measurement of the triple-differential cross section for photon + jets production in proton-proton collisions at = 7 TeV in JOURNAL OF HIGH ENERGY PHYSICS
• 2014-06 Study of the production of charged pions, kaons, and protons in pPb collisions at sNN=5.02TeV in THE EUROPEAN PHYSICAL JOURNAL C
• 2014-06 Probing color coherence effects in pp collisions at s=7TeV in THE EUROPEAN PHYSICAL JOURNAL C
• 2014-06 Search for pair production of excited top quarks in the lepton + jets final state in JOURNAL OF HIGH ENERGY PHYSICS
• 2014-06 Measurement of the production cross sections for a Z boson and one or more b jets in pp collisions at s = 7 TeV in JOURNAL OF HIGH ENERGY PHYSICS
• 2014-06 Search for new physics in the multijet and missing transverse momentum final state in proton-proton collisions at = 8 TeV in JOURNAL OF HIGH ENERGY PHYSICS
• 2014-05 Search for W′ → tb decays in the lepton + jets final state in pp collisions at = 8 TeV in JOURNAL OF HIGH ENERGY PHYSICS
• 2014-05 Evidence for the 125 GeV Higgs boson decaying to a pair of τ leptons in JOURNAL OF HIGH ENERGY PHYSICS
• 2014-04 Event activity dependence of (nS) production in = 5.02 TeV pPb and = 2.76 TeV pp collisions in JOURNAL OF HIGH ENERGY PHYSICS
• 2014-04 Measurement of the top-quark mass in all-jets tt¯ events in pp collisions at s=7 TeV in THE EUROPEAN PHYSICAL JOURNAL C
• 2014-04 Measurements of the charge asymmetry using the dilepton decay channel in pp collisions at = 7 TeV in JOURNAL OF HIGH ENERGY PHYSICS
• 2014-03 Erratum: Search for anomalous production in the highly-boosted all-hadronic final state in JOURNAL OF HIGH ENERGY PHYSICS
• 2014-03 Study of double parton scattering using W + 2-jet events in proton-proton collisions at = 7 TeV in JOURNAL OF HIGH ENERGY PHYSICS
• 2014-02 Erratum: Measurement of the production cross section in the dilepton channel in pp collisions at = 8 TeV in JOURNAL OF HIGH ENERGY PHYSICS
• 2014-02 Measurement of associated W + charm production in pp collisions at = 7 TeV in JOURNAL OF HIGH ENERGY PHYSICS
• 2014-02 Measurement of the production cross section in the dilepton channel in pp collisions at = 8 TeV in JOURNAL OF HIGH ENERGY PHYSICS
• 2014-02 Studies of azimuthal dihadron correlations in ultra-central PbPb collisions at =2.76 TeV in JOURNAL OF HIGH ENERGY PHYSICS
• 2014-01 Measurement of Higgs boson production and properties in the WW decay channel with leptonic final states in JOURNAL OF HIGH ENERGY PHYSICS
• 2014-01 Search for new physics in events with same-sign dileptons and jets in pp collisions at = 8 TeV in JOURNAL OF HIGH ENERGY PHYSICS
• 2013-12 Measurement of the cross section and angular correlations for associated production of a Z boson with b hadrons in pp collisions at = 7 TeV in JOURNAL OF HIGH ENERGY PHYSICS
• 2013-12 Measurement of the differential and double-differential Drell-Yan cross sections in proton-proton collisions at = 7 TeV in JOURNAL OF HIGH ENERGY PHYSICS
• 2013-12 Search for top-squark pair production in the single-lepton final state in pp collisions at in THE EUROPEAN PHYSICAL JOURNAL C
• 2013-12 Jet and underlying event properties as a function of charged-particle multiplicity in proton–proton collisions at in THE EUROPEAN PHYSICAL JOURNAL C
• 2013-10 Measurement of the W-boson helicity in top-quark decays from production in lepton+jets events in pp collisions at TeV in JOURNAL OF HIGH ENERGY PHYSICS
• 2013-10 Measurement of the W+W− cross section in pp collisions at and limits on anomalous WWγ and WWZ couplings in THE EUROPEAN PHYSICAL JOURNAL C
• 2013-10 Measurement of the production cross section for in pp collisions at TeV and limits on ZZγ and Zγγ triple gauge boson couplings in JOURNAL OF HIGH ENERGY PHYSICS
• 2013-10 Measurement of the ratio of the inclusive 3-jet cross section to the inclusive 2-jet cross section in pp collisions at and first determination of the strong coupling constant in the TeV range in THE EUROPEAN PHYSICAL JOURNAL C
• 2013-10 Measurement of the hadronic activity in events with a Z and two jets and extraction of the cross section for the electroweak production of a Z with two jets in pp collisions at TeV in JOURNAL OF HIGH ENERGY PHYSICS
• 2013-09 Search for supersymmetry in hadronic final states with missing transverse energy using the variables αT and b-quark multiplicity in pp collisions at in THE EUROPEAN PHYSICAL JOURNAL C
• 2013-07 Search for physics beyond the standard model in events with τ leptons, jets, and large transverse momentum imbalance in pp collisions at in THE EUROPEAN PHYSICAL JOURNAL C
• 2013-07 Search for microscopic black holes in pp collisions at TeV in JOURNAL OF HIGH ENERGY PHYSICS
• 2013-07 Measurement of the lifetime in pp collisions at TeV in JOURNAL OF HIGH ENERGY PHYSICS
• 2013-07 Searches for long-lived charged particles in pp collisions at =7 and 8 TeV in JOURNAL OF HIGH ENERGY PHYSICS
• 2013-07 Study of exclusive two-photon production of W+W− in pp collisions at TeV and constraints on anomalous quartic gauge couplings in JOURNAL OF HIGH ENERGY PHYSICS

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text:
# Math Help - Two divergent sequences whose product converges

1. ## Two divergent sequences whose product converges

I spend what was probably way too long trying to figure this out:

"Provide two divergent sequences X and Y whose product converges."

