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# Prove the sum from $1$ to $n$ is congruent to half $n$ or $0$ for even or odd values of $n$

I am trying to prove this for any natural number $$n$$:

$$\sum_{k=1}^{n} k \equiv \left\{ \begin{array}{ll} \frac{n}{2} \pmod n & \quad 2\mid n \\ 0 \pmod n & \quad 2\nmid n \end{array} \right.$$

So essentially the sum of all natural numbers 1 to $$n$$ is congruent to either 0 when $$n$$ is odd or half of $$n$$ for even $$n$$ values. I haven't worked much with sums within the congruence relationship so I'm not sure where to start on this one.

• Note that the sum is $n(n+1)/2$ – J. W. Tanner Feb 13 '19 at 20:51
• A comment on consistency: why not get rid of $m$ and rewrite the top line $\frac n2\pmod n\quad2\mid n$? That way, there is an easy comparison between the two conditions $2\mid n$ and $2\nmid n$. – Théophile Feb 13 '19 at 20:52

Since is well known that $$\sum_{k=1}^{n} k = \frac{n}{2} (n+1)$$, then if $$n$$ is even so $$n=2m$$, then the sum is $$m(n+1)$$ which is $$mn +m \equiv m \;(mod \;n)$$. If $$n$$ is odd, then $$(n+1)$$ is even, so there exist $$c\in \mathbb{N}$$ such $$c=\frac{n+1}{2}$$, so then the sum will be $$c*n \equiv 0 \; (mod \; n)$$

• I think you meant $mn+m$ – J. W. Tanner Feb 13 '19 at 21:27
• Oh thanks for that – JoseSquare Feb 13 '19 at 21:33

Note that $$1+2+\cdots+n$$ is also equal to $$0+1+2+\cdots+n.$$ The latter sum can be rearranged as $$(0+n)+(1+(n-1))+(2+(n-2))+\cdots\qquad\qquad(*)$$ where every written summand is equal to $$n$$.

But what is the last summand in (*)?.

If $$n$$ is odd, taking $$\{0,1,2,...,n\}$$ in pairs will exhaust the set, so that the total sum is a multiple of $$n$$, hence congruent to $$0\bmod n$$.

But if $$n=2m$$ is even, after taking pairs you are left with just $$m$$, so the total is congruent to $$m\bmod n$$.

I think this method could be called "baby Gauss $$+0$$".

• Clever. +1 from me – J. W. Tanner Feb 13 '19 at 21:29

It's well known that $$\sum_{i=1}^n i = \frac {n(n+1)}2,$$ which means $$2|n(n+1),$$ which, if you are naive, might seem like a coincidence. (very naive). But if $$n$$ is even then $$2|n$$ and $$\frac {n(n+1)}2 = \frac n2(n+1)$$ and if $$n$$ is odd then $$n+1$$ is even and $$2|(n+1)$$ and $$\frac {n(n+1)}2 = n\frac {n+1}2$$.

So the result follows pretty simply:

If $$n$$ is odd, then $$\frac {n(n+1)}2 = n\frac {n+1}2 \equiv 0 \pmod n$$.

If $$n$$ is even, then $$\frac {n(n+1)}2 = \frac n2(n+1) = m(n+1)\equiv m* 1 \equiv m \pmod n$$

....

As a side note I think there are 3) types of people.

1) Those who learn $$\sum_{i=1}^n i = \frac {n(n+1)}2$$ because someone told them.

2) Those who figure on their own that

$$N = 1 + 2 + ....... + n$$

$$N = n + (n-1) + ..... + 1$$

$$2N = (n+1) + (n+1) + ..... + (n+1)$$

$$N = \frac {n(n+1)}2$$

And

3) Those who figure out on their own that

$$N = \underbrace{1 + \underbrace{2 + \underbrace{....}+(n-1)}+1}=$$

$$\begin{cases}\frac n2\times (n+1)&n \text{ is even}\\\frac {n-1}2\times (n+1) + \text{middle of }(1...n)=\frac{n-1}2\times (n+1) + \frac {n+1}2 = \frac n2(n+1)_{\text{a fraction times an even number}}&n\text{ is odd}\end{cases}$$


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1. ## Proof by induction

Let a1=3 and a(n+1) =($\displaystyle {a}^{2}n+10$)/7 for every n$\displaystyle \geq$1

Prove by induction that 2
$\displaystyle \leq$an$\displaystyle \leq$5 for every n$\displaystyle \geq$1

Deduce that an+1$\displaystyle \leq$an for every n$\displaystyle \geq$1

Show the sequence {an} converges and determine its limit.

2. Originally Posted by Mathman87
Let a1=3 and a(n+1) =($\displaystyle {a}^{2}n+10$)/7 for every n$\displaystyle \geq$1

Prove by induction that 2$\displaystyle \leq$an$\displaystyle \leq$5 for every n$\displaystyle \geq$1

Deduce that an+1$\displaystyle \leq$an for every n$\displaystyle \geq$1

Show the sequence {an} converges and determine its limit.

Induction on $\displaystyle n$ , $\displaystyle a_1 =3 \in[2,5]$ and assume it is true that $\displaystyle 2 \leq a_n \leq 5$ and we have

$\displaystyle a_{n+1} = \frac{a_n^2 + 10}{7} \leq \frac{ 5^2 + 10}{7} = \frac{35}{7} = 5$

Also ,

$\displaystyle a_{n+1} = \frac{a_n^2 + 10}{7} \geq \frac{2^2 + 10}{7} = \frac{14}{7} = 2$

Therefore , $\displaystyle 2\leq a_{n+1} \leq 5$

Hence we deduce it is true that $\displaystyle 2\leq a_n \leq 5 ~~\forall n \in \mathbb{N}$

Consider $\displaystyle a_{n+1} - a_n = \frac{a_n^2 + 10}{7} -a_n = \frac{a_n^2 - 7a_n + 10 }{7} = \frac{(a_n - 2)(a_n -5 )}{7}$

$\displaystyle a_n - 2 \geq 0$ because $\displaystyle a_n \geq 2$

but $\displaystyle a_n - 5 \leq 0$ from $\displaystyle a_n \leq 5$

Therefore , $\displaystyle a_{n+1} - a_n$ is a non-positive number or $\displaystyle a_{n+1} - a_n \leq 0 ~,~ a_{n+1} \leq a_n$

Note that the sequence is bounded and it is monotonic decreasing , use some theorems to show that the limit exists and find it .

format: plain


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## The Riemann hypothesis in various settings

The Riemann hypothesis is arguably the most important and famous unsolved problem in number theory. It is usually phrased in terms of the Riemann zeta function $latex {\zeta}&fg=000000$, defined by

$latex \displaystyle \zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}&fg=000000$

for $latex {\hbox{Re}(s)>1}&fg=000000$ and extended meromorphically to other values of $latex {s}&fg=000000$, and asserts that the only zeroes of $latex {\zeta}&fg=000000$ in the critical strip $latex {\{ s: 0 \leq \hbox{Re}(s) \leq 1 \}}&fg=000000$ lie on the critical line $latex {\{ s: \hbox{Re}(s)=\frac{1}{2} \}}&fg=000000$.

Just another mathematician
This entry was posted in Singular integral operators. Bookmark the permalink.

### 5 Responses to The Riemann hypothesis in various settings

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Correlation is the relationship between two variables (Schwartz, 2015). Linear correlation

can be described as the measure of intensity of association between two random variables

(Statlect.com, 2016). In other words, correlation quantifies the linear association between two

variables (The Math Works, 2016). When describing this relationship several factors have to be

considered. Two variables that have a small or no linear correlation can have a strong nonlinear

association. Also, when there is no correlation between the two variables, the values of the

variables will not have a tendency to decrease or increase in tandem (The Math Works, 2016).

Correlation does not indicate or suggest existence of a statistically significant or casual

relationship between conduct and result. There are set of principles of casual interference that

need to be satisfied to imply cause and effect (Schwartz, 2015). Here the dose-response effect

has to be considered to establish the cause-and effect, i.e., the value of response variable has to

change in a meaningful way with the dose or level of the suspected casual agent. Also, the

hypothesis that there is a casual relationship between number of cigarettes and pulse rate is not

consistent with the current biological or theoretical knowledge. In the given conclusion the error

is that the pulse rate increases as the number of cigarettes smoke increases does not meet the

dose-response criteria, is inconsistent finding, biologically incredible, and has no coherence of

evidence.

I NEED YOU TO COMMENT FROM THIS POST NO MORE THAN 150 WORDS NEEDED AND A REFERENCE PLEASE


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# Homework Help: Charge flow in a RC circuit

Tags:
1. Jan 14, 2018

### sumit_1

• Member advised to use the formatting template for all homework help requests; Thread moved to Introductory Physics Homework from Advanced Physics Homework.
I have a RC circuit which also includes a battery, the capacitor has some charge initially before we put all the components in the circuit. Now as I connect all of them then charge will start to flow, but wheather this charge will be the sum of initial charge on the capacitor and the new charge produced by the battery or it will be just the new charge produced by the battery?

According to me the charge flowing in the circuit should be the sum of the initial charge of the capacitor and the charge produced by the battery.
Is this correct?

2. Jan 14, 2018

### scottdave

Look at how much charge (and polarity) will be on the capacitor after a long time. How does this compare with the charge and polarity before they're connected?

3. Jan 14, 2018

### sumit_1

I connected the positive side of capacitor to the positive terminal of the battery.

4. Jan 14, 2018

### scottdave

So can you calculate how much charge will be on the cap after a long time (if the cap was initially "empty")?

This ending number will be the same for a cap with some initial charge. But the amount supplied by the battery will be different.

5. Jan 14, 2018

### scottdave

Do you know how much initial charge is on the capacitor?

6. Jan 14, 2018

### sumit_1

I know the initial charge on the capacitor.

7. Jan 14, 2018

### scottdave

Maybe think of it as adding money to your bank account. If you started with $10 and end with$50, how much money was added?

8. Jan 14, 2018

### sumit_1

\$40 is added. But i want to know how much amount of charge will flow in the circuit as i have described in the problem?

9. Jan 14, 2018

### scottdave

Do you have details of the setup that you can share, along with what you have worked out so far?


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Codimension of linear subspace.

What is the codimension of X? X is subset of space of $2\times 2$ matrices $M_{2\times 2}$. $$X=\left\{\left( \begin{array}{cc} 0 & 0 \\ 0 & x \end{array}\right)\bigg| x \in \mathbb{R}\right\}$$ I think that the codimension is two, because dimension of $M_{2\times 2}$ is four, and dimension of $X$ is two, then the codimension is $4-2$. What do you think? thank you very much

-

The dimension of $X$ is $1$, since its elements only depend on the single parameter $x$. The dimension of $M_{2x2}$ is, as you said, 4. The codimension is therefore $4 - 1 = 3$.
$\left( \begin{array}{cc} 0 & 0 \\ 0 & 1 \end{array}\right)$ is a basis of $X$. Hence the dimension of $X$ is 1. Since the dimension of $M_{2\times 2}$ is 4, the codimension of $X$ is 3.


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# 9.1 Current (new)  (Page 2/7)

 Page 2 / 7

Note that the direction of current flow in [link] is from positive to negative. The direction of conventional current is the direction that positive charge would flow . Depending on the situation, positive charges, negative charges, or both may move. In metal wires, for example, current is carried by electrons—that is, negative charges move. In ionic solutions, such as salt water, both positive and negative charges move. This is also true in nerve cells. A Van de Graaff generator used for nuclear research can produce a current of pure positive charges, such as protons. [link] illustrates the movement of charged particles that compose a current. The fact that conventional current is taken to be in the direction that positive charge would flow can be traced back to American politician and scientist Benjamin Franklin in the 1700s. He named the type of charge associated with electrons negative, long before they were known to carry current in so many situations. Franklin, in fact, was totally unaware of the small-scale structure of electricity.

It is important to realize that there is an electric field in conductors responsible for producing the current, as illustrated in [link] . Unlike static electricity, where a conductor in equilibrium cannot have an electric field in it, conductors carrying a current have an electric field and are not in static equilibrium. An electric field is needed to supply energy to move the charges.

## Making connections: take-home investigation—electric current illustration

Find a straw and little peas that can move freely in the straw. Place the straw flat on a table and fill the straw with peas. When you pop one pea in at one end, a different pea should pop out the other end. This demonstration is an analogy for an electric current. Identify what compares to the electrons and what compares to the supply of energy. What other analogies can you find for an electric current?

Note that the flow of peas is based on the peas physically bumping into each other; electrons flow due to mutually repulsive electrostatic forces.

## Calculating the number of electrons that move through a calculator

If the 0.300-mA current through the calculator mentioned in the [link] example is carried by electrons, how many electrons per second pass through it?

Strategy

The current calculated in the previous example was defined for the flow of positive charge. For electrons, the magnitude is the same, but the sign is opposite, ${I}_{\text{electrons}}=-0.300×{\text{10}}^{-3\phantom{\rule{0.25em}{0ex}}}\text{C/s}$ .Since each electron $\left({e}^{-}\right)$ has a charge of $–1\text{.}\text{60}×{\text{10}}^{-\text{19}}\phantom{\rule{0.25em}{0ex}}\text{C}$ , we can convert the current in coulombs per second to electrons per second.

Solution

Starting with the definition of current, we have

${I}_{\text{electrons}}=\frac{\text{Δ}{Q}_{\text{electrons}}}{\text{Δ}t}=\frac{–0\text{.}\text{300}×{\text{10}}^{-3}\phantom{\rule{0.25em}{0ex}}\text{C}}{\text{s}}\text{.}$

We divide this by the charge per electron, so that

$\begin{array}{lll}\frac{{e}^{\text{–}}}{\text{s}}& =& \frac{–0\text{.}\text{300}×{\text{10}}^{–3}\phantom{\rule{0.25em}{0ex}}\text{C}}{\text{s}}×\frac{\text{1}\phantom{\rule{0.25em}{0ex}}{e}^{\text{–}}}{–1\text{.60}×{\text{10}}^{-\text{19}}\phantom{\rule{0.25em}{0ex}}\text{C}}\\ & =& \text{1.88}×{\text{10}}^{\text{15}}\phantom{\rule{0.25em}{0ex}}\frac{{e}^{\text{–}}}{\text{s}}\text{.}\end{array}$

Discussion

There are so many charged particles moving, even in small currents, that individual charges are not noticed, just as individual water molecules are not noticed in water flow. Even more amazing is that they do not always keep moving forward like soldiers in a parade. Rather they are like a crowd of people with movement in different directions but a general trend to move forward. There are lots of collisions with atoms in the metal wire and, of course, with other electrons.

how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Do somebody tell me a best nano engineering book for beginners?
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
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Cied
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I start with an easy one. carbon nanotubes woven into a long filament like a string
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Porter
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Yasmin
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I'm interested in nanotube
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preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
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Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
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Damian
silver nanoparticles could handle the job?
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Azam
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I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
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can nanotechnology change the direction of the face of the world
how did you get the value of 2000N.What calculations are needed to arrive at it
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# Resistance experienced by capacitors in circuit

what is the resistance experienced by the capacitors in the circuit (diagram) ?

i am actually supposed calculate the time taken during the discharging of capacitor using this formula -

$$V=V_oe^{-\frac{t}{CR}}$$

what should the R there be ?

Since they are connected in parallel , would it be

$$\frac{1}{R_e}=\frac{1}{R_1}+\frac{1}{R_2}$$ ?

where R=R_e

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rl.bhat
Homework Helper

The two resistances are in series.

The two resistances are in series.
thanks , i only recognise this (diagram) to be series , how do you see that ?

#### Attachments

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rl.bhat
Homework Helper

thanks , i only recognise this (diagram) to be series , how do you see that ?
It is in series. If you bend it in the middle, still it is in series.


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## Files in this item

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## Description

 Title: Tractive Performance From Axle Instrumentation Author(s): Snyder, Karl Alan Doctoral Committee Chair(s): Buck, Nelson, Department / Program: Agricultural Engineering Discipline: Agricultural Engineering Degree Granting Institution: University of Illinois at Urbana-Champaign Degree: Ph.D. Genre: Dissertation Subject(s): Engineering, Agricultural Engineering, Electronics and Electrical Engineering, Mechanical Abstract: The forces associated with the traction of a wheeled farm tractor can be measured in the axles driving the wheels. An instrumentation system was developed to measure the bending forces in an unreduced section of axle. Two four-gage, constant current, semiconductor strain gage bridge circuits were developed to measure the strains associated with the bending forces. The analog strain signals were converted to FM signals on the axle. A multi-channel capacitor with rotating and stationary plates was developed to couple the signals to stationary receiving and processing circuitry.An axle simulator was constructed to provide strain levels in a shaft equivalent to those expected in tractor axles, yet at considerably lower levels of loading. Tests were devised to separate effects of the instrumentation system from effects due to the mechanical system used to load the shaft under test.The transmission accuracy through the coupling capacitor was found to be independent of the shaft velocities tested, which spanned 0 to 140 rev/min. The instrumentation was found to be sensitive to mechanical vibrations which were due, in this case, to the apparatus used to load the shaft. Averaging 730 points collected over one and one-half shaft revolutions reduced the errors due to vibration variations. The average values at a 4.3 $\mu$strain level, which corresponds to a drawbar pull of only 2 kN for an Allis-Chalmers One-Ninety tractor, were measured with less than 6.5% error. The vertical load corresponding to this pull, 19.3 kN, which would cause 87.4 $\mu$strain in the sample tractor, was measured with less than 0.6% error. The angle of the vector sum of these two forces was measured with less than 0.3% error. Single point readings, however, contained errors equal to the minimum pull itself, up to 6.5% of the vertical load and up to 2$\sp\circ$ in the resultant angle. Issue Date: 1988 Type: Text Description: 517 p.Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1988. URI: http://hdl.handle.net/2142/70963 Other Identifier(s): (UMI)AAI8908852 Date Available in IDEALS: 2014-12-16 Date Deposited: 1988


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# How to find derivative of $[1-F(v)]^{n}$ and $1-[1-F(v)]^{n}$?