I was also asking to find the same, but whose sum converges, which I think I got. I don't necessarily want you to tell me two sequences satisfy this, but a hint or two would be nice. Thank you!

2. Originally Posted by mremwo
"Provide two divergent sequences X and Y whose product converges.

$x_n=(-1)^n,\;y_n=(-1)^{n+1}$

Fernando Revilla

P.S. I have used the term divergent as equivalent to not convergent.


text:
# The image of a meromorphic function is dense

Claim 1: The image of the complex plane under a nonconstant entire function is dense in the complex plane.

Claim 2: The image of any small enough punctured neighborhood around an essential singularity is dense in the complex plane.

Claim 3: The image of the complex plane under a nonconstant meromorphic function is dense in the complex plane.

I know the proof to the first claim and it uses Liouville's theorem. I know the proof to the second claim (the Casorati-Weierstrass theorem)

I'm trying to prove claim 3 but I don't have any direction. I have tried proving it by way of contradiction, just like how the first 2 claims are proved but got stuck.. Please don't give me a full solution, hints will suffice :)

• If $f$ avoids the open ball $B_r(z_0)$, then $z\mapsto \frac1{f(z)-z_0}$ is bounded by $\frac1r$ - and what happens to the poles? – Hagen von Eitzen May 22 '17 at 11:29
• The poles become zeros of the new function. So can we say the new function is entire, bounded and therefore constant? – zokomoko May 22 '17 at 13:50


text:
Learning tasks encapsulate the data set and further relevant information about a machine learning problem, for example the name of the target variable for supervised problems.

The tasks are organized in a hierarchy, with the generic Task at the top. The following tasks can be instantiated and all inherit from the virtual superclass Task:

To create a task, just call make<TaskType>, e.g., makeClassifTask. All tasks require an identifier (argument id) and a data.frame (argument data). If no ID is provided it is automatically generated using the variable name of the data. The ID will be later used to name results, for example of benchmark experiments, and to annotate plots. Depending on the nature of the learning problem, additional arguments may be required and are discussed in the following sections.

### Regression

For supervised learning like regression (as well as classification and survival analysis) we, in addition to data, have to specify the name of the target variable.

data(BostonHousing, package = "mlbench")
#> Type: regr
#> Target: medv
#> Observations: 506
#> Features:
#> numerics  factors  ordered
#>       12        1        0
#> Missings: FALSE
#> Has weights: FALSE
#> Has blocking: FALSE

As you can see, the Task records the type of the learning problem and basic information about the data set, e.g., the types of the features (numeric vectors, factors or ordered factors), the number of observations, or whether missing values are present.

Creating tasks for classification and survival analysis follows the same scheme, the data type of the target variables included in data is simply different. For each of these learning problems some specifics are described below.

### Classification

For classification the target column has to be a factor.

In the following example we define a classification task for the BreastCancer data set and exclude the variable Id from all further model fitting and evaluation.

data(BreastCancer, package = "mlbench")
df = BreastCancer
df$Species = NULL costsens.task = makeCostSensTask(data = df, cost = cost) costsens.task #> Supervised task: df #> Type: costsens #> Observations: 150 #> Features: #> numerics factors ordered #> 4 0 0 #> Missings: FALSE #> Has blocking: FALSE #> Classes: 3 #> y1, y2, y3  For more details see the page about cost-sensitive classification. ## Further settings The Task help page also lists several other arguments to describe further details of the learning problem. For example, we could include a blocking factor in the task. This would indicate that some observations "belong together" and should not be separated when splitting the data into training and test sets for resampling. Another option is to assign weights to observations. These can simply indicate observation frequencies or result from the sampling scheme used to collect the data. Note that you should use this option only if the weights really belong to the task. If you plan to train some learning algorithms with different weights on the same Task, mlr offers several other ways to set observation or class weights (for supervised classification). See for example the tutorial page about training or function makeWeightedClassesWrapper. ## Accessing a learning task We provide many operators to access the elements stored in a Task. The most important ones are listed in the documentation of Task and getTaskData. To access the task description that contains basic information about the task you can use: getTaskDesc(classif.task) #>$id
#> [1] "BreastCancer"
#>
#> $type #> [1] "classif" #> #>$target
#> [1] "Class"
#>
#> $size #> [1] 699 #> #>$n.feat
#> numerics  factors  ordered
#>        0        4        5
#>
#> $has.missings #> [1] TRUE #> #>$has.weights
#> [1] FALSE
#>
#> $has.blocking #> [1] FALSE #> #>$class.levels
#> [1] "benign"    "malignant"
#>
#> positive #> [1] "malignant" #> #>negative
#> [1] "benign"
#>
#> attr(,"class")

Note that task descriptions have slightly different elements for different types of Tasks. Frequently required elements can also be accessed directly.