How can I find the derivative with respect to $v$ for the following two equations: $$[1-F(v)]^{n} \tag{1}$$ and $$1-[1-F(v)]^{n} \tag{2}$$ where $$F'(v)=f(v)$$

-
Have you studied the chain rule? – David Mitra Feb 16 '12 at 22:26
Well, you have to show that by induction. – checkmath Feb 16 '12 at 22:41
Note: An "equation" must have an "equal" sign in it (hence the name). What you give are not "equations", they are expressions. – Arturo Magidin Feb 16 '12 at 22:43

By using the chain rule: $$\Big((1-F(v))^n\Big)'=(1-F(v))'n(1-F(v))^{n-1}=-f(v)n(1-F(v))^{n-1}$$ Similarly: $$\Big(1-(1-F(v))^n\Big)'=0-(-f(v)n(1-F(v))^{n-1})=f(v)n(1-F(v))^{n-1}$$


text:
Eurasian Mathematical Journal
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 Eurasian Math. J., 2014, Volume 5, Number 1, Pages 7–60 (Mi emj148)

Hölder analysis and geometry on Banach spaces: homogeneous homeomorphisms and commutative group structures, approximation and Tzar’kov’s phenomenon. Part I

S. S. Ajiev

School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, NSW, 2522, Australia

Abstract: In an explicit quantitative and often precise manner, we construct the homogeneous Hölder homeomorphisms and study the approximation of uniformly continuous mappings by the Hölder–Lipschitz ones between the pairs of abstract and concrete metric and (quasi) Banach spaces including, in particular, Banach lattices, general non-commutative $L_p$-spaces, the classes $IG$ and $IG_+$ of independently generated spaces (for example, non-commutative-valued Bochner–Lebesgue spaces) and anisotropic Sobolev, Nikol’skii–Besov and Lizorkin–Triebel spaces of functions on an open subset or a class of domains of an Euclidean space defined with underlying mixed $L_p$-norms in terms of differences, local approximations by polynomials, wavelet decompositions and systems of closed operators, such as holomorphic functional calculus and Fourier multipliers of smooth Littlewood–Paley decompositions. Our approach also allows to treat both the finite (as in the initial and/or boundary value problems in PDE) and infinite $l_p$-sums of these spaces, their duals and “Bochnerizations”. Many results are automatically extended to the setting of the function spaces with variable smoothness, including the weighted ones. The sharpness of the approximation results, shown for the majority of the pairs under some mild conditions and underpinning the corresponding sharpness of the Hölder continuity exponents of the homogeneous homeomorphisms, indicates that the range of the exponents is often a proper subset of $(0,1]$, that is the presence of Tsar’kov’s phenomenon. We also consider the approximation by the mappings taking the values in the convex envelope of the range of the original approximated mapping. The negative results on the absence of uniform embeddings of the balls of some function spaces, particularly including $BMO$, $VMO$, Nikol’skii–Besov and Lizorkin–Triebel spaces with $q=\infty$ and their $VMO$-like separable subspaces, into any Hilbert space are established. Relying on the solution to the problem of the global Hölder continuity of metric projections and the existence of the Hölder continuous homogeneous right inverses of closed surjective operators and retractions onto closed convex subsets, as well as our results on the bounded extendability of the Hölder–Lipschitz mappings and re-homogenisation technique, we develop and employ our key explicit quantitative tools, such as the global (on arbitrary bounded subsets) Hölder continuity of the duality mapping and Lozanovskii factorisation, the answer to the three-space problem for the Hölder classification of infinite-dimensional spheres, the Hölder continuous counterpart of the Kalton–Pełczyńki decomposition method, the Hölder continuity of the homogeneous homeomorphism induced by the complex interpolation method and such counterparts of the classical Mazur mappings as the abstract and simple Mazur ascent and complex Mazur descent. Important role is also played by the study of the local unconditional structure and other complementability results, as well as the existence of equivalent geometrically friendly norms.

Keywords and phrases: Hölder classification of spheres, Hölder–Lipschitz mappings, approximation of uniformly continuous mappings, Tsar’kov’s phenomenon, Mazur mappings, Lozanovskii factorisation, homogeneous right inverses, metric projection, asymmetric uniform convexity and smoothness, dimension-free estimates, $\lambda$-horn condition, local unconditional structure, Nikol’skii–Besov, Lizorkin–Triebel, Sobolev, non-commutative $L_p$ and $IG$-spaces, Banach lattices, wavelets, three-space problem, Kalton–Pełczyńki decomposition, bounded extension of Hölder mappings, Markov type and cotype, complemented subspaces, UMD.

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Citation: S. S. Ajiev, “Hölder analysis and geometry on Banach spaces: homogeneous homeomorphisms and commutative group structures, approximation and Tzar’kov’s phenomenon. Part I”, Eurasian Math. J., 5:1 (2014), 7–60

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\Bibitem{Aji14} \by S.~S.~Ajiev \paper H\"older analysis and geometry on Banach spaces: homogeneous homeomorphisms and commutative group structures, approximation and Tzar’kov’s phenomenon. Part I \jour Eurasian Math. J. \yr 2014 \vol 5 \issue 1 \pages 7--60 \mathnet{http://mi.mathnet.ru/emj148}

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# Issues ascertaining current x time for battery life

It is simple to calculate a continuously-on device's theoretical battery life based on the battery datasheet capacity (assuming always nominal voltage, etc.)

An LED @ 20mA..

2800mAh / 20mA = 240 hours.

The issue I am having is verifying calculation of intermittent use of such a device.

Lets say it consumes 20mA for ten seconds three times per day. Would a calculation such as this make sense?

20 * 10 * 3 = 600mAs? / 60 / 60 = 0.166mAh/day

To be novel, resembles $$0.166 \centerdot 10^{-3} \ C \centerdot s \centerdot hour \centerdot day^{-1}$$

Now to extract days from that I would possibly do this:

2800mAh / 0.166mAhd? = 16867 days / 365 = 46 years battery life

Does this seem at all coherent? The only proof I can muster:

16867 days * (30 * 10 * 2 / 60 / 60) = ~2800mAh

Had never seen anyone use this before, maybe I have pioneered some new units.

Battery capacity with time varies with usage and with battery chemistry.
For very long lifetimes (years) information about battery chemistry is essential to lifetime considerations.

You mention a 2800 mAh capacity - what sort of battery type is that? If it is an AA (14500) of any sort the capacity claim will be higher than actual regardless of load or manufacturer). If it is a LiIon 18650 cell it is also probably higher than actual. For very long lifetime quality and manufacturer experience matter and brand is often but not always an indicator of quality. Batteries with claimed capacities at the top end of the range for their size and chemistry usually have lower capacities than quality competitors with lower claimed capacities.

Primary (non rechargeable) and secondary (rechargeable) batteries have somewhat different factors affecting lifetime. A rechargeable battery has a self discharge rate but can be "refilled". A non-rechargeable battery loses available capacity with time but this is usually seen as a shelf life or degradation rater than loss of charge as such. The two continue to increasingly overlap with the introduction of better performing low self discharge batteries such as the recent versions of the Sanyo Eneloop which retain most of their charge for several years.
The following is biased toward non-rechargeable batteries as these are usually used for very long lifetimes, but to some extent applies to rechargeables.

It is only possible to obtain approximate capacity results for batteries which are subject to occasional small loads.
Where the result indicates an extremely long lifetime, such as the 46 years that you calculate, the self discharge of shelf life of the battery will be the main factor affecting lifetime.
The longest battery lifetimes that you are liable to be able to obtain with commercial products are around 10 years although longer or much longer may be possible in special cases.

Your method of dividing battery capacity by mAh/day to get days of lifetime is valid providing the mAh capacity that you use is correct. However, for very light loads and long lifetimes the mAh capacity will be greater than the manufacturer's rated value.
For extremely long lifetimes the user-utilised mAh will fall because an increasingly large part of the true mAh capacity will begin to be consumed by non load related factors such as self discharge.
In addition to true self discharge, some battery chemistries have "calendar lives" where the battery slowly degrades in capacity due to irreversible chemical and mechanical changes in the battery structure. A good example are standard Lithium Ion batteries which typically have "calendar lives" in the 2 to 5 year range regardless of amount of use.