## Get the ID
#> [1] "BreastCancer"

## Get the type of task
#> [1] "classif"

## Get the names of the target columns
#> [1] "Class"

## Get the number of observations
#> [1] 699

## Get the number of input variables
#> [1] 9

## Get the class levels in classif.task
#> [1] "benign"    "malignant"

Moreover, mlr provides several functions to extract data from a Task.

## Accessing the data set in classif.task
#> 'data.frame':    699 obs. of  10 variables:
#>  $Cl.thickness : Ord.factor w/ 10 levels "1"<"2"<"3"<"4"<..: 5 5 3 6 4 8 1 2 2 4 ... #>$ Cell.size      : Ord.factor w/ 10 levels "1"<"2"<"3"<"4"<..: 1 4 1 8 1 10 1 1 1 2 ...
#>  $Cell.shape : Ord.factor w/ 10 levels "1"<"2"<"3"<"4"<..: 1 4 1 8 1 10 1 2 1 1 ... #>$ Marg.adhesion  : Ord.factor w/ 10 levels "1"<"2"<"3"<"4"<..: 1 5 1 1 3 8 1 1 1 1 ...
#>  $Epith.c.size : Ord.factor w/ 10 levels "1"<"2"<"3"<"4"<..: 2 7 2 3 2 7 2 2 2 2 ... #>$ Bare.nuclei    : Factor w/ 10 levels "1","2","3","4",..: 1 10 2 4 1 10 10 1 1 1 ...
#>  $Bl.cromatin : Factor w/ 10 levels "1","2","3","4",..: 3 3 3 3 3 9 3 3 1 2 ... #>$ Normal.nucleoli: Factor w/ 10 levels "1","2","3","4",..: 1 2 1 7 1 7 1 1 1 1 ...
#>  $Mitoses : Factor w/ 9 levels "1","2","3","4",..: 1 1 1 1 1 1 1 1 5 1 ... #>$ Class          : Factor w/ 2 levels "benign","malignant": 1 1 1 1 1 2 1 1 1 1 ...

## Get the names of the input variables in cluster.task
#>  [1] "mpg"  "cyl"  "disp" "hp"   "drat" "wt"   "qsec" "vs"   "am"   "gear"
#> [11] "carb"

## Get the values of the target variables in surv.task
#>   time status
#> 1  306   TRUE
#> 2  455   TRUE
#> 3 1010  FALSE
#> 4  210   TRUE
#> 5  883   TRUE
#> 6 1022  FALSE

## Get the cost matrix in costsens.task
#> NULL

Note that getTaskData offers many options for converting the data set into a convenient format. This especially comes in handy when you integrate a new learner from another R package into mlr. In this regard function getTaskFormula is also useful.

mlr provides several functions to alter an existing Task, which is often more convenient than creating a new Task from scratch. Here are some examples.

## Select observations and/or features

## It may happen, especially after selecting observations, that features are constant.
## These should be removed.
#> Removing 1 columns: am
#> Type: cluster
#> Observations: 14
#> Features:
#> numerics  factors  ordered
#>       10        0        0
#> Missings: FALSE
#> Has weights: FALSE
#> Has blocking: FALSE

## Remove selected features
#> Type: surv
#> Target: time,status
#> Events: 165
#> Observations: 228
#> Features:
#> numerics  factors  ordered
#>        6        0        0
#> Missings: TRUE
#> Has weights: FALSE
#> Has blocking: FALSE

## Standardize numerical features
#>       mpg              cyl              disp              hp
#>  Min.   :0.0000   Min.   :0.0000   Min.   :0.0000   Min.   :0.0000
#>  1st Qu.:0.3161   1st Qu.:0.5000   1st Qu.:0.1242   1st Qu.:0.2801
#>  Median :0.5107   Median :1.0000   Median :0.4076   Median :0.6311
#>  Mean   :0.4872   Mean   :0.7143   Mean   :0.4430   Mean   :0.5308
#>  3rd Qu.:0.6196   3rd Qu.:1.0000   3rd Qu.:0.6618   3rd Qu.:0.7473
#>  Max.   :1.0000   Max.   :1.0000   Max.   :1.0000   Max.   :1.0000
#>       drat              wt              qsec              vs
#>  Min.   :0.0000   Min.   :0.0000   Min.   :0.0000   Min.   :0.0000
#>  1st Qu.:0.2672   1st Qu.:0.1275   1st Qu.:0.2302   1st Qu.:0.0000
#>  Median :0.3060   Median :0.1605   Median :0.3045   Median :0.0000
#>  Mean   :0.4544   Mean   :0.3268   Mean   :0.3752   Mean   :0.4286
#>  3rd Qu.:0.7026   3rd Qu.:0.3727   3rd Qu.:0.4908   3rd Qu.:1.0000
#>  Max.   :1.0000   Max.   :1.0000   Max.   :1.0000   Max.   :1.0000
#>        am           gear             carb
#>  Min.   :0.5   Min.   :0.0000   Min.   :0.0000
#>  1st Qu.:0.5   1st Qu.:0.0000   1st Qu.:0.3333
#>  Median :0.5   Median :0.0000   Median :0.6667
#>  Mean   :0.5   Mean   :0.2857   Mean   :0.6429
#>  3rd Qu.:0.5   3rd Qu.:0.7500   3rd Qu.:1.0000
#>  Max.   :0.5   Max.   :1.0000   Max.   :1.0000

For more functions and more detailed explanations have a look at the data preprocessing page.