Manufacturers data sheets are available in some cases that show the effect of loading on both lifetime and capacity. The very wide range of possible loads and usage patterns means that these are only a guide. Very low and very high temperatures, especially those outside the manufacturers stated range, may have very significant effects on available capacity.

Here is a datasheet for Evereday CR2032 coil cell with nominal 240 mAh capacity.
It does not answer your questij but gives some indication of how loading affects capacity.


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# Another combinatorial proof

• July 4th 2010, 08:28 AM
oldguynewstudent
Another combinatorial proof
Give combinatorial or bijective proofs of the following. Part of your job is to determine all values of n,k, and/or m for which the identities are valid.

c) $\left({0\atop m}\right)+\left({1\atop m}\right)+\left({2\atop m}\right)+\ldots+\left({n\atop m}\right)=\left({n+1\atop m+1}\right)$

I can't get very far with this one either. I notice that all terms on the left hand side are zero until we get to $\left({m\atop m}\right)$. Then I calculate all terms equal 1, so we have n-m terms which makes the left hand side equal n-m.

The right hand side I get $\frac{(n+1)!}{(m+1)!(n+1-m-1)!}=\frac{(n+1)!}{n!(m+1)}=\frac{n+1}{m+1}$

I can get no further.

Can someone help or tell me if I have the whole combinatorial proof thing wrong.
• July 4th 2010, 07:38 PM
kompik
Quote:

Originally Posted by oldguynewstudent
Give combinatorial or bijective proofs of the following. Part of your job is to determine all values of n,k, and/or m for which the identities are valid.

c) $\left({0\atop m}\right)+\left({1\atop m}\right)+\left({2\atop m}\right)+\ldots+\left({n\atop m}\right)=\left({n+1\atop m+1}\right)$

I can't get very far with this one either. I notice that all terms on the left hand side are zero until we get to $\left({m\atop m}\right)$. Then I calculate all terms equal 1, so we have n-m terms which makes the left hand side equal n-m.

The right hand side I get $\frac{(n+1)!}{(m+1)!(n+1-m-1)!}=\frac{(n+1)!}{n!(m+1)}=\frac{n+1}{m+1}$

I can get no further.

Can someone help or tell me if I have the whole combinatorial proof thing wrong.

Use induction. Inductive step: Suppose that
$\binom0m+\binom1m+\dots+\binom nm=\binom{n+1}{m+1}$.
We show that the same is true for n+1.
$\binom0m+\binom1m+\dots+\binom nm+\binom{n+1}m=\binom{n+1}{m+1}+\binom{n+1}m=\bin om{n+2}{m+1}$.

See Sum of k Choose m up to n - ProofWiki
• July 5th 2010, 06:23 AM
chaoticmindsnsync
Remember that the point of a combinatorial proof is that they are easier to write than simply grinding out the identity, as you seem do be doing. In a combinatorial proof, you have to ask a question, and then answer it by counting it in two different ways.

Therefore, consider this.

Remember, the RHS basically is asking the number of ways to choose $m+1$ elements from a set with $n+1$ elements. So, you have one answer you need: $\binom{n+1}{m+1}$. Now on the left hand side, you have a sum. What does that imply? Or in other words, what conditions must you apply to get a sum?