## Example tasks and convenience functions

For your convenience mlr provides pre-defined Tasks for each type of learning problem. These are also used throughout this tutorial in order to get shorter and more readable code. A list of all Tasks can be found in the Appendix.

Moreover, mlr's function convertMLBenchObjToTask can generate Tasks from the data sets and data generating functions in package mlbench.


text:
# Applying a stylesheet to a notebook using a setterbar - palette hiccup

I built a setterbar that allows me to quickly apply stylesheets to a notebook. Here is the code:

SetterBar[Dynamic[temp], {"Lecture", "Testing"}],

Dynamic[If[temp == "Lecture",
SetOptions[InputNotebook[],
StyleDefinitions ->
"/Users/michaelmccain/Library/Mathematica/SystemFiles/FrontEnd/\
StyleSheets/MyLectureStyles.nb"],
SetOptions[InputNotebook[],
StyleDefinitions ->
"/Users/michaelmccain/Library/Mathematica/SystemFiles/FrontEnd/\
StyleSheets/Testing.nb"]]]

When the code is located in the target notebook it works perfectly. See gif below.

However, if I put the code above into a palette, then I get an annoying hiccup when I try and use it... The styles in the notebook don't update immediately. If I type at least one character in any cell, then the styles update. Strange. Below is gif illustrating the issue.

I'm not sure why. What is going on here? Is there a fix you can think of?

• A workaround is to add NotebookWrite[InputNotebook[],""] after SetOptions[]. This resets everything. I'm interested if anyone thinks this is a bug. – Michael McCain Dec 19 '16 at 0:17
• I've faced it too. I'm doing SetOptions[nb, StyleDefinitions->"Default.nb"] as an intermediate step and it works. That day it was one of minor problems so I forgot to report it. You can do that. – Kuba Dec 19 '16 at 7:36


text:
# Particle oscillation in a potensial field

Problem by zainul abidin
30 Nov 1999 00:00 WIB
A particle of mass m is located in a unidimensional potential field where the potential energy of the particle depends on the coordinate x as $U(x)=(1-a\cos bx)$, where a and b are constants. Find the period of small oscillations that the particle performs about the equilibrium position.


text:
“Be kind, for everyone you meet is fighting a hard battle” - Often attributed to Plato but likely from Ian McLaren (pseudonym of Reverend John Watson)

## Saturday, May 31, 2014

### Stopping time

 Image credit: http://www.mastermarf.com
I recently completed a business trip to Thailand entailing some 33 total hours in the air. My client won't reimburse business class, let alone first class, so "economy plus" was the best that I could do. On Delta, that gives me 4" of extra legroom and a few extra degrees of recline in the seat. That doesn't sound like a lot but it makes a huge difference!

In any case, while on such flights, I bring along work and recreational reading and, typically, attempt to get some sleep, I also tend to give some thought to something to do with the flight. For example, I've made several posts about using the accelerometer in my smart phone to determine acceleration and takeoff speeds and distances. On another trip, I used the pressure sensor in my Galaxy Note 3 to estimate the cabin altitude of the aircraft in cruise. While there are apps that will automatically give you altitude based on the pressure sensor, I hadn't (and haven't) yet installed one so it was an interesting little exercise in calculus, fluid mechanics, and the ideal gas law to estimate that the cabin altitude was about 6,700 feet.

On the most recent trip, I was wondering about my relationship to the sun as we chased it (westbound) or rushed toward it (eastbound). It occurred to me that, for any speed up to about 1040 m.p.h., there's some latitude where that westbound speed would keep the sun at the same position in the sky. At the north or south pole, if the sun were visible at all, you wouldn't even have to move, you'd only have to rotate.

In any event, it was a pretty trivial exercise in spherical coordinates to find the appropriate latitude at a speed of 500 m.p.h. (a reasonable speed for the B747-400 at altitude flying into a typical not too strong headwind). For any speed, the equation is $\theta =\arccos\left(\frac{24v}{2 \pi R}\right)$, where v is the speed in miles per hour and R is the Earth's radius in miles (of course, any units of speed, i.e., length/time, and length will work so long as they're consistent and the "24" is changed to the number of whatever time unit that's used that comprises a day). Substituting 500 for v and 3,957 for the radius, I find that a direct westbound path at about 61.14 degrees north (or south) latitude will make time stand still. Or, at least, it will make the sun stand still in the sky. Were I to walk at 3 m.p.h., I'd need to be at about 89.8 degrees north (or south) latitude. This would be around 11 miles from the north (or south) pole.

Of course, I'd be jolted forward a whole 24 hours each time I crossed the International Date Line so, sadly, I can't use this method to stay Forever Young.

.


text:
# Review: Godel’s Proof

This was something that I had to stop reading because of the exam season, and finally finished within 24 hrs of the exam getting over.

It is a rightly acknowledged classic in the field of popular mathematics (Stuff which makes abstruse mathematical ideas accessible to the (almost) general public).

Like the title says, it deals with a set of proofs that Godel published which demolished one of the most ambitious projects of modern times, to axiomatize all of mathematics.

It was believed that given a finite set of axioms and a set of rules to operate on them, one can mindlessly manipulate symbols to generate every known theorem. A simple example would be translating geometry into algebra ($x^2 + y^2 = c^2$ is how a circle would look in algebraic terms ). So, we can say that if  algebra is consistent, then geometry is consistent.