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# 104 Tau

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 Local subgiants and time-scales of disc formationDetailed evolutionary tracks and basic stellar parameters for the nearbysubgiants 104 Tau and HD 168443 are presented and discussed. Both serveas invaluable key stars for accurate age-datings, especially as theypossess the chemical signatures intermediate to the stars of the thick-and thin-disc population. We derive τ =9.7-0.6+0.8 Gyr for 104 Tau and τ =9.8-0.8+1.0 Gyr for HD 168443, very much in linewith their local sibling HR 7569 at τ = 9.1 +/- 1.0 Gyr that wasformerly presented in Bernkopf, Fiedler & Fuhrmann. Both, 104 Tauand HD 168443, thus add to our local sample of the rare but veryrelevant subgiants for the disc formation time-scales and support thenotion of a thick disc as an extremely old τ >= 12 Gyrsingle-burst population, and a thin disc that has a local age of τ ~8 Gyr. 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Our spectroscopicallydetermined masses have a median fractional precision of 15%, but theyare systematically 10% higher than masses obtained by interpolatingisochrones. Our spectroscopic radii have a median fractional precisionof 3%. Our ages from isochrones have a precision that variesdramatically with location in the Hertzsprung-Russell diagram. We planto extend the catalog by applying our automated analysis technique toother large stellar samples. Radial Velocities of Late-Type Field Subgiant StarsHigh-dispersion coudé spectra were observed and measured for 43field subgiants of the G and K spectral types, with the intent ofsearching for hitherto undiscovered spectroscopic binaries. Statisticalanalysis of the measurements revealed no definite but two possible newbinaries. We present the data and discuss the statistical analysis usedto test for new binaries. Additionally, we discuss the techniques forhigh precision of radial-velocity measurement, and the systematic errorsthat interfere with that goal. Measurements of CH3D and CH4 in Titan from Infrared SpectroscopyWe measured the CH4 column abundance in Titan's atmospherethrough an analysis of Titan's monodeuterated methane (CH3D)spectral features. CH3D is several orders of magnitude lessabundant in Titan's atmosphere than CH4. Thus, unlikeCH4, the strong and well-studied CH3D3ν2 lines are not saturated and provide a sensitivemeasure of its column abundance. We recorded the CH3D3ν2 lines at 1.55 μm at NASA's Infrared TelescopeFacility (IRTF) equipped with the Cryogenic Echelle Spectrograph. Wederive a total integrated column abundance of 2.1+/-0.1 m amagat forCH3D. We also measured stratospheric emission lines of bothCH3D and CH4 at 8.6 μm at higher resolutionthan previously possible to better constrain theCH3D/CH4 ratio. 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From this subsetof stars, we determine that fewer than 3% of stars with-0.5<[Fe/H]<0.0 have Doppler-detected planets. Above solarmetallicity, there is a smooth and rapid rise in the fraction of starswith planets. At [Fe/H]>+0.3 dex, 25% of observed stars have detectedgas giant planets. A power-law fit to these data relates the formationprobability for gas giant planets to the square of the number of metalatoms. High stellar metallicity also appears to be correlated with thepresence of multiple-planet systems and with the total detected planetmass. This data set was examined to better understand the origin of highmetallicity in stars with planets. 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This explanation neglects other sources of diversity betweendiscs (such as density and planetesimal composition and orbits), but itdoes have the merit of matching the observational results. Chromospheric Ca II Emission in Nearby F, G, K, and M StarsWe present chromospheric Ca II H and K activity measurements, rotationperiods, and ages for ~1200 F, G, K, and M type main-sequence stars from~18,000 archival spectra taken at Keck and Lick Observatories as a partof the California and Carnegie Planet Search Project. We have calibratedour chromospheric S-values against the Mount Wilson chromosphericactivity data. From these measurements we have calculated medianactivity levels and derived R'HK, stellar ages,and rotation periods from general parameterizations for 1228 stars,~1000 of which have no previously published S-values. 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A clear kinematic distinction betweensolar-scaled and α-enhanced stars was found, although a one-to-onecorrespondence to thin disk'' and thick disk'' components cannot besupported with the present data. Nearby stars of the Galactic disk and halo. III.High-resolution spectroscopic observations of about 150 nearby stars orstar systems are presented and discussed. The study of these and another100 objects of the previous papers of this series implies that theGalaxy became reality 13 or 14 Gyr ago with the implementation of amassive, rotationally-supported population of thick-disk stars. The veryhigh star formation rate in that phase gave rise to a rapid metalenrichment and an expulsion of gas in supernovae-driven Galactic winds,but was followed by a star formation gap for no less than three billionyears at the Sun's galactocentric distance. In a second phase, then, thethin disk - our familiar Milky Way'' - came on stage. 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Although the significance of thiscorrelation certainly remains a matter of debate for years to come, wepoint in this context to the principal importance of the thick disk fora complete census with respect to the local surface and volumedensities. Other important effects that can be ascribed to this darkstellar population comprise (i) the observed plateau in the shape of theluminosity function of the local FGK stars, (ii) a small thoughsystematic effect on the basic solar motion, (iii) a reassessment of theterm asymmetrical drift velocity'' for the remainder (i.e. the thindisk) of the stellar objects, (iv) its ability to account for the bulkof the recently discovered high-velocity blue white dwarfs, (v) itsmajor contribution to the Sun's 220 km s-1 rotationalvelocity around the Galactic center, and (vi) the significant flatteningthat it imposes on the Milky Way's rotation curve. Finally we note ahigh multiplicity fraction in the small but volume-complete local sampleof stars of this ancient population. This in turn is highly suggestivefor a star formation scenario wherein the few existing single stellarobjects might only arise from either late mergers or the dynamicalejection of former triple or higher level star systems. The Geneva-Copenhagen survey of the Solar neighbourhood. Ages, metallicities, and kinematic properties of 14 000 F and G dwarfsWe present and discuss new determinations of metallicity, rotation, age,kinematics, and Galactic orbits for a complete, magnitude-limited, andkinematically unbiased sample of 16 682 nearby F and G dwarf stars. Our63 000 new, accurate radial-velocity observations for nearly 13 500stars allow identification of most of the binary stars in the sampleand, together with published uvbyβ photometry, Hipparcosparallaxes, Tycho-2 proper motions, and a few earlier radial velocities,complete the kinematic information for 14 139 stars. These high-qualityvelocity data are supplemented by effective temperatures andmetallicities newly derived from recent and/or revised calibrations. Theremaining stars either lack Hipparcos data or have fast rotation. Amajor effort has been devoted to the determination of new isochrone agesfor all stars for which this is possible. Particular attention has beengiven to a realistic treatment of statistical biases and errorestimates, as standard techniques tend to underestimate these effectsand introduce spurious features in the age distributions. Our ages agreewell with those by Edvardsson et al. (\cite{edv93}), despite severalastrophysical and computational improvements since then. We demonstrate,however, how strong observational and theoretical biases cause thedistribution of the observed ages to be very different from that of thetrue age distribution of the sample. Among the many basic relations ofthe Galactic disk that can be reinvestigated from the data presentedhere, we revisit the metallicity distribution of the G dwarfs and theage-metallicity, age-velocity, and metallicity-velocity relations of theSolar neighbourhood. Our first results confirm the lack of metal-poor Gdwarfs relative to closed-box model predictions (the G dwarfproblem''), the existence of radial metallicity gradients in the disk,the small change in mean metallicity of the thin disk since itsformation and the substantial scatter in metallicity at all ages, andthe continuing kinematic heating of the thin disk with an efficiencyconsistent with that expected for a combination of spiral arms and giantmolecular clouds. Distinct features in the distribution of the Vcomponent of the space motion are extended in age and metallicity,corresponding to the effects of stochastic spiral waves rather thanclassical moving groups, and may complicate the identification ofthick-disk stars from kinematic criteria. More advanced analyses of thisrich material will require careful simulations of the selection criteriafor the sample and the distribution of observational errors.Based on observations made with the Danish 1.5-m telescope at ESO, LaSilla, Chile, and with the Swiss 1-m telescope at Observatoire deHaute-Provence, France.Complete Tables 1 and 2 are only available in electronic form at the CDSvia anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or viahttp://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/418/989 Some anomalies in the occurrence of debris discs around main-sequence A and G starsDebris discs consist of large dust grains that are generated bycollisions of comets or asteroids around main-sequence stars, and thequantity and distribution of debris may be used to detect the presenceof perturbing planets akin to Neptune. We use stellar and disc surveysto compare the material seen around A- and G-type main-sequence stars.Debris is detected much more commonly towards A stars, even when acomparison is made only with G stars of comparable age. Detection ratesare consistent with disc durations of ~0.5 Gyr, which may occur at anytime during the main sequence. The higher detection rate for A stars canresult from this duration being a larger fraction of the main-sequencelifetime, possibly boosted by a globally slightly larger disc mass thanfor the G-type counterparts. The disc mass range at any given age is afactor of at least ~100 and any systematic decline with time is slow,with a power law estimated to not be steeper than t-1/2.Comparison with models shows that dust can be expected as late as a fewGyr when perturbing planetesimals form slowly at large orbital radii.Currently, the Solar system has little dust because the radius of theKuiper Belt is small and hence the time-scale to produce planetesimalswas less than 1 Gyr. However, the apparently constant duration of ~0.5Gyr when dust is visible is not predicted by the models. Improved Astrometry and Photometry for the Luyten Catalog. II. Faint Stars and the Revised CatalogWe complete construction of a catalog containing improved astrometry andnew optical/infrared photometry for the vast majority of NLTT starslying in the overlap of regions covered by POSS I and by the secondincremental Two Micron All Sky Survey (2MASS) release, approximately 44%of the sky. The epoch 2000 positions are typically accurate to 130 mas,the proper motions to 5.5 mas yr-1, and the V-J colors to0.25 mag. Relative proper motions of binary components are measured to 3mas yr-1. The false-identification rate is ~1% for11<~V<~18 and substantially less at brighter magnitudes. Theseimprovements permit the construction of a reduced proper-motion diagramthat, for the first time, allows one to classify NLTT stars intomain-sequence (MS) stars, subdwarfs (SDs), and white dwarfs (WDs). We inturn use this diagram to analyze the properties of both our catalog andthe NLTT catalog on which it is based. In sharp contrast to popularbelief, we find that NLTT incompleteness in the plane is almostcompletely concentrated in MS stars, and that SDs and WDs are detectedalmost uniformly over the sky δ>-33deg. Our catalogwill therefore provide a powerful tool to probe these populationsstatistically, as well as to reliably identify individual SDs and WDs. Contributions to the Nearby Stars (NStars) Project: Spectroscopy of Stars Earlier than M0 within 40 Parsecs: The Northern Sample. I.We have embarked on a project, under the aegis of the Nearby Stars(NStars)/Space Interferometry Mission Preparatory Science Program, toobtain spectra, spectral types, and, where feasible, basic physicalparameters for the 3600 dwarf and giant stars earlier than M0 within 40pc of the Sun. In this paper, we report on the results of this projectfor the first 664 stars in the northern hemisphere. These resultsinclude