Hilbert developed a method to provide proofs of consistency, which was then taken over by the logicians of the time to try and provide similar proofs to certain parts and eventually all of  mathematics.

That is, until Godel came along. How he did it is explained beautifully in the book and is unecessary to reproduce here. Godel proved that any system for number theory that is consistent is incomplete, i.e,  you can produce a true theorem which cannot be proved within the system. Thus, we cannot have a formal system describing all of mathematics. We can have formal systems with finite set of axioms, but not for something like number theory, where Godel proved it was impossible to generate a finite set of axioms.

The book also has a section on the philosophical implications of this proof, which essentially says that present day A.I is a far cry from imitating the human brain, and unless we can actually describe the human brain as a logical system with a finite set of axioms.

There are many approaches to imitate human intelligence, like neural networks, but they run on a system with a very limited set of axioms and rules of inference commonly known as a digital computer. Maybe we need some new kind of ‘computer’. Researchers have a lot of work to do!!

## 4 thoughts on “Review: Godel’s Proof”

1. I took a class about Godel’s theorem and it is a pretty powerful blow not only against Strong AI, but also against empiricism. He believed that although finite formal systems were incomplete, there are abstract entities that are complete, and we are able to understand how to complete mathematical truths beyond a finite formal system. Even if there aren’t mathematical entities, it is still implausible that numbers are nothing more than a made-up human invention.

2. Yup, I remember reading that Godel himself believed that something like the Platonic Ideal existed which can be manifest only imperfectly.

3. This sounds a bit similar to another book I have read.

Douglas Hofstadter wrote “I Am a Strange Loop”, which tries to explain how human’s develop conciousness despite being made up of physical inanimate material. He also looks at why we have fallen short in developing AI etc.

What Hofstadter does is try to explain the human mind as a self-referential system, based on Gödel’s Incompleteness Theorem.

To be honest, I didn’t particularly enjoy the book as I am not a fan of Hofstadter’s indirect, meandering and repetitive writing style, but it did raise some interesting points and made me think.

So it MAY be something you want to check out if you enjoyed this one, as they sound similar. I’ll have to find myself a copy of “Gödel’s Proof”

4. Well, this one just has a intro by Hofstadter, and is not as much about epistemology as about the proof itself, which was fleshed out by the original authors. I have GEB by hofstadter waiting in my lib, so will try him out anyways :)


text:
Data
Lorenz_attractor_regime_changes

# Lorenz_attractor_regime_changes

active ARFF Public Domain (CC0) Visibility: public Uploaded 27-10-2019 by Matteo Caorsi
Issue #Downvotes for this reason By

This dataset contains a simulation of the Lorenz attractor with the parameter $\rho$ varying in time. The stable and chaotic regimes alternate.

### 4 features

 x numeric 4942 unique values 0 missing y numeric 4941 unique values 0 missing z numeric 4941 unique values 0 missing rho numeric 101 unique values 0 missing

### 62 properties

4942
Number of instances (rows) of the dataset.
4
Number of attributes (columns) of the dataset.
Number of distinct values of the target attribute (if it is nominal).
0
Number of missing values in the dataset.
0
Number of instances with at least one value missing.
4
Number of numeric attributes.
0
Number of nominal attributes.
4.62
Mean standard deviation of attributes of the numeric type.
Second quartile (Median) of mutual information between the nominal attributes and the target attribute.
Percentage of instances belonging to the most frequent class.
Minimal entropy among attributes.
0.57
Second quartile (Median) of skewness among attributes of the numeric type.
Number of instances belonging to the most frequent class.
-0.91
Minimum kurtosis among attributes of the numeric type.
0
Percentage of binary attributes.
4.77
Second quartile (Median) of standard deviation of attributes of the numeric type.
Maximum entropy among attributes.
-2.19
Minimum of means among attributes of the numeric type.
0
Percentage of instances having missing values.
Third quartile of entropy among attributes.
1.18
Maximum kurtosis among attributes of the numeric type.
Minimal mutual information between the nominal attributes and the target attribute.
0
Percentage of missing values.
1.07
Third quartile of kurtosis among attributes of the numeric type.
12
Maximum of means among attributes of the numeric type.
The minimal number of distinct values among attributes of the nominal type.
100
Percentage of numeric attributes.
11.4
Third quartile of means among attributes of the numeric type.
Maximum mutual information between the nominal attributes and the target attribute.
-0.72
Minimum skewness among attributes of the numeric type.
0
Percentage of nominal attributes.
Third quartile of mutual information between the nominal attributes and the target attribute.
The maximum number of distinct values among attributes of the nominal type.
3.33
Minimum standard deviation of attributes of the numeric type.
First quartile of entropy among attributes.
0.72
Third quartile of skewness among attributes of the numeric type.
0.72
Maximum skewness among attributes of the numeric type.
Percentage of instances belonging to the least frequent class.
-0.84
First quartile of kurtosis among attributes of the numeric type.
5.44
Third quartile of standard deviation of attributes of the numeric type.
5.61
Maximum standard deviation of attributes of the numeric type.
Average entropy of the attributes.
Number of instances belonging to the least frequent class.
-2.19
First quartile of means among attributes of the numeric type.
Standard deviation of the number of distinct values among attributes of the nominal type.
0.09
Mean kurtosis among attributes of the numeric type.
0
Number of binary attributes.
First quartile of mutual information between the nominal attributes and the target attribute.
4.31
Mean of means among attributes of the numeric type.
-0.43
First quartile of skewness among attributes of the numeric type.
Average mutual information between the nominal attributes and the target attribute.
3.64
First quartile of standard deviation of attributes of the numeric type.
Average class difference between consecutive instances.
An estimate of the amount of irrelevant information in the attributes regarding the class. Equals (MeanAttributeEntropy - MeanMutualInformation) divided by MeanMutualInformation.
Second quartile (Median) of entropy among attributes.
Entropy of the target attribute values.
Average number of distinct values among the attributes of the nominal type.
0.04
Second quartile (Median) of kurtosis among attributes of the numeric type.
0
Number of attributes divided by the number of instances.
0.29
Mean skewness among attributes of the numeric type.
3.71
Second quartile (Median) of means among attributes of the numeric type.
Number of attributes needed to optimally describe the class (under the assumption of independence among attributes). Equals ClassEntropy divided by MeanMutualInformation.