precise, homogeneous spectral types, basic physical parameters(including the effective temperature, surface gravity, and overallmetallicity [M/H]), and measures of the chromospheric activity of ourprogram stars. Observed and derived data presented in this paper arealso available on the project's Web site. A Survey of Nearby Main-Sequence Stars for Submillimeter EmissionWe searched for submillimeter emission around 10 Vega-type stars and oneHerbig Ae star with the four-color bolometer at 1300 μm and the 19channel bolometer array at 870 μm using the Heinrich Hertz Telescopeat the Submillimeter Telescope Observatory. All of our sources wereundetected at 870 μm. In the case of HD 131156, we have a 3 σdetection at 1300 μm. We report a flux of 6.25+/-1.88 mJy for the HD131156 disk and a corresponding dust mass of 2.4+/-0.7 lunar masses.However, we did not detect HD 131156 at 870 μm, so we are cautiousabout the 1300 μm detection. We performed follow-up infraredobservations of HD 131156 using MIRLIN at the Palomar 200 inchtelescope, which resolved both components of the binary. The data arephotospheric, implying that the system does not have a hot, inner dustcomponent. We report submillimeter upper limits on fluxes for theremaining systems. STELIB: A library of stellar spectra at R ~ 2000We present STELIB, a new spectroscopic stellar library, available athttp://webast.ast.obs-mip.fr/stelib. STELIB consists of an homogeneouslibrary of 249 stellar spectra in the visible range (3200 to 9500Å), with an intermediate spectral resolution (la 3 Å) andsampling (1 Å). This library includes stars of various spectraltypes and luminosity classes, spanning a relatively wide range inmetallicity. The spectral resolution, wavelength and spectral typecoverage of this library represents a substantial improvement overprevious libraries used in population synthesis models. The overallabsolute photometric uncertainty is 3%.Based on observations collected with the Jacobus Kaptein Telescope,(owned and operated jointly by the Particle Physics and AstronomyResearch Council of the UK, The Nederlandse Organisatie voorWetenschappelijk Onderzoek of The Netherlands and the Instituto deAstrofísica de Canarias of Spain and located in the SpanishObservatorio del Roque de Los Muchachos on La Palma which is operated bythe Instituto de AstrofÃsica de Canarias), the 2.3 mtelescope of the Australian National University at Siding Spring,Australia, and the VLT-UT1 Antu Telescope (ESO).Tables \ref{cat1} to \ref{cat6} and \ref{antab1} to A.7 are onlyavailable in electronic form at http://www.edpsciences.org. The StellarLibrary STELIB library is also available at the CDS, via anonymous ftpto cdsarc.u-strasbg.fr (130.79.128.5) or viahttp://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/402/433 Statistical cataloging of archival data for luminosity class IV-V stars. II. The epoch 2001 [Fe/H] catalogThis paper describes the derivation of an updated statistical catalog ofmetallicities. The stars for which those metallicities apply are ofspectral types F, G, and K, and are on or near the main sequence. Theinput data for the catalog are values of [Fe/H] published before 2002February and derived from lines of weak and moderate strength. Theanalyses used to derive the data have been based on one-dimensional LTEmodel atmospheres. Initial adjustments which are applied to the datainclude corrections to a uniform temperature scale which is given in acompanion paper (see Taylor \cite{t02}). After correction, the data aresubjected to a statistical analysis. For each of 941 stars considered,the results of that analysis include a mean value of [Fe/H], an rmserror, an associated number of degrees of freedom, and one or moreidentification numbers for source papers. The catalog of these resultssupersedes an earlier version given by Taylor (\cite{t94b}).Catalog is only available in electronic form at the CDS via anonymousftp cdsarc.u-strasbg.fr (130.79.128.5) or viahttp://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/398/731 Statistical cataloging of archival data for luminosity class IV-V stars. I. The epoch 2001 temperature catalogThis paper is one of a pair in which temperatures and metallicitycatalogs for class IV-V stars are considered. The temperature catalogdescribed here is derived from a calibration based on stellar angulardiameters. If published calibrations of this kind are compared by usingcolor-index transformations, temperature-dependent differences among thecalibrations are commonly found. However, such differences are minimizedif attention is restricted to calibrations based on Johnson V-K. Acalibration of this sort from Di Benedetto (\cite{dib98}) is thereforetested and adopted. That calibration is then applied to spectroscopicand photometric data, with the latter predominating. Cousins R-Iphotometry receives special attention because of its high precision andlow metallicity sensitivity. Testing of temperatures derived from thecalibration suggests that their accuracy and precision are satisfactory,though further testing will be warranted as new results appear. Thesetemperatures appear in the catalog as values of theta equiv5040/T(effective). Most of these entries are accompanied by measured orderived values of Cousins R-I. Entries are given for 951 stars.Catalog is only available in electronic form at the CDS via anonymousftp to cdsarc.u-strasbg.fr (130.79.128.5) or viahttp://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/398/721 Limb profiles of the Moon from grazing occultation observations collected at RGOFrom lunar grazing occultation observations it is shown that the presentlunar limb profile data have sometimes large errors, and therefore it isapparent that they need to be modified. For that purpose observations ofgrazing occultations collected at the Royal Greenwich Observatory until1980 were analyzed and lunar limb profile data were obtained. As aresult the number of the luanr limb profile data obtained from grazingoccultations was almost doubled. These profile data are being used forthe predictions of lunar grazing occultations in order to locateobservers at better positions, so that they can also get good grazingoccultation data to improve the lunar limb profiles. These profile datawill be used in the future analyses of solar eclipse observations and ofthe possible errors of the Hipparcos proper motion system. Dating the Almagest star catalogue using proper motions : a reconsideration.Not Available Radial Velocities for 889 Late-Type StarsWe report radial velocities for 844 FGKM-type main-sequence and subgiantstars and 45 K giants, most of which had either low-precision velocitymeasurements or none at all. These velocities differ from the standardstars of Udry et al. by 0.035 km s-1 (rms) for the 26 FGKstandard stars in common. The zero point of our velocities differs fromthat of Udry et al.: =+0.053km s-1. Thus, these new velocities agree with the best knownstandard stars both in precision and zero point, to well within 0.1 kms-1. Nonetheless, both these velocities and the standardssuffer from three sources of systematic error, namely, convectiveblueshift, gravitational redshift, and spectral type mismatch of thereference spectrum. These systematic errors are here forced to be zerofor G2 V stars by using the Sun as reference, with Vesta and day sky asproxies. But for spectral types departing from solar, the systematicerrors reach 0.3 km s-1 in the F and K stars and 0.4 kms-1 in M dwarfs. Multiple spectra were obtained for all 889stars during 4 years, and 782 of them exhibit velocity scatter less than0.1 km s-1. These stars may serve as radial velocitystandards if they remain constant in velocity. We found 11 newspectroscopic binaries and report orbital parameters for them. Based onobservations obtained at the W. M. Keck Observatory, which is operatedjointly by the University of California and the California Institute ofTechnology, and on observations obtained at the Lick Observatory, whichis operated by the University of California. Revised Coordinates and Proper Motions of the Stars in the Luyten Half-Second CatalogWe present refined coordinates and proper-motion data for the highproper-motion (HPM) stars in the Luyten Half-Second (LHS) catalog. Thepositional uncertainty in the original Luyten catalog is typicallygreater than 10" and is often greater than 30". We have used the digitalscans of the POSS I and POSS II plates to derive more accurate positionsand proper motions of the objects. Out of the 4470 candidates in the LHScatalog, 4323 objects were manually reidentified in the POSS I and POSSII scans. A small fraction of the stars were not found because of thelack of finder charts and digitized POSS II scans. The uncertainties inthe revised positions are typically ~2" but can be as high as ~8" in afew cases, which is a large improvement over the original data.Cross-correlation with the Tycho-2 and Hipparcos catalogs yielded 819candidates (with mR<~12). For these brighter sources, theposition and proper-motion data were replaced with the more accurateTycho-2/Hipparcos data. In total, we have revised proper-motionmeasurements and coordinates for 4040 stars and revised coordinates for4330 stars. The electronic version of the paper5 contains the updated information on all 4470stars in the LHS catalog. Statistics of spectroscopic sub-systems in visual multiple starsA large sample of visual multiples of spectral types F5-M has beensurveyed for the presence of spectroscopic sub-systems. Some 4200 radialvelocities of 574 components were measured in 1994-2000 with thecorrelation radial velocity meter. A total of 46 new spectroscopicorbits were computed for this sample. Physical relations are establishedfor most of the visual systems and several optical components areidentified as well. The period distribution of sub-systems has a maximumat periods from 2 to 7 days, likely explained by a combination of tidaldissipation with triple-star dynamics. The fraction of spectroscopicsub-systems among the dwarf components of close visual binaries withknown orbits is similar to that of field dwarfs, from 11% to 18% percomponent. Sub-systems are more frequent among the components of widevisual binaries and among wide tertiary components to the known visualor spectroscopic binaries - 20% and 30%, respectively. In triple systemswith both outer (visual) and inner (spectroscopic) orbits known, we findan anti-correlation between the periods of inner sub-systems and theeccentricities of outer orbits which must be related to dynamicalstability constraints. Tables 1, 2, and 6 are only available inelectronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr(130.79.128.5) or viahttp://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/382/118 A revision of the solar neighbourhood metallicity distributionWe present a revised metallicity distribution of dwarfs in the solarneighbourhood. This distribution is centred on solar metallicity. Weshow that previous metallicity distributions, selected on the basis ofspectral type, are biased against stars with solar metallicity orhigher. A selection of G-dwarf stars is inherently biased againstmetal-rich stars and is not representative of the solar neighbourhoodmetallicity distribution. Using a sample selected on colour, we obtain adistribution where approximately half the stars in the solarneighbourhood have metallicities higher than [Fe/H]=0. The percentage ofmid-metal-poor stars ([Fe/H]<-0.5) is approximately 4 per cent, inagreement with present estimates of the thick disc. In order to have ametallicity distribution comparable to chemical evolution modelpredictions, we convert the star fraction to mass fraction, and showthat another bias against metal-rich stars affects dwarf metallicitydistributions, due to the colour (or spectral type) limits of thesamples. Reconsidering the corrections resulting from the increasingthickness of the stellar disc with age, we show that the simpleclosed-box model with no instantaneous recycling approximation gives areasonable fit to the observed distribution. Comparisons with theage-metallicity relation and abundance ratios suggest that the simpleclosed-box model may be a viable model of the chemical evolution of theGalaxy at solar radius. Catalogue of Apparent Diameters and Absolute Radii of Stars (CADARS) - Third edition - Comments and statisticsThe Catalogue, available at the Centre de Données Stellaires deStrasbourg, consists of 13 573 records concerning the results obtainedfrom different methods for 7778 stars, reported in the literature. Thefollowing data are listed for each star: identifications, apparentmagnitude, spectral type, apparent diameter in arcsec, absolute radiusin solar units, method of determination, reference, remarks. Commentsand statistics obtained from CADARS are given. The Catalogue isavailable in electronic form at the CDS via anonymous ftp tocdsarc.u-strasbg.fr (130.79.128.5) or viahttp://cdsweb.u-strasbg.fr/cgi-bin/qcar?J/A+A/367/521
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# Also in the Article