text:
# help me

• October 15th 2005, 04:30 PM
math
help me
a,b,c>0 prove the equation have 1 root :
d^3 - d(ab+bc+ca) - 2abc =0 .

• October 15th 2005, 11:47 PM
rgep
If a cubic f has three real roots then it has a (real) maximum and a minimum between the roots. So the derivative f' has to have two real roots, and the value of f at the maximum must be positive and the value at the minimum must be negative. In this case f' is 3d^2 - (ab+bc+ca). At d = sqrt((ab+bc+ca)/3) we have f = d(d^2 - (ab+bc+ca)) - 2abc = -(2/3)(ab+bc+ba)^(3/2) - 2abc and this is indeed negative. At d = -sqrt((ab+bc+ca)/2) we have f = d(d^2 - (ab+bc+ca)) - 2abc = (2/3)(ab+bc+ba)^(3/2) - 2abc. Now (ab+bc+bc)/3 >= (ab.bc.ca)^(1/3) by the AM-GM inequality and this shows that the value at the maximum is negative unless there is equality in the AM-GM which happens only when a,b,c are all equal. We conclude that if a=b=c there are three real roots, at -a,-a and 2a; if a,b,c are not all equal there is one real root which lies to the right of sqrt((ab+bc+ca)/3).
• October 15th 2005, 11:58 PM
math
thansk to rgeq you are very wonderful , and second question
from the equation and x,y,z >0 . ax+by+cz=xyz . Prove the inequality

x+y+z >= 2/d . (\sqrt[(a+d)(b+d)(c+d)])
• October 17th 2005, 02:59 AM
math
Quote:

Originally Posted by rgep
If a cubic f has three real roots then it has a (real) maximum and a minimum between the roots. So the derivative f' has to have two real roots, and the value of f at the maximum must be positive and the value at the minimum must be negative. In this case f' is 3d^2 - (ab+bc+ca). At d = sqrt((ab+bc+ca)/3) we have f = d(d^2 - (ab+bc+ca)) - 2abc = -(2/3)(ab+bc+ba)^(3/2) - 2abc and this is indeed negative. At d = -sqrt((ab+bc+ca)/2) we have f = d(d^2 - (ab+bc+ca)) - 2abc = (2/3)(ab+bc+ba)^(3/2) - 2abc. Now (ab+bc+bc)/3 >= (ab.bc.ca)^(1/3) by the AM-GM inequality and this shows that the value at the maximum is negative unless there is equality in the AM-GM which happens only when a,b,c are all equal. We conclude that if a=b=c there are three real roots, at -a,-a and 2a; if a,b,c are not all equal there is one real root which lies to the right of sqrt((ab+bc+ca)/3).

the root of equation ???? i don't understand why we get d = sqrt((ab+bc+ca)/3) and d = -sqrt((ab+bc+ca)/2)
• October 17th 2005, 01:09 PM
rgep
Typo: it's + or - sqrt((ab+bc+ca)/3)
• October 20th 2005, 09:23 PM
math
Quote:

Originally Posted by math
thansk to rgeq you are very wonderful , and second question
from the equation and x,y,z >0 . ax+by+cz=xyz . Prove the inequality

x+y+z >= 2/d . (\sqrt[(a+d)(b+d)(c+d)])

• October 20th 2005, 09:27 PM
math
x,y,z >0 . ax+by+cz=xyz . with 2/d = 1/(a+d)+1/(b+d)+1/(c+d)Prove the inequality
x+y+z >= 2/d . (\sqrt[(a+d)(b+d)(c+d)]) :D :D :D
• October 26th 2005, 03:44 PM
math
:( :( :( :(
next week i will give the proof to my teacher !!! please
• October 26th 2005, 04:01 PM
MathGuru
Maybe you should make a new thread for your second question because people usually try to answer posts with 0 replies.
• October 27th 2005, 06:16 AM
math
all member can't help me !!! hix hix hix
• October 27th 2005, 07:29 AM
MathGuru
Seriously,

Start a new thread and more people will see it. Or maybe private message rgep for some more help. Few will be reading this because it is an old thread with many replies.
• October 27th 2005, 03:40 PM
math
okie i know .
put 2/d = 1/a+d + 1/b+d + 1/c+d
and x,y,z>0 such that : ax+by+cz=xyz . Prove the inequality
(x+y+z)>=\sqrt{(a+d).(b+d).(c+d)} . 2/d
:o
• October 27th 2005, 04:53 PM
Jameson
Quote:

Originally Posted by math
okie i know .
put 2/d = 1/a+d + 1/b+d + 1/c+d
and x,y,z>0 such that : ax+by+cz=xyz . Prove the inequality
(x+y+z)>=\sqrt{(a+d).(b+d).(c+d)} . 2/d
:o

Here, I'll do you a favor and put some of this into Latex so people can read it and want to help you more.

You didn't have parantheses, so I'll have to assume what you meant.

Put $\frac{2}{d}=\frac{1}{a+d}+\frac{1}{b+d}+\frac{1}{c +d}$

and $x,y,z>0$ such that: $ax+by+cz=xyz$.

Prove the inequality $x+y+z\ge \frac{2\sqrt{(a+d)(b+d)(c+d)}}{d}$

Is that what you meant?
• October 27th 2005, 09:03 PM
math
yes , that's right . do you have proof ????
• January 15th 2006, 02:11 AM
konlsy
it is difficult. but i will try to sovel it. i hope i can sovel this puzzl ;)


text:
# Question regarding diameter of subsets of a metric space

The question is :

Find a condition on a metric space$(X,d)$ that ensures that there exist subsets $A$ and $B$ of $X$ with $A \subset B$ such that $diam(A)$ = $diam(B)$.

I know that if $X$ is a metric space and $A$ and $B$ are subsets of $X$ with $A \subset B$ then

$diam(A)$ <= $diam(B)$.

If i assume that the metric $d$ on $X$ is the discrete metric then then diameters of $A$ and $B$ will be the same.

Is this good enough ? or can there be some other condition on the metric space $X$ ?

• HINT: >A: [0, 1] >B: [-1, 0, 1]. Yes you only need the discrete metric. Why are you trying to overcomplicate it? – Don Larynx Oct 9 '13 at 11:31


text:
# How do you find all the zeros of x^3 + x^2 + 9x + 9  given zero 3i?

May 21, 2016

#### Answer:

The zeros are $- 1$, $3 i$ and $- 3 i$.

#### Explanation:

We don't really need to be told that $3 i$ is a zero, except that it does inform us that we should include Complex zeros in the answer.

This cubic factors by grouping:

${x}^{3} + {x}^{2} + 9 x + 9$

$= \left({x}^{3} + {x}^{2}\right) + \left(9 x + 9\right)$

$= {x}^{2} \left(x + 1\right) + 9 \left(x + 1\right)$

$= \left({x}^{2} + 9\right) \left(x + 1\right)$

$= \left(x - 3 i\right) \left(x + 3 i\right) \left(x + 1\right)$

So the zeros are $- 1$, $3 i$ and $- 3 i$.


text:
# CoSiF

Interface Algorithms for Distributed Power System Co-Simulation

# CoSif: Co-Simulation Interface Algorithms

CoSif is a collection of co-simulation Interface Algorithms (IA) for (geographically) distributed real-time simulation. It has been developed by the Institute for Automation of Complex Power Systems at RWTH Aachen University with the goal to simplify the execution of distributed co-simulations by using using re-usable library blocks within the simulations models.

We aim to provide a set of tested and interoperable implementations of IA targeted at the following platforms:

• VILLASnode / DPsim

At the moment we aim to provide interface algorithms for the following domains:

• HVDC-VSC
• Vrms / Freq / Phase
• PQ
• Dynamic Phasor ITM

The latest version of the CoSiF libraries can be found at RWTH’s GitLab server:

## Publications

• S. Vogel, V. R. Raijkumar, H. T. Nguyen, M. Stevic, R. Bhandia, K. Heussen, P. Palensky, A. Monti, “Improvements to the Co-simulation Interface forGeographically Distributed Real-time Simulation,” 45th Annual Conference of the IEEE Industrial Electronics Society (IECON), Lisbon Portugal, October 2019.

## Documentation

The documentation for CoSiF is not yet complete. The current working state can be found in the Git repo:

For further question please consider joining our Slack channel: FEIN e.V. Slack.

This project is released under the terms of the GPL version 3.

This program is free software: you can redistribute it and/or modify
the Free Software Foundation, either version 3 of the License, or
any later version.

This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
GNU General Public License for more details.

You should have received a copy of the GNU General Public License
along with this program.  If not, see <http://www.gnu.org/licenses/>.

For other licensing options please consult Prof. Antonello Monti.


text:
Find Paper, Faster
Example：10.1021/acsami.1c06204 or Chem. Rev., 2007, 107, 2411-2502
Review of Nanocomposite Dielectric Materials With High Thermal Conductivity
Proceedings of the IEEE  (IF10.961),  Pub Date : 2021-06-24, DOI: 10.1109/jproc.2021.3085836
Manojkumar Lokanathan, Palash V. Acharya, Abdelhamid Ouroua, Shannon M. Strank, Robert E. Hebner, Vaibhav Bahadur