Procedure

We built the model in three steps. First we used the model of Raftery et al.1 for probabilistic forecasting of CO2 emissions. This involves Bayesian hierarchical models for fertility and mortality, and hence population, for GDP, and for carbon intensity for each country. We then built Bayesian time series models of CMIP 5 models forecasts and historical simulations, together with the actual historical temperature anomalies, in order to estimate the bias and measurement error variance of the CMIP 5 models. Finally we took the probabilistic forecast of CO2 emissions from the first step as input, and linked the cumulative emissions to the CMIP 5 model forecasts.

The Coupled Model Intercomparison Project Phase 5 (CMIP 5)8, is a standard experimental protocol for studying the output of coupled ocean-atmosphere general circulation models (GCMs). GCMs take inputs such as future forcings, land use, and make corresponding forecasts of global climate. CMIP 5 models take inputs with the RCPs, converting them into radiative forcings and other climate factors, and make forecasts correspondingly. The data are illustrated in Figure 6.

CMIP 5 models. Both model simulations and the observed data are adjusted such that the mean anomaly between 1861–1880 for each model and the observed data is 0. The black line represents the observed data, while the colored lines are for the CMIP 5 model simulations. In total there are 39 CMIP 5 models, but we include only 10 models in this plot for visual clarity.

In this paper, we use our forecasting emissions trajectories as inputs to the GCMs. Instead of running each GCM for each input trajectory of emissions forecasts for probabilistic forecasts, which would not be feasible, we used the existing forecasts with different scenarios of emissions forecasts and developed statistical models of the relationship between CO2 emissions and model forecasts in CMIP 5 models. We found that the forecasts of each CMIP 5 model match linearly with the cumulative emissions for different RCP scenarios, with different correlation and scale. We took our CO2 emissions forecast as input, and used the linear relationship between global mean temperature and the cumulative emissions to generate probabilistic forecasts of future global mean temperature anomalies.

We generated forecasts based on each CMIP 5 model, and then combined these forecasts as an ensemble. For each model, our forecast is based on two parts. The first part takes CO2 emissions trajectories as input, and uses the linear relationship between cumulative CO2 emissions and global mean temperature predictions to generate the CMIP 5 model temperature forecast. The second part takes the uncertainty and bias of the CMIP 5 model forecast directly, based on the difference between historical simulations and the historical temperature anomalies.

For each CMIP 5 model, the historical simulations are generated similarly under different scenarios. We denote the historical simulation of the global mean temperature anomaly in year t from CMIP 5 model i by xi,t. We take the HadCRUT 4 observed temperature anomalies as the observations of the global surface-mean temperature, and denote them by $y˜t$. Then we denote the difference between the CMIP model backcast and the observed value by $yi,t=xi,t−y˜t$. We denote the true temperature anomalies by $z˜t$. Then $zi,t=xi,t−z˜t$ is the difference between the unobserved true global surface-mean temperature anomaly and the backcasts from the CMIP models. Analysis of the autocorrelation and partial autocorrelation functions of the estimated zi,t time series shows that it is well represented by a first-order autoregressive, or AR(1) model. This leads us to specify the following model:

where

where $Vt2$ is the variance of the measurement error of HadCRUT 4 observations, and is provided in the HadCRUT 4 dataset.

The forecast anomalies are modeled as linearly related to the cumulative emissions. Therefore, we collected the input cumulative CO2 emissions, denoted by cj,t for scenario or RCP j and year t, and the forecast anomalies from each model xi,j,t, where i indexes models, j indexes scenarios and t indexes years. Then we model the historical simulations as:

This relationship is close to being perfectly linear, as can be seen from the simple linear regression results in Table 2.

Note: The content above has been extracted from a research article, so it may not display correctly.

Q&A


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Home > PC3 > Chapter 8 > Lesson 8.3.1 > Problem8-89

8-89.
1. Rewrite each of the following expressions in terms of x, where x is any real number. Use the unit circle to justify your answers. Homework Help ✎

1. sin(π + x) = _____

2. cos(π + x) = _____

3. tan(π + x) = _____

4. sin(2π x) = _____

5. cos(2π x) = _____

6. tan(2π x) = _____

tangent = slope

tangent = slope


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# An Introduction to Stock Options

## Stock Options Defined

Call Option:

• A call option grants the owner the right to buy a share of an underlying stock for the strike price anytime before expiration.

• A call-option seller (also known as a writer) has the obligation to sell a share of the underlying stock for the strike price, if the buyer exercises the option.

Put Option:

• A put option grants the owner the right to sell a share of an underlying stock for the strike price anytime before expiration.

• A put-option seller (also known as a writer) has the obligation to buy a share of the underlying stock for the strike price, if the buyer exercises the option.

## American vs. European

The previous definitions were for American options. These are the type of stock options traded on most exchanges in the US.

In a European option, you can only exercise the option at expiration.

The intrinsic value is what the option would be worth if you had to exercise it immediately. If exercising the option would result in a negative amount, intrinsic is set to $0. Denoting I for intrinsic value, and S and X for the stock and strike price respectively, we can write: • Call option: I=max(S-X, 0) • Put option: I=max(X-S, 0) For example, if S=\$53, X=\$50 the intrinsic value of a call is \$3 and the intrinsic value of a put is \$0. ## Additional Terms • For both calls and puts, if I > 0 then we say the option is ‘in-the-money.’ • If I < 0 the option is ‘out-of-the-money.’ • The option with a strike nearest to the present stock price is termed the ‘at-the-money.’ These terms are useful because in/at/out-of-the-money options have similarities regardless of whether they are calls or puts. For example, out-of-the-money option premiums are not sensitive to changes in the underlying stock price. • The market price of an option is termed the option’s premium. • An option’s time value is the option’s premium less its intrinsic value. ## Practical Notes • Option contracts listed on US exchanges (such as the Chicago Board Options Exchange (CBOE)) are for 100 options. • Option contracts traded on US exchanges are cleared, which means you don’t have to worry about counterparty risk. This also means you’ll put up margin for each option trade. • Option strike prices are adjusted for stock splits, as well as for stock dividends of more than 10%. • Option contracts are not adjusted for cash dividends. This means call/put option premiums are lower/higher for high-dividend-paying stocks. Remember, if a stock pays a$1 dividend, its stock price is lowered by $1. ## What Can We Say So Far? An American option’s premium (both call and put) can never be less than the option’s intrinsic. Why not? • Arbitrage: If the intrinsic were$5, and the option premium was $3, you could buy the call option and immediately exercise it earning a riskless$2/share.
• However, a European put’s premium could be less than the intrinsic. Also a European call option on a dividend-paying stock can have a premium less than its intrinsic.
• In the next slide, you can graph a European call option’s intrinsic, premium, and time value for any set of inputs. Note that as you increase the stock’s dividend yield, the option premium drops below the intrinsic.

• The risk-free rate is the rate on a zero-coupon Treasury security of the same maturity as the option

## Time Value

In the previous chart, you saw that the time value is greatest for at-the-money options. Can you guess why?

• Very in- or very out- of-the-money options are not really options. You are almost certain to not exercise, or exercise, respectively.
• An at-the-money option roughly has a 50% chance of being exercised and a 50% chance of not being exercised. In other words, it is where there is the most option in the option.

## European Put Values

We mentioned earlier that put-option premiums on European options could be less than the option’s intrinsic value. You can use the previous app to see this. Can you guess why this is so? As a hint, set both the time-to-maturity and risk-free rates very low. Then set them very high and note the difference.

• Imagine the stock price has fallen to \$0, then of course you would want to immediately exercise it. The option can never be worth more, and you would rather have your money now than later. But because it is European, you have to wait until expiration. By the simple time value of money, the value of the option is the present value of the intrinsic. As time passes and the risk-free rate increases, the option premium decreases.

The following slide will allow you to calculate both call and put option premiums for your given set of inputs (via the Black-Scholes equation).

• Use the app to see how the underlying variables affect the option premium. For example, how does an increase in interest rates effect call and put values? How about an increase in volatility?

• Don’t worry yet about how to derive the option pricing equation – we’ll cover that in later presentations.

## Credits and Collaboration

Click the following links to see the codeline-by-line contributions to this presentation, and all the collaborators who have contributed to 5-Minute Finance via GitHub.


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# How does the formula for azimuth to another point work

If I want to know the azimuth (initial heading) to another point on a sphere I use the formula $$\tag{1} \tan(\theta) = \frac{\sin(\Delta\lambda)\cos(\varphi_2)}{\cos(\varphi_1)\sin(\varphi_2)-\sin(\varphi_1)\cos(\varphi_2)\cos(\Delta\lambda)}$$ where $\theta$ is the initial bearing to point 2, $\varphi_1$ and $\varphi_2$ are the latitudes of point one and two respectively and $\Delta\lambda$ is the difference in longitudes of the two points. To get the azimuth I then use the $atan2$-function and insert $$\tag{2} \sin(\Delta\lambda)\cos(\varphi_2)$$ for $X$ and $$\tag{3} \cos(\varphi_1)\sin(\varphi_2)-\sin(\varphi_1)\cos(\varphi_2)\cos(\Delta\lambda)$$ for $Y$. I know that $\tan(\alpha) = \frac{X}{Y}$ is used to calculate the heading $\alpha$ of a vector. However, I can't figure out why equation 2 is my $X$ and equation 3 my $Y$. Where do these formulae come from?