Dielectric materials with high thermal conductivity (TC) can enable disruptive performance enhancement in the areas of electronics packaging, thermal management, energy storage, cabling, and heat sinks. There have been widespread global efforts over the past decade on developing novel nanocomposite dielectric materials. As a baseline, legacy polymers and epoxies used in the abovementioned applications have very low thermal conductivities ranging from 0.1–0.5 $\text{W}\cdot \text{m}^{-1} \cdot \text{K}^{-1}$ . Recent advances have led to the commercial availability of polymeric materials with thermal conductivities approaching 10 $\text{W}\cdot \text{m}^{-1} \cdot \text{K}^{-1}$ . Importantly, several fundamental studies report novel nanocomposites with thermal conductivities $>50\,\,\text{W}\cdot \text{m}^{-1} \cdot \text{K}^{-1}$ . This article summarizes progress in the development of such materials with a focus on developments that show promise for improved practical dielectrics. This review highlights that high TC alone is inadequate to characterize the suitability of any material for the above applications. Other thermal properties, such as thermal diffusivity, glass transition temperature, and the ratio of the in-plane to out-of-plane TC, are important to quantify the thermal performance of novel nanocomposites. In addition, characterization and understanding of mechanical properties (coefficient of thermal expansion (CTE), tensile strength and elastic modulus) and electrical properties (dielectric strength and dielectric permittivity) are critical for holistic multifunctional assessments of these materials. There are other parameters and properties that influence performance, life, and manufacturability, such as viscosity and moisture absorption. This study reviews all the above aspects of nanocomposite dielectric materials reported in the literature. More specifically, we analyze various filler-polymer combinations, and the influence of approaches to incorporate fillers in the polymer on the thermal, mechanical, and electrical properties. While the addition of fillers leads to huge enhancements in TC, the TC is highly anisotropic, with out-of-plane TC lower than in-plane TC by an order of magnitude. It is seen that most present-day materials are still inadequate for future applications due to their low glass transition temperatures; specific promising materials are highlighted. While the addition of fillers reduces the CTE, further reduction is needed to favorably improve the mechanical performance of these materials. While the electrical insulating properties of these composite materials are adequate, there is very little data reported on other electrical properties. In summary, while there is an understandable focus on enhancing the TC, other properties are underreported, and there is insufficient information to support the assessment of most novel materials for practical applications. Overall, this study summarizes the state-of-the-art dielectric nanocomposites and outlines directions for future research to bridge the gap between basic materials science and applications.


text:
## sleestak Group Title Given the following, x1=2 x2=-1 x3=-2 x4=5 x5=-4 x6=8 Find: 3 ∑ ( x subi - 1 ) 2 i=1 2 years ago 2 years ago

1. dpflan Group Title

I think I can handle this ;)

2. dpflan Group Title

$\sum_{i=1}^{i=3} (x_i - 1) ^2$

3. amistre64 Group Title

the suspense is killing me :)

4. dpflan Group Title

You given 6 x_i values. The equation only wants you to consider x_i for (i=1 until i=3), that would be x1, x2, x3. The sigma is the symbol for sum. So the problem is to find the sum for the following events: plugging in each x_i subtracting 1 then squaring that. Hence you'll have 3 values that you then add to together to get the final value. $i=1, x_1 = 2, (2-1)^2 = 1$ $i=2, x_2 = -1, ((-1) -1)^2 = ...$ ... and so on

5. amistre64 Group Title

could we expand the binomial and pick it apart sum for sum?

6. sleestak Group Title

OH! THANK YOU. DPFLAN, you are the best <3

7. dpflan Group Title

@amistre64 for sure, that would create even more suspense!

8. amistre64 Group Title

the butler did it, im sure of it ....

9. dpflan Group Title

with the candlestick?

10. dpflan Group Title

@sleestak No problem, glad I could help!

11. amistre64 Group Title

yes, with the candlestick lol ... get a clue

12. dpflan Group Title

@sleestak I imagine you may have some more problems that are similar to this that you need to work on, so good luck

13. sleestak Group Title

thanks. I'll tell you what I come up with

14. sleestak Group Title

9

15. sleestak Group Title

oh no. sorry one sec

16. sleestak Group Title

14. OK. I think I can do the other problem. THANKS :D

17. dpflan Group Title

x1 -> 1 x2 -> 4 x3 -> 9 SUM = 14, yeah, you're now an expert it seems

18. sleestak Group Title

you guys need to list finance as the next subject :D

19. dpflan Group Title
20. dpflan Group Title

It actually had/has a pretty strong following/support from a professor at NYU - http://pages.stern.nyu.edu/~adamodar/

21. dpflan Group Title

Math is the most active subject

22. sleestak Group Title

oh. excellent.

23. dpflan Group Title

So if your questions are mathematical, you'd probably be best served to ask them here because there is a greater mass of users in this group, so you're more likely get an answer.

24. sleestak Group Title

yeah totally. this is awesome! I am going to use this all the time in school

25. dpflan Group Title

26. dpflan Group Title

*got

27. sleestak Group Title

oh okay. let me post

28. sleestak Group Title

should I now 'close' this question?

29. dpflan Group Title

Yep


text:
## MathJax Incompatibility to LaTex

Issue:

“Error: Missing argument to \begin” when using \color command and together with \boxed command in WordPress with MathJax plugin as if they are under LaTex syntax.

Solution:

Adding {} to embrace the content under the \color command. Actually, in LaTex syntax, \color command does not need to use {} as all content after the \color is coloured according to the defined color, e.g. \color{red}. However, for MathJax, you cannot add another LaTex command immediately after \color command, and you need to add {} to embrace the content to be coloured. For example, \color{red}{\boxed{f(x)}}

Config:

• WordPress:         2.6.1
• MathJax plugin:  1.3.4