Let $p$, $p'$ be two unit vectors, directed from the Earth center to points $P$ and $P'$ on the sphere, and $n$ the analogous unit vector for the North Pole.
To compute the heading from $P$ to $P'$ you must set up a coordinate system in a plane perpendicular to $p$, with the $y$-axis pointing towards the North Pole. This can be simply done by constructing two unit vectors $x$ and $y$ as follows: $$y={p\times n\over|p\times n|}\times p,\quad x=y\times p.$$ A vector $t$ in the same plane pointing towards $P'$ is: $$t=(p\times p')\times p$$ and its coordinates are then its projections along vectors $x$ and $y$: $$t_x=t\cdot x,\quad t_y=t\cdot y.$$ If you now express the coordinates of $p$, $p'$ as a function of their latitude and longitude (and of course $n=(0,0,1)$), you can find explicit expressions for $t_x$ and $t_y$ and should recover your formulas.
If $\phi$, $\phi'$ are the latitudes, and $\lambda$, $\lambda'$ the longitudes of points $P$ and $P'$, we have: $$p=(\cos\phi\cos\lambda,\cos\phi\sin\lambda,\sin\phi),\quad p'=(\cos\phi'\cos\lambda',\cos\phi'\sin\lambda',\sin\phi'),\quad n=(0,0,1).$$ It follows that: $$p\times n=(\cos\phi\sin\lambda,-\cos\phi\cos\lambda,0), \quad |p\times n|=\cos\phi,$$ whence $$y=(-\cos\lambda\sin\phi,-\sin\lambda\sin\phi,\cos\phi), \quad x=(-\sin\lambda,\cos\lambda,0).$$ Moreover: $$t=(p\times p')\times p=-p(p'\cdot p)+p'=\\ \big( \sin ^2\phi \cos \phi' \cos \lambda' +\cos \phi^2\cos \phi' \sin \lambda \sin (\lambda-\lambda') -\sin\phi\cos \phi \sin \phi' \cos \lambda,\\ \sin ^2\phi\cos \phi' \sin \lambda' -\cos^2 \phi\cos \phi' \cos \lambda \sin (\lambda-\lambda') -\sin\phi \cos \phi \sin \phi' \sin \lambda,\\ \cos \phi (\cos \phi \sin \phi'-\sin \phi \cos \phi' \cos (\lambda-\lambda'))\big),$$ and finally: $$t_x=-\cos\phi'\sin(\lambda-\lambda'), \quad t_y=-\cos\phi'\sin\phi\cos(\lambda-\lambda')+\cos\phi\sin\phi'.$$


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## anonymous 5 years ago The ymca has collected 7875 dollars which is 63% of its goal,what is the amount of the goal? How much more money must the ymca collect?

You can always set up questions like this by saying `7875 of what is 63%', which you can put in equation form as: $\frac{7875}{x} = \frac{63}{100}$ Then you can cross multiply to get x. Does that make sense?


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# Dipole moment and Berry Phase¶

By default, Band computes (and prints to the output) only the components of the dipole moment orthogonal to the periodic direction(s).

Since the position operator is ill-defined in periodic systems, computing the longitudinal components of the dipole moment (i.e. the components along periodic directions) is not completely trivial. To obtain the longitudinal components of the dipole moment, you can perform a Berry Phase calculation:

BerryPhase Yes/No

BerryPhase
Type: Bool No Boolean that determines whether the dipole as determined through the Berry phase approach should be calculated.

In the graphical user interface (AMSinput) you can find the Berry Phase checkbox in the Details → Expert BAND panel.

In a Berry phase calculation, the dipole moment of a unit cell is calculated with the help of the geometric phase within the unit cell. The theoretical framework of the calculations originates from the ‘Modern theory of polarization’ that was come up with in the early 1990s by King-Smith, Vanderbilt and Resta. [1] [2] The implementation in BAND is a generalization of the one-dimensional Berry phase approach for quantum chemistry codes with local basis sets devised by Kudin and Car. [3]

Warning

The Berry phase implementation in BAND has been thoroughly for 1D systems. Nonetheless, the implementation seems to break down for 2D and 3D systems, thus requiring careful testing and validation of the calculations for such systems.

Warning

• Berry phase calculations require orthorhombic unit cells, differently shaped unit cells are not currently supported.
• For Berry Phase calculations you should use an all electron basis set (i.e. Set the core to none)
• Using a good k-space sampling is recommended for Berry Phase calculations

An option that can be useful when validating the Berry Phase calculation is ShiftCoordinates in the AMS System block:

System header
ShiftCoordinates float_list
End

System
ShiftCoordinates
Type: Float List Bohr Translate the atoms by the specified shift (three numbers).

References

 [1] R. King-Smith, D. Vanderbilt, Theory of polarization of crystalline solids. Physical Review B 47, 1651 (1993).
 [2] R. Resta, Macroscopic polarization in crystalline dielectrics: the geometric phase approach. Reviews of Modern Physics 66, 899 (1994).
 [3] K. Kuding, R. Car, Berry phase approach to longitudinal dipole moments of infinite chains in electronic-structure methods with local basis sets Journal of Chemical Physics 126, 234101 (2007).


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Solutions of Differential Equations

The value of the dependent variable which along with its derivatives is involved in the differential equation and satisfies the differential equation is called the solution of that equation.

For example, consider the first order differential equation:
$\frac{{dy}}{{dx}} – y = 0\,\,\,\,{\text{ – – – }}\left( {\text{i}} \right)$

Let $y = c{e^x}$, then $\frac{{dy}}{{dx}} = c{e^x}$. Putting these values of $y$ and $\frac{{dy}}{{dx}}$ in equation (i), we have
$\begin{gathered} c{e^x} – c{e^x} = 0 \\ \Rightarrow 0 = 0 \\ \end{gathered}$

This shows that $y = c{e^x}$ satisfies equation (i), so $y = c{e^x}$ is the solution of equation (i). Since $c$ is an arbitrary constant, this solution is called the general solution of the differential equation. In fact, the general solution of a second order differential equation has two arbitrary constants.

Among the simpler differential equations of the first order are those in which the two variables involved in the differential equations are separated. The solution of such a differential equation can be obtained by simple integration.


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# Find the volume of a sphere whose radius is(i) $7 \mathrm{~cm}$(ii) $063 \mathrm{~m}$.

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Given:

(i) $7 \mathrm{~cm}$
(ii) $063 \mathrm{~m}$.

To do:

We have to find the volumes of the sphere in each case.

Solution:

We know that,

Volume of a sphere of radius $r$ is $\frac{4}{3} \pi r^3$

Therefore,

(i) Volume of the sphere of radius $7\ cm= \frac{4}{3} \pi (7)^3$

$=\frac{4}{3} \times \frac{22}{7} \times (7)^3$

$= \frac{4312}{3}\ cm^3$

Hence, the volume of the sphere is $\frac{4312}{3}\ cm^3$

(ii) Volume of the sphere of radius $0.63\ m= \frac{4}{3} \pi (0.63)^3$

$=\frac{4}{3} \times \frac{22}{7} \times (0.63)^3$

$=\frac{4}{3} \times 22 \times (0.63)^2 \times 0.09$

$=88 \times 0.3969 \times 0.03$

$= 1.0478\ m^3$

$\approx 1.05\ m^3$

Hence, the volume of the sphere is $1.05\ m^3$.

Updated on 10-Oct-2022 13:46:39


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# Irrational Conformal Field Theory v.s. Non-Unitary Conformal Field Theory?

Unitary conformal field theories (CFTs) with irrational (or including the special case of rational) central charge is called irrational conformal field theory (ICFT).

Irrational conformal field theory (ICFT) is said to include all conformal field theories, and, in particular, ICFT includes rational conformal field theory (RCFT).

ICFT $$\supset$$ RCFT.

RCFT is only a tiny subset inside ICFT.

However, do we have examples of non-unitary Conformal Field Theory (NUCFT)? If so, what are the relations between:

• unitary RCFT
• unitary ICFT
• non-unitary RCFT
• non-unitary ICFT

Are there good quantum mechanical lattice realizations of these CFTs?

What are the relations of their sets?

• unitarity at $c<1$ implies rationality, no? – AccidentalFourierTransform Sep 5 '19 at 1:06
• Rational theories are, generally speaking, theories which have finite number of primaries of their respective chiral algebras (in the simplest case of Virasoro algebra, but one can have Kac-Moody algebras, etc.), although there may be people with stronger and/or more abstract opinions on this term. Also, there exist non-unitary theories, which can be rational or non-rational. I don't know whether anything useful can be said beyond these two facts. – Peter Kravchuk Sep 5 '19 at 2:01
• Also, your definitions are not consistent. You define ICFT to have irrational central charge, and then you say that RCFT is a special case of ICFT. It can't be since RCFT's have rational central charges. – Peter Kravchuk Sep 5 '19 at 2:03
• Minimal models and Liouville theory provide examples of rational and non-rational CFTs, which can be unitary or not depending on the central charge. See Wikipedia. – Sylvain Ribault Sep 5 '19 at 14